The Project Gutenberg eBook of The organisation of thought
This ebook is for the use of anyone anywhere in the United States and
most other parts of the world at no cost and with almost no restrictions
whatsoever. You may copy it, give it away or re-use it under the terms
of the Project Gutenberg License included with this ebook or online
at www.gutenberg.org. If you are not located in the United States,
you will have to check the laws of the country where you are located
before using this eBook.
Title: The organisation of thought
Educational and scientific
Author: Alfred North Whitehead
Release date: October 8, 2025 [eBook #77011]
Language: English
Original publication: London: Williams and Norgate, 1917
Credits: Jamie Brydone-Jack, Laura Natal and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive)
*** START OF THE PROJECT GUTENBERG EBOOK THE ORGANISATION OF THOUGHT ***
THE ORGANISATION
OF THOUGHT
THE ORGANISATION
OF THOUGHT
EDUCATIONAL AND SCIENTIFIC
BY
A. N. WHITEHEAD, Sc.D., F.R.S.
FELLOW OF TRINITY COLLEGE, CAMBRIDGE, AND PROFESSOR OF
APPLIED MATHEMATICS AT THE IMPERIAL COLLEGE OF
SCIENCE AND TECHNOLOGY
LONDON
WILLIAMS AND NORGATE
14 HENRIETTA STREET, COVENT GARDEN, W.C.2
1917
_The rights of translation are reserved_
PREFACE
THE discourses included in this volume have been delivered as addresses
on various occasions which are duly noted; the only exception is
_The Anatomy of Some Scientific Ideas_, which is now published
for the first time. These discourses fall into two sections, the first
five chapters deal with education, and the remaining three embody
discussions on certain points arising in the philosophy of science.
But a common line of reflection extends through the whole, and the two
sections influence each other.
I have left in each chapter the reference to the particular occasion of
its first production, and I have not sought for a verbal consistency
covering perplexity. But the various parts of the book were in fact
composed with express reference to each other, so as to form one whole.
I have to thank the Syndics of the Cambridge University Press for
permission to republish the contents of Chapter V.
_Imperial College of Science and Technology,_
_April 1917._
TABLE OF CONTENTS
CHAP. PAGE
PREFACE v
I. THE AIMS OF EDUCATION--A PLEA FOR REFORM 1
(Presidential Address to the Mathematical Association,
January, 1916.)
II. TECHNICAL EDUCATION AND ITS RELATION TO
SCIENCE AND LITERATURE 29
(Presidential Address to the Mathematical Association,
January, 1917.)
III. A POLYTECHNIC IN WAR-TIME 58
(Address at the Prize Distribution, Borough Polytechnic
Institute, Southwark, February, 1917.)
IV. THE MATHEMATICAL CURRICULUM 69
(Presidential Address to the London Branch of the
Mathematical Association, 1912.)
V. THE PRINCIPLES OF MATHEMATICS IN RELATION
TO ELEMENTARY TEACHING 92
(Paper read in the Educational Section of the International
Congress of Mathematicians, Cambridge,
1912.)
VI. THE ORGANISATION OF THOUGHT 105
(Presidential Address to Section A, British Association,
Newcastle, 1916; also read subsequently before the
Aristotelian Society.)
VII. THE ANATOMY OF SOME SCIENTIFIC IDEAS 134
VIII. SPACE, TIME, AND RELATIVITY 191
(Paper read to Section A at the Manchester Meeting of
Section A, 1915; also, with the appended Commentary,
read subsequently before the Aristotelian
Society.)
ORGANISATION OF THOUGHT
CHAPTER I
THE AIMS OF EDUCATION--A PLEA FOR REFORM
(_Presidential Address to the Mathematical Association, January,
1916_)
WHEN I had the honour of being made President of the Mathematical
Association, I did not foresee the unusual responsibility which it
entailed. It was my intention to take as the theme of a presidential
address the consideration of some aspect of those special subjects to
which my own researches have principally been directed. Events have
forced me to abandon that intention. It is useless to discuss abstract
questions in the midst of dominant practical preoccupation. We cannot
disregard the present crisis in European civilisation. It affects every
function of life. In the harder struggle for existence which lies
before the nation, all departments of national effort will be reviewed
for judgment. The mere necessity for economy in resources will provoke
this reformation.
We are concerned with education. This Association, so rich in its
membership of educationalists, with the conception of reform as the
very reason of its being, is among those bodies which must take the
lead in guiding that educational reconstruction which by a sociological
law follows every social revolution. We do not want impracticable
ideals, only to be realised beyond the clouds in
"Some wild, weird clime,
Out of Space, and out of Time."
We require to know what is possible now in England, a nation conscious
of its high achievements, and of great failures, shaken to its
foundations, distrustful of the old ways, and dreading fantastic
novelties.
I will take my courage in both hands, and put before you an outline of
educational principles. What I am going to say is of course entirely
without your authority, and does not pledge or prejudge any action of
the Association. We are primarily concerned only with the intellectual
side of education, and, as mathematicians, are naturally concerned to
illustrate details more particularly by reference to mathematics. Thus
much to explain deliberate omissions in what follows.
Consider now the general and special education of two types of boys,
namely those in secondary schools who in after life must form the
professional and directing classes in commerce, industry, and public
administration, and again those in junior technical schools and later
in advanced continuation classes, who are going to form the class of
skilled artisans and foremen of workshops. These two sets compose the
educated strength of the nation. We must form no ideals which include
less than these entire classes within their scope. What I shall say,
will in phraseology apply more directly to the secondary schools, but
with unessential changes it will apply equally to the other group.
What is the first commandment to be obeyed in any educational scheme?
It is this: Do not teach too many subjects. The second command is this:
What you teach, teach thoroughly. The devil in the scholastic world
has assumed the form of a general education consisting of scraps of a
large number of disconnected subjects; and, with the artfulness of the
serpent, he has entrenched himself behind the matriculation examination
of the University of London, with a wire entanglement formed by the
Oxford and Cambridge schools' examination.
Culture is activity of thought, and receptiveness to beauty, and humane
feeling. Scraps of information have nothing to do with it. A merely
well-informed man is the most useless bore on God's earth. What we
should aim at producing is men who possess both culture and expert
knowledge in some special direction. Their expert knowledge will give
them the ground to start from, and their culture will lead them as deep
as philosophy and as high as art. We have to remember that the valuable
intellectual development is self-development, and that it mostly takes
place between the ages of sixteen and thirty. As to training, the most
important part is given by mothers before the age of twelve. A saying
due to Archbishop Temple illustrates my meaning. Surprise was expressed
at the success in after-life of a man, who as a boy at Rugby had been
somewhat undistinguished. He answered, "It is not what they are at
eighteen, it is what they become afterwards that matters."
In training a child to activity of thought, above all things we must
beware of what I will call "inert ideas"--that is to say, ideas that
are merely received into the mind without being utilised, or tested, or
thrown into fresh combinations.
In the history of education, the most striking phenomenon is that
schools of learning, which at one epoch are alive with a ferment of
genius, in a succeeding generation exhibit merely pedantry and routine.
The reason is, that they are overladen with inert ideas. Education
with inert ideas is not only useless: it is, above all things,
harmful--_Corruptio optimi, pessima_. Except at rare intervals
of intellectual ferment, education in the past has been radically
infected with inert ideas. That is the reason why uneducated clever
women, who have seen much of the world, are in middle life so much
the most cultured part of the community. They have been saved from
this horrible burden of inert ideas. Every intellectual revolution
which has ever stirred humanity into greatness has been a passionate
protest against inert ideas. Then, alas, with pathetic ignorance of
human psychology, it has proceeded by some educational scheme to bind
humanity afresh with inert ideas of its own fashioning.
Let us now ask how in our system of education we are to guard against
this mental dry rot. We recur to our two educational commandments,
"Do not teach too many subjects," and again, "What you teach, teach
thoroughly."
The result of teaching small parts of a large number of subjects is the
passive reception of disconnected ideas, not illumined with any spark
of vitality. Let the main ideas which are introduced into a child's
education be few and important, and let them be thrown into every
combination possible. The child should make them his own, and should
understand their application here and now in the circumstances of his
actual life. From the very beginning of his education, the child should
experience the joy of discovery. The discovery which he has to make,
is that general ideas give an understanding of that stream of events
which pours through his life, which is his life. By understanding I
mean more than a mere logical analysis, though that is included. I mean
"understanding" in the sense in which it is used in the French proverb,
"To understand all, is to forgive all." Pedants sneer at an education
which is useful. But if education is not useful, what is it? Is it a
talent, to be hidden away in a napkin? Of course, education should be
useful, whatever your aim in life. It was useful to Saint Augustine
and it was useful to Napoleon. It is useful, because understanding is
useful.
I pass lightly over that understanding which should be given by the
literary side of education. It is not peculiarly the function of this
Association to consider it. Nor do I wish to be supposed to pronounce
on the relative merits of a classical or a modern curriculum. I would
only remark that the understanding which we want is an understanding
of an insistent present. The only use of a knowledge of the past
is to equip us for the present. No more deadly harm can be done to
young minds than by depreciation of the present. The present contains
all that there is. It is holy ground; for it is the past, and it is
the future. At the same time it must be observed that an age is no
less past if it existed two hundred years ago than if it existed two
thousand years ago. Do not be deceived by the pedantry of dates. The
ages of Shakespeare and of Molière are no less past than are the ages
of Sophocles and of Virgil. The communion of saints is a great and
inspiring assemblage, but it has only one possible hall of meeting,
and that is, the present; and the mere lapse of time through which any
particular group of saints must travel to reach that meeting-place,
makes very little difference.
Passing now to the scientific and logical side of education, we
remember that here also ideas which are not utilised are positively
harmful. By utilising an idea, I mean relating it to that stream,
compounded of sense perceptions, feelings, hopes, desires, and of
mental activities relating thought to thought, which forms our life.
I can imagine a set of beings which might fortify their souls by
passively reviewing disconnected ideas. Humanity is not built that
way--except perhaps some editors of newspapers.
In scientific training, the first thing to do with an idea is to prove
it. But allow me for one moment to extend the meaning of "prove"; I
mean--to prove its worth. Now an idea is not worth much unless the
propositions in which it is embodied are true. Accordingly an essential
part of the proof of an idea is the proof, either by experiment or by
logic, of the truth of the propositions. But it is not essential that
this proof of the truth should constitute the first introduction to
the idea. After all, its assertion by the authority of respectable
teachers is sufficient evidence to begin with. In our first contact
with a set of propositions, we commence by appreciating their
importance. That is what we all do in after-life. We do not attempt,
in the strict sense, to prove or to disprove anything, unless its
importance makes it worthy of that honour. These two processes of
proof, in the narrow sense, and of appreciation do not require a rigid
separation in time. Both can be proceeded with nearly concurrently. But
in so far as either process must have the priority, it should be that
of appreciation by use.
Furthermore, we should not endeavour to use propositions in isolation.
Emphatically I do not mean, a neat little set of experiments to
illustrate Proposition I and then the proof of Proposition I, a neat
little set of experiments to illustrate Proposition II and then the
proof of Proposition II, and so on to the end of the book. Nothing
could be more boring. Inter-related truths are utilised _en bloc_,
and the various propositions are employed in any order, and with any
reiteration. Choose some important applications of your theoretical
subject; and study them concurrently with the systematic theoretical
exposition. Keep the theoretical exposition short and simple, but
let it be strict and rigid so far as it goes. It should not be too
long for it easily to be known with thoroughness and accuracy. The
consequences of a plethora of half-digested theoretical knowledge are
deplorable. Also the theory should not be muddled up with the practice.
The child should have no doubt when it is proving and when it is
utilising. My point is that what is proved should be utilised, and that
what is utilised should--so far as is practicable--be proved. I am far
from asserting that proof and utilisation are the same thing.
At this point of my discourse, I can most directly carry forward
my argument in the outward form of a digression. We are only just
realising that the art and science of education require a genius and
a study of their own; and that this genius and this science are more
than a bare knowledge of some branch of science or of literature. This
truth was partially perceived in the past generation; and headmasters,
somewhat crudely, were apt to supersede learning in their colleagues
by requiring left-hand bowling and a taste for football. But culture
is more than cricket, and more than football, and more than extent of
knowledge.
Education is the acquisition of the art of the utilisation of
knowledge. This is an art very difficult to impart. Whenever a
text-book is written of real educational worth, you may be quite
certain that some reviewer will say that it will be difficult to
teach from it. Of course it will be difficult to teach from it. If it
were easy, the book ought to be burned; for it cannot be educational.
In education, as elsewhere, the broad primrose path leads to a nasty
place. This evil path is represented by a book or a set of lectures
which will practically enable the student to learn by heart all the
questions likely to be asked at the next external examination. And I
may say in passing that no educational system is possible unless every
question directly asked of a pupil at any examination is either framed
or modified by the actual teacher of that pupil in that subject. The
external assessor may report on the curriculum or on the performance
of the pupils, but never should be allowed to ask the pupil a question
which has not been strictly supervised by the actual teacher, or
at least inspired by a long conference with him. There are a few
exceptions to this rule, but they are exceptions, and could easily be
allowed for under the general rule.
We now return to my previous point, that theoretical ideas should
always find important applications within the pupil's curriculum. This
is not an easy doctrine to apply, but a very hard one. It contains
within itself the problem of keeping knowledge alive, of preventing it
from becoming inert, which is the central problem of all education.
The best procedure will depend on several factors, none of which can
be neglected, namely, the genius of the teacher, the intellectual type
of the pupils, their prospects in life, the opportunities offered by
the immediate surroundings of the school, and allied factors of this
sort. It is for this reason that the uniform external examination
is so deadly. We do not denounce it because we are cranks, and like
denouncing established things. We are not so childish. Also, of course,
such examinations have their use in testing slackness. Our reason of
dislike is very definite and very practical. It kills the best part of
culture. When you analyse in the light of experience the central task
of education, you find that its successful accomplishment depends on a
delicate adjustment of many variable factors. The reason is that we are
dealing with human minds, and not with dead matter. The evocation of
curiosity, of judgment, of the power of mastering a complicated tangle
of circumstances, the use of theory in giving foresight in special
cases--all these powers are not to be imparted by a set rule embodied
in one schedule of examination subjects.
I appeal to you, as practical teachers. With good discipline, it is
always possible to pump into the minds of a class a certain quantity of
inert knowledge. You take a text-book and make them learn it. So far,
so good. The child then knows how to solve a quadratic equation. But
what is the point of teaching a child to solve a quadratic equation?
There is a traditional answer to this question. It runs thus: The
mind is an instrument, you first sharpen it, and then use it; the
acquisition of the power of solving a quadratic equation is part of
the process of sharpening the mind. Now there is just enough truth in
this answer to have made it live through the ages. But for all its
half-truth, it embodies a radical error which bids fair to stifle the
genius of the modern world. I do not know who was first responsible
for this analogy of the mind to a dead instrument. For aught I know,
it may have been one of the seven wise men of Greece, or a committee
of the whole lot of them. Whoever was the originator, there can be no
doubt of the authority which it has acquired by the continuous approval
which it has received from eminent persons. But whatever its weight of
authority, whatever the high approval which it can quote, I have no
hesitation in denouncing it as one of the most fatal, erroneous, and
dangerous conceptions ever introduced into the theory of education. The
mind is never passive; it is a perpetual activity, delicate, receptive,
responsive to stimulus. You cannot postpone its life until you have
sharpened it. Whatever interest attaches to your subject-matter, must
be evoked here and now; whatever powers you are strengthening in the
pupil, must be exercised here and now; whatever possibilities of mental
life your teaching should impart, must be exhibited here and now. That
is the golden rule of education, and a very difficult rule to follow.
The difficulty is just this: the apprehension of general ideas,
intellectual habits of mind, and pleasurable interest in mental
achievement can be evoked by no form of words, however accurately
adjusted. All practical teachers know that education is a patient
process of the mastery of details, minute by minute, hour by hour, day
by day. There is no royal road to learning through an airy path of
brilliant generalisations. There is a proverb about the difficulty of
seeing the wood because of the trees. That difficulty is exactly the
point which I am enforcing. The problem of education is to make the
pupil see the wood by means of the trees.
The solution which I am urging, is to eradicate the fatal disconnection
of subjects which kills the vitality of our modern curriculum.
There is only one subject-matter for education, and that is Life
in all its manifestations. Instead of this single unity, we offer
children--Algebra, from which nothing follows; Geometry, from which
nothing follows; Science, from which nothing follows; History, from
which nothing follows; a Couple of Languages, never mastered; and
lastly, most dreary of all, Literature, represented by plays of
Shakespeare, with philological notes and short analyses of plot and
character to be in substance committed to memory. Can such a list be
said to represent Life, as it is known in the midst of the living of
it? The best that can be said of it is, that it is a rapid table of
contents which a deity might run over in his mind while he was thinking
of creating a world, and had not yet determined how to put it together.
Let us now return to quadratic equations. We still have on hand the
unanswered question. Why should children be taught their solution?
Unless quadratic equations fit into a connected curriculum, of course
there is no reason to teach anything about them. Furthermore, extensive
as should be the place of mathematics in a complete culture, I am a
little doubtful whether for many types of boys algebraic solutions of
quadratic equations do not lie on the specialist side of mathematics.
I may here remind you that as yet I have not said anything of the
psychology or the content of the specialism, which is so necessary a
part of an ideal education. But all that is an evasion of our real
question, and I merely state it in order to avoid being misunderstood
in my answer.
Quadratic equations are part of algebra, and algebra is the
intellectual instrument which has been created for rendering clear
the quantitative aspects of the world. There is no getting out of
it. Through and through the world is infected with quantity. To talk
sense, is to talk in quantities. It is no use saying that the nation
is large,--How large? It is no use saying that radium is scarce,--How
scarce? You cannot evade quantity. You may fly to poetry and to music,
and quantity and number will face you in your rhythms and your octaves.
Elegant intellects which despise the theory of quantity, are but half
developed. They are more to be pitied than blamed. The scraps of
gibberish, which in their school-days were taught to them in the name
of algebra, deserve some contempt.
This question of the degeneration of algebra into gibberish, both in
word and in fact, affords a pathetic instance of the uselessness of
reforming educational schedules without a clear conception of the
attributes which you wish to evoke in the living minds of the children.
A few years ago there was an outcry that school algebra was in need
of reform, but there was a general agreement that graphs would put
everything right. So all sorts of things were extruded, and graphs were
introduced. So far as I can see, with no sort of idea behind them,
but just graphs. Now every examination paper has one or two questions
on graphs. Personally, I am an enthusiastic adherent of graphs. But I
wonder whether as yet we have gained very much. You cannot put life
into any schedule of general education unless you succeed in exhibiting
its relation to some essential characteristic of all intelligent or
emotional perception. It is a hard saying, but it is true; and I do
not see how to make it any easier. In making these little formal
alterations you are beaten by the very nature of things. You are pitted
against too skilful an adversary, who will see to it that the pea is
always under the other thimble.
Reformation must begin at the other end. First, you must make up your
mind as to those quantitative aspects of the world which are simple
enough to be introduced into general education; then a schedule of
algebra should be framed which will about find its exemplification in
these applications. We need not fear for our pet graphs, they will
be there in plenty when we once begin to treat algebra as a serious
means of studying the world. Some of the simplest applications will be
found in the quantities which occur in the simplest study of society.
The curves of history are more vivid and more informing than the dry
catalogues of names and dates which comprise the greater part of
that arid school study. What purpose is effected by a catalogue of
undistinguished kings and queens? Tom, Dick, or Harry, they are all
dead. General resurrections are failures, and are better postponed.
The quantitative flux of the forces of modern society are capable
of very simple exhibition. Meanwhile, the idea of the variable, of
the function, of rate of change, of equations and their solution, of
elimination, are being studied as an abstract science for their own
sake. Not, of course, in the pompous phrases with which I am alluding
to them here, but with that iteration of simple special cases proper to
teaching.
If this course be followed, the route from Chaucer to the Black Death,
from the Black Death to modern Labour troubles, will connect the tales
of the mediæval pilgrims with the abstract science of algebra, both
yielding diverse aspects of that single theme, Life. I know what most
of you are thinking at this point. It is that the exact course which
I have sketched out is not the particular one which you would have
chosen, or even see how to work. I quite agree. I am not claiming that
I could do it myself. But your objection is the precise reason why a
common external examination system is fatal to education. The process
of exhibiting the applications of knowledge must, for its success,
essentially depend on the character of the pupils and the genius of the
teacher. Of course I have left out the easiest applications with which
most of us are more at home. I mean the quantitative sides of sciences,
such as mechanics and physics.
My meaning can be illustrated by looking more closely into a special
case of this type of application. In my rough catalogue of the sort
of subjects which should form the schedule for algebra, I mentioned
Elimination. It was not put there by accident, for it covers a very
important body of thought.
In the first place, there is the abstract process of algebraic
elimination for suitable simple cases. The pupil acquires a firm grasp
of this by the process, inevitable in education, of working an adequate
number of examples. Again, there are the graphical solutions of the
same problem. Then we consider the significance in the external world.
We consider the velocity, time, space, acceleration diagrams. We take
uniform acceleration; we eliminate "_t_" between
_v_ = _u_ + _ƒt_, and _s_ = _ut_ + ½_ƒt_^2,
and eliminate "_s_" between
_v_^2 = _u_^2 + 2_ƒs_, and _s_ = _ut_ + ½_ƒt_^2.
Then we remember that constant acceleration is a very special case,
and we consider graphical solutions or empirically given variations of
_v_ or of _ƒ_. In preference, we use those empirical formulæ
which occur in the pupil's experimental work. We compare the strong and
weak points of the algebraic and graphical solutions.
Again, in the same connection we plot the statistics of social
phenomena against the time. We then eliminate the time between suitable
pairs. We can speculate how far we have exhibited a real casual
connection, or how far a mere temporal coincidence. We notice that
we might have plotted against the time one set of statistics for one
country and another set for another country, and thus, with suitable
choice of subjects, have obtained graphs which certainly exhibited mere
coincidence. Also other graphs exhibit obvious casual connections. We
wonder how to discriminate. And so are drawn on as far as we will.
But in considering this description, I must beg you to remember what I
have been insisting on above. In the first place, one train of thought
will not suit all groups of children. For example, I should expect that
artisan children will want something more concrete and, in a sense,
swifter than I have set down here. Perhaps I am wrong, but that is
what I should guess. In the second place, I am not contemplating one
beautiful lecture stimulating, once and for all, an admiring class.
That is not the way in which education proceeds. No; all the time the
pupils are hard at work solving examples, drawing graphs, and making
experiments, until they have a thorough hold on the whole subject.
I am describing the interspersed explanations, the directions which
should be given to their thoughts. The pupils have got to be made to
feel that they are studying something, and are not merely executing
intellectual minuets.
In this connection the excellence of some of the most recent text-books
on elementary algebra emanating from members of this Association,
should create an epoch in the teaching of the subject.
Finally, if you are teaching pupils for some general examination,
the problem of sound teaching is greatly complicated. Have you ever
noticed the zig-zag moulding round a Norman arch? The ancient work is
beautiful, the modern work is hideous. The reason is, that the modern
work is done to exact measure, the ancient work is varied according
to the idiosyncrasy of the workman. Here it is crowded, and there it
is expanded. Now the essence of getting pupils through examinations
is to give equal weight to all parts of the schedule. But mankind is
naturally specialist. One man sees a whole subject, where another can
find only a few detached examples. I know that it seems contradictory
to allow for specialism in a curriculum especially designed for a
broad culture. Without contradictions the world would be simpler, and
perhaps duller. But I am certain that in education wherever you exclude
specialism you destroy life.
We now come to the other great branch of a general mathematical
education, namely Geometry. The same principles apply. The theoretical
part should be clear-cut, rigid, short, and important. Every
proposition not absolutely necessary to exhibit the main connection
of ideas should be cut out, but the great fundamental ideas should
be all there. No omission of concepts, such as those of Similarity
and Proportion. We must remember that, owing to the aid rendered by
the visual presence of a figure, Geometry is a field of unequalled
excellence for the exercise of the deductive faculties of reasoning.
Then, of course, there follows Geometrical Drawing, with its training
for the hand and eye.
But, like Algebra, Geometry and Geometrical Drawing must be extended
beyond the mere circle of geometrical ideas. In an industrial
neighbourhood, machinery and workshop practice form the appropriate
extension. For example, in the London Polytechnics this has been
achieved with conspicuous success. For many secondary schools I suggest
that surveying and maps are the natural applications. In particular,
plane-table surveying should lead pupils to a vivid apprehension of the
immediate application of geometric truths. Simple drawing apparatus, a
surveyor's chain, and a surveyor's compass, should enable the pupils
to rise from the survey and mensuration of a field to the construction
of the map of a small district. The best education is to be found
in gaining the utmost information from the simplest apparatus. The
provision of elaborate instruments is greatly to be deprecated. To
have constructed the map of a small district, to have considered its
roads, its contours, its geology, its climate, its relation to other
districts, the effects on the status of its inhabitants, will teach
more history and geography than any knowledge of Perkin Warbeck or
of Behren's Straits. I mean not a nebulous lecture on the subject,
but a serious investigation in which the real facts are definitely
ascertained by the aid of accurate theoretical knowledge. A typical
mathematical problem should be: Survey such and such a field, draw a
plan of it to such and such a scale, and find the area. It would be
quite a good procedure to impart the necessary geometrical propositions
without their proofs. Then, concurrently in the same term, the proofs
of the propositions would be learnt while the survey was being made.
Fortunately, the specialist side of education presents an easier
problem than does the provision of a general culture. For this there
are many reasons. One is that many of the principles of procedure
to be observed are the same in both cases, and it is unnecessary
to recapitulate. Another reason is that specialist training takes
place--or should take place--at a more advanced stage of the pupil's
course, and thus there is easier material to work upon. But undoubtedly
the chief reason is that the specialist study is normally a study
of peculiar interest to the student. He is studying it because, for
some reason, he wants to know it. This makes all the difference.
The general culture is designed to foster an activity of mind; the
specialist course utilises this activity. But it does not do to lay too
much stress on these neat antitheses. As we have already seen, in the
general course foci of special interest will arise; and similarly in
the special study, the external connections of the subject drag thought
outwards.
Again, there is not one course of study which merely gives general
culture, and another which gives special knowledge. The subjects
pursued for the sake of a general education are special subjects
specially studied; and, on the other hand, one of the ways of
encouraging general mental activity is to foster a special devotion.
You may not divide the seamless coat of learning. What education has
to impart is an intimate sense for the power of ideas, for the beauty
of ideas, and for the structure of ideas, together with a particular
body of knowledge which has peculiar reference to the life of the being
possessing it.
The appreciation of the structure of ideas is that side of a cultured
mind which can only grow under the influence of a special study. I
mean that eye for the whole chess-board, for the bearing of one set of
ideas on another. Nothing but a special study can give any appreciation
for the exact formulation of general ideas, for their relations when
formulated, for their service in the comprehension of life. A mind so
disciplined should be both more abstract and more concrete. It has been
trained in the comprehension of abstract thought and in the analysis of
facts.
Finally, there should grow the most austere of all mental qualities; I
mean the sense for style. It is an æsthetic sense, based on admiration
for the direct attainment of a foreseen end, simply and without waste.
Style in art, style in literature, style in science, style in logic,
style in practical execution have fundamentally the same æsthetic
qualities, namely, attainment and restraint. The love of a subject in
itself and for itself, where it is not the sleepy pleasure of pacing a
mental quarter-deck, is the love of style as manifested in that study.
Here we are brought back to the position from which we started,
the utility of education. Style, in its finest sense, is the last
acquirement of the educated mind; it is also the most useful. It
pervades the whole being. The administrator with a sense for style,
hates waste; the engineer with a sense for style, economises his
material; the artisan with a sense for style, prefers good work. Style
is the ultimate morality of mind.
But above style, and above knowledge, there is something, a vague
shape like fate above the Greek gods. That something is Power. Style
is the fashioning of power, the restraining of power. But, after all,
the power of attainment of the desired end is fundamental. The first
thing is to get there. Do not bother about your style, but solve your
problem, justify the ways of God to man, administer your province, or
do whatever else is set before you.
Where, then, does style help? In this, with style the end is attained
without side issues, without raising undesirable inflammations. With
style you attain your end and nothing but your end. With style the
effect of your activity is calculable, and foresight is the last gift
of gods to men. With style your power is increased, for your mind is
not distracted with irrelevancies, and you are more likely to attain
your object. Now style is the exclusive privilege of the expert.
Whoever heard of the style of an amateur painter, of the style of an
amateur poet? Style is always the product of specialist study, the
peculiar contribution of specialism to culture.
English education in its present phase suffers from a lack of definite
aim, and from an external machinery which kills its vitality. Hitherto
in this address I have been considering the aims which should govern
education. In this respect England halts between two opinions. It has
not decided whether to produce amateurs or experts. The profound change
in the world which the nineteenth century has produced is that the
growth of knowledge has given foresight. The amateur is essentially a
man with appreciation and with immense versatility in mastering a given
routine. But he lacks the foresight which comes from special knowledge.
The object of this address is to suggest how to produce the expert
without loss of the essential virtues of the amateur. The machinery
of our secondary education is rigid where it should be yielding,
and lax where it should be rigid. Every school is bound on pain of
extinction to train its boys for a small set of definite examinations.
No headmaster has a free hand to develop his general education or his
specialist studies in accordance with the opportunities of his school,
which are created by its staff, its environment, its class of boys, and
its endowments. I suggest that no system of external tests which aims
primarily at examining individual scholars can result in anything but
educational waste.
Primarily it is the schools and not the scholars which should be
inspected. Each school should grant its own leaving certificates, based
on its own curriculum. The standards of these schools should be sampled
and corrected. But the first requisite for educational reform is the
school as a unit, with its approved curriculum based on its own needs,
and evolved by its own staff. If we fail to secure that, we simply fall
from one formalism into another, from one dung-hill of inert ideas into
another.
In stating that the school is the true educational unit in any national
system for the safe-guarding of efficiency, I have conceived the
alternative system as being the external examination of the individual
scholar. But every Scylla is faced by its Charybdis--or, in more homely
language, there is a ditch on both sides of the road. It will be
equally fatal to education if we fall into the hands of a supervising
department which is under the impression that it can divide all schools
into two or three rigid categories, each type being forced to adopt
a rigid curriculum. When I say that the school is the educational
unit, I mean exactly what I say, no larger unit, no smaller unit.
Each school must have the claim to be considered in relation to its
special circumstances. The classifying of schools for some purposes is
necessary. But no absolutely rigid curriculum, not modified by its own
staff, should be permissible. Exactly the same principles apply, with
the proper modifications, to universities and to technical colleges.
When one considers in its length and in its breadth the importance
of this question of the education of a nation's young, the broken
lives, the defeated hopes, the national failures, which result from the
frivolous inertia with which it is treated, it is difficult to restrain
within oneself a savage rage. In the conditions of modern life the ride
is absolute, the race which does not value trained intelligence is
doomed. Not all your heroism, not all your social charm, not all your
wit, not all your victories on land or at sea, can move back the finger
of fate. To-day we maintain ourselves. To-morrow science will have
moved forward yet one more step, and there will be no appeal from the
judgment which will then be pronounced on the uneducated.
We can be content with no less than the old summary of educational
ideal which has been current at any time from the dawn of our
civilisation. The essence of education is that it be religious.
Pray, what is religious education?
A religious education is an education which inculcates duty and
reverence. Duty arises from our potential control over the course
of events. Where attainable knowledge could have changed the issue,
ignorance has the guilt of vice. And the foundation of reverence is
this perception, that the present holds within itself the complete sum
of existence, backwards and forwards, that whole amplitude of time,
which is eternity.
CHAPTER II
TECHNICAL EDUCATION AND ITS RELATION TO SCIENCE AND LITERATURE
(_Presidential Address to the Mathematical Association, January,
1917_)
THE subject of this address is Technical Education. I wish to examine
its essential nature and also its relation to a liberal education. Such
an inquiry may help us to realise the conditions for the successful
working of a national system of technical training. It is also a very
burning question among mathematical teachers; for mathematics is
included in most technological courses.
Now it is unpractical to plunge into such a discussion without framing
in our own minds the best ideal towards which we desire to work,
however modestly we may frame our hopes as to the result which in the
near future is likely to be achieved.
People are shy of ideals; and accordingly we find a formulation of the
ideal state of mankind placed by a modern dramatist[1] in the mouth of
a mad priest: "In my dreams it is a country where the State is the
Church and the Church the people: three in one and one in three. It is
a commonwealth in which work is play and play is life: three in one and
one in three. It is a temple in which the priest is the worshipper and
the worshipper the worshipped: three in one and one in three. It is a
godhead in which all life is human and all humanity divine: three in
one and one in three. It is, in short, the dream of a madman."
Now the part of this speech to which I would direct attention is
embodied in the phrase, "It is a commonwealth in which work is play
and play is life." This is the ideal of technical education. It sounds
very mystical when we confront it with the actual facts, the toiling
millions, tired, discontented, mentally indifferent, and then the
employers---- I am not undertaking a social analysis, but I shall carry
you with me when I admit that the present facts of society are a long
way off this ideal. Furthermore, we are agreed that an employer who
conducted his workshop on the principle that "work should be play"
would be ruined in a week.
The curse that has been laid on humanity, in fable and in fact, is,
that by the sweat of its brow shall it live. But reason and moral
intuition have seen in this curse the foundation for advance. The early
Benedictine monks rejoiced in their labours because they conceived
themselves as thereby made fellow-workers with Christ.
Stripped of its theological trappings, the essential idea remains,
that work should be transfused with intellectual and moral vision and
thereby turned into a joy, triumphing over its weariness and its pain.
Each of us will re-state this abstract formulation in a more concrete
shape in accordance with his private outlook. State it how you like,
so long as you do not lose the main point in your details. However you
phrase it, it remains the sole real hope of toiling humanity; and it
is in the hands of technical teachers, and of those who control their
spheres of activity, so to mould the nation that daily it may pass to
its labours in the spirit of the monks of old.
The immediate need of the nation is a large supply of skilled workmen,
of men with inventive genius, and of employers alert in the development
of new ideas.
There is one--and only one--way to obtain these admirable results.
It is by producing workmen, men of science, and employers who enjoy
their work. View the matter practically in the light of our knowledge
of average human nature. Is it likely that a tired, bored workman,
however skilful his hands, will produce a large output of first-class
work? He will limit his production, will scamp his work, and be an
adept at evading inspection; he will be slow in adapting himself to
new methods; he will be a focus of discontent, full of unpractical
revolutionary ideas, controlled by no sympathetic apprehension of the
real working of trade conditions. If, in the troubled times which may
be before us, you wish appreciably to increase the chance of some
savage upheaval, introduce widespread technical education and ignore
the Benedictine ideal. Society will then get what it deserves.
Again, inventive genius requires pleasurable mental activity as a
condition for its vigorous exercise. "Necessity is the mother of
invention" is a silly proverb. "Necessity is the mother of futile
dodges" is much nearer to the truth. The basis of the growth of modern
invention is science, and science is almost wholly the outgrowth of
pleasurable intellectual curiosity.
The third class are the employers, who are to be enterprising. Now
it is to be observed that it is the successful employers who are the
important people to get at, the men with business connections all over
the world, men who are already rich. No doubt there will always be a
continuous process of rise and fall of businesses. But it is futile
to expect flourishing trade, if in the mass the successful houses of
business are suffering from atrophy. Now if these men conceive their
businesses as merely indifferent means for acquiring other disconnected
opportunities of life, they have no spur to alertness. They are
already doing very well, the mere momentum of their present business
engagements will carry them on for their time. They are not at all
likely to bother themselves with the doubtful chances of new methods.
Their real soul is in the other side of their life. Desire for money
will produce hard-fistedness and not enterprise. There is much more
hope for humanity from manufacturers who enjoy their work than from
those who continue in irksome business with the object of founding
hospitals.
Finally, there can be no prospect of industrial peace so long as
masters and men in the mass conceive themselves as engaged in a
soulless operation of extracting money from the public. Enlarged views
of the work performed, and of the communal service thereby rendered,
can be the only basis on which to found sympathetic co-operation.
The conclusion to be drawn from this discussion is, that alike for
masters and for men a technical or technological education, which is to
have any chance of satisfying the practical needs of the nation, must
be conceived in a liberal spirit as a real intellectual enlightenment
in regard to principles applied and services rendered. In such an
education geometry and poetry are as essential as turning lathes.
The mythical figure of Plato may stand for modern liberal education
as does that of St. Benedict for technical education. We need not
entangle ourselves in the qualifications necessary for a balanced
representation of the actual thoughts of the actual men. They are used
here as symbolic figures typical of antithetical notions. We consider
Plato in the light of the type of culture he now inspires.
In its essence a liberal education is an education for thought and
for æsthetic appreciation. It proceeds by imparting a knowledge of
the masterpieces of thought, of imaginative literature, and of art.
The action which it contemplates is command. It is an aristocratic
education implying leisure. This Platonic ideal has rendered
imperishable services to European civilisation. It has encouraged art,
it has fostered that spirit of disinterested curiosity which is the
origin of science, it has maintained the dignity of mind in the face of
material force, a dignity which claims freedom of thought. Plato did
not, like St. Benedict, bother himself to be a fellow-worker with his
slaves; but he must rank among the emancipators of mankind. His type
of culture is the peculiar inspiration of the liberal aristocrat, the
class from which Europe derives what ordered liberty it now possesses.
For centuries, from Pope Nicholas V to the schools of the Jesuits, and
from the Jesuits to the modern headmasters of English public schools,
this educational ideal has had the strenuous support of the clergy.
For certain people it is a very good education. It suits their type
of mind and the circumstances amid which their life is passed. But
more has been claimed for it than this. All education has been judged
adequate or defective according to its approximation to this sole type.
The essence of the type is a large discursive knowledge of the best
literature. The ideal product of the type is the man who is acquainted
with the best that has been written. He will have acquired the chief
languages, he will have considered the histories of the rise and fall
of nations, the poetic expression of human feeling, and have read the
great dramas and novels. He will also be well grounded in the chief
philosophies, and have attentively read those philosophic authors who
are distinguished for lucidity of style.
It is obvious that, except at the close of a long life, he will not
have much time for anything else if any approximation is to be made to
the fulfilment of this programme. One is reminded of the calculation
in a dialogue of Lucian that, before a man could be justified in
practising any one of the current ethical systems, he should have spent
a hundred and fifty years in examining their credentials.
Such ideals are not for human beings. What is meant by a liberal
culture is nothing so ambitious as a full acquaintance with the varied
literary expression of civilised mankind from Asia to Europe, and from
Europe to America. A small selection only is required; but then, as
we are told, it is a selection of the very best. I have my doubts of
a selection which includes Xenophon and omits Confucius, but then I
have read neither in the original. The ambitious programme of a liberal
education really shrinks to a study of some fragments of literature
included in a couple of important languages.
But the expression of the human spirit is not confined to literature.
There are the other arts, and there are the sciences. Also education
must pass beyond the passive reception of the ideas of others. Powers
of initiative must be strengthened. Unfortunately initiative does not
mean just one acquirement--there is initiative in thought, initiative
in action, and the imaginative initiative of art; and these three
categories require many subdivisions.
The field of acquirement is large, and the individual so fleeting and
so fragmentary: classical scholars, scientists, headmasters are alike
ignoramuses.
There is a curious illusion that a more complete culture was possible
when there was less to know. Surely the only gain was, that it was more
possible to remain unconscious of ignorance. It cannot have been a gain
to Plato to have read neither Shakespeare, nor Newton, nor Darwin.
The achievements of a liberal education have in recent times not been
worsened. The change is that its pretensions have been found out.
My point is, that no course of study can claim any position of ideal
completeness. Nor are the omitted factors of subordinate importance.
The insistence in the Platonic culture on disinterested intellectual
appreciation is a psychological error. Action and our implication in
the transition of events amid the evitable bond of cause to effect
are fundamental. An education which strives to divorce intellectual
or æsthetic life from these fundamental facts carries with it the
decadence of civilisation. Essentially culture should be for action,
and its effect should be to divest labour from the associations of
aimless toil. Art exists that we may know the deliverances of our
senses as good. It heightens the sense-world.
Disinterested scientific curiosity is a passion for an ordered
intellectual vision of the connection of events. But the goal of
such curiosity is the marriage of action to thought. This essential
intervention of action even in abstract science is often overlooked. No
man of science wants merely to know. He acquires knowledge to appease
his passion for discovery. He does not discover in order to know, he
knows in order to discover. The pleasure which art and science can
give to toil is the enjoyment which arises from successfully directed
intention. Also it is the same pleasure which is yielded to the
scientist and to the artist.
The antithesis between a technical and a liberal education is
fallacious. There can be no adequate technical education which is not
liberal, and no liberal education which is not technical: that is, no
education which does not impart both technique and intellectual vision.
In simpler language, education should turn out the pupil with something
he knows well and something he can do well. This intimate union of
practice and theory aids both. The intellect does not work best in a
vacuum. The stimulation of creative impulse requires, especially in
the case of a child, the quick transition to practice. Geometry and
mechanics, followed by workshop practice, gain that reality without
which mathematics is verbiage.
There are three main methods which are required in a national system of
education, namely, the literary curriculum, the scientific curriculum,
the technical curriculum. But each of these curricula should include
the other two. What I mean is, that every form of education should give
the pupil a technique, a science, an assortment of general ideas, and
æsthetic appreciation, and that each of these sides of his training
should be illuminated by the others. Lack of time, even for the most
favoured pupil, makes it impossible to develop fully each curriculum.
Always there must be a dominant emphasis. The most direct æsthetic
training naturally falls in the technical curriculum in those cases
when the training is that requisite for some art or artistic craft. But
it is of high importance in both a literary and a scientific education.
The educational method of the literary curriculum is the study of
language, that is, the study of our most habitual method of conveying
to others our states of mind. The technique which should be acquired
is the technique of verbal expression, the science is the study of the
structure of language and the analysis of the relations of language
to the states of mind conveyed. Furthermore, the subtle relations of
language to feeling, and the high development of the sense organs
to which written and spoken words appeal, lead to keen æsthetic
appreciations being aroused by the successful employment of language.
Finally, the wisdom of the world is preserved in the masterpieces of
linguistic composition.
This curriculum has the merit of homogeneity. All its various parts
are co-ordinated and play into each other's hands. We can hardly be
surprised that such a curriculum, when once broadly established, should
have claimed the position of the sole perfect type of education. Its
defect is unduly to emphasise the importance of language. Indeed the
varied importance of verbal expression is so overwhelming that its
sober estimation is difficult. Recent generations have been witnessing
the retreat of literature, and of literary forms of expression, from
their position of unique importance in intellectual life. In order
truly to become a servant and a minister of nature something more is
required than literary aptitudes.
A scientific education is primarily a training in the art of observing
natural phenomena, and in the knowledge and deduction of laws
concerning the sequence of such phenomena. But here, as in the case of
a liberal education, we are met by the limitations imposed by shortness
of time. There are many types of natural phenomena, and to each type
there corresponds a science with its peculiar modes of observation,
and its peculiar types of thought employed in the deduction of laws. A
study of science in general is impossible in education, all that can be
achieved is the study of two or three allied sciences. Hence the charge
of narrow specialism urged against any education which is primarily
scientific. It is obvious that the charge is apt to be well-founded;
and it is worth considering how, within the limits of a scientific
education and to the advantage of such an education, the danger can be
avoided.
Such a discussion requires the consideration of technical education. A
technical education is in the main a training in the art of utilising
knowledge for the manufacture of material products. Such a training
emphasises manual skill, and the co-ordinated action of hand and
eye, and judgment in the control of the process of construction. But
judgment necessitates knowledge of those natural processes of which the
manufacture is the utilisation. Thus somewhere in technical training
an education in scientific knowledge is required. If you minimise
the scientific side, you will confine it to the scientific experts;
if you maximise it, you will impart it in some measure to the men,
and--what is of no less importance--to the directors and managers of
the businesses.
Technical education is not necessarily allied exclusively to science
on its mental side. It may be an education for an artist or for
apprentices to an artistic craft. In that case æsthetic appreciation
will have to be cultivated in connection with it.
An evil side of the Platonic culture has been its total neglect of
technical education as an ingredient in the complete development
of ideal human beings. This neglect has arisen from two disastrous
antitheses, namely, that between mind and body, and that between
thought and action. I will here interject, solely to avoid criticism,
that I am well aware that the Greeks highly valued physical beauty and
physical activity. They had, however, that perverted sense of values
which is the nemesis of slave-owning.
I lay it down as an educational axiom that in teaching you will come
to grief as soon as you forget that your pupils have bodies. This is
exactly the mistake of the post-renaissance Platonic curriculum. But
nature can be kept at bay by no pitchfork; so in English education,
being expelled from the classroom, she returned with a cap and bells in
the form of all-conquering athleticism.
The connections between intellectual activity and the body, though
diffused in every bodily feeling, are focussed in the eyes, the ears,
the voice, and the hands. There is a co-ordination of senses and
thought, and also a reciprocal influence between brain activity and
material creative activity. In this reaction the hands are peculiarly
important. It is a moot point whether the human hand created the human
brain, or the brain created the hand. Certainly the connection is
intimate and reciprocal. Such deep-seated relations are not widely
atrophied by a few hundred years of disuse in exceptional families.
The disuse of hand-craft is a contributory cause to the brain-lethargy
of aristocracies, which is only mitigated by sport where the concurrent
brain-activity is reduced to a minimum and the hand-craft lacks
subtlety. The necessity for constant writing and vocal exposition is
some slight stimulus to the thought-power of the professional classes.
Great readers, who exclude other activities, are not distinguished by
subtlety of brain. They tend to be timid conventional thinkers. No
doubt this is partly due to their excessive knowledge outrunning their
powers of thought; but it is partly due to the lack of brain-stimulus
from the productive activities of hand or voice.
In estimating the importance of technical education we must rise above
the exclusive association of learning with book-learning. First-hand
knowledge is the ultimate basis of intellectual life. To a large
extent book-learning conveys second-hand information, and as such can
never rise to the importance of immediate practice. Our goal is to see
the immediate event of our lives as instances of our general ideas.
What the learned world tends to offer is one second-hand scrap of
information illustrating ideas derived from another second-hand scrap
of information. The second-handedness of the learned world is the
secret of its mediocrity. It is tame because it has never been scared
by facts. The main importance of Francis Bacon's influence does not
lie in any peculiar theory of inductive reasoning which he happened to
express, but in the revolt against second-hand information of which he
was a leader.
The peculiar merit of a scientific education should be, that it bases
thought upon first-hand observation; and the corresponding merit of a
technical education is, that it follows our deep natural instinct to
translate thought into manual skill, and manual activity into thought.
We are a Mathematical Association, and it is natural to ask: Where do
we come in? We come in just at this point.
The thought which science evokes is logical thought. Now logic is of
two kinds: the logic of discovery and the logic of the discovered.
The logic of discovery consists in the weighing of probabilities, in
discarding details deemed to be irrelevant, in divining the general
rules according to which events occur, and in testing hypotheses by
devising suitable experiments. This is inductive logic.
The logic of the discovered is the deduction of the special events
which, under certain circumstances, would happen in obedience to the
assumed laws of nature. Thus when the laws are discovered or assumed,
their utilisation entirely depends on deductive logic. Without
deductive logic science would be entirely useless. It is merely a
barren game to ascend from the particular to the general, unless
afterwards we can reverse the process and descend from the general to
the particular, ascending and descending like the angels on Jacob's
ladder. When Newton had divined the law of gravitation he at once
proceeded to calculate the earth's attractions on an apple at its
surface and on the moon. We may note in passing that inductive logic
would be impossible without deductive logic. Thus Newton's calculations
were an essential step in his inductive verification of the great law.
Now mathematics is nothing else than the more complicated parts of
the art of deductive reasoning, especially where it concerns number,
quantity, and space.
In the teaching of science, the art of thought should be taught:
namely, the art of forming clear conceptions applying to first-hand
experience, the art of divining the general truths which apply, the
art of testing divinations, and the art of utilising general truths
by reasoning to more particular cases of some peculiar importance.
Furthermore, a power of scientific exposition is necessary, so that the
relevant issues from a confused mass of ideas can be stated clearly,
with due emphasis on important points.
By the time a science, or a small group of sciences, has been taught
thus amply, with due regard to the general art of thought, we have gone
a long way towards correcting the specialism of science. The worst
of a scientific education based, as necessarily must be the case, on
one or two particular branches of science, is that the teachers under
the influence of the examination system are apt merely to stuff
their pupils with the narrow results of these special sciences. It is
essential that the generality of the method be continually brought to
light and contrasted with the speciality of the particular application.
A man who only knows his own science, as a routine peculiar to that
science, does not even know that. He has no fertility of thought, no
power of quickly seizing the bearing of alien ideas. He will discover
nothing, and be stupid in practical applications.
This exhibition of the general in the particular is extremely difficult
to effect, especially in the case of younger pupils. The art of
education is never easy. To surmount its difficulties, especially those
of elementary education, is a task worthy of the highest genius. It is
the training of human souls.
Mathematics, well taught, should be the most powerful instrument
in gradually implanting this generality of idea. The essence of
mathematics is perpetually to be discarding more special ideas in
favour of more general ideas, and special methods in favour of general
methods. We express the conditions of a special problem in the form of
an equation, but that equation will serve for a hundred other problems,
scattered through diverse sciences. The general reasoning is always
the powerful reasoning, because deductive cogency is the property of
abstract form.
Here, again, we must be careful. We shall ruin mathematical education
if we use it merely to impress general truths. The general ideas are
the means of connecting particular results. After all, it is the
concrete special cases which are important. Thus in the handling of
mathematics in your results you cannot be too concrete, and in your
methods you cannot be too general. The essential course of reasoning
is to generalise what is particular, and then to particularise what is
general. Without generality there is no reasoning, without concreteness
there is no importance.
Concreteness is the strength of technical education. I would
remind you that truths which lack the highest generality are not
necessarily concrete facts. For example, _x_ + _y_ = _y_ + _x_ is an
algebraic truth more general than 2 + 2 = 4. But "two and two make
four" is itself a highly general proposition lacking any element of
concreteness. To obtain a concrete proposition immediate intuition of a
truth concerning particular objects is requisite; for example, "these
two apples and those apples together make four apples" is a concrete
proposition, if you have direct perception or immediate memory of the
apples.
In order to obtain the full realisation of truths as applying, and not
as empty formulæ, there is no alternative to technical education. Mere
passive observation is not sufficient. In creation only is there vivid
insight into the properties of the object thereby produced. If you
want to understand anything, make it yourself, is a sound rule. Your
faculties will be alive, your thoughts gain vividness by an immediate
translation into acts. Your ideas gain that reality which comes from
seeing the limits of their application.
In elementary education this doctrine has long been put into practice.
Young children are taught to familiarise themselves with shapes and
colours by simple manual operations of cutting out and of sorting. But
good though this is, it is not quite what I mean. That is practical
experience before you think, experience antecedent to thought in order
to create ideas, a very excellent discipline. But technical education
should be much more than that: it is creative experience while you
think, experience which realises your thought, experience which teaches
you to co-ordinate act and thought, experience leading you to associate
thought with foresight and foresight with achievement. Technical
education gives theory, and a shrewd insight as to where theory fails.
A technical education is not to be conceived as a maimed alternative
to the perfect Platonic culture: namely, as a defective training
unfortunately made necessary by cramped conditions of life. No
human being can attain to anything but fragmentary knowledge and a
fragmentary training of his capacities. There are, however, three main
roads along which we can proceed with good hope of advancing towards
the best balance of intellect and character: these are the way of
literary culture, the way of scientific culture, the way of technical
culture. No one of these methods can be exclusively followed without
grave loss of intellectual activity and of character. But a mere
mechanical mixture of the three curricula will produce bad results in
the shape of scraps of information never interconnected or utilised.
We have already noted as one of the strong points of the traditional
literary culture that all its parts are co-ordinated. The problem
of education is to retain the dominant emphasis, whether literary,
scientific, or technical, and without loss of co-ordination to infuse
into each way of education something of the other two.
To make definite the problem of technical education fix attention on
two ages: one thirteen, when elementary education ends; and the other
seventeen, when technical education ends so far as it is compressed
within a school curriculum. I am aware that for artisans in junior
technical schools a three-years' course would be more usual. On the
other hand, for naval officers, and for directing classes generally,
a longer time can be afforded. We want to consider the principles
to govern a curriculum which shall land these children at the age of
seventeen in the position of having technical skill useful to the
community.
Their technical manual training should start at thirteen, bearing a
modest proportion to the rest of their work, and should increase in
each year finally to attain to a substantial proportion. Above all
things it should not be too specialised. Workshop finish and workshop
dodges, adapted to one particular job, should be taught in the
commercial workshop, and should form no essential part of the school
course. A properly trained worker would pick them up in no time. In all
education the main cause of failure is staleness. Technical education
is doomed if we conceive it as a system for catching children young and
for giving them one highly specialised manual aptitude. The nation has
need of a fluidity of labour, not merely from place to place, but also
within reasonable limits of allied aptitudes, from one special type of
work to another special type. I know that here I am on delicate ground,
and I am not claiming that men while they are specialising on one sort
of work should spasmodically be set to other kinds. That is a question
of trade organisation with which educationalists have no concern. I am
only asserting the principles that training should be broader than the
ultimate specialisation, and that the resulting power of adaptation to
varying demands is advantageous to the workers, to the employers, and
to the nation.
In considering the intellectual side of the curriculum we must be
guided by the principle of the co-ordination of studies. In general,
the intellectual studies most immediately related to manual training
will be some branches of science. More than one branch will, in fact,
be concerned; and even if that be not the case, it is impossible to
narrow down scientific study to a single thin line of thought. It is
possible, however, provided that we do not press the classification
too far, roughly to classify technical pursuits according to the
dominant science involved. We thus find a sixfold division, namely,
(1) Geometrical techniques, (2) Mechanical techniques, (3) Physical
techniques, (4) Chemical techniques, (5) Biological techniques, (6)
Techniques of commerce and of social service.
By this division, it is meant that apart from auxiliary sciences
some particular science requires emphasis in the training for most
occupations. We can, for example, reckon carpentry, ironmongery,
and many artistic crafts among geometrical techniques. Similarly
agriculture is a biological technique. Probably cookery, if it includes
food catering, would fall midway between biological, physical, and
chemical sciences, though of this I am not sure.
The sciences associated with commerce and social service would be
partly algebra, including arithmetic and statistics, and partly
geography and history. But this section is somewhat heterogeneous in
its scientific affinities. Anyhow the exact way in which technical
pursuits are classified in relation to science is a detail. The
essential point is, that with some thought it is possible to find
scientific courses which illuminate most occupations. Furthermore, the
problem is well understood, and has been brilliantly solved in many of
the schools of technology and junior technical schools throughout the
country.
In passing from science to literature, in our review of the
intellectual elements of technical education, we note that many studies
hover between the two: for example, history and geography. They are
both of them very essential in education, provided that they are the
right history and the right geography. Also books giving descriptive
accounts of general results, and trains of thought in various sciences
fall in the same category. Such books should be partly historical
and partly expository of the main ideas which have finally arisen.
Prof. R. A. Gregory's recent book, _Discovery_, and the _Home
University Library_ series illustrate my meaning. Their value in
education depends on their quality as mental stimulants. They must not
be inflated with gas on the wonders of science, and must be informed
with a broad outlook.
It is unfortunate that the literary element in education has rarely
been considered apart from grammatical study. The historical reason is,
that when the modern Platonic curriculum was being formed Latin and
Greek were the sole keys which rendered great literature accessible.
But there is no necessary connection between literature and grammar.
The great age of Greek literature was already past before the arrival
of the grammarians of Alexandria. Of all types of men to-day existing,
classical scholars are the most remote from the Greeks of the Periclean
times.
Mere literary knowledge is of slight importance. The only thing that
matters is, how it is known. The facts related are nothing. Literature
only exists to express and develop that imaginative world which is our
life, the kingdom which is within us. It follows that the literary side
of a technical education should consist in an effort to make the pupils
enjoy literature. It does not matter what they know, but the enjoyment
is vital. The great English Universities, under whose direct authority
school-children are examined in plays of Shakespeare, to the certain
destruction of their enjoyment, should be prosecuted for soul-murder.
Now there are two kinds of intellectual enjoyment: the enjoyment of
creation, and the enjoyment of relaxation. They are not necessarily
separated. A change of occupation may give the full tide of happiness
which comes from the concurrence of both forms of pleasure. The
appreciation of literature is really creation. The written word, its
music, and its associations, are only the stimuli. The vision which
they evoke is our own doing. No one, no genius other than our own,
can make our own life live. But except for those engaged in literary
occupations, literature is also a relaxation. It gives exercise to that
other side which any occupation must suppress during the working hours.
Art also has the same function in life as has literature.
To obtain the pleasure of relaxation requires no help. The pleasure
is merely to cease doing. Some such pure relaxation is a necessary
condition of health. Its dangers are notorious, and to the greater
part of the necessary relaxation nature has affixed, not enjoyment,
but the oblivion of sleep. Creative enjoyment is the outcome of
successful effort and requires help for its initiation. Such enjoyment
is necessary for high-speed work and for original achievement.
To speed up production with unrefreshed workmen is a disastrous
economic policy. Temporary success will be at the expense of the
nation, which, for long years of their lives, will have to support
worn-out artisans--unemployables. Equally disastrous is the
alternation of spasms of effort with periods of pure relaxation.
Such periods are the seed-times of degeneration, unless rigorously
curtailed. The normal recreation should be change of activity,
satisfying the cravings of instincts. Games afford such activity. Their
disconnection emphasises the relaxation, but their excess leaves us
empty.
It is here that literature and art should play an essential part in
a healthily organised nation. Their services to economic production
would be only second to those of sleep or of food. I am not now talking
of the training of an artist, but of the use of art as a condition of
healthy life. It is analogous to sunshine in the physical world.
When we have once rid our minds of the idea that knowledge is to
be exacted, there is no especial difficulty or expense involved in
helping the growth of artistic enjoyment. All school-children could
be sent at regular intervals to neighbouring theatres where suitable
plays could be subsidised. Similarly for concerts and cinema films.
Pictures are more doubtful in their popular attraction; but interesting
representations of scenes or ideas which the children have read about
would probably appeal. The pupils themselves should be encouraged
in artistic efforts. Above all the art of reading aloud should be
cultivated. The Roger de Coverley essays of Addison are perfect
examples of readable prose.
Art and literature have not merely an indirect effect on the main
energies of life. Directly, they give vision. The world spreads wide
beyond the deliverances of material sense, with subtleties of reaction
and with pulses of emotion. Vision is the necessary antecedent to
control and to direction. In the contest of races which in its final
issues will be decided in the workshops and not on the battle-field,
the victory will belong to those who are masters of stores of trained
nervous energy, working under conditions favourable to growth. One such
essential condition is Art.
If there had been time, there are other things which I should like
to have said: for example, to advocate the inclusion of one foreign
language in all education. From direct observation I know this to be
possible for artisan children. But enough has been put before you,
to make plain the principles with which we should undertake national
education.
In conclusion, I recur to the thought of the Benedictines, who
saved for mankind the vanishing civilisation of the ancient world
by linking together knowledge, labour, and moral energy. Our danger
is to conceive practical affairs as the kingdom of evil, in which
success is only possible by the extrusion of ideal aims. I believe
that such a conception is a fallacy directly negatived by practical
experience. In education this error takes the form of a mean view of
technical training. Our forefathers in the dark ages saved themselves
by embodying high ideals in great organisations. It is our task,
without servile imitation, boldly to exercise our creative energies,
remembering amid discouragements that the coldest hour immediately
precedes the dawn.
FOOTNOTES:
[Footnote 1: _Cf._ BERNARD SHAW: _John Bull's Other
Island_.]
CHAPTER III
A POLYTECHNIC IN WAR-TIME
_Address at the Prize Distribution, Borough Polytechnic Institute,
Southwark, 16th February, 1917_
I WILL commence by drawing your attention to some of the satisfactory
features of the Principal's report on the work of the Institute during
the past year. It has been a year of great difficulties. Some of our
staff are serving with the colours, and our classes have been depleted.
But in spite of everything, we have done very well. First, the average
result in the examinations has been good, surprisingly good in view of
the present circumstances. The Governors attach great importance to
the maintenance of a high average result; it is the best single test
of efficiency. Again, our individual successes have been notable. We
have gained--I say _we_ because we are all one in our pleasure at
these successes--we have gained two £80 L.C.C. scholarships, nineteen
exhibitions, in addition to a first-place, and medals, prizes and
certificates. All this is very satisfactory. It tells of efficient
teaching, and of hard work and regular attendance on the part of the
students. We know that we are keeping up the standard of efficiency
which in the past has been a source of pride to every one connected
with this Institute.
Now all this good work does not come about by itself without any one
making an effort. Such a record requires our skilled staff of teachers
and organisers. They have worked very hard during the last session
under great difficulties, in order to create the successful result
which we are here to celebrate. I know something about teaching. It
is very exacting work, and can be made successful only by continual
devotion. I am sure that I am voicing your feelings, and I know that
I am expressing those of the Governors, when I thank the ladies and
gentlemen of the staff very heartily for their services during the last
session.
Prize-givings are always pleasant occasions. We have come here to think
about our successes, and to congratulate our students. There is no more
satisfactory Governors' Meeting in the course of the year than when we
meet on this occasion, and face our friends and tell them how pleased
we are at the successful result of their hard work. This evening I am
in a doubly happy position, for my colleagues have asked me to be their
spokesman in tendering our good wishes to the prize-winners. You have
worked hard and you have done well, and I am sure that you all deserve
your successes; they are a pleasure not only to you, but in your homes
and to your companions and fellow-students.
Successful work here will enable you to acquire skill in your trades,
and thereby the better to earn your living. Earning a living is on the
average no bad test of service rendered to the community. A man who
has made himself skilful in his trade and has done well for himself in
his walk of life, has in general good reason to believe that he is a
citizen who has benefited his country. It is an evil day for a nation
when it loses respect for success in industry.
But if you steer your lives by the compass which points steadily to
the North Pole of personal success, you will have missed your greatest
chances in life. The genial climate is in the south.
What I mean is this: you must make up your mind to find the best part
of your happiness in kindly helpful relations with others. It should be
our ambition to leave our own small corner of the world a little tidier
and a little happier than when we entered it. I am well aware that
this is an old story; but old stories are sometimes true, and this is
the biggest truth in the whole world. The warm kindly feelings are the
happy feelings. The fortunate people are those whose minds are filled
with thoughts in which they forget themselves and remember others. It
is not true that nature is a mere scene of struggle in which every one
competes with his neighbour. Those communities thrive best and last
longest which are filled with a spirit of mutual help.
The future of the country lies with you. The crown of your success
is the promise of future work, often unrecognised work, done under
discouragement, but done steadily and cheerfully. It is on you that
the country depends for the maintenance and the growth of those ideals
without which a race withers. Do not be discouraged by difficulties
which seem unsurmountable. The conditions of life which mould us all
are modified by our will, by our energy, and by the purity of our
intentions.
If we may judge of intensity of feeling by length of memory, the
enjoyment of receiving a prize bites very deep. Across the space of
more than forty years, before many of your parents were born, or when
they were being carried about in long clothes, I can remember, as if it
were yesterday, the occasion when I received my first prize at school.
I can see the mediæval school-room, the headmaster, and my companions.
Perhaps some of you, when a generation has passed by, will remember the
scene to-night--this Stanley Gymnasium recalling the memory of Miss
Maude Stanley, who devoted to our welfare so much of her energy and
her thought--the adjoining Edric Hall associated with the name of Mr.
Edric Bayley, the Father of the Institute; Mr. Millis and Miss Smith,
the first Principal and the first Lady Superintendent, the architects
of our prosperity; Mr. Leonard Spicer, our Chairman and member of a
family and of a firm known throughout the world, and respected in
proportion as they are known. And the cause why to-night we are a small
gathering is one more reason why this assembly can never slip from your
memory. We meet at a moment when England stands in as deadly a peril as
in any previous moment of her history--such peril as when the Spanish
ships of the Armada rode in the English Channel, or when Napoleon
watched our coast across the Strait of Dover. The present danger can be
overcome only by the same courage as that which saved our freedom in
those former times.
Therefore, to-night, in recalling the activities of the various
sections of our society which form this great Polytechnic Institute,
our thoughts go further afield. They travel by land and by sea, till
they bring before our minds the gallant band whom this Institute has
sent to the Front--more than 800 of our members are with the Colours.
What our fighting men have done for us, for the world in general, and
for the future of England, is so overwhelming that words cannot praise
them enough. I will just say one thing to you: When you read of great
deeds done in past times, of perils encountered, of adventures, of
undaunted courage, of patriotism, of self-sacrifice, of suffering
endured for noble cause, you each can say--I, too, have known such
heroes; they are among my countrymen, they are among my fellow-workers,
they are among my fellow-students and companions, they are among the
dear inmates of my home. And for those who have fallen, it is for us
to erect a monument sufficient to transmit to future ages the memory
of their sacrifice. For this purpose there is only one memorial which
can suffice, namely, the cause for which they died. The greatness of
England, the future of England, has been left by them to our keeping.
Guard it well.
The greatness of a country is nothing else than the greatness of the
lives of the men and of the women who compose it. Do not look round and
think who ought to be great Englishmen--be great yourselves--you are
the people to achieve it, you who are sitting here to-night. There can
be no substitute service for this purpose. It is the collective energy
of the whole people that will be needed to fashion a new England worthy
of the sufferings which for its sake have been endured.
A few days ago I asked a man who has worked in Egypt for many years
under Lord Kitchener, what he would pick out as the best sign of Lord
Kitchener's greatness. He answered, whatever Kitchener set himself to
do, thereby became important. Now that is the secret of it all--take
hold of your opportunities and make them important.
Here we are in this Borough Polytechnic. What an opportunity it
represents. This Institute is a centre for social meeting, a centre
for recreation, a centre for education, a centre for discussion. We
will not sacrifice any one of our sides. They must all be part of the
greatness which we claim. Make them all first-rate.
Consider first the social and recreative sides. For heaven's sake don't
think that you must be dull in order to be great. There is no finer
test of a nation than the way in which it fashions its amusements.
Three centuries ago after the Armada we made a good start in Southwark.
Shakespeare had his theatre here and wrote his plays to be acted in
this borough. He has walked these streets, and if you had met him in
Westminster he would, quite likely, have told you that he was going
down to the "Elephant." And even now the performances given at the
"Old Vic" are among the best in London for the purpose of seeing his
plays properly acted. What Southwark has done for the drama, she can
do for the other arts, by using this Institute as the instrument for
her energies. Why should we not be a centre for artistic enterprise--I
mean for our own art and our own enterprise, thought of by ourselves
and enjoyed by ourselves and carried through by ourselves? We shall not
always enjoy each others' creations, but the great point is to make our
own efforts. Of course all efforts require preparation and stimulus and
knowledge of what others are doing.
At the present time--interrupted for the moment by the war--a great
revolution in the art of painting is in progress throughout the world.
Its centres are Paris and Italy and London and Munich, and its origin
in the far east, in China and Japan. There are two sides as in every
revolution, the Conservatives and the Revolutionists. Our own frescoes
in a neighbouring room represent an early stage of the movement in
London. Why should we not know all about it--obtain loans of pictures
which illustrate its phases and its cross currents, and compare these
with examples of the old style?
But pictures are only one phase of art, and not the sort of art which
we ourselves can produce most easily. There are music, dancing,
recitation, literature, carving and modelling, and the various
decorative arts, such as embroidery, bookbinding, dress-making and
upholstery. This list, incomplete as it is, tells us two great
truths--you cannot separate art and recreation, and you cannot
separate art and business. The list includes items which we consider
as amusements, and items which we think of as business. We began with
dancing and ended with upholstery. Make them all beautiful.
Beautiful things have dignity. Enjoy the rhythm of your dancing and
admire the beauty of your embroidery or your bookbinding. In whatever
you do, have an ideal of excellence. Any separation between art and
work is not only an error, but it is very bad business. Our brave
allies, the French, have made Paris the art centre of the world. They
have built up and maintain their large and lucrative trade in the
decorative products of France, mainly by reason of three qualities
which they possess. In the first place, they enjoy art themselves, and
reverence it. In the second place, they have a tremendous power of hard
work. And in the third place, every Frenchman, and still more every
Frenchwoman, have within them an immense fund of common sense. The
threefold secret is, Love of Art, Industry, and Common Sense.
To make available our industry and common sense in the trades where
they are wanted, rigorous training in schools of design and technique
are necessary. We have such departments here. But all such training of
you will be a failure unless you yourselves enjoy art and beauty as
a natural recreation. A technical school of training is like a deep,
narrow well, sunk with careful labour to tap the underground river of
water which flows below the surface of our natures. But your well will
be dry unless the bright warm sun has first drawn up the vapour from
the wide ocean, and the free untrammelled breezes have carried the
clouds hither and thither, until at length they break, as it were by
chance over the distant hills and soak the land with their downpour.
What I have said about art is a parable which applies to other
occupations and other studies. It is more than a parable; it is the
literal truth. The love of art is the love of excellence, it is the
enjoyment of the triumph of design over the shapeless products of
chance forces. An engineer, who is worth his salt, loves the beauty of
his machines, shown in their adjustment of parts and in their swift,
smooth motions. He loves also the sense of foresight and of insight
which knowledge can give him. People say that machinery and commerce
are driving beauty out of the modern world. I do not believe it. A new
beauty is being added, a more intellectual beauty, appealing to the
understanding as much as to the eye.
The wonder of London ever takes the mind with fresh astonishment. The
city possesses parks, and palaces, and cathedrals. But no other parts
of it surpass in wonder its houses of business and its workshops and
its factories.
In the next few years the future of the world will be decided for
centuries to come. The battles of this war are only the first part of
the contest between races, and between the ways of life for which
those races stand. We believe that England, with its various peoples
and communities scattered in islands and continents beyond the seas,
stands for ways of life infinitely precious, the way of humanity, the
way of liberty, the way of self-government, the way of good order based
on toleration and kindly feeling, the way of peaceful industry. The
final decision in this struggle will be found in the workshops of the
world. It lies in your hands. Statesmen and emperors will only register
the results which you have achieved. Your weapons will be skill, and
energy, and knowledge. You will require a sane understanding of your
own rights, and a sane understanding of the rights and the difficulties
of other classes. The greatness of England will be your greatness, and
its success your success.
The arsenal for war is at Woolwich. This Polytechnic Institute is an
arsenal for peace, where you can find the weapons for the conquest of
your lives.
CHAPTER IV
THE MATHEMATICAL CURRICULUM
(_Presidential Address to the London Branch of the Mathematical
Association, 1912_)
THE situation in regard to education at the present time cannot find
its parallel without going back for some centuries to the break-up of
the mediæval traditions of learning. Then, as now, the traditional
intellectual outlook, despite the authority which it had justly
acquired from its notable triumphs, had grown to be too narrow for the
interests of mankind. The result of this shifting of human interest
was a demand for a parallel shifting of the basis of education, so
as to fit the pupils for the ideas which later in life would in fact
occupy their minds. Any serious fundamental change in the intellectual
outlook of human society must necessarily be followed by an educational
revolution. It may be delayed for a generation by vested interests or
by the passionate attachment of some leaders of thought to the cycle
of ideas within which they received their own mental stimulus at an
impressionable age. But the law is inexorable that education to be
living and effective must be directed to informing pupils with those
ideas, and to creating for them those capacities which will enable them
to appreciate the current thought of their epoch.
There is no such thing as a successful system of education in a vacuum,
that is to say, a system which is divorced from immediate contact with
the existing intellectual atmosphere. Education which is not modern
shares the fate of all organic things which are kept too long.
But the blessed word "modern" does not really solve our difficulties.
What we mean is, relevant to modern thought, either in the ideas
imparted or in the aptitudes produced. Something found out only
yesterday may not really be modern in this sense. It may belong to
some bygone system of thought prevalent in a previous age, or, what
is very much more likely, it may be too recondite. When we demand
that education should be relevant to modern thought, we are referring
to thoughts broadly spread throughout cultivated society. It is this
question of the unfitness of recondite subjects for use in general
education which I wish to make the keynote of my address this afternoon.
It is in fact rather a delicate subject for us mathematicians.
Outsiders are apt to accuse our subject of being recondite. Let us
grasp the nettle at once and frankly admit that in general opinion it
is the very typical example of reconditeness. By this word I do not
mean difficulty, but that the ideas involved are of highly special
application, and rarely influence thought.
This liability to reconditeness is the characteristic evil which is
apt to destroy the utility of mathematics in liberal education. So
far as it clings to the educational use of the subject, so far we
must acquiesce in a miserably low level of mathematical attainment
among cultivated people in general. I yield to no one in my anxiety to
increase the educational scope of mathematics. The way to achieve this
end is not by a mere blind demand for more mathematics. We must face
the real difficulty which obstructs its extended use.
Is the subject recondite? Now, viewed as a whole, I think it is.
_Securus judicat orbis terrarum_--the general judgment of mankind
is sure.
The subject as it exists in the minds and in the books of students of
mathematics _is_ recondite. It proceeds by deducing innumerable
special results from general ideas, each result more recondite than the
preceding. It is not my task this afternoon to defend mathematics as a
subject for profound study. It can very well take care of itself. What
I want to emphasise is, that the very reasons which make this science
a delight to its students are reasons which obstruct its use as an
educational instrument--namely, the boundless wealth of deductions from
the interplay of general theorems, their complication, their apparent
remoteness from the ideas from which the argument started, the variety
of methods, and their purely abstract character which brings, as its
gift, eternal truth.
Of course, all these characteristics are of priceless value to
students; for ages they have fascinated some of the keenest intellects.
My only remark is that, except for a highly selected class, they are
fatal in education. The pupils are bewildered by a multiplicity of
detail, without apparent relevance either to great ideas or to ordinary
thoughts. The extension of this sort of training in the direction
of acquiring more detail is the last measure to be desired in the
interests of education.
The conclusion at which we arrive is, that mathematics, if it is
to be used in general education, must be subjected to a rigorous
process of selection and adaptation. I do not mean, what is of course
obvious, that however much time we devote to the subject the average
pupil will not get very far. But that, however limited the progress,
certain characteristics of the subject, natural at any stage, must be
rigorously excluded. The science as presented to young pupils must
lose its aspect of reconditeness. It must, on the face of it, deal
directly and simply with a few general ideas of far-reaching importance.
Now, in this matter of the reform of mathematical instruction, the
present generation of teachers may take a very legitimate pride in
its achievements. It has shown immense energy in reform, and has
accomplished more than would have been thought possible in so short a
time. It is not always recognised how difficult is the task of changing
a well-established curriculum entrenched behind public examinations.
But for all that, great progress has been made, and, to put the matter
at its lowest, the old dead tradition has been broken up. I want to
indicate this afternoon the guiding idea which should direct our
efforts at reconstruction. I have already summed it up in a phrase,
namely, we must aim at the elimination of reconditeness from the
educational use of the subject.
Our courses of instruction should be planned to illustrate simply a
succession of ideas of obvious importance. All pretty divagations
should be rigorously excluded. The goal to be aimed at is that the
pupil should acquire familiarity with abstract thought, should realise
how it applies to particular concrete circumstances, and should know
how to apply general methods to its logical investigation. With this
educational ideal nothing can be worse than the aimless accretion
of theorems in our text-books, which acquire their position merely
because the children can be made to learn them and examiners can set
neat questions on them. The bookwork to be learnt should all be very
important as illustrating ideas. The examples set--and let there
be as many examples as teachers find necessary--should be direct
illustrations of the theorems, either by way of abstract particular
cases or by way of application to concrete phenomena. Here it is worth
remarking that it is quite useless to simplify the bookwork, if the
examples set in examinations in fact require an extended knowledge of
recondite details. There is a mistaken idea that problems test ability
and genius, and that bookwork tests cram. This is not my experience.
Only boys who have been specially crammed for scholarships can ever
do a problem paper successfully. Bookwork properly set, not in mere
snippets according to the usual bad plan, is a far better test of
ability, provided that it is supplemented by direct examples. But this
is a digression on the bad influence of examinations on teaching.
The main ideas which lie at the base of mathematics are not at all
recondite. They are abstract. But one of the main objects of the
inclusion of mathematics in a liberal education is to train the pupils
to handle abstract ideas. The science constitutes the first large
group of abstract ideas which naturally occur to the mind in any
precise form. For the purposes of education, mathematics consists of
the relations of number, the relations of quantity, and the relations
of space. This is not a general definition of mathematics, which, in
my opinion, is a much more general science. But we are now discussing
the use of mathematics in education. These three groups of relations,
concerning number, quantity, and space, are interconnected.
Now, in education we proceed from the particular to the general.
Accordingly, children should be taught the use of these ideas by
practice among simple examples. My point is this: The goal should be,
not an aimless accumulation of special mathematical theorems, but the
final recognition that the preceding years of work have illustrated
those relations of number, and of quantity, and of space, which are of
fundamental importance. Such a training should lie at the base of all
philosophical thought. In fact elementary mathematics rightly conceived
would give just that philosophical discipline of which the ordinary
mind is capable. But what at all costs we ought to avoid, is the
pointless accumulation of details. As many examples as you like; let
the children work at them for terms, or for years. But these examples
should be direct illustrations of the main ideas. In this way, and
this only, can the fatal reconditeness be avoided.
I am not now speaking in particular of those who are to be professional
mathematicians, or of those who for professional reasons require a
knowledge of certain mathematical details. We are considering the
liberal education of all students, including these two classes. This
general use of mathematics should be the simple study of a few general
truths, well illustrated by practical examples. This study should
be conceived by itself, and completely separated in idea from the
professional study mentioned above, for which it would make a most
excellent preparation. Its final stage should be the recognition of
the general truths which the work done has illustrated. As far as I
can make out, at present the final stage is the proof of some property
of circles connected with triangles. Such properties are immensely
interesting to mathematicians. But are they not rather recondite,
and what is the precise relation of such theorems to the ideal of
a liberal education? The end of all the grammatical studies of the
student in classics is to read Virgil and Horace--the greatest thoughts
of the greatest men. Are we content, when pleading for the adequate
representation in education of our own science, to say that the end of
a mathematical training is that the student should know the properties
of the nine-point circle? I ask you frankly, is it not rather a "come
down"?
This generation of mathematical teachers has done so much strenuous
work in the way of reorganising mathematical instruction that there is
no need to despair of its being able to elaborate a curriculum which
shall leave in the minds of the pupils something even nobler than "the
ambiguous case."
Let us think how this final review, closing the elementary course,
might be conducted for the more intelligent pupils. Partly no doubt
it requires a general oversight of the whole work done, considered
without undue detail so as to emphasise the general ideas used, and
their possibilities of importance when subjected to further study. Also
the analytical and geometrical ideas find immediate application in the
physical laboratory where a course of simple experimental mechanics
should have been worked through. Here the point of view is twofold, the
physical ideas and the mathematical ideas illustrate each other.
The mathematical ideas are essential to the precise formulation of the
mechanical laws. The idea of a precise law of nature, the extent to
which such laws are in fact verified in our experience, and the rôle of
abstract thought in their formulation, then become practically apparent
to the pupil. The whole topic of course requires detailed development
with full particular illustration, and is not suggested as requiring
merely a few bare abstract statements.
It would, however, be a grave error to put too much emphasis on the
mere process of direct explanation of the previous work by way of
final review. My point is, that the latter end of the course should
be so selected that in fact the general ideas underlying all the
previous mathematical work should be brought into prominence. This may
well be done by apparently entering on a new subject. For example,
the ideas of quantity and the ideas of number are fundamental to
all precise thought. In the previous stages they will not have been
sharply separated; and children are, rightly enough, pushed on to
algebra without too much bother and quantity. But the more intelligent
among them at the end of their curriculum would gain immensely by a
careful consideration of those fundamental properties of quantity in
general which lead to the introduction of numerical measurement. This
is a topic which also has the advantage that the necessary books are
actually to hand. Euclid's fifth book is regarded by those qualified
to judge as one of the triumphs of Greek mathematics. It deals with
this very point. Nothing can be more characteristic of the hopelessly
illiberal character of the traditional mathematical education than the
fact that this book has always been omitted. It deals with ideas,
and therefore was ostracised. Of course a careful selection of the
more important propositions and a careful revision of the argument are
required. This also is to hand in the publications of my immediate
predecessor in the office of president, Prof. Hill. Furthermore, in Sir
T. L. Heath's complete edition of Euclid, there is a full commentary
embodying most of what has been said and thought on the point. Thus
it is perfectly easy for teachers to inform themselves generally
on the topic. The whole book would not be wanted, but just the few
propositions which embody the fundamental ideas. The subject is not
fit for backward pupils; but certainly it could be made interesting to
the more advanced class. There would be great scope for interesting
discussion as to the nature of quantity, and the tests which we should
apply to ascertain when we are dealing with quantities. The work
would not be at all in the air, but would be illustrated at every
stage by reference to actual examples of cases where the quantitative
character is absent, or obscure, or doubtful, or evident. Temperature,
heat, electricity, pleasure and pain, mass and distance could all be
considered.
Another idea which requires illustration is that of functionality.
A function in analysis is the counterpart of a law in the physical
universe, and of a curve in geometry. Children have studied the
relations of functions to curves from the first beginning of their
study of algebra, namely in drawing graphs. Of recent years there has
been a great reform in respect to graphs. But at its present stage it
has either gone too far or not far enough. It is not enough merely
to draw a graph. The idea behind the graph--like the man behind the
gun--is essential in order to make it effective. At present there is
some tendency merely to set the children to draw curves, and there to
leave the whole question.
In the study of simple algebraic functions and of trigonometrical
functions we are initiating the study of the precise expression of
physical laws. Curves are another way of representing these laws. The
simple fundamental laws--such as the inverse square and the direct
distance--should be passed under review, and the applications of the
simple functions to express important concrete cases of physical laws
considered. I cannot help thinking that the final review of this topic
might well take the form of a study of some of the main ideas of the
differential calculus applied to simple curves. There is nothing
particularly difficult about the conception of a rate of change; and
the differentiation of a few powers of _x_, such as _x_^2,
_x_^3, etc., could easily be effected; perhaps by the aid of
geometry even sin _x_ and cos _x_ could be differentiated.
If we once abandon our fatal habit of cramming the children with
theorems which they do not understand, and will never use, there will
be plenty of time to concentrate their attention on really important
topics. We can give them familiarity with conceptions which really
influence thought.
Before leaving this topic of physical laws and mathematical functions,
there are other points to be noticed. The fact that the precise law is
never really verified by observation in its full precision is capable
of easy illustration and of affording excellent examples. Again,
statistical laws, namely laws which are only satisfied on the average
by large numbers, can easily be studied and illustrated. In fact a
slight study of statistical methods and their application to social
phenomena affords one of the simplest examples of the application of
algebraic ideas.
Another way in which the students' ideas can be generalised is by the
use of the History of Mathematics, conceived not as a mere assemblage
of the dates and names of men, but as an exposition of the general
current of thought which occasioned the subjects to be objects of
interest at the time of their first elaboration. The use of the History
of Mathematics is to be considered at a later stage of our proceedings
this afternoon. Accordingly I merely draw attention to it now, to
point out that perhaps it is the very subject which may best obtain the
results for which I am pleading.
We have indicated two main topics, namely general ideas of quantity
and of laws of nature, which should be an object of study in the
mathematical curriculum of a liberal education. But there is another
side to mathematics which must not be overlooked. It is the chief
instrument for discipline in logical method.
Now, what is logical method, and how can any one be trained in it?
Logical method is more than the mere knowledge of valid types of
reasoning and practice in the concentration of mind necessary to
follow them. If it were only this, it would still be very important;
for the human mind was not evolved in the bygone ages for the sake of
reasoning, but merely to enable mankind with more art to hunt between
meals for fresh food supplies. Accordingly few people can follow close
reasoning without considerable practice.
More than this is wanted to make a good reasoner, or even to enlighten
ordinary people with knowledge of what constitutes the essence of the
art. The art of reasoning consists in getting hold of the subject at
the right end, of seizing on the few general ideas which illuminate
the whole, and of persistently marshalling all subsidiary facts round
them. Nobody can be a good reasoner unless by constant practice he
has realised the importance of getting hold of the big ideas and of
hanging on to them like grim death. For this sort of training geometry
is, I think, better than algebra. The field of thought of algebra is
rather obscure, whereas space is an obvious insistent thing evident to
all. Then the process of simplification, or abstraction, by which all
irrelevant properties of matter, such as colour, taste, and weight, are
put aside is an education in itself. Again, the definitions and the
propositions assumed without proof illustrate the necessity of forming
clear notions of the fundamental facts of the subject-matter and of the
relations between them. All this belongs to the mere prolegomena of the
subject. When we come to its development, its excellence increases. The
learner is not initially confronted with any symbolism which bothers
the memory by its rules, however simple they may be. Also, from the
very beginning the reasoning, if properly conducted, is dominated by
well-marked ideas which guide each stage of development. Accordingly
the essence of logical method receives immediate exemplification.
Let us now put aside for the moment the limitations introduced by
the dullness of average pupils and the pressure on time due to other
subjects, and consider what geometry has to offer in the way of a
liberal education. I will indicate some stages in the subject,
without meaning that necessarily they are to be studied in this
exclusive order. The first stage is the study of _congruence_.
Our perception of congruence is in practice dependent on our judgments
of the invariability of the intrinsic properties of bodies when their
external circumstances are varying. But however it arises, congruence
is in essence the correlation of two regions of space, point by point,
so that all homologous distances and all homologous angles are equal.
It is to be noticed that the definition of the equality of lengths and
angles is their congruence, and all tests of equality, such as the use
of the yard measure, are merely devices for making immediate judgments
of congruence easy. I make these remarks to suggest that apart from the
reasoning connected with it, congruence, both as an example of a larger
and very far-reaching idea and also for its own sake, is well worthy
of attentive consideration. The propositions concerning it elucidate
the elementary properties of the triangle, the parallelogram, and the
circle, and of the relations of two planes to each other. It is very
desirable to restrict the proved propositions of this part within the
narrowest bounds, partly by assuming redundant axiomatic propositions,
and partly by introducing only those propositions of absolutely
fundamental importance.
The second stage is the study of similarity. This can be reduced to
three or four fundamental propositions. Similarity is an enlargement of
the idea of congruence, and, like that idea, is another example of a
one-to-one correlation of points of spaces. Any extension of study of
this subject might well be in the direction of the investigation of one
or two simple properties of similar and similarly situated rectilinear
figures. The whole subject receives its immediate applications in plans
and maps. It is important, however, to remember that trigonometry is
really the method by which the main theorems are made available for use.
The third stage is the study of the elements of trigonometry. This is
the study of the periodicity introduced by rotation and of properties
preserved in a correlation of similar figures. Here for the first time
we introduce a slight use of the algebraic analysis founded on the
study of number and quantity. The importance of the periodic character
of the functions requires full illustration. The simplest properties of
the functions are the only ones required for the solution of triangles,
and the consequent applications to surveying. The wealth of formulæ,
often important in themselves, but entirely useless for this type of
study, which crowd our books should be rigorously excluded, except
so far as they are capable of being proved by the pupils as direct
examples of the bookwork.
This question of the exclusion of formulæ is best illustrated by
considering this example of Trigonometry, though of course I may well
have hit on an unfortunate case in which my judgment is at fault. A
great part of the educational advantage of the subject can be obtained
by confining study to Trigonometry of one angle and by exclusion of the
addition formulæ for the sine and cosine of the sum of two angles. The
functions can be graphed, and the solution of triangles effected. Thus
the aspects of the science as (1) embodying analytically the immediate
results of some of the theorems deduced from congruence and similarity,
(2) as a solution of the main problem of surveying, (3) as a study of
the fundamental functions required to express periodicity and wave
motion, will all be impressed on the pupils' minds both by bookwork and
example.
If it be desired to extend this course, the addition formulæ should
be added. But great care should be taken to exclude specialising
the pupils in the wealth of formulæ which comes in their train. By
"exclude" is meant that the pupils should not have spent time or energy
in acquiring any facility in their deduction. The teacher may find
it interesting to work a few such examples before a class. But such
results are not among those which learners need retain. Also, I would
exclude the whole subject of circumscribed and inscribed circles both
from Trigonometry and from the previous geometrical courses. It is
all very pretty, but I do not understand what its function is in an
elementary non-professional curriculum.
Accordingly, the actual bookwork of the subject is reduced to very
manageable proportions. I was told the other day of an American college
where the students are expected to know by heart ninety formulæ or
results in Trigonometry alone. We are not quite so bad as that. In
fact, in Trigonometry we have nearly approached the ideal here sketched
out as far as our elementary courses are concerned.
The fourth stage introduces Analytical Geometry. The study of graphs in
algebra has already employed the fundamental notions, and all that is
now required is a rigorously pruned course on the straight line, the
circle, and the three types of conic sections, defined by the forms of
their equations. At this point there are two remarks to be made. It is
often desirable to give our pupils mathematical information which we do
not prove. For example, in co-ordinate geometry, the reduction of the
general equation of the second degree is probably beyond the capacities
of most of the type of students whom we are considering. But that need
not prevent us from explaining the fundamental position of conics, as
exhausting the possible types of such curves.
The second remark is to advocate the entire sweeping away of
geometrical conics as a separate subject. Naturally, on suitable
occasions the analysis of analytical geometry will be lightened by
the use of direct deduction from some simple figure. But geometrical
conics, as developed from the definition of a conic section by the
focus and directrix property, suffers from glaring defects. It is
hopelessly recondite. The fundamental definition of a conic, _SP_ = _e_
· _PM_, usual in this subject at this stage, is thoroughly bad. It is
very recondite, and has no obvious importance. Why should such curves
be studied at all, any more than those defined by an indefinite number
of other formulæ? But when we have commenced the study of the Cartesian
methods, the equations of the first and second degrees are naturally
the first things to think about.
In this ideal course of Geometry, the fifth stage is occupied with
the elements of Projective Geometry. The general ideas of cross ratio
and of projection are here fundamental. Projection is yet a more
general instance of that one-to-one correlation which we have already
considered under congruence and similarity. Here again we must avoid
the danger of being led into a bewildering wealth of detail.
The intellectual idea which projective geometry is to illustrate is
the importance in reasoning of the correlation of all cases which
can be proved to possess in common certain identical properties. The
preservation of the projective properties in projection is the one
important educational idea of the subject. Cross ratio only enters
as the fundamental metrical property which is preserved. The few
propositions considered are selected to illustrate the two allied
processes which are made possible by this procedure. One is proof
by simplification. Here the simplification is psychological and not
logical--for the general case is logically the simplest. What is meant
is: Proof by considering the case which is in fact the most familiar to
us, or the easiest to think about. The other procedure is the deduction
of particular cases from known general truths, as soon as we have a
means of discovering such cases or a criterion for testing them.
The projective definition of conic sections and the identity of the
results obtained with the curves derived from the general equation
of the second degree are capable of simple exposition, but lie on
the border-line of the subject. It is the sort of topic on which
information can be given, and the proofs suppressed.
The course of geometry as here conceived in its complete ideal--and
ideals can never be realised--is not a long one. The actual amount
of mathematical deduction at each stage in the form of bookwork is
very slight. But much more explanation would be given, the importance
of each proposition being illustrated by examples, either worked out
or for students to work, so selected as to indicate the fields of
thought to which it applies. By such a course the student would gain an
analysis of the leading properties of space, and of the chief methods
by which they are investigated.
The study of the elements of mathematics, conceived in this spirit,
would constitute a training in logical method together with an
acquisition of the precise ideas which lie at the base of the
scientific and philosophical investigations of the universe. Would it
be easy to continue the excellent reforms in mathematical instruction
which this generation has already achieved, so as to include in the
curriculum this wider and more philosophic spirit? Frankly, I think
that this result would be very hard to achieve as the result of
single individual efforts. For reasons which I have already briefly
indicated, all reforms in education are very difficult to effect. But
the continued pressure of combined effort, provided that the ideal is
really present in the minds of the mass of teachers, can do much, and
effects in the end surprising modification. Gradually the requisite
books get written, still more gradually the examinations are reformed
so as to give weight to the less technical aspects of the subject, and
then all recent experience has shown that the majority of teachers are
only too ready to welcome any practicable means of rescuing the subject
from the reproach of being a mechanical discipline.
CHAPTER V
THE PRINCIPLES OF MATHEMATICS IN RELATION TO ELEMENTARY TEACHING
(_International Congress of Mathematicians, Cambridge, August,
1912_)
WE are concerned not with the advanced teaching of a few specialist
mathematical students, but with the mathematical education of the
majority of boys in our secondary schools. Again these boys must be
divided into two sections: one section consists of those who desire to
restrict their mathematical education; the other section is made up of
those who will require some mathematical training for their subsequent
professional careers, either in the form of definite mathematical
results or in the form of mathematically trained minds.
I shall call the latter of these two sections the mathematical section,
and the former the non-mathematical section. But I must repeat that by
the mathematical section is meant that large number of boys who desire
to learn more than the minimum amount of mathematics. Furthermore, most
of my remarks about these sections of boys apply also to elementary
classes of our University students.
The subject of this paper is the investigation of the place which
should be occupied by modern investigations respecting mathematical
principles in the education of both of these sections of school-boys.
To find a foothold from which to start the inquiry, let us ask why the
non-mathematical section should be taught any mathematics at all beyond
the barest elements of arithmetic. What are the qualities of mind which
a mathematical training is designed to produce when it is employed as
an element in a liberal education?
My answer, which applies equally to both sections of students, is
that there are two allied forms of mental discipline which should be
acquired by a well-designed mathematical course. These two forms though
closely allied are perfectly distinct.
The first form of discipline is not in its essence logical at all.
It is the power of clearly grasping abstract ideas, and of relating
them to particular circumstances. In other words, the first use of
mathematics is to strengthen the power of abstract thought. I repeat
that in its essence this has nothing to do with logic, though as a
matter of fact a logical discipline is the best method of producing the
desired effect. It is not the philosophical theory of abstract ideas
which is to be acquired, but the habit and the power of using them.
There is one and only way of acquiring the habit and the power of using
anything, that is by the simple common-place method of habitually using
it. There is no other short cut. If in education we desire to produce
a certain conformation of mind, we must day by day, and year by year,
accustom the students' minds to grow into the desired structural shape.
Thus to teach the power of grasping abstract ideas and the habit of
using them, we must select a group of such ideas, which are important
and are also easy to think about because they are clear and definite.
The fundamental mathematical truths concerning geometry, ratio,
quantity, and number, satisfy these conditions as do no others. Hence,
the fundamental universal position held by mathematics as an element of
a liberal education.
But what are the fundamental mathematical truths concerning geometry,
quantity and number?
At this point we come to the great question of the relation between
the modern science of the principles of mathematics and a mathematical
education.
My answer to the question as to these fundamental mathematical truths
is, that in any absolute sense there are none. There is no unique
small body of independent primitive unproved propositions which are
the necessary starting points of all mathematical reasoning on these
subjects. In mathematical reasoning the only absolute necessary
pre-suppositions are those which make logical deduction possible.
Between these absolute logical truths and so-called fundamental truths
concerning geometry, quantity and number, there is a whole new world of
mathematical subjects concerning the logic of propositions, of classes,
and of relations.
But this subject is too abstract to form an elementary training ground
in the difficult art of abstract thought.
It is for this reason that we have to make a compromise and start from
such obvious general ideas as naturally occur to all men when they
perceive objects with their senses.
In geometry, the ideas elaborated by the Greeks and presented by Euclid
are, roughly speaking, those adapted for our purpose, namely, ideas
of volumes, surfaces, lines, of straightness and of curvature, of
intersection and of congruence, of greater and less, of similarity,
shape, and scale. In fact, we use in education those general ideas of
spatial properties which must be habitually present in the mind of any
one who is to observe the world of phenomena with understanding.
Thus we come back to Plato's opinion that for a liberal education,
geometry, as he knew it, is the queen of sciences.
In addition to geometry, there remain the ideas of quantity, ratio,
and of number. This in practice means, elementary algebra. Here the
prominent ideas are those of "any number," in other words, the use
of the familiar _x_, _y_, _z_, and of the dependence of variables on
each other, or otherwise, the idea of functionality. All this is to be
gradually acquired by the continual use of the very simplest functions
which we can devise: of linear functions, graphically represented
by straight lines; of quadratic functions, graphically represented
by parabolas; and of those simple implicit functions, graphically
represented by conic sections. Thence, with good fortune and a willing
class, we can advance to the ideas of rates of increase, still
confining ourselves to the simplest possible cases.
I wish here emphatically to remind you that both in geometry and in
algebra a clear grasp of these general ideas is not what the pupil
starts from, it is the goal at which he is to arrive. The method of
progression is continual practice in the consideration of the simplest
particular cases, and the goal is not philosophical analysis but the
power of use.
But how is he to practise himself in their use? He cannot simply sit
down and think of the relation _y_ = _x_ + 1, he must employ
it in some simple obvious way.
This brings us to the second power of mind which is to be produced by
a mathematical training, namely, the power of logical reasoning. Here
again, it is not the knowledge of the philosophy of logic which it is
essential to teach, but the habit of thinking logically. By logic, I
mean deductive logic.
Deductive logic is the science of certain relations, such as
implication, etc., between general ideas. When logic begins, definite
particular individual things have been banished. I cannot relate
logically this thing to that thing, for example this pen to that pen,
except by the indirect way of relating some general idea which applies
to this pen to some general idea which applies to that pen. And the
individualities of the two pens are quite irrelevant to the logical
process. This process is entirely concerned with the two general ideas.
Thus the practice of logic is a certain way of employing the mind
in the consideration of such ideas; and an elementary mathematical
training is in fact nothing else but the logical use of the general
ideas respecting geometry and algebra which we have enumerated above.
It has therefore, as I started this paper by stating, a double
advantage. It makes the mind capable of abstract thought, and it
achieves this object by training the mind in the most important kind of
abstract thought, namely, deductive logic.
I may remind you that other choices of a type of abstract thought
might be made. We might train the children to contemplate directly the
beauty of abstract moral ideas, in the hope of making them religious
mystics. The general practice of education has decided in favour of
logic, as exemplified in elementary mathematics.
We have now to answer the further question, what is the rôle of
logical precision in the teaching of mathematics? Our general answer
to the implied question is obvious: logical precision is one of the
two objects of the teaching of mathematics, and it is the only weapon
by which the teaching of mathematics can achieve its other object. To
teach mathematics is to teach logical precision. A mathematical teacher
who has not taught that has taught nothing.
But having stated this thesis in this unqualified way, its meaning must
be carefully explained; for otherwise its real bearing on the problem
will be entirely misunderstood.
Logical precision is the faculty to be acquired. It is the quality of
mind which it is the object of the teaching to impart. Thus the habit
of reading great literature is the goal at which a literary education
aims. But we do not expect a child to start its first lesson by reading
for itself Shakespeare. We recognise that reading is impossible till
the pupil has learnt its alphabet and can spell, and then we start it
with books of one syllable.
In the same way, a mathematical education should grow in logical
precision. It is folly to expect the same careful logical analysis at
the commencement of the training as would be appropriate at the end.
It is an entire misconception of my thesis to construe it as meaning
that a mathematical training should assume in the pupil a power of
concentrated logical thought. My thesis is in fact the exact opposite,
namely, that this power cannot be assumed, and has got to be acquired,
and that a mathematical training is nothing else than the process of
acquiring it. My whole groundwork of assumption is that this power does
not initially exist in a fully developed state. Of course like every
other power which is acquired, it must be developed gradually.
The various stages of development must be guided by the judgment and
the genius of the teacher. But what is essential is, that the teacher
should keep clearly in his mind that it is just this power of logical
precise reasoning which is the whole object of his efforts. If his
pupils have in any measure gained this, they have gained all.
We have not yet, however, fully considered this part of our subject.
Logical precision is the full realisation of the steps of the argument.
But what are the steps of the argument? The full statement of all the
steps is far too elaborate and difficult an operation to be introduced
into the mathematical reasoning of an educational curriculum. Such a
statement involves the introduction of abstract logical ideas which
are very difficult to grasp, because there is so rarely any need to
make them explicit in ordinary thought. They are therefore not a fit
subject-ground for an elementary education.
I do not think that it is possible to draw any theoretical line
between those logical steps which form a theoretically full logical
investigation, and those which are full enough for most practical
purposes, including that of education. The question is one of
psychology, to be solved by a process of experiment. The object to be
attained is to gain that amount of logical alertness which will enable
its possessors to detect fallacy and to know the types of sound logical
deduction. The objects of going further are partly philosophical, and
also partly to lay bare abstract ideas whose investigation is in itself
important. But both these objects are foreign to education.
My opinion is, that, on the whole, the type of logical precision handed
down to us by the Greek mathematicians is, roughly speaking, what we
want. In geometry, this means the sort of precision which we find in
Euclid. I do not mean that we should use his famous _Elements_ as
a text-book, nor that here and there a certain compression in his mode
of exposition is not advisable. All this is mere detail. What I do mean
is, that the sort of logical transition which he made explicit, we
should make explicit, and that the sort of transition which he omits,
we should omit.
I doubt, however, whether it is desirable to plunge the student into
the full rigour of euclidean geometry without some mitigation. It is
for this reason that the modern habit, at least in England, of laying
great stress in the initial stages on training the pupil in simple
constructions from numerical data is to be praised. It means that after
a slight amount of reasoning on the euclidean basis of accuracy, the
mind of the learner is relieved by doing the things in various special
cases, and noting by rough measurements that the desired results are
actually attained. It is important, however, that the measurements
be not mistaken for the proofs. Their object is to make the beginner
apprehend what the abstract ideas really mean.
Again in algebra, the notation and the practical use of the symbols
should be acquired in the simplest cases, and the more theoretical
treatment of the symbolism reserved to a suitable later stage. My
rule would be initially to learn the meaning of the ideas by a crude
practice in simple ways, and to refine the logical procedure in
preparation for an advance to greater generality. In fact the thesis of
my paper can be put in another way thus, the object of a mathematical
education is, to acquire the powers of analysis, of generalisation,
and of reasoning. The two processes of analysis and generalisation were
in my previous statement put together as the power of grasping abstract
ideas.
But in order to analyse and to generalise, we must commence with the
crude material of ideas which are to be analysed and generalised.
Accordingly it is a positive error in education to start with the
ultimate products of this process, namely the ideas in their refined
analysed and generalised forms. We are thereby skipping an important
part of the training, which is to take the ideas as they actually exist
in the child's mind, and to exercise the child in the difficult art of
civilising them and clothing them.
The schoolmaster is in fact a missionary, the savages are the ideas in
the child's mind; and the missionary shirks his main task if he refuse
to risk his body among the cannibals.
At this point I should like to turn your attention to those pupils
forming the mathematical section. There is an idea, widely prevalent,
that it is possible to teach mathematics of a relatively advanced
type--such as differential calculus, for instance--in a way useful to
physicists and engineers without any attention to its logic or its
theory.
This seems to me to be a profound mistake. It implies that a merely
mechanical knowledge without understanding of ways of arriving at
mathematical results is useful in applied science. It is of no use
whatever. The results themselves can all be found stated in the
appropriate pocket books and in other elementary works of reference. No
one when applying a result need bother himself as to why it is true. He
accepts it and applies it. What is of supreme importance in physics and
in engineering is a mathematically trained mind, and such a mind can
only be acquired by a proper mathematical discipline.
I fully admit that the proper way to start such a subject as the
differential calculus is to plunge quickly into the use of the notation
in a few absurdly simple cases, with a crude explanation of the idea
of rates of increase. The notation as thus known can then be used by
the lecturers in the Physical and Engineering Laboratories. But the
mathematical training of the applied scientists consists in making
these ideas precise and the proofs accurate.
I hope that the thesis of this paper respecting the position of logical
precision in the teaching of mathematics has been rendered plain. The
habit of logical precision with its necessary concentration of thought
upon abstract ideas is not wholly possible in the initial stages of
learning. It is the ideal at which the teacher should aim. Also logical
precision, in the sense of logical explicitness, is not an absolute
thing: it is an affair of more or less. Accordingly the quantity of
explicitness to be introduced at each stage of progress must depend
upon the practical judgment of the teacher. Lastly, in a sense, the
instructed mind is less explicit; for it travels more quickly over a
well-remembered path, and may save the trouble of putting into words
trains of thought which are very obvious to it. But on the other
hand it atones for this rapidity by a concentration on every subtle
point where a fallacy can lurk. The habit of logical precision is the
instinct for the subtle difficulty.
CHAPTER VI
THE ORGANISATION OF THOUGHT
(_Presidential Address to Section A, British Association, Newcastle,
1916_)
THE subject of this address is the organisation of thought, a topic
evidently capable of many diverse modes of treatment. I intend more
particularly to give some account of that department of logical science
with which some of my own studies have been connected. But I am
anxious, if I can succeed in so doing, to handle this account so as to
exhibit the relation with certain considerations which underlie general
scientific activities.
It is no accident that an age of science has developed into an age
of organisation. Organised thought is the basis of organised action.
Organisation is the adjustment of diverse elements so that their
mutual relations may exhibit some predetermined quality. An epic poem
is a triumph of organisation, that is to say, it is a triumph in the
unlikely event of its being a good epic poem. It is the successful
organisation of multitudinous sounds of words, associations of words,
pictorial memories of diverse events and feelings ordinarily occurring
in life, combined with a special narrative of great events: the whole
so disposed as to excite emotions which, as defined by Milton, are
simple, sensuous, and passionate. The number of successful epic poems
is commensurate, or rather, is inversely commensurate, with the obvious
difficulty of the task of organisation.
Science is the organisation of thought. But the example of the epic
poem warns us that science is not any organisation of thought. It is
an organisation of a certain definite type which we will endeavour to
determine.
Science is a river with two sources, the practical source and the
theoretical source. The practical source is the desire to direct our
actions to achieve predetermined ends. For example, the British nation,
fighting for justice, turns to science, which teaches it the importance
of compounds of nitrogen. The theoretical source is the desire to
understand. Now I am going to emphasise the importance of theory in
science. But to avoid misconception I most emphatically state that I
do not consider one source as in any sense nobler than the other, or
intrinsically more interesting. I cannot see why it is nobler to strive
to understand than to busy oneself with the right ordering of one's
actions. Both have their bad sides; there are evil ends directing
actions, and there are ignoble curiosities of the understanding.
The importance, even in practice, of the theoretical side of science
arises from the fact that action must be immediate, and takes place
under circumstances which are excessively complicated. If we wait for
the necessities of action before we commence to arrange our ideas, in
peace we shall have lost our trade, and in war we shall have lost the
battle. Success in practice depends on theorists who, led by other
motives of exploration, have been there before, and by some good chance
have hit upon the relevant ideas. By a theorist I do not mean a man who
is up in the clouds, but a man whose motive for thought is the desire
to formulate correctly the rules according to which events occur. A
successful theorist should be excessively interested in immediate
events, otherwise he is not at all likely to formulate correctly
anything about them. Of course, both sources of science exist in all
men.
Now, what is this thought organisation which we call science? The
first aspect of modern science which struck thoughtful observers was
its inductive character. The nature of induction, its importance, and
the rules of inductive logic have been considered by a long series of
thinkers, especially English thinkers: Bacon, Herschel, J. S. Mill,
Venn, Jevons, and others. I am not going to plunge into an analysis
of the process of induction. Induction is the machinery and not the
product, and it is the product which I want to consider. When we
understand the product we shall be in a stronger position to improve
the machinery.
First, there is one point which it is necessary to emphasise. There
is a tendency in analysing scientific processes to assume a given
assemblage of concepts applying to nature, and to imagine that the
discovery of laws of nature consists in selecting by means of inductive
logic some one out of a definite set of possible alternative relations
which may hold between the things in nature answering to these obvious
concepts. In a sense this assumption is fairly correct, especially
in regard to the earlier stages of science. Mankind found itself in
possession of certain concepts respecting nature--for example, the
concept of fairly permanent material bodies--and proceeded to determine
laws which related the corresponding percepts in nature. But the
formulation of laws changed the concepts, sometimes gently by an added
precision, sometimes violently. At first this process was not much
noticed, or at least was felt to be a process curbed within narrow
bounds, not touching fundamental ideas. At the stage where we now
are, the formulation of the concepts can be seen to be as important
as the formulation of the empirical laws connecting the events in the
universe as thus conceived by us. For example, the concepts of life,
of heredity, of a material body, of a molecule, of an atom, of an
electron, of energy, of space, of time, of quantity, and of number. I
am not dogmatising about the best way of getting such ideas straight.
Certainly it will only be done by those who have devoted themselves to
a special study of the facts in question. Success is never absolute,
and progress in the right direction is the result of a slow, gradual
process of continual comparison of ideas with facts. The criterion of
success is that we should be able to formulate empirical laws, that is,
statements of relations, connecting the various parts of the universe
as thus conceived, laws with the property that we can interpret the
actual events of our lives as being our fragmentary knowledge of this
conceived interrelated whole.
But, for the purpose of science, what is the actual world? Has science
to wait for the termination of the metaphysical debate till it can
determine its own subject-matter? I suggest that science has a much
more homely starting-ground. Its task is the discovery of the relations
which exist within that flux of perceptions, sensations, and emotions
which forms our experience of life. The panorama yielded by sight,
sound, taste, smell, touch, and by more inchoate sensible feelings,
is the sole field of activity. It is in this way that science is the
thought organisation of experience. The most obvious aspect of this
field of actual experience is its disorderly character. It is for
each person a _continuum_, fragmentary, and with elements not
clearly differentiated. The comparison of the sensible experiences of
diverse people brings its own difficulties. I insist on the radically
untidy, ill-adjusted character of the fields of actual experience from
which science starts. To grasp this fundamental truth is the first
step in wisdom, when constructing a philosophy of science. This fact
is concealed by the influence of language, moulded by science, which
foists on us exact concepts as though they represented the immediate
deliverances of experience. The result is, that we imagine that we have
immediate experience of a world of perfectly defined objects implicated
in perfectly defined events which, as known to us by the direct
deliverance of our senses, happen at exact instants of time, in a space
formed by exact points, without parts and without magnitude: the neat,
trim, tidy, exact world which is the goal of scientific thought.
My contention is, that this world is a world of ideas, and that its
internal relations are relations between abstract concepts, and that
the elucidation of the precise connection between this world and the
feelings of actual experience is the fundamental question of scientific
philosophy. The question which I am inviting you to consider is this:
How does exact thought apply to the fragmentary, vague _continua_
of experience? I am not saying that it does not apply: quite the
contrary. But I want to know how it applies. The solution I am
asking for is not a phrase, however brilliant, but a solid branch of
science, constructed with slow patience, showing in detail how the
correspondence is effected.
The first great steps in the organisation of thought were due
exclusively to the practical source of scientific activity, without
any admixture of theoretical impulse. Their slow accomplishment was
the cause and also the effect of the gradual evolution of moderately
rational beings. I mean the formation of the concepts of definite
material objects, of the determinate lapse of time, of simultaneity,
of recurrence, of definite relative position, and of analogous
fundamental ideas, according to which the flux of our experience is
mentally arranged for handy reference: in fact, the whole apparatus of
commonsense thought. Consider in your mind some definite chair. The
concept of that chair is simply the concept of all the interrelated
experiences connected with that chair--namely, of the experience of the
folk who made it, of the folk who sold it, of the folk who have seen
it or used it, of the man who is now experiencing a comfortable sense
of support, combined with our expectations of an analogous future,
terminated finally by a different set of experiences when the chair
collapses and becomes firewood. The formation of that type of concept
was a tremendous job, and zoologists and geologists tell us that it
took many tens of millions of years. I can well believe it.
I now emphasise two points. In the first place, science is rooted in
what I have just called the whole apparatus of commonsense thought.
That is the _datum_ from which it starts, and to which it
must recur. We may speculate, if it amuses us, of other beings in
other planets who have arranged analogous experiences according to
an entirely different conceptual code--namely, who have directed
their chief attention to different relations between their various
experiences. But the task is too complex, too gigantic, to be revised
in its main outlines. You may polish up commonsense, you may contradict
it in detail, you may surprise it. But ultimately your whole task is to
satisfy it.
In the second place, neither commonsense nor science can proceed with
their task of thought organisation without departing in some respect
from the strict consideration of what is actual in experience. Think
again of the chair. Among the experiences upon which its concept is
based, I included our expectations of its future history. I should
have gone further and included our imagination of all the possible
experiences which in ordinary language we should call perceptions of
the chair which might have occurred. This is a difficult question, and
I do not see my way through it. But, at present, in the construction of
a theory of space and of time there seem insuperable difficulties if we
refuse to admit ideal experiences.
This imaginative perception of experiences, which, if they occurred,
would be coherent with our actual experiences, seems fundamental in our
lives. It is neither wholly arbitrary, nor yet fully determined. It
is a vague background which is only made in part definite by isolated
activities of thought. Consider, for example, our thoughts of the
unseen flora of Brazil.
Ideal experiences are closely connected with our imaginative
reproduction of the actual experiences of other people, and also
with our almost inevitable conception of ourselves as receiving our
impressions from an external complex reality beyond ourselves. It
may be that an adequate analysis of every source and every type of
experience yields demonstrative proof of such a reality and of its
nature. Indeed, it is hardly to be doubted that this is the case. The
precise elucidation of this question is the problem of metaphysics. One
of the points which I am urging in this address is, that the basis of
science does not depend on the assumption of any of the conclusions of
metaphysics; but that both science and metaphysics start from the same
given groundwork of immediate experience, and in the main proceed in
opposite directions on their diverse tasks.
For example, metaphysics inquires how our perceptions of the chair
relate us to some true reality. Science gathers up these perceptions
into a determinate class, adds to them ideal perceptions of analogous
sort, which under assignable circumstances would be obtained, and this
single concept of that set of perceptions is all that science needs;
unless indeed you prefer that thought find its origin in some legend of
those great twin brethren, the Cock and Bull.
My immediate problem is to inquire into the nature of the texture of
science. Science is essentially logical. The nexus between its concepts
is a logical nexus, and the grounds for its detailed assertions
are logical grounds. King James said, "No bishops, no king." With
greater confidence we can say, "No logic, no science." The reason for
the instinctive dislike which most men of science feel towards the
recognition of this truth is, I think, the barren failure of logical
theory during the past three or four centuries. We may trace this
failure back to the worship of authority, which in some respects
increased in the learned world at the time of the Renaissance. Mankind
then changed its authority, and this fact temporally acted as an
emancipation. But the main fact, and we can find complaints[2] of it
at the very commencement of the modern movement, was the establishment
of a reverential attitude towards any statement made by a classical
author. Scholars became commentators on truths too fragile to bear
translation. A science which hesitates to forget its founders is lost.
To this hesitation I ascribe the barrenness of logic. Another reason
for distrust of logical theory and of mathematics is the belief that
deductive reasoning can give you nothing new. Your conclusions are
contained in your premises, which by hypothesis are known to you.
In the first place this last condemnation of logic neglects the
fragmentary, disconnected character of human knowledge. To know one
premise on Monday, and another premise on Tuesday, is useless to you
on Wednesday. Science is a permanent record of premises, deductions,
and conclusions, verified all along the line by its correspondence
with facts. Secondly, it is untrue that when we know the premises we
also know the conclusions. In arithmetic, for example, mankind are not
calculating boys. Any theory which proves that they are conversant with
the consequences of their assumptions must be wrong. We can imagine
beings who possess such insight. But we are not such creatures.
Both these answers are, I think, true and relevant. But they are
not satisfactory. They are too much in the nature of bludgeons,
too external. We want something more explanatory of the very real
difficulty which the question suggests. In fact, the true answer is
embedded in the discussion of our main problem of the relation of logic
to natural science.
It will be necessary to sketch in broad outline some relevant features
of modern logic. In doing so I shall try to avoid the profound general
discussions and the minute technical classifications which occupy the
main part of traditional logic. It is characteristic of a science in
its earlier stages--and logic has become fossilised in such a stage--to
be both ambitiously profound in its aims and trivial in its handling of
details.
We can discern four departments of logical theory. By an analogy
which is not so very remote I will call these departments or sections
the arithmetic section, the algebraic section, the section of
general-function theory, the analytical section. I do not mean that
arithmetic arises in the first section, algebra in the second section,
and so on; but the names are suggestive of certain qualities of
thought in each section which are reminiscent of analogous qualities
in arithmetic, in algebra, in the general theory of a mathematical
function, and in the mathematical analysis of the properties of
particular functions.
The first section--namely, the arithmetic stage--deals with the
relations of definite propositions to each other, just as arithmetic
deals with definite numbers. Consider any definite proposition; call
it "_p_." We conceive that there is always another proposition which
is the direct contradictory to "_p_"; call it "not-_p_." When we have
got two propositions, _p_ and _q_, we can form derivative propositions
from them, and from their contradictories. We can say, "At last one of
_p_ or _q_ is true, and perhaps both." Let us call this proposition
"_p_ or _q_." I may mention as an aside that one of the greatest living
philosophers has stated that this use of the word "or"--namely, "_p_ or
_q_" in the sense that either or both may be true--makes him despair of
exact expression. We must brave his wrath, which is unintelligible to
me.
We have thus got hold of four new propositions, namely, "_p_ or _q_,"
and "not-_p_ or _q_," and "_p_ or not-_q_," and "not-_p_ or not-_q_."
Call these the set of disjunctive derivatives. There are, so far,
in all eight propositions, _p_, not-_p_, _q_, not-_q_, and the four
disjunctive derivatives. Any pair of these eight propositions can be
taken, and substituted for _p_ and _q_ in the foregoing treatment.
Thus each pair yields eight propositions, some of which may have been
obtained before. By proceeding in this way we arrive at an unending set
of propositions of growing complexity, ultimately derived from the two
original propositions _p_ or _q_. Of course, only a few are important.
Similarly we can start from three propositions, _p_, _q_, _r_, or
from four propositions, _p_, _q_, _r_, _s_, and so on. Any one of the
propositions of these aggregates may be true or false. It has no other
alternative. Whichever it is, true or false, call it the "truth-value"
of the proposition.
The first section of logical inquiry is to settle what we know of the
truth-values of these propositions, when we know the truth-values of
some of them. The inquiry, so far as it is worth while carrying it,
is not very abstruse, and the best way of expressing its results is a
detail which I will not now consider. This inquiry forms the arithmetic
stage.
The next section of logic is the algebraic stage. Now, the difference
between arithmetic and algebra is, that in arithmetic definite
numbers are considered, and in algebra symbols--namely, letters--are
introduced which stand for any numbers. The idea of a number is also
enlarged. These letters, standing for any numbers, are called sometimes
variables and sometimes parameters. Their essential characteristic is
that they are undetermined, unless, indeed, the algebraic conditions
which they satisfy implicitly determine them. Then they are sometimes
called unknowns. An algebraic formula with letters is a blank form. It
becomes a determinate arithmetic statement when definite numbers are
substituted for the letters. The importance of algebra is a tribute to
the study of form. Consider now the following proposition--
The specific heat of mercury is 0·033.
This is a definite proposition which, with certain limitations, is
true. But the truth-value of the proposition does not immediately
concern us. Instead of mercury put a mere letter which is the name of
some undetermined thing: we get--
The specific heat of _x_ is 0·033.
This is not a proposition; it has been called by Russell a
propositional function. It is the logical analogy of an algebraic
expression. Let us write ƒ(_x_) for any propositional function.
We could also generalise still further, and say,
The specific heat of _x_ is _y_.
We thus get another propositional function, F(_x_, _y_),
of two arguments _x_ and _y_, and so on for any number of
arguments.
Now, consider ƒ(_x_). There is the range of values of _x_,
for which ƒ(_x_) is a proposition, true or false. For values of
_x_ outside this range, ƒ(_x_) is not a proposition at all,
and is neither true nor false. It may have vague suggestions for us,
but it has no unit meaning of definite assertion. For example,
The specific heat of water is 0·033
is a proposition which is false; and--
The specific heat of virtue is 0·033
is, I should imagine, not a proposition at all; so that it is neither
true nor false, though its component parts raise various associations
in our minds. This range of values, for which ƒ(_x_) has sense, is
called the "type" of the argument _x_.
But there is also a range of values of _x_ for which ƒ(_x_)
is a true proposition. This is the class of those values of the
argument which _satisfy_ ƒ(_x_). This class may have no
members, or, in the other extreme, the class may be the whole type of
the arguments.
We thus conceive two general propositions respecting the indefinite
number of propositions which share in the same logical form, that is,
which are values of the same propositional function. One of these
propositions is,
ƒ(_x_) yields a true proposition for each value of _x_ of
the proper type;
the other proposition is,
There is a value of _x_ for which ƒ(_x_) is true.
Given two, or more, propositional functions ƒ(_x_) and ϕ(_x_)
with the same argument _x_, we form derivative propositional
functions, namely,
ƒ(_x_) or ϕ(_x_), ƒ(_x_) or not-ϕ(_x_),
and so on with the contradictories, obtaining, as in the arithmetical
stage, an unending aggregate of propositional functions. Also each
propositional function yields two general propositions. The theory
of the interconnection between the truth-values of the general
propositions arising from any such aggregate of propositional functions
forms a simple and elegant chapter of mathematical logic.
In this algebraic section of logic the theory of types crops up, as we
have already noted. It cannot be neglected without the introduction of
error. Its theory has to be settled at least by some safe hypothesis,
even if it does not go to the philosophic basis of the question. This
part of the subject is obscure and difficult, and has not been finally
elucidated, though Russell's brilliant work has opened out the subject.
The final impulse to modern logic comes from the independent discovery
of the importance of the logic variable by Frege and Peano. Frege went
further than Peano, but by an unfortunate symbolism rendered his work
so obscure that no one fully recognised his meaning who had not found
it out for himself. But the movement has a large history reaching back
to Leibniz and even to Aristotle. Among English contributors are De
Morgan, Boole, and Sir Alfred Kempe; their work is of the first rank.
The third logical section is the stage of general-function theory.
In logical language, we perform in this stage the transition from
intension to extension, and investigate the theory of denotation. Take
the propositional function, ƒ(_x_). There is the class, or range
of values for _x_, whose members satisfy ƒ(_x_). But the same
range may be the class whose members satisfy another propositional
function ϕ(_x_). It is necessary to investigate how to indicate
the class by a way which is indifferent as between the various
propositional functions which are satisfied by any member of it, and of
it only. What has to be done is to analyse the nature of propositions
about a class--namely, those propositions whose truth-values depend on
the class itself and not on the particular meaning by which the class
is indicated.
Furthermore, there are propositions about alleged individuals
indicated by descriptive phrases: for example, propositions about "the
present King of England," who does exist, and "the present Emperor
of Brazil," who does not exist. More complicated, but analogous,
questions involving propositional functions of two variables involve
the notion of "correlation," just as functions of one argument involve
classes. Similarly functions of three arguments yield three-cornered
correlations, and so on. This logical section is one which Russell has
made peculiarly his own by work which must always remain fundamental.
I have called this the section of functional theory, because its
ideas are essential to the construction of logical denoting functions
which include as a special case ordinary mathematical functions, such
as sine, logarithm, etc. In each of these three stages it will be
necessary gradually to introduce an appropriate symbolism, if we are to
pass on to the fourth stage.
The fourth logical section, the analytic stage, is concerned with the
investigation of the properties of special logical constructions,
that is, of classes and correlations of special sorts. The whole of
mathematics is included here. So the section is a large one. In fact,
it is mathematics, neither more nor less, but it includes an analysis
of mathematical ideas not hitherto included in the scope of that
science, nor, indeed, contemplated at all. The essence of this stage is
construction. It is by means of suitable constructions that the great
framework of applied mathematics, comprising the theories of number,
quantity, time, and space, is elaborated.
It is impossible, even in brief outline, to explain how mathematics
is developed from the concepts of class and correlation, including
many-cornered correlations, which are established in the third section.
I can only allude to the headings of the process, which is fully
developed in the work, _Principia Mathematica_, by Mr. Russell
and myself. There are in this process of development seven special
sorts of correlations which are of peculiar interest. The first sort
comprises one-to-many, many-to-one, and one-to-one correlations.
The second sort comprises serial relations, that is, correlations
by which the members of some field are arranged in serial order, so
that, in the sense defined by the relation, any member of the field
is either before or after any other member. The third class comprises
inductive relations, that is, correlations on which the theory of
mathematical induction depends. The fourth class comprises selective
relations, which are required for the general theory of arithmetic
operations, and elsewhere. It is in connection with such relations that
the famous multiplicative axiom arises for consideration. The fifth
class comprises vector relations, from which the theory of quantity
arises. The sixth class comprises ratio relations, which interconnect
number and quantity. The seventh class comprises three-cornered and
four-cornered relations which occur in geometry.
A bare enumeration of technical names, such as the above, is not very
illuminating, though it may help to a comprehension of the demarcations
of the subject. Please remember that the names are technical names,
meant, no doubt, to be suggestive, but used in strictly defined
senses. We have suffered much from critics who consider it sufficient
to criticise our procedure on the slender basis of a knowledge of
the dictionary meanings of such terms. For example, a one-to-one
correlation depends on the notion of a class with only one member, and
this notion is defined without appeal to the concept of the number one.
The notion of diversity is all that is wanted. Thus the class α has
only one member, if (1) the class of values of _x_ which satisfies
the propositional function,
_x_ is not a member of α,
is not the whole type of relevant values of _x_, and if (2) the
propositional function,
_x_ and _y_ are members of α, and _x_ is diverse from
_y_
is false, whatever be the values of _x_ and _y_ in the
relevant type.
Analogous procedures are obviously possible for higher finite cardinal
members. Thus, step by step, the whole cycle of current mathematical
ideas is capable of logical definition. The process is detailed and
laborious, and, like all science, knows nothing of a royal road of
airy phrases. The essence of the process is, first, to construct
the notion in terms of the forms of propositions, that is, in terms
of the relevant propositional functions, and secondly, to prove the
fundamental truths which hold about the notion by reference to the
results obtained in the algebraic section of logic.
It will be seen that in this process the whole apparatus of special
indefinable mathematical concepts, and special _a priori_
mathematical premises, respecting number, quantity, and space, has
vanished. Mathematics is merely an apparatus for analysing the
deductions which can be drawn from any particular premises, supplied
by commonsense, or by more refined scientific observation, so far as
these deductions depend on the forms of the propositions. Propositions
of certain forms are continually occurring in thought. Our existing
mathematics is the analysis of deductions which concern those forms and
in some way are important, either from practical utility or theoretical
interest. Here I am speaking of the science as it in fact exists. A
theoretical definition of mathematics must include in its scope any
deductions depending on the mere forms of propositions. But, of course
no one would wish to develop that part of mathematics which in no sense
is of importance.
This hasty summary of logical ideas suggests some reflections. The
question arises, How many forms of propositions are there? The
answer is, An unending number. The reason for the supposed sterility
of logical science can thus be discerned. Aristotle founded the
science by conceiving the idea of the form of a proposition, and by
conceiving deduction as taking place in virtue of the forms. But he
confined propositions to four forms, now named A, I, E, O. So long as
logicians were obsessed by this unfortunate restriction, real progress
was impossible. Again, in their theory of form, both Aristotle and
subsequent logicians came very near to the theory of the logical
variable. But to come very near to a true theory, and to grasp its
precise application, are two very different things, as the history of
science teaches us. Everything of importance has been said before by
somebody who did not discover it.
Again, one reason why logical deductions are not obvious is, that
logical form is not a subject which ordinarily enters into thought.
Commonsense deduction probably moves by blind instinct from concrete
proposition to concrete proposition, guided by some habitual
association of ideas. Thus commonsense fails in the presence of a
wealth of material.
A more important question is the relation of induction, based on
observation, to deductive logic. There is a tradition of opposition
between adherents of induction and of deduction. In my view, it
would be just as sensible for the two ends of a worm to quarrel.
Both observation and deduction are necessary for any knowledge worth
having. We cannot get at an inductive law without having recourse to a
propositional function. For example, take the statement of observed
fact,
This body is mercury, and its specific heat is 0·033.
The propositional function is formed,
Either _x_ is not mercury, or its specific heat is 0·033.
The inductive law is the assumption of the truth of the general
proposition, that the above propositional function is true for every
value of _x_ in the relevant type.
But it is objected that this process and its consequences are so simple
that an elaborate science is out of place. In the same way, a British
sailor knows the salt sea when he sails over it. What, then, is the use
of an elaborate chemical analysis of sea-water? There is the general
answer, that you cannot know too much of methods which you always
employ; and there is the special answer, that logical forms and logical
implications are not so very simple, and that the whole of mathematics
is evidence to this effect.
One great use of the study of logical method is not in the region of
elaborate deduction, but to guide us in the study of the formation of
the main concepts of science. Consider geometry, for example. What are
the points which compose space? Euclid tells us that they are without
parts and without magnitude. But how is the notion of a point derived
from the sense-perceptions from which science starts? Certainly points
are not direct deliverances of the senses. Here and there we may see or
unpleasantly feel something suggestive of a point. But this is a rare
phenomenon, and certainly does not warrant the conception of space as
composed of points. Our knowledge of space properties is not based on
any observations of relations between points. It arises from experience
of relations between bodies. Now a fundamental space-relation between
bodies is that one body may be part of another. We are tempted to
define the "whole and part" relation by saying that the points occupied
by the part are some of the points occupied by the whole. But "whole
and part" being more fundamental than the notion of "point," this
definition is really circular and vicious.
We accordingly ask whether any other definition of "spatial whole and
part" can be given. I think that it can be done in this way, though,
if I be mistaken, it is unessential to my general argument. We have
come to the conclusion that an extended body is nothing else than the
class of perception of it by all its percipients, actual or ideal. Of
course, it is not any class of perceptions, but a certain definite sort
of class which I have not defined here, except by the vicious method
of saying that they are perceptions of body. Now, the perceptions of a
part of a body are among the perceptions which compose the whole body.
Thus two bodies _a_ and _b_ are both classes of perceptions; and _b_ is
part of _a_ when the class which is _b_ is contained in the class which
is _a_. It immediately follows from the logical form of this definition
that if _b_ is part of _a_, and _c_ is part of _b_, then _c_ is part of
_a_. Thus the relation "whole to part" is transitive. Again, it will
be convenient to allow that a body is part of itself. This is a mere
question of how you draw the definition. With this understanding, the
relation is reflexive. Finally, if _a_ is part of _b_, and _b_ is part
of _a_, then _a_ and _b_ must be identical. These properties of "whole
and part" are not fresh assumptions, they follow from the logical form
of our definition.
One assumption has to be made if we assume the ideal infinite
divisibility of space. Namely, we assume that every class of
perceptions which is an extended body contains other classes of
perceptions which are extended bodies diverse from itself. This
assumption makes rather a large draft on the theory of ideal
perceptions. Geometry vanishes unless in some form you make it. The
assumption is not peculiar to my exposition.
It is then possible to define what we mean by a point. A point is the
class of extended objects which, in ordinary language, contain that
point. The definition, without presupposing the idea of a point, is
rather elaborate, and I have not now time for its statement.
The advantage of introducing points into geometry is the simplicity
of the logical expression of their mutual relations. For science,
simplicity of definition is of slight importance, but simplicity of
mutual relations is essential. Another example of this law is the way
physicists and chemists have dissolved the simple idea of an extended
body, say of a chair, which a child understands, into a bewildering
notion of a complex dance of molecules and atoms and electrons and
waves of light. They have thereby gained notions with simpler logical
relations.
Space as thus conceived is the exact formulation of the properties of
the apparent space of the commonsense world of experience. It is not
necessarily the best mode of conceiving the space of the physicist.
The one essential requisite is that the correspondence between the
commonsense world in its space and the physicists' world in its space
should be definite and reciprocal.
I will now break off the exposition of the function of logic in
connection with the science of natural phenomena. I have endeavoured
to exhibit it as the organising principle, analysing the derivation
of the concepts from the immediate phenomena, examining the structure
of the general propositions which are the assumed laws of nature,
establishing their relations to each other in respect to reciprocal
implications, deducing the phenomena we may expect under given
circumstances.
Logic, properly used, does not shackle thought. It gives freedom, and
above all, boldness. Illogical thought hesitates to draw conclusions,
because it never knows either what it means, or what it assumes, or
how far it trusts its own assumptions, or what will be the effect of
any modification of assumptions. Also the mind untrained in that part
of constructive logic which is relevant to the subject in hand will be
ignorant of the sort of conclusions which follow from various sorts of
assumptions, and will be correspondingly dull in divining the inductive
laws. The fundamental training in this relevant logic is, undoubtedly,
to ponder with an active mind over the known facts of the case,
directly observed. But where elaborate deductions are possible, this
mental activity requires for its full exercise the direct study of the
abstract logical relations. This is applied mathematics.
Neither logic without observation, nor observation without logic, can
move one step in the formation of science. We may conceive humanity
as engaged in an internecine conflict between youth and age. Youth is
not defined by years but by the creative impulse to make something. The
aged are those who, before all things, desire not to make a mistake.
Logic is the olive branch from the old to the young, the wand which in
the hands of youth has the magic property of creating science.
FOOTNOTES:
[Footnote 2: _E.g._ in 1551 by Italian schoolmen; cf. Sarpi's
_History of the Council of Trent_, under that date.]
CHAPTER VII
THE ANATOMY OF SOME SCIENTIFIC IDEAS
_I. Fact_
THE characteristic of physical science is, that it ignores all
judgments of value: for example, æsthetic or moral judgments. It is
purely matter-of-fact, and this is the sense in which we must interpret
the sonorous phrase, "Man, the servant and the minister of Nature."
The sphere of thought which is thus left is even then too wide for
physical science. It would include Ontology, namely, the determination
of the nature of what truly exists; in other words, Metaphysics. From
an abstract point of view this exclusion of metaphysical inquiry
is a pity. Such an inquiry is a necessary critique of the worth of
science, to tell us what it all comes to. The reasons for its careful
separation from scientific thought are purely practical; namely,
because we can agree about science--after due debate--whereas in
respect to metaphysics debate has hitherto accentuated disagreement.
These characteristics of science and metaphysics were unexpected in the
early days of civilised thought. The Greeks thought that metaphysics
was easier than physics, and tended to deduce scientific principles
from _a priori_ conceptions of the nature of things. They were
restrained in this disastrous tendency by their vivid naturalism, their
delight in first-hand perception. Mediæval Europe shared the tendency
without the restraint. It is possible that some distant generations
may arrive at unanimous conclusions on ontological questions, whereas
scientific progress may have led to ingrained opposing veins of
thought which can neither be reconciled nor abandoned. In such times
metaphysics and physical science will exchange their rôles. Meanwhile
we must take the case as we find it.
But a problem remains. How can mankind agree about science without
a preliminary determination of what really is? The answer must be
found in an analysis of the facts which form the field of scientific
activity. Mankind perceives, and finds itself thinking about its
perceptions. It is the thought that matters and not that element of
perception which is not thought. When the immediate judgment has been
formed--Hullo, red!--it does not matter if we can imagine that in other
circumstances--in better circumstances, perhaps--the judgment would
have been--Hullo, blue!--or even--Hullo, nothing! For all intents and
purposes, at the time it was red. Everything else is hypothetical
reconstruction. The field of physical science is composed of these
primary thoughts, and of thoughts about these thoughts.
But--to avoid confusion--a false simplicity has been introduced above
into the example given of a primary perceptive thought. "Hullo, red!"
is not really a primary perceptive thought, though it often is the
first thought which finds verbal expression even silently in the mind.
Nothing is in isolation. The perception of red is of a red object in
its relations to the whole content of the perceiving consciousness.
Among the most easily analysed of such relations are the space
relations. Again the red object is in immediate perception nothing else
than a red object. It is better termed an "object of redness." Thus a
better approximation to an immediate perceptive judgment is, "Hullo,
object of redness there!" But, of course, in this formulation other
more complex relations are omitted.
This tendency towards a false simplicity in scientific analysis, to
an excessive abstraction, to an over-universalising of universals,
is derived from the earlier metaphysical stage. It arises from the
implicit belief that we are endeavouring to qualify the real with
appropriate adjectives. In conformity with this tendency we think,
"this real thing is red." Whereas our true goal is to make explicit
our perception of the apparent in terms of its relations. What we
perceive is redness related to other apparents. Our object is the
analysis of the relations.
One aim of science is the harmony of thought, that is, to secure
that judgments which are logical contraries should not be
thought-expressions of consciousness. Another aim is the extension of
such harmonised thought.
Some thoughts arise directly from sense-presentation, and are part of
the state of consciousness which is perception. Such a thought is, "An
object of redness is there." But in general the thought is not verbal,
but is a direct apprehension of qualities and relations within the
content of consciousness.
Amid such thoughts there can be no lack of harmony. For direct
apprehension is in its essence unique, and it is impossible to
apprehend an object as both red and blue. Subsequently it may be judged
that if other elements of the consciousness had been different, the
apprehension would have been of a blue object. Then--under certain
circumstances--the original apprehension will be called an error. But
for all that the fact remains, there was an apprehension of a red
object.
When we speak of sense-presentation, we mean these primary thoughts
essentially involved in its perception. But there are thoughts about
thoughts, and thoughts derived from other thoughts. These are secondary
thoughts. At this point it is well explicitly to discriminate between
an actual thought-expression, namely, a judgment actually made, and a
mere proposition which is a hypothetical thought-expression, namely,
an imagined possibility of thought-expression. Note that the actual
complete thought-content of the consciousness is explicitly neither
affirmed or denied. It is just what _is_ thought. Thus, to think
"two and two make four" is distinct from affirming that two and two
make four. In the first case the proposition is the thought-expression,
in the second case the affirmation of the proposition is the
thought-expression, and the proposition has been degraded to a mere
proposition, namely, to a hypothetical thought-expression which is
reflected upon.
A distinction is sometimes made between facts and thoughts. So far as
physical science is concerned, the facts are thoughts, and thoughts
are facts. Namely, the facts of sense-presentation as they affect
science are those elements in the immediate apprehensions which are
thoughts. Also, actual thought-expressions, primary or secondary, are
the material facts which science interprets.
The distinction that facts are given, but thoughts are free, is
not absolute. We can select and modify our sense-presentation, so
that facts--in the narrower sense of immediate apprehension of
sense-presentation--are to some degree subject to volition. Again, our
stream of thought-expression is only partially modified by explicit
volition. We can choose our physical experience, and we find ourselves
thinking; namely, on the one hand there is selection amid the dominant
necessity of sense, and on the other hand, the thought-content of
consciousness (so far as secondary thoughts are concerned) is not
wholly constituted by the selection of will.
Thus, on the whole there is a large primary region of secondary
thought, as well as of the primary thoughts of sense-presentation,
which is given in type. That is the way in which we do think of things,
not wholly from any abstract necessity, so far as we know, but because
we have inherited the method from an environment. It is the way we find
ourselves thinking, a way which can only be fundamentally laid aside by
an immense effort, and then only for isolated short periods of time.
This is what I have called the "whole apparatus of commonsense thought."
It is this body of thought which is assumed in science. It is a way
of thinking rather than a set of axioms. It is, in fact, the set of
concepts which commonsense has found useful in sorting out human
experience. It is modified in detail, but assumed in gross. The
explanations of science are directed to finding conceptions and
propositions concerning nature which explain the importance of these
common sense notions. For example, a chair is a common sense notion,
molecules and electrons explain our vision of chairs.
Now science aims at harmonising our reflective and derivative thoughts
with the primary thoughts involved in the immediate apprehension of
sense-presentation. It also aims at producing such derivative thoughts,
logically knit together. This is scientific theory; and the harmony to
be achieved is the agreement of theory with observation, which is the
apprehension of sense-presentation.
Thus there is a twofold scientific aim: (1) the production of theory
which agrees with experience; and (2) the explanation of commonsense
concepts of nature, at least in their main outlines. This explanation
consists in the preservation of the concepts in a scientific theory of
harmonised thought.
It is not asserted that this is what scientists in the past meant to
achieve, or thought that they could achieve. It is suggested as the
actual result of scientific effort, so far as that effort has had
any measure of success. In short, we are here discussing the natural
history of ideas and not volitions of scientists.
_II. Objects_
We perceive things in space. For example, among such things are dogs,
chairs, curtains, drops of water, gusts of air, flames, rainbows,
chimes of bells, odours, aches and pains. There is a scientific
explanation of the origin of these perceptions. This explanation
is given in terms of molecules, atoms, electrons, and their mutual
relations, in particular of their space-relations, and waves of
disturbance of these space-relations which are propagated through
space. The primary elements of the scientific explanation--molecules,
etc.--are not the things directly perceived. For example, we do not
perceive a wave of light; the sensation of sight is the resultant
effect of the impact of millions of such waves through a stretch of
time. Thus the object directly perceived corresponds to a series of
events in the physical world, events which are prolonged through
a stretch of time. Nor is it true that a perceived object always
corresponds to the same group of molecules. After a few years we
recognise the same cat, but we are thereby related to different
molecules.
Again, neglecting for a moment the scientific explanation, the
perceived object is largely the supposition of our imagination. When
we recognised the cat, we also recognised that it was glad to see
us. But we merely heard its mewing, saw it arch its back, and felt it
rubbing itself against us. We must distinguish, therefore, between the
many direct objects of sense, and the single indirect object of thought
which is the cat.
Thus, when we say that we perceived the cat and understood its
feelings, we mean that we heard a sense-object of sound, that we saw a
sense-object of sight, that we felt a sense-object of touch, and that
we thought of a cat and imagined its feelings.
Sense-objects are correlated by time-relations and space-relations.
Three simultaneous sense-objects which are also spatially coincident,
are combined by thought into the perception of one cat. Such
combination of sense-objects is an instinctive immediate judgment in
general without effort of reasoning. Sometimes only one sense-object
is present. For example, we hear mewing and say there must be a cat
in the room. The transition from the sense-object to the cat has
then been made, by deliberate ratiocination. Even the concurrence of
sense-objects may provoke such a self-conscious effort. For example, in
the dark we feel something, and hear mewing from the same place, and
think, Surely this is a cat. Sight is more bold; when we see a cat, we
do not think further. We identify the sight with the cat, whereas the
cat and the mew are separate. But such immediate identification of
a sight object and an object of thought may lead to error; the birds
pecked at the grapes of Apelles.
A single sense-object is a complex entity. The sight-object of a
tile on the hearth may remain unchanged as we watch it in a steady
light, remaining ourselves unchanged in position. Even then it is
prolonged in time, and has parts in space. Also it is somewhat
arbitrarily distinguished from a larger whole of which it forms part.
But the glancing fire-light and a change in our position alters the
sight-object. We judge that the tile thought-object remains unchanged.
The sight-object of the coal on the fire gradually modifies, though
within short intervals it remains unchanged. We judge that the coal
thought-object is changing. The flame is never the same, and its shape
is only vaguely distinguishable.
We conclude that a single self-identical sight-object is already a
phantasy of thought. Consider the unchanging sight-object of the tile,
as we remain still in a steady light. Now a sense-object perceived at
one time is a distinct object from a sense-object seen at another time.
Thus the sight of the tile at noon is distinct from its sight at 12.30.
But there is no such thing as a sense-object at an instant. As we stare
at the tile, a minute, or a second, or a tenth of a second, has flown
by: essentially there is a duration. There is a stream of sight, and
we can distinguish its parts. But the parts also are streams, and
it is only in thought that the stream separates into a succession of
elements. The stream may be "steady" as in the case of the unchanging
sight-tile, or may be "turbulent" as in the case of the glancing
sight-flame. In either case a sight-object is some arbitrarily small
part of the stream.
Again, the stream which forms the succession of sight-tiles is merely a
distinguishable part of the whole stream of sight-presentation.
So, finally, we conceive ourselves each experiencing a complete
time-flux (or stream) of sense-presentation. This stream is
distinguishable into parts. The grounds of distinction are differences
of sense--including within that term, differences of types of sense,
and differences of quality and of intensity within the same type
of sense--and differences of time-relations, and differences of
space-relations. Also the parts are not mutually exclusive and exist in
unbounded variety.
The time-relation between the parts raises the questions of memory and
recognition, subjects too complex for discussion here. One remark must
be made. If it be admitted, as stated above, that we live in durations
and not in instants, namely, that the present essentially occupies
a stretch of time, the distinction between memory and immediate
presentation cannot be quite fundamental; for always we have with us
the fading present as it becomes the immediate past. This region of our
consciousness is neither pure memory nor pure immediate presentation.
Anyhow, memory is also a presentation in consciousness.
Another point is to be noted in connection with memory. There is no
directly perceived time-relation between a present event and a past
event. The present event is only related to the memory of the past
event. But the memory of a past event is itself a present element
in consciousness. We assert the principle that directly perceived
relations can only exist between elements of consciousness, both in
that present during which the perception occurs. All other relations
between elements of perception are inferential constructions. It thus
becomes necessary to explain how the time stream of events establishes
itself in thought, and how the apparent world fails to collapse into
one single present. The solution of the difficulty is arrived at
by observing that the present is itself a duration, and therefore
includes directly perceived time-relations between events contained
within it. In other words we put the present on the same footing as
the past and the future in respect to the inclusion within it of
antecedent and succeeding events, so that past, present, and future
are in this respect exactly analogous ideas. Thus there will be two
events _a_ and _b_, both in the same present, but the event
_a_ will be directly perceived to precede the event _b_.
Again time flows on, and the event _a_ fades into the past, and
in the new present duration events _b_ and _c_ occur, event
_b_ preceding event _c_, also in the same present duration
there is the memory of the time-relation between _a_ and _b_.
Then by an inferential construction the event _a_ in the past
precedes the event _c_ in the present. By proceeding according to
this principle the time-relations between elements of consciousness,
not in the same present, are established. The method of procedure
here explained is a first example of what we will call the Principle
of Aggregation. This is one of the fundamental principles of mental
construction according to which our conception of the external physical
world is constructed. Other examples will later on be met with.
The space-relations between the parts are confused and fluctuating,
and in general lack determinate precision. The master-key by which
we confine our attention to such parts as possess mutual relations
sufficiently simple for our intellects to consider is the principle of
convergence to simplicity with diminution of extent. We will call it
the "principle of convergence." This principle extends throughout the
whole field of sense-presentation.
The first application of the principle occurs in respect to time.
The shorter the stretch of time, the simpler are the aspects of the
sense-presentation contained within it. The perplexing effects of
change are diminished and in many cases can be neglected. Nature has
restricted the acts of thought which endeavour to realise the content
of the present, to stretches of time sufficiently short to secure this
static simplicity over the greater part of the sense-stream.
Spatial relations become simplified within the approximately static
sense-world of the short time. A further simplicity is gained by
partitioning this static world into parts of restricted space-content.
The various parts thus obtained have simpler mutual space-relations,
and again the principle of convergence holds.
Finally, the last simplicity is obtained by partitioning the
parts, already restricted as to space and time, into further parts
characterised by homogeneity in type of sense, and homogeneity in
quality and intensity of sense. These three processes of restriction
yield, finally, the sense-objects which have been mentioned above. Thus
the sense-object is the result of an active process of discrimination
made in virtue of the principle of convergence. It is the result of
the quest for simplicity of relations within the complete stream of
sense-presentation.
The thought-objects of perception are instances of a fundamental
law of nature, the law of objective stability. It is the law of the
coherence of sense-objects. This law of stability has an application
to time and an application to space; also it must be applied in
conjunction with that other law, the principle of convergence to
simplicity from which sense-objects are derived.
Some composite partial streams of sense-presentation can be
distinguished with the following characteristics: (1) the
time-succession of sense-objects, belonging to a single sense, involved
in any such a composite partial stream, is composed of very similar
objects whose modifications increase only gradually, and thus forms
a homogeneous component stream within the composite stream; (2) the
space-relations of those sense-objects (of various senses) of such
a composite stream which are confined within any sufficiently short
time are identical so far as they are definitely apprehended, and thus
these various component streams, each homogeneous, "cohere" to form the
whole composite partial stream; (3) there are other sense-presentations
occurring in association with that composite partial stream which
can be determined by rules derived from analogous composite partial
streams, with other space and time relations, provided that the analogy
be sufficiently close. Call these the "associated sense-presentations."
A partial stream of this sort, viewed as a whole, is here called a
"first crude thought-object of perception."
For example, we look at an orange for half a minute, handle it, and
smell it, note its position in the fruit-basket, and then turn away.
The stream of sense-presentation of the orange during that half-minute
is a first crude thought-object of perception. Among the associated
sense presentations are those of the fruit-basket which we conceive as
supporting the orange.
The essential ground of the association of sense-objects of
various types, perceived within one short duration, into a first
crude thought-object of perception is the coincidence of their
space-relations, that is, in general an approximate coincidence of
such relations perhaps only vaguely apprehended. Thus coincident
space-relations associate sense-objects into a first crude
thought-object, and diverse space-relations dissociate sense-objects
from aggregation into a first crude thought-object. In respect to
some groups of sense-objects the association may be an immediate
judgment devoid of all inference, so that the primary perceptual
thought is that of the first crude thought-object, and the separate
sense-objects are the result of reflective analysis acting on memory.
For example sense-objects of sight and sense-objects of touch are
often thus primarily associated and only secondarily dissociated in
thought. But sometimes the association is wavering and indeterminate,
for example, that between the sound-object of the mew of the cat and
the sight-object of the cat. Thus to sum up, the partial stream of
sense-perceptions coalesces into that first crude thought-object of
perception which is the momentary cat because the sense-perceptions
belonging to this stream are in the same place, but equally it would
be true to say that they are in the same place because they belong
to the same momentary cat. This analysis of the complete stream of
sense-presentation in any small present duration into a variety of
first crude thought-objects only partially fits the facts; for one
reason because many sense-objects, such as sound for instance, have
vague and indeterminate space-relations, for example vaguely those
space-relations which we associate with our organs of sense and also
vaguely those of the origin from which (in the scientific explanation)
they proceed.
The procedure by which the orange of half a minute is elaborated into
the orange in the ordinary sense of the term involves in addition the
two principles of aggregation and of hypothetical sense-presentation.
The principle of aggregation, as here employed, takes the form that
many distinct first crude thought-objects of perception are conceived
as one thought-object of perception, if the many partial streams
forming these objects are sufficiently analogous, if their times of
occurrence are distinct, and if the associated sense-presentations are
sufficiently analogous.
For example, after leaving the orange, in five minutes we return.
A new first crude thought-object of perception presents itself to
us, indistinguishable from the half-minute orange we previously
experienced; it is in the same fruit-basket. We aggregate the two
presentations of an orange into the same orange. By such aggregations
we obtain "second crude thought-objects of perception." But however far
we can proceed with aggregation of this type, the orange is more than
that. For example, what do we mean when we say, The orange is in the
cupboard, if Tom has not eaten it?
The world of present fact is more than a stream of sense-presentation.
We find ourselves with emotions, volitions, imaginations, conceptions,
and judgments. No factor which enters into consciousness is by itself
or even can exist in isolation. We are analysing certain relations
between sense-presentation and other factors of consciousness.
Hitherto we have taken into account merely the factors of concept and
judgment. Imagination is necessary to complete the orange, namely,
the imagination of hypothetical sense-presentations. It is beside
the point to argue whether we ought to have such imaginations, or to
discuss what are the metaphysical truths concerning reality to which
they correspond. We are here only concerned with the fact that such
imaginations exist and essentially enter into the formation of the
concepts of the thought-objects of perception which are the first
data of science. We conceive the orange as a permanent collection of
sense-presentations existing as if they were an actual element in our
consciousness, which they are not. The orange is thus conceived as
in the cupboard with its shape, odour, colour, and other qualities.
Namely, we imagine hypothetical possibilities of sense-presentation,
and conceive their want of actuality in our consciousness as immaterial
to their existence in fact. The fact which is essential for science is
our conception; its meaning in regard to the metaphysics of reality is
of no scientific importance, so far as physical science is concerned.
The orange completed in this way is the thought-object of perception.
It must be remembered that the judgments and concepts arising in the
formation of thought-objects of perception are in the main instinctive
judgments, and instinctive concepts, and are not concepts and judgments
consciously sought for and consciously criticised before adoption.
Their adoption is facilitated by and interwoven with the expectation of
the future in which the hypothetical passes into the actual, and also
with the further judgment of the existence of other consciousnesses,
so that much that is hypothetical to one consciousness is judged to be
actual to others.
The thought-object of perception is, in fact, a device to make plain
to our reflective consciousness relations which hold within the
complete stream of sense-presentation. Concerning the utility of this
weapon there can be no question; it is the rock upon which the whole
structure of commonsense thought is erected. But when we consider the
limits of its application the evidence is confused. A great part of our
sense-presentation can be construed as perception of various persistent
thought-objects. But hardly at any time can the sense-presentations
be construed wholly in that way. Sights lend themselves easily to
this construction, but sight can be baffled: for example, consider
reflections in looking-glasses, apparently bent sticks half in and half
out of water, rainbows, brilliant patches of light which conceal the
object from which they emanate, and many analogous phenomena. Sound
is more difficult; it tends largely to disengage itself from any such
object. For example, we see the bell, but we hear the sound which
comes from the bell; yet we also say that we hear the bell. Again, a
toothache is largely by itself, and is only indirectly a perception
of the nerve of the tooth. Illustrations to the same effect can be
accumulated from every type of sensation.
Another difficulty arises from the fact of change. The thought-object
is conceived as one thing, wholly actual at each instant. But since
the meat has been bought it has been cooked, the grass grows and then
withers, the coal burns in the fire, the pyramids of Egypt remain
unchanged for ages, but even the pyramids are not wholly unchanged. The
difficulty of change is merely evaded by affixing a technical Latin
name to a supposed logical fallacy. A slight cooking leaves the meat
the same object, but two days in the oven burns it to a cinder. When
does the meat cease to be? Now the chief use of the thought-object
is the concept of it as one thing, here and now, which later can be
recognised, there and then. This concept applies sufficiently well to
most things for short times, and to many things for long times. But
sense-presentation as a whole entirely refuses to be patient of the
concept.
We have now come to the reflective region of explanation, which is
science.
A great part of the difficulty is at once removed by applying the
principle of convergence to simplicity. We habitually make our
thought-objects too large; we should think in smaller parts. For
example, the Sphinx has changed by its nose becoming chipped, but by
proper inquiry we could find the missing part in some private house
of Western Europe or Northern America. Thus, either part, the rest
of the Sphinx or the chip, regains its permanence. Furthermore, we
enlarge this explanation by conceiving parts so small that they can
only be observed under the most favourable circumstances. This is a
wide extension of the principle of convergence in its application to
nature; but it is a principle amply supported by the history of exact
observation.
Thus, change in thought-objects of perception is largely explained as
a disintegration into smaller parts, themselves thought-objects of
perception. The thought-objects of perception which are presupposed in
the common thought of civilised beings are almost wholly hypothetical.
The material universe is largely a concept of the imagination which
rests on a slender basis of direct sense-presentation. But none the
less it is a fact; for it is a fact that actually we imagine it. Thus
it is actual in our consciousness just as sense-presentation also is
actual there. The effort of reflective criticism is to make these two
factors in our consciousness agree where they are related, namely,
to construe our sense-presentation as actual realisation of the
hypothetical thought-objects of perception.
The wholesale employment of purely hypothetical thought-objects of
perception enables science to explain some of the stray sense-objects
which cannot be construed as perceptions of a thought-object of
perception: for example, sounds. But the phenomena as a whole defy
explanation on these lines until a further fundamental step is taken,
which transforms the whole concept of the material universe. Namely,
the thought-object of perception is superseded by the thought-object of
science.
The thought-objects of science are molecules, atoms, and electrons. The
peculiarity of these objects is that they have shed all the qualities
which are capable of direct sense-representation in consciousness.
They are known to us only by their associated phenomena, namely,
series of events in which they are implicated are represented in our
consciousness by sense-presentations. In this way, the thought-objects
of science are conceived as the causes of sense-representation. The
transition from thought-objects of perception to thought-objects of
science is decently veiled by an elaborate theory concerning primary
and secondary qualities of bodies.
This device, by which sense-presentations are represented in thought
as our perception of events in which thought-objects of science are
implicated, is the fundamental means by which a bridge is formed
between the fluid vagueness of sense and the exact definition of
thought. In thought a proposition is either true or false, an entity
is exactly what it is, and relations between entities are expressible
(in idea) by definite propositions about distinctly conceived entities.
Sense-perception knows none of these things, except by courtesy.
Accuracy essentially collapses at some stage of inquiry.
_III. Time and Space_
_Recapitulation._--Relations of time and relations of space
hold between sense-objects of perception. These sense-objects are
distinguished as separate objects by the recognition of either (1)
differences of sense-content, or (2) time-relations between them
other than simultaneity, or (3) space-relations between them other
than coincidence. Thus sense-objects arise from the recognition of
contrast within the complete stream of sense-presentation, namely, from
the recognition of the objects as related terms, by relations which
contrast them. Differences of sense-content are infinitely complex in
their variety. Their analysis under the heading of general ideas is the
unending task of physical science. Time-relations and space-relations
are comparatively simple, and the general ideas according to which
their analysis should proceed are obvious.
This simplicity of time and space is perhaps the reason why thought
chooses them as the permanent ground for objectival distinction,
throwing the various sense-objects thus obtainable into one heap, as
a first crude thought-object of perception, and thence, as described
above, obtaining a thought-object of perception. Thus a thought-object
of perception conceived as in the present of a short duration is a
first crude thought-object of perception either actual or hypothetical.
Such a thought-object of perception, confined within a short duration,
takes on the space-relations of its component sense-objects within that
same duration. Accordingly thought-objects of perception, conceived in
their whole extents, have to each other the time-relationships of their
complete existences, and within any small duration have to each other
the space-relationships of their component sense-objects which lie
within that duration.
Relations bind together: thus thought-objects of perception are
connected in time and in space. The genesis of the objectival analysis
of sense-presentation is the recognition of sense-objects as distinct
terms in time-relations and space-relations: thus thought-objects of
perception are separated by time and by space.
_Whole and Part._--A sense-object is part of the complete stream
of presentation. This concept of being a part is merely the statement
of the relation of the sense-object to the complete sense-presentation
for that consciousness. Also a sense-object can be part of another
sense-object. It can be a part in two ways, namely, a part in time
and a part in space. It seems probable that both these concepts of
time-part and space-part are fundamental; that is, are concepts
expressing relations which are directly presented to us, and are not
concepts about concepts. In that case no further definition of the
actual presentation is possible. It may even then be possible to
define an adequate criterion of the occurrence of such a presentation.
For example, adopting for the moment a realist metaphysic as to the
existence of the physical world of molecules and electrons, the vision
of a chair as occurring for some definite person at some definite
time is essentially indefinable. It is his vision, though each of us
guesses that it must be uncommonly like our vision under analogous
circumstances. But the existence of the definable molecules and
waves of light in certain definable relations to his bodily organs
of sense, his body also being in a certain definable state, forms an
adequate criterion of the occurrence of the vision, a criterion which
is accepted in Courts of Law and for physical science is tacitly
substituted for the vision.
The connection between the relations "whole and part" and "all and
some" is intimate. It can be explained thus so far as concerns directly
presented sense-objects. Call two sense-objects "separated" if there
is no third sense-object which is a part of both of them. Then an
object A is composed of the two objects B and C, if (1) B and C are
both parts of A, (2) B and C are separated, and (3) there is no part of
A which is separated both from B and from C. In such a case the class
α which is composed of the two objects B and C is often substituted
in thought for the sense-object A. But this process presupposes the
fundamental relation "whole and part." Conversely the objects B and C
may be actual sense-objects, but the sense-object A which corresponds
to the class α may remain hypothetical. For example, the round world
on which we live remains a conception corresponding to no single
sense-object at any time presented in any human being's consciousness.
It is possible, however, that some mode of conceiving the
whole-and-part relation between extended objects as the all-and-some
relation of logical classes can be found. But in this case the
extended objects as here conceived cannot be the true sense-objects
which are present to consciousness. For as here conceived a part of a
sense-object is another sense-object of the same type; and therefore
one sense-object cannot be a class of other sense-objects, just as
a tea-spoon cannot be a class of other tea-spoons. The ordinary way
in thought by which whole-and-part is reduced to all-and-some is by
the device of points, namely, the part of an object occupies some
of the points occupied by the whole object. If any one holds that
in his consciousness the sense-presentation is a presentation of
point-objects, and that an extended object is merely a class of such
point-objects collected together in thought, then this ordinary method
is completely satisfactory. We shall proceed on the assumption that
this conception of directly perceived point-objects has no relation to
the facts.
In the preceding address on "The Organisation of Thought," another mode
is suggested. But this method would apply only to the thought-object
of perception, and has no reference to the primary sense-objects here
considered. Accordingly it must reckon as a subordinate device for a
later stage of thought.
Thus the point-object in time and the point-object in space, and the
double point-object both in time and space, must be conceived as
intellectual constructions. The fundamental fact is the sense-object,
extended both in time and space, with the fundamental relation of
whole-to-part to other such objects, and subject to the law of
convergence to simplicity as we proceed in thought through a series of
successively contained parts.
The relation whole-to-part is a temporal or spatial relation, and
is therefore primarily a relation holding between sense-objects of
perception, and it is only derivatively ascribed to the thought-objects
of perception of which they are components. More generally, space and
time relations hold primarily between sense-objects of perception and
derivatively between thought-objects of perception.
_Definition of Points._--The genesis of points of time and of
space can now be studied. We must distinguish (1) sense-time and
sense-space, and (2) thought-time of perception and thought-space of
perception.
Sense-time and sense-space are the actually observed time-relations
and space-relations between sense-objects. Sense-time and sense-space
have no points except, perhaps, a few sparse instances, sufficient
to suggest the logical idea; also, sense-time and sense-space are
discontinuous and fragmentary.
Thought-time of perception and thought-space of perception are the time
and space relations which hold between thought-objects of perception.
Thought-time of perception and thought-space of perception are each
continuous. By "continuous" is here meant that all thought-objects of
perception have to each other a time (or space) relation.
The origin of points is the effort to take full advantage of
the principle of convergence to simplicity. In so far as this
principle does not apply, a point is merely a cumbrous way of
directing attention to a set of relations between a certain set of
thought-objects of perception, which set of relations, though actual so
far as a thought-object is actual, is (under this supposition) of no
particular importance. Thus the proved importance in physical science
of the concepts of points in time and points in space is a tribute to
the wide applicability of this principle of convergence.
Euclid defines a point as without parts and without magnitude. In
modern language a point is often described as an ideal limit by
indefinitely continuing the process of diminishing a volume (or
area). Points as thus conceived are often called convenient fictions.
This language is ambiguous. What is meant by a fiction? If it means
a conception which does not correspond to any fact, there is some
difficulty in understanding how it can be of any use in physical
science. For example, the fiction of a red man in a green coat
inhabiting the moon can never be of the slightest scientific service,
simply because--as we may presume--it corresponds to no fact. By
calling the concept of points a convenient fiction, it must be meant
that the concept does correspond to some important facts. It is, then,
requisite, in the place of such vague allusiveness, to explain exactly
what are the facts to which the concept corresponds.
We are not much helped by explaining that a point is an ideal limit.
What is a limit? The idea of a limit has a precise meaning in the
theory of series, and in the theory of the values of functions; but
neither of these meanings apply here. It may be observed that, before
the ordinary mathematical meanings of limit had received a precise
explanation, the idea of a point as a limit might be considered as
one among other examples of an idea only to be apprehended by direct
intuition. This view is not now open to us. Thus, again, we are
confronted with the question: What are the precise properties meant
when a point is described as an ideal limit? The discussion which now
follows is an attempt to express the concept of a point in terms of
thought-objects of perception related together by the whole-and-part
relation, considered either as a time-relation or as a space-relation.
If it is so preferred, it may be considered that the discussion is
directed towards a precise elucidation of the term "ideal limit" as
often used in this connection.
The subsequent explanations can be made easier to follow by a small
piece of symbolism: Let _aEb_ mean that "_b_ is part of _a_." We need
not decide whether we are talking of time-parts or space-parts, but
whichever choice is supposed to be made must be conceived as adhered to
throughout any connected discussion. The symbol _E_ may be considered
as the initial letter of "encloses," so we read "_aEb_" as "_a_
encloses _b_." Again the "field of _E_" is the set of things which
either enclose or are enclosed, _i. e._ everything "_a_," which is such
that _x_ can be found so that either _aEx_ or _xEa_. A member of the
field of _E_ is called "an enclosure-object."
Now, we assume that this relation of whole-to-part, which in the future
we will call "enclosure," always satisfies the conditions in that the
relation _E_ is (1) transitive, (2) asymmetrical, and (3) with its
domain including its converse domain.
These four conditions deserve some slight consideration; only the first
two of them embody hypotheses which enter vitally into the reasoning.
Condition (1) may be stated as the condition that _aEb_ and _bEc_
always implies _aEc_. The fact that an entity _b_ can be found such
that _aEb_ and _bEc_ may be conceived as a relation between _a_ and
_c_. It is natural to write _E_^2 for this relation. Thus the condition
is now written: If _aE_^2_c_, then _aEc_. This can be still otherwise
expressed by saying that the relation _E_^2 implies, whenever it holds,
that the relation _E_ also holds.
Condition (2) is partly a mere question of trivial definition, and
partly a substantial assumption. The asymmetrical relation (_E_) is
such that _aEb_ and _bEa_ can never hold simultaneously. This property
splits up into two parts: (1) that no instance of _aEb_ and _bEa_ and
"_a_ diverse from _b_," can occur, and (2) that _aEa_ cannot occur.
The first part is a substantial assumption, the second part (so far as
we are concerned) reduces to the trivial convention that we shall not
consider an object as part of itself, but will confine attention to
"proper parts."
Condition (3) means that _aEb_ always implies that _c_ can be
found such that _bEc_. This condition, taken in conjunction with
the fact that we are only considering proper parts, is the assertion of
the principle of the indefinite divisibility of extended objects, both
in space and in time.
An indivisible part will lack duration in time, and extension in
space, and is thus an entity of essentially a different character
to a divisible part. If we admit such indivisibles as the only true
sense-objects, our subsequent procedure is an unnecessary elaboration.
It will be found that a fourth condition is necessary owing to
logical difficulties connected with the theory of an infinite number
of choices. It will not be necessary for us to enter further on
this question, which involves difficult considerations of abstract
logic. The outcome is, that apart from hypothesis we cannot prove the
existence of the sets, each containing an infinite number of objects,
which are here called points, as will be explained immediately.
Now consider a set of enclosure objects which is such that (1)
of any two of its members one encloses the other, and (2) there
is no member which is enclosed by all the others, and (3) there
is no enclosure-object, not a member of the set which is enclosed
by every member of the set. Call such a set a "convergent set of
enclosure-objects." As we pass along the series from larger to smaller
members, evidently we converge towards an ideal simplicity to any
degree of approximation to which we like to proceed, and the series as
a whole embodies the complete ideal along that route of approximation.
In fact, to repeat, the series is a _route of approximation_.
We have now to inquire if the principle of convergence to simplicity
may be expected to yield the same type of simplicity for every such
convergent route. The answer is, as we might expect, namely, that this
depends upon the nature of the properties which are to be simplified.
For example, consider the application to time. Now, time is
one-dimensional; so when this property of one-dimensionality has been
expressed by the proper conditions, not here stated, a convergent set
of enclosure-objects must, considered as a route of approximation,
exhibit the properties of one unique instant of time, as ordinarily
conceived by the euclidean definition. Accordingly, whatever simplicity
is to be achieved by the application to time of the principle of
convergence to simplicity must be exhibited among the properties of
any such route of approximation.
For space, different considerations arise. Owing to its multiple
dimensions, we can show that different convergent sets of
enclosure-objects, indicating different routes of approximation, may
exhibit convergence to different types of simplicity, some more complex
than others.
For example, consider a rectangular box of height _h_ ft., breadth
_b_ ft., and thickness _c_ ft. Now, keep _h_ and _b_ constant, and
let the central plane (height _h_, breadth _b_) perpendicular to the
thickness be fixed, then make _c_ diminish indefinitely. We thus obtain
a convergent series of an indefinitely large number of boxes, and there
is no smallest box. Thus this convergent series exhibits the route of
approximation towards the type of simplicity expressed as being a plane
area of height _h_, breadth _b_, and no thickness.
Again, by keeping the central line of height _h_ fixed, and by making
_b_ and _c_ diminish indefinitely, the series converges to the segment
of a straight line of length _h_.
Finally, by keeping only the central point fixed, and by making _h_,
_b_, and _c_ diminish indefinitely, the series converges to a point.
Furthermore, we have introduced as yet no concept which would prevent
an enclosure-object being formed of detached fragments in space. Thus
we can easily imagine a convergent set which converges to a number of
points in space. For example, each object of the set might be formed of
two not overlapping spheres of radius _r_, with centres _A_ and _B_.
Then by diminishing _r_ indefinitely, and keeping _A_ and _B_ fixed, we
have convergence to the pair of points _A_ and _B_.
It remains now to consider how those convergent sets which converge to
a single point can be discriminated from all the other types of such
sets, merely by utilising concepts founded on the relation of enclosure.
Let us name convergent sets by Greek letters; by proceeding "forward"
along any such set let us understand the process of continually passing
from the larger to the smaller enclosure-objects which form the set.
The convergent set α will be said to "cover" the convergent set β, if
every member of α encloses some members of β. We notice that if an
enclosure-object _x_ encloses any member (_y_) of β, then every member
of the "tail-end" of β, found by proceeding forward along β from _y_,
must be enclosed by _x_. Thus if α covers β, every member of α encloses
every member of the tail-end of β, starting from the largest member of
β which is enclosed by that member of α.
It is possible for each of two convergent sets to cover the other. For
example, let one set (α) be a set of concentric spheres converging to
their centre _A_, and the other set (β) be a set of concentric
cubes, similarly situated, converging to the same centre _A_. Then
α and β will each cover the other.
Let two convergent sets which are such that each covers the other be
called "equal."
Then it is a sufficient condition to secure that a convergent set
α possesses the point type of convergence, if every convergent set
covered by it is also equal to it, namely, α is a convergent set with
the punctual type of convergence, if "α covers β" always implies that β
covers α.
It can easily be seen by simple examples that the other types of
convergence to surfaces or lines or sets of points cannot possess this
property. Consider, for example, the three convergent sets of boxes in
the preceding illustration, which converge respectively to a central
plane, a central line in the central plane, and the central point in
the central line. The first set covers the second and third sets, and
the second set covers the third set, but no two of the sets are equal.
It is a more difficult question to determine whether the condition here
indicated as sufficient to secure the punctual type of convergence
is also necessary. The question turns on how far thought-objects
of perception possess exact boundaries prior to the elaboration of
exact mathematical concepts of space. If they are to be conceived as
possessing such exact boundaries, then convergent sets converging to
points on such boundaries must be allowed for. The procedure necessary
for the specification of the complete punctual condition becomes then
very elaborate,[3] and will not be considered here.
But such exact determination as is involved in the conception of
an exact spatial boundary does not seem to belong to the true
thought-object of perception. The ascription of an exact boundary
really belongs to the transition stage of thought as it passes from
the thought-object of perception to the thought-object of science.
The transition from the sense-object immediately presented to the
thought-object of perception is historically made in a wavering
indeterminate line of thought. The definite stages here marked out
simply serve to prove that a logically explicable transition is
possible.
We accordingly assume that the condition laid down above to secure the
punctual convergence of a convergent set of enclosure-objects is not
only sufficient, but necessary.
It can be proved that, if two convergent sets of enclosure-objects are
both equal to a third convergent set, they are equal to each other.
Consider now any punctual convergent set (α). We want to define the
"point" to which α is a route of approximation in a way which is
neutral between α and all the convergent sets which are equal to α.
Each of these sets is a route of approximation to the same "point"
as α. This definition is secured if we define the point as the class
formed by all the enclosure-objects which belong either to α or to
any convergent set which is equal to α. Let _P_ be this class
of enclosure-objects. Then any convergent set (β) which consists of
enclosure-objects entirely selected from members of the class _P_
must be a route of approximation to the same "point" as does the
original punctual set α; namely, provided that we choose a small enough
enclosure-object in β, we can always find a member of α which encloses
it; and provided that we choose a small enough enclosure-object in
α, we can always find a member of β which encloses it. Thus _P_
only includes convergent sets of the punctual type, and the route
of approximation indicated by any two convergent sets selected from
_P_ converges to identical results.
_The Uses of Points._--The sole use of points is to facilitate the
employment of the principle of Convergence to Simplicity. By this
principle some simple relations in appropriate circumstances become
true, when objects are considered which are sufficiently restricted in
time or in space. The introduction of points enables this principle to
be carried through to its ideal limit. For example, suppose _g_ (_a_,
_b_, _c_) represents some statement concerning three enclosure-objects,
_a_, _b_, _c_, which may be true if the objects are sufficiently
restricted in extent. Let _A_, _B_, _C_ be three given points,
then we define _g_ (_A_, _B_, _C_) to mean that _whatever_ three
enclosure-objects _a_, _b_, _c_ are chosen, such that _a_ is a member
of _A_, _b_ of _B_, and _c_ of _C_, it is _always possible_ to find
three other members of _A_, _B_, _C_, namely, _x_ a member of _A_, _y_
of _B_, and _z_ of _C_, such that _aEx_, _bEy_, _cEz_, and _g_ (_x_,
_y_, _z_). So by going far enough down in the tail-ends of _A_, _B_,
_C_ we can always secure three objects _x_, _y_, _z_ for which _g_
(_x_, _y_, _z_) is true.
For example, let _g_ (_A_, _B_, _C_) mean "_A_, _B_, _C_ are three
points in a linear row." This must be construed to mean that whatever
three objects _a_, _b_, _c_ we choose, members of _A_, _B_, _C_
respectively, we can always find three objects _x_, _y_, _z_, also
members of _A_, _B_, _C_ respectively, and such that _a_ encloses _x_,
_b_ encloses _y_, _c_ encloses _z_, and also such that _x_, _y_, _z_
are in a linear row.
Sometimes a double convergence is necessary, namely, a convergence of
conditions as well as a convergence of objects. For example, consider
the statement, "the points _A_ and _B_ are two feet apart." Now, the
exact statement "two feet apart" does not apply to objects. For objects
_x_ and _y_ we must substitute the statement, "the distance between
_x_ and _y_ lies between the limits (2 ± _e_) feet." Here _e_ is some
number, less than two, which we have chosen for this statement. Then
the points _A_ and _B_ are two feet apart; if, _however we choose the
number e_, whatever enclosure-objects _a_ and _b_, members of _A_ and
_B_ respectively, we consider, we can always find enclosure-objects _x_
and _y_, members of _A_ and _B_ respectively, such that _a_ encloses
_x_ and _b_ encloses _y_, and also such that the distance between _x_
and _y_ lies between the limits (2 ± _e_) feet. It is evident, since
_e_ can be chosen as small as we please, that this statement exactly
expresses the condition that _A_ and _B_ are two feet apart.
_Straight Lines and Planes._--But the problem of the intellectual
construction of straight lines and planes is not yet sufficiently
analysed. We have interpreted the meaning of the statement that three
or more points are collinear, and can similarly see how to interpret
the meaning of the statement that four or more points are coplanar,
in either case deriving the exact geometrical statements from vaguer
statements respecting extended objects.
This procedure only contemplates groups of finite numbers of points.
But straight lines and planes are conceived as containing infinite
numbers of points. This completion of lines and planes is obtained
by a renewed application of the principle of aggregation, just as
a set of first crude thought-objects of perception are aggregated
into one complete thought-object of perception. In this way repeated
judgments of the collinearity of sets of points are finally, when
certain conditions of interlacing are fulfilled, aggregated in the
single judgment of all the points of the groups as forming one whole
collinear group. Similarly for judgments of coplanarity. This process
of logical aggregation can be exhibited in its exact logical analysis.
But it is unnecessary here to proceed to such details. Thus we conceive
our points as sorted into planes and straight lines, concerning which
the various axioms of geometry hold. These axioms, in so far as they
essentially require the conception of points, are capable of being
exhibited as the outcome of vaguer, less exact judgments respecting the
relations of extended objects.
_Empty Space._--It must be observed that the points, hitherto
defined, necessarily involve thought-objects of perception, and lie
within the space-extension occupied by such objects. It is true that
such objects are largely hypothetical, and that we can bring into our
hypotheses sufficient objects to complete our lines and planes. But
every such hypothesis weakens the connection between our scientific
concept of nature and the actual observed facts which are involved in
the actual sense-presentations.
Occam's razor, _Entia non multiplicanda præter necessitatem_,
is not an arbitrary rule based on mere logical elegancy. Nor is its
application purely confined to metaphysical speculation. I am ignorant
of the precise reason for its metaphysical validity, but its scientific
validity is obvious, namely, every use of hypothetical entities
diminishes the claim of scientific reasoning to be the necessary
outcome of a harmony between thought and sense-presentation. As
hypothesis increases, necessity diminishes.
Commonsense thought also supports this refusal to conceive of all
space as essentially depending on hypothetical objects which fill it.
We think of material objects as filling space, but we ask whether any
objects exist between the Earth and the Sun, between the stars, or
beyond the stars. For us, space is there; the only question is whether
or not it be full. But this form of question presupposes the meaning of
empty space, namely, of space not containing hypothetical objects.
This brings up a wider use of the concept of points, necessitating
a wider definition. Hitherto we have conceived points as indicating
relations of enclosure between objects. We thus arrive at what now
we will term "material points." But the idea of points can now be
transformed so as to indicate the possibilities of external relations
not those of enclosure. This is effected by an enlargement of the
concept of ideal points, already known to geometers.
Define "material lines" to be complete collinear classes of collinear
points. Consider now the set of material lines which contain a certain
material point. Call such a set of lines an ideal point. This set of
lines indicates a possibility of position, which is in fact occupied
by that material point common to all the material lines. So this ideal
point is an occupied ideal point. Now consider a set of three material
lines, such that any two are coplanar, but not the whole three, and
further consider the complete set of material lines such that each is
coplanar with each of the three material lines first chosen. The axioms
which hold for the material lines will enable us to prove that any two
lines of this set are coplanar. Then the whole set of lines, including
the three original lines, forms an ideal point, according to the
definition in its full generality. Such an ideal point may be occupied.
In that case there is a material point common to all the lines of the
set, but it may be unoccupied. Then the ideal point merely indicates
a possibility of spatial relations which has not been realised. It is
the point of empty space. Thus the ideal points, which may or may not
be occupied, are the points of geometry viewed as an applied science.
These points are distributed into straight lines and planes. But any
further discussion of this question will lead us into the technical
subject of the axioms of geometry and their immediate consequences.
Enough has been said to show how geometry arises according to the
relational theory of space.
Space as thus conceived is the thought-space of the material world.
_IV. Fields of Force_
The thought-objects of science are conceived as directly related to
this thought-space. Their spatial relations are among those indicated
by the points of the thought-space. Their emergence in science has
been merely a further development of processes already inherent in
commonsense thought.
Relations within the complete sense-presentation were represented
in thought by the concept of thought-objects of perception. All
sense-presentation could not be represented in this way; also the
change and disappearance of thought-objects occasioned confusion
of thought. A reduction to order of this confusion was attempted
by the concepts of permanent matter with primary and secondary
qualities. Finally, this has issued in the secondary qualities being
traced as perception of events generated by the objects, but--as
perceived--entirely disconnected with them. Also the thought-objects
of perception have been replaced by molecules and electrons and
ether-waves, until at length it is never the thought-object of science
which is perceived, but complicated series of events in which they are
implicated. If science be right, nobody ever perceived a thing, but
only an event. The result is, that the older language of philosophy
which still survives in many quarters is now thoroughly confusing
when brought into connection with the modern concepts of science.
Philosophy--that is, the older philosophy--conceives the thing as
directly perceived. According to scientific thought, the ultimate thing
is never perceived, perception essentially issuing from a series of
events. It is impossible to reconcile the two points of view.
The advantage of the modern scientific concept is that it is enabled
to "explain" the fluid vague outlines of sense-presentation. The
thought-object of perception is now conceived as a fairly stable
state of motion of a huge group of molecules, constantly changing,
but preserving a certain identity of characteristics. Also stray
sense-objects, not immediately given as part of a thought-object of
perception, are now explicable: the dancing light-reflection, the
vaguely heard sound, the smell. In fact, the perceived events of
the scientific world have the same general definition and lack of
definition, and the same general stability and lack of stability,
as the sense-objects of the complete sense-presentation or as the
thought-objects of perception.
The thought-objects of science, namely, molecules, atoms, and
electrons, have gained in permanence. The events are reduced to changes
in space-configuration. The laws determining these changes are the
ultimate laws of nature.
The laws of change in the physical universe proceed on the assumption
that the preceding states of the universe determine the character of
the change. Thus, to know the configurations and events of the universe
up to and including any instant would involve sufficient data from
which to determine the succession of events throughout all time.
But in tracing the antecedents of events, commonsense thought, dealing
with the world of thought-objects of perception, habitually assumes
that the greater number of antecedent events can be neglected as
irrelevant. Consideration of causes is restricted to a few events
during a short preceding interval. Finally, in scientific thought it
has been assumed that the events in an arbitrarily small preceding
duration are sufficient. Thus physical quantities and their successive
differential co-efficients up to any order at the instant, but with
their limiting values just before that instant, are on this theory
sufficient to determine the state of the universe at all times after
the instant. More particular laws are assumed. But the search for
them is guided by this general principle. Also it is assumed that the
greater number of events in the physical universe are irrelevant to
the production of any particular effect, which is assumed to issue
from relatively few antecedents. These assumptions have grown out of
the experience of mankind. The first lesson of life is to concentrate
attention on few factors of sense-presentations, and on still fewer of
the universe of thought-objects of perception.
The principle by which--consciously or unconsciously--thought has been
guided is that in searching for particular causes, remoteness in time
and remoteness in space are evidences of comparative disconnection
of influence. The extreme form of this principle is the denial of
any action at a distance either in time or space. The difficulty in
accepting this principle in its crude form is, that since there are no
contiguous points, only coincident bodies can act on each other. I can
see no answer to this difficulty--namely, either bodies have the same
location and are thus coincident, or they have different locations and
are thus at a distance and do not act on each other.
This difficulty is not evaded by the hypothesis of an ether,
continuously distributed. For two reasons: in the first place, the
continuity of the ether does not avoid the dilemma; and secondly, the
difficulty applies to time as well as to space, and the dilemma would
prove that causation producing change is impossible, namely, no changed
condition could be the result of antecedent circumstances.
On the other hand, a direct interaction between two bodies separated
in space undoubtedly offends the conception of distance as implying
physical disconnection as well as spatial relation. There is no logical
difficulty in the assumption of action at a distance as in the case
of its denial, but it is contradictory to persistent assumptions of
that apparatus of commonsense thought which it is the main business
of science to harmonise with sense-presentation, employing only the
minimum of modification.
Modern science is really unconcerned with this debate. Its
(unacknowledged) conceptions are really quite different, though the
verbal explanations retain the form of a previous epoch. The point of
the change in conception is that the old thought-object of science was
conceived as possessing a simplicity not belonging to the material
universe as a whole. It was secluded within a finite region of space,
and changes in its circumstances could only arise from forces which
formed no essential part of its nature. An ether was called into
existence to explain the active relations between these passive
thought-objects. The whole conception suffers from the logical
difficulties noted above. Also no clear conception can be formed of
the sense in which the ether is explanatory. It is to possess a type
of activity denied to the original thought-object, namely, it carries
potential energy, whereas the atom possessed only kinetic energy,
the so-called potential energy of an atom belonging really to the
surrounding ether. The truth is, that ether is really excepted from the
axiom "no action at a distance," and the axiom thereby is robbed of all
its force.
The modern thought-object of science--not yet explicitly
acknowledged--has the complexity of the whole material universe. In
physics, as elsewhere, the hopeless endeavour to derive complexity
from simplicity has been tacitly abandoned. What is aimed at is not
simplicity, but persistence and regularity. In a sense regularity
is a sort of simplicity. But it is the simplicity of stable mutual
relations, and not the simplicity of absence of types of internal
structure or of type of relation. This thought-object fills all space.
It is a "field"; that is to say, it is a certain distribution of
scalar and vector quantities throughout space, these quantities having
each its value for each point of space at each point of time, being
continuously distributed throughout space and throughout time, possibly
with some exceptional discontinuities. The various types of quantity
which form the field have fixed relations to each other at each point
of time and space. These relations are the ultimate laws of nature.
For example, consider an electron. There is a scalar distribution of
electricity, which is what is ordinarily called the electron. This
scalar distribution has a volume-density ρ at the time _t_ at any
point (_x_, _y_, _z_). Thus ρ is a function of (_x_, _y_, _z_, _t_),
which is zero except within a restricted region. Furthermore, at
any time _t_, as an essential adjunct, there is a continuous space
distribution at each point of the two vectors (_X_, _Y_, _Z_), which is
the electric force, and (α, β, γ), which is the magnetic force. Lastly,
individuality is ascribed to the scalar electric distribution, so that
in addition to its conservation of quantity--involved in the assumed
laws--it is also possible to assign the velocities with which the
various individual parts of the distribution are moving. Let (_u_, _v_,
_w_) be this velocity at (_x_, _y_, _z_, _t_).
This whole scheme of scalar and vector quantities, namely, ρ, (_X_,
_Y_, _Z_), (α, β, γ), (_u_, _v_, _w_) is interconnected by the
electromagnetic laws. It follows from these laws that the electron, in
the sense of the scalar distribution ρ, is to be conceived as at each
instant propagating from itself an emanation which travels outwards
with the velocity of light _in vacuo_, and from which (_X_, _Y_,
_Z_) and (α, β, γ) can be calculated, so far as they are due to that
electron. Thus the field, at any time, due to the electron as a whole
depends on the previous history of the electron, the nearer to the
electron the more recent being the relevant history. The whole scheme
of such a field is one single thought-object of science: the electron
and its emanations form one essential whole, namely one thought-object
of science, essentially complex and essentially filling all space. The
electron proper, namely, the scalar distribution ρ, is the focus of the
whole, the essential focal property being that the field at any instant
is completely determined by the previous history of the focus and of
its space relations through all previous time. But the field and the
focus are not independent concepts, they are essentially correlated in
one organised unity, namely, they are essentially correlated terms in
the field of one relation in virtue of which the entities enter into
our thoughts.
The fields of a group of electrons are superposed according to the
linear law for aggregation, namely, pure addition for analogous scalar
quantities and the parallelogram law for analogous vectors. The changes
in motion of each electron depend entirely on the resultant field in
the region it occupies. Thus a field can be viewed as a possibility of
action, but a possibility which represents an actuality.
It is to be noted that the two alternative views of causation are
here both included. The complete field within any region of space
depends on the past histories of all the electrons, histories extending
backwards in proportion to their distances. Also this dependence can
be conceived as a transmission. But viewing the cause which effects
changes on the electron within that region, it is solely that field
within the region, which field is coincident with that electron both in
time and in space.
This process of conceiving the actuality underlying a possibility is
the uniform process by which regularity and permanence is introduced
into scientific thought, namely, we proceed from the actuality of the
fact to the actuality of possibility.
In conformity with this principle, propositions are the outgrowth
from actual thought-expressions, thought-objects of perceptions from
crude sense-objects, hypothetical thought-objects of perception from
actual thought-objects of perception, material points from hypothetical
infinite suites of hypothetical thought-objects of perception,
ideal points from material points, thought-objects of science from
thought-objects of perception, fields of electrons from actual mutual
reactions of actual electrons.
The process is a research for permanence, uniformity, and simplicity
of logical relation. But it does not issue in simplicity of internal
structure. Each ultimate thought-object of science retains every
quality attributed to the whole scientific universe, but retains them
in a form characterised by permanence and uniformity.
_V. Conclusion_
We commenced by excluding judgments of worth and ontological judgments.
We conclude by recalling them. Judgments of worth are no part of
the texture of physical science, but they are part of the motive of
its production. Mankind have raised the edifice of science, because
they have judged it worth while. In other words, the motives involve
innumerable judgments of value. Again, there has been conscious
selection of the parts of the scientific field to be cultivated, and
this conscious selection involves judgments of value. These values may
be æsthetic, or moral, or utilitarian, namely, judgments as to the
beauty of the structure, or as to the duty of exploring the truth, or
as to utility in the satisfaction of physical wants. But whatever the
motive, without judgments of value there would have been no science.
Again, ontological judgments were not excluded by reason of any lack
of interest. They are in fact presupposed in every act of life: in our
affections, in our self-restraints, and in our constructive efforts.
They are presupposed in moral judgments. The difficulty about them is
the absence of agreement as to the method of harmonising the crude
judgments of commonsense.
Science does not diminish the need of a metaphysic. Where this need is
most insistent is in connection with what above has been termed "the
actuality underlying a possibility." A few words of explanation may
render the argument clearer, although they involve a rash approach
to metaphysical heights which it is not the purpose of this paper to
explore.
The conception of subject and object in careless discussion covers
two distinct relations. There is the relation of the whole perceiving
consciousness to part of its own content, for example, the relation
of a perceiving consciousness to an object of redness apparent to it.
There is also the relation of a perceiving consciousness to an entity
which does not exist in virtue of being part of the content of that
consciousness. Such a relation, so far as known to the perceiving
consciousness, must be an inferred relation, the inference being
derived from an analysis of the content of the perceiving consciousness.
The bases for such inferences must be elements in consciousness
directly known as transcending their immediate presentation in
consciousness. Such elements are universal logical truths, moral and
æsthetic truths, and truths embodied in hypothetical propositions.
These are the immediate objects of perception which are other than the
mere affections of the perceiving subject. They have the property of
being parts of the immediate presentations for individual subjects and
yet more than such parts. All other existence is inferred existence.
In this chapter we are more directly concerned with truths embodied
in hypothetical propositions. Such truths must not be confused with
any doubtfulness which attaches to our judgments of the future course
of natural phenomena. A hypothetical proposition, like a categorical
judgment, may or may not be doubtful. Also like a categorial judgment,
it expresses a fact. This fact is twofold: as a presentation in
consciousness, it is just this hypothetical judgment; as expressing a
categorical fact, it states a relation which lies beyond consciousness,
holding between entities thereby inferred.
But this metaphysical analysis, short though it be, is probably wrong,
and at the best will only command very partial assent. Certainly;
and this admission brings out the very point which I wished to make.
Physical science is based on elements of thought, such as judgments
registering actual perceptions, and judgments registering hypothetical
perceptions which under certain circumstances would be realised.
These elements form the agreed content of the apparatus of commonsense
thought. They require metaphysical analysis; but they are among the
data from which metaphysics starts. A metaphysic which rejects them
has failed, in the same way as physical science has failed when it is
unable to harmonise them into its theory.
Science only renders the metaphysical need more urgent. In itself
it contributes little directly to the solution of the metaphysical
problem. But it does contribute something, namely, the exposition of
the fact that our experience of sensible apparent things is capable of
being analysed into a scientific theory, a theory not indeed complete,
but giving every promise of indefinite expansion. This achievement
emphasises the intimate relation between our logical thought and the
facts of sensible apprehension. Also the special form of scientific
theory is bound to have some influence. In the past false science
has been the parent of bad metaphysics. After all, science embodies
a rigorous scrutiny of one part of the whole evidence from which
metaphysicians deduce their conclusions.
FOOTNOTES:
[Footnote 3: Cf. _Révue de Métaphysique et de Morale_, May 1916,
where this question is dealt with by the author at the end of an
article, "La théorie relationniste de l'espace."]
CHAPTER VIII
SPACE, TIME, AND RELATIVITY
(_Paper read to Section A at the Manchester Meeting of the British
Association, 1915, and later before the Aristotelian Society_)
FUNDAMENTAL Problems concerning space and time have been considered
from the standpoints created by many different sciences. The object of
this paper is the humble one of bringing some of these standpoints into
relation with each other. This necessitates a very cursory treatment of
each point of view.
Mathematical physicists have evolved their theory of relativity to
explain the negative results of the Morley-Michelson experiment and of
the Trouton experiment. Experimental psychologists have considered the
evolution of spatial ideas from the crude sense-data of experience.
Metaphysicians have considered the majestic uniformity of space and
time, without beginning and without end, without boundaries, and
without exception in the truths concerning them; all these qualities
the more arresting to our attention from the confused accidental
nature of the empirical universe which is conditioned by them.
Mathematicians have studied the axioms of geometry, and can now deduce
all that is believed to be universally true of space and of time by the
strictest logic from a limited number of assumptions.
These various lines of thought have been evolved with surprisingly
little interconnection. Perhaps it is as well. The results of science
are never quite true. By a healthy independence of thought perhaps we
sometimes avoid adding other people's errors to our own. But there can
be no doubt that the normal method of cross-fertilising thought is by
considering the same, or allied problems to our own, in the form which
they assume in other sciences.
Here I do not propose to enter into a systematic study of these various
chapters of science. I have neither the knowledge nor the time.
First, let us take the ultimate basis of any theory of relativity.
All space measurement is from stuff in space to stuff in space. The
geometrical entities of empty space never appear. The only geometrical
properties of which we have any direct knowledge are properties of
those shifting, changeable appearances which we call things in space.
It is the sun which is distant, and the ball which is round, and the
lamp-posts which are in linear order. Wherever mankind may have got its
idea of an infinite unchangeable space from, it is safe to say that it
is not an immediate deliverance of direct observation.
There are two antagonistic philosophical ways of recognising this
conclusion.
One is to affirm that space and time are conditions for sensible
experience, that without projection into space and time sensible
experience would not exist. Thus, although it may be true to say that
our knowledge of space and time is given in experience, it is not true
to say that it is deduced from experience in the same sense that the
Law of Gravitation is so deduced. It is not deduced, because in the act
of experiencing we are necessarily made aware of space as an infinite
given whole, and of time as an unending uniform succession. This
philosophical position is expressed by saying that space and time are
_a priori_ forms of sensibility.
The opposed philosophical method of dealing with the question is
to affirm that our concepts of time and space are deductions from
experience, in exactly the same way as the Law of Gravitation is such
a deduction. If we form exact concepts of points, lines and surfaces,
and of successive instants of time, and assume them to be related as
expressed by the axioms of geometry and the axioms for time, then we
find that we have framed a concept which, with all the exactness of
which our observations are capable, expresses the facts of experience.
These two philosophic positions are each designed to explain a
certain difficulty. The _a priori_ theory explains the absolute
universality ascribed to the laws of space and time, a universality
not ascribed to any deduction from experience. The experiential theory
explains the derivation of the space-time concepts without introducing
any other factors beyond those which are admittedly present in framing
the other concepts of physical science.
But we have not yet done with the distinctions which in any discussion
of space or time must essentially be kept in mind. Put aside the
above question as to how these space-time concepts are related to
experience--What are they when they are formed?
We may conceive of the points of space as self-subsistent entities
which have the indefinable relation of being occupied by the ultimate
stuff (matter, I will call it) which is there. Thus, to say that the
sun is _there_ (wherever it is) is to affirm the relation of
occupation between the set of positive and negative electrons which
we call the sun and a certain set of points, the points having an
existence essentially independent of the sun. This is the absolute
theory of space. The absolute theory is not popular just now, but it
has very respectable authority on its side--Newton, for one--so treat
it tenderly.
The other theory is associated with Leibniz. Our space concepts are
concepts of relations between things in space. Thus there is no such
entity as a self-subsistent point. A point is merely the name for some
peculiarity of the relations between the matter which is, in common
language, said to be in space.
It follows from the relative theory that a point should be definable in
terms of the relations between material things. So far as I am aware,
this outcome of the theory has escaped the notice of mathematicians,
who have invariably assumed the point as the ultimate starting ground
of their reasoning. Many years ago I explained some types of ways in
which we might achieve such a definition, and more recently have added
some others. Similar explanations apply to time. Before the theories
of space and time have been carried to a satisfactory conclusion on
the relational basis, a long and careful scrutiny of the definitions
of points of space and instants of time will have to be undertaken,
and many ways of effecting these definitions will have to be tried and
compared. This is an unwritten chapter of mathematics, in much the same
state as was the theory of parallels in the eighteenth century.
In this connection I should like to draw attention to the analogy
between time and space. In analysing our experience we distinguish
events, and we also distinguish things whose changing relations form
the events. If I had time it would be interesting to consider more
closely these concepts of events and of things. It must suffice now to
point out that things have certain relations to each other which we
consider as relations between the space extensions of the things; for
example, one space can contain the other, or exclude it, or overlap
it. A point in space is nothing else than a certain set of relations
between spatial extensions.
Analogously, there are certain relations between events which we
express by saying that they are relations between the temporal
durations of these events, that is, between the temporal extensions of
the events. [The durations of two events A and B may one precede the
other, or may partially overlap, or may one contain the other, giving
in all six possibilities.] The properties of the extension of an event
in time are largely analogous to the extension of an object in space.
Spatial extensions are expressed by relations between objects, temporal
extensions by relations between events.
The point in time is a set of relations between temporal extensions.
It needs very little reflection to convince us that a point in time
is no direct deliverance of experience. We live in durations, and not
in points. But what community, beyond the mere name, is there between
extension in time and extension in space? In view of the intimate
connection between time and space revealed by the modern theory of
relativity, this question has taken on a new importance.
I have not thought out an answer to this question. I suggest, however,
that time and space embody those relations between objects on which
depends our judgment of their externality to ourselves. Namely,
location in space and location in time both embody and perhaps
necessitate a judgment of externality. This suggestion is very vague,
and I must leave it in this crude form.
_Diverse Euclidean Measure Systems_
Turning now to the mathematical investigations on the axioms of
geometry, the outcome, which is most important for us to remember,
is the great separation which it discloses between non-metrical
projective geometry, and metrical geometry. Non-metrical projective
geometry is by far the more fundamental. Starting with the concepts
of points, straight lines, and planes (of which not all three need
be taken as indefinable), and with certain very simple non-metrical
properties of these entities--such as, for instance, that two points
uniquely determine a straight line--nearly the whole of geometry can
be constructed. Even quantitative coordinates can be introduced, to
facilitate the reasoning. But no mention of distance, area, or volume,
need have been introduced. Points will have an order on the line, but
order does not imply any settled distance.
When we now inquire what measurements of distance are possible, we find
that there are different systems of measurement all equally possible.
There are three main types of system: any system of one type gives
Euclidean geometry, any system of another type gives Hyperbolic (or
Lobatchewskian) geometry, any system of the third type gives Elliptic
geometry. Also different beings, or the same being if he chooses,
may reckon in different systems of the same type, or in systems of
different types. Consider the example which will interest us later.
Two beings, A and B, agree to use the same three intersecting lines
as axes of _x_, _y_, _z_. They both employ a system of measurement of
the Euclidean type, and (what is not necessarily the case) agree as to
the plane at infinity. That is, they agree as to the lines which are
parallel. Then with the usual method of rectangular Cartesian axes,
they agree that the coordinates of P are the lengths ON, NM, MP. So
far all is harmony. A fixes on the segment OU_{1}, on O_x_, as being
the unit length, and B on the segment OV_{1}, on O_x_. A calls his
coordinates (_x_, _y_, _z_), and B calls them (X, Y, Z).
Then it is found [since both systems are Euclidean] that, whatever
point P be taken,
X = β_x_, Y = γ_y_, Z = δ_z_. [β ≠ γ ≠ δ.]
They proceed to adjust their differences, and first take the
_x_-coordinates. Obviously they have taken different units of
length along O_x_. The length OU_{1}, which A calls one unit, B
calls β units. B changes his unit length to OU_{1}, from its original
length OV_{1}, and obtains X = _x_. But now, as he must use the
same unit for all his measurements, his other coordinates are altered
in the same ratio. Thus we now have
X = _x_, Y = γ_y_/β, Z = δ_z_/β.
The fundamental divergence is now evident. A and B agree as to their
units along O_x_. They settled that by taking along that axis a
given segment OU_{1} as having the unit length. But they cannot agree
as to what segment along O_y_ is equal to OU_{1}. A says it is
OU_{2}, and B that it is OU_{2}′. Similarly for lengths along OZ.
The result is that A's spheres
_x_^2 + _y_^2 + _z_^2 = _r_^2,
are B's ellipsoids,
X^2 + β^2Y^2/γ^2 + β^2Z^2/δ^2 = _r_^2,
_i. e._ X^2/β^2 + Y^2/γ^2 + Z^2/δ^2 = _r_^2/β^2.
Thus the measurement of angles by the two is hopelessly at variance.
If β ≠ γ ≠ δ, there is one, and only one, set of common rectangular
axes at O, namely that from which they started. If γ = δ, but β ≠ γ,
then there are a singly infinite number of common rectangular axes
found by rotating the axes round O_x_. This is, for us, the
interesting case. The same phenomena are reproduced by transferring to
any parallel axes.
The root of the difficulty is, that A's measuring rod, which for him is
a rigid invariable body, appears to B as changing in length when turned
in different directions. Similarly all measuring rods, satisfactory
to A, violate B's immediate judgment of invariability, and change
according to the same law. There is no way out of the difficulty.
Two rods ρ and σ coincide whenever laid one on the other; ρ is held
still, and both men agree that it does not change. But σ is turned
round. A says it is invariable, B says it changes. To test the matter,
ρ is turned round to measure it, and exactly fits it. But while A is
satisfied, B declares that ρ has changed in exactly the same way as did
σ. Meanwhile B has procured two material rods satisfactory to him as
invariable, and A makes exactly the same objections.
We shall say that A and B employ diverse Euclidean metrical systems.
The most extraordinary fact of human life is that all beings seem to
form their judgments of spatial quantity according to the same metrical
system.
_Relativity in Modern Physics_
Owing to the fact that points of space are incapable of direct
recognition, there is a difficulty--apart from any abstract question
of the nature of space--in deciding on the motion to be ascribed to
any body. Even if there be such a thing as absolute position, it is
impossible in practice to decide directly whether a body's absolute
position has changed. All spatial measurement is relative to matter.
Newton's laws of motion in their modern dress evade this difficulty
by asserting that a framework of axes of coordinates can be defined
by their relations to matter such that, assuming these axes to be at
rest, and all velocities to be measured relatively to them, the laws
hold. The same expedient has to be employed for time, namely, the laws
hold when the measurement of the flow of time is made by the proper
reference to periodic events. Thus the laws assert that the framework
and the natural clock adapted for their use have been successfully
found.
But, if one framework will do, an infinity of others serve equally
well; namely, not only--as is of course the case--all those at rest
relatively to the first framework, but also all those which move
without relative rotation with uniform velocity relatively to the
first. This whole set of frameworks is on a level in respect to
Newton's laws. We will call them Dynamical frameworks.
Now, suppose there are two observers, A and B. They agree in their
non-metrical projective geometry, _e.g._, what A calls a straight
line so does B. They also both apply a Euclidean metrical system of
measurement to this space. Their two metrical systems also agree in
having the same plane at infinity, that is, lines which are parallel
for A are also parallel for B. Furthermore, they have both successfully
applied Newton's laws to the movement of matter, and agree in having
the same sets of dynamical axes. But the framework (among these sets)
which A chooses to regard as at rest is different from the frame (among
the same sets) which B so regards.
Without alteration of their respective judgment of rest, they choose
their co-ordinate axes so that the origins (O for A, and O′ for B) are
in relative motion along OO′, which is the axis of _x_ for both.
Further, since OO′ is the line of symmetry of their diverse Euclidean
systems, we assume that the two measure-systems agree for planes
perpendicular to OO′, _i.e._, we assume a symmetry round OO′.
Then if, for A at O, the distance OO′ be ξ, the relations at any
instant between A's coordinates (_x_, _y_, _z_) and B's
coordinates (_x′_, _y′_, _z′_) for the same point P are
given by
_x′_ = β(_x_ - ξ), _y′_ = _y_, _z′_ = _z_.
Also, according to A's clock, O′ is moving forward with a uniform
velocity _v_, and we measure A's time from the instant of the
coincidence of O and O′.
Thus
ξ = _vt_,
and
_x′_ = β(_x_ - _vt_), _y′_ = _y_, _z′_ = _z_.
We now consider B's clock, and ask for the most general supposition
which is consistent with the fact that their judgments as to the fact
of uniform motion are in agreement.
We do not assume that events in various parts of space which A
considers to be simultaneous are so considered by B. But we assume that
at any point P, with coordinates (_x_, _y_, _z_) for A, there is a
determinate relation between B's time T and _x_, _y_, _z_, _t_.
Put
T = ƒ(_x_, _y_, _z_, _t_).
Write
P = δT/δ_x_, Q = δT/δ_y_, R = δT/δ_z_, S = δT/δ_t_.
Now suppose that the point P is moving, and that (_u_{1},
_u_{2}, _u_{3}) is its set of component velocities along
the axes according to A's "space and clock" system, and (U_{1}, U_{2},
U_{3}) is its set of component velocities according to B's "space and
clock" system. Then by mere differentiation it follows after a short
mathematical deduction that
U_{1} = {(_d_β/_dt_)(_x_ - _vt_) + β(_u_{1} - _v_)}/{P_u_{1} + Q_u_{2} + R_u_{3} + S},
U_{2} = _u_{2}/{P_u_{1} + Q_u_{2} + R_u_{3} + S},
U_{3} = _u_{3}/{P_u_{1} + Q_u_{2} + R_u_{3} + S}.
But we have assumed that, whatever the direction of the resultant
velocity (_u_{1}, _u_{2}, _u_{3}), the velocities (U_{1}, U_{2}, U_{3})
and (_u_{1}, _u_{2}, _u_{3}) are both uniform when either is uniform.
Hence it is easily proved that β, P, Q, R, S are independent of the
coordinates (_x_, _y_, _z_) and of the time _t_. In other words, they
are constant.
Hence we obtain
U_{1} = β(_u_{1} - _v_)/{P_u_{1} + Q_u_{2} + R_u_{3} + S},
and
T = P_x_ + Q_y_ + R_z_ + S_t_.
But we assumed that OO′, _i.e._, O_x_, is an axis of
symmetry. It follows from this assumption that
Q = R = 0.
We thus obtain the simplified results
T = P_x_ + S_t_, }
U_{1} = β(_u_{1} - _v_)/(P_u_{1} + S), } (I)
U_{2} = _u_{2}/(P_u_{1} + S), }
U_{3} = _u_{3}/(P_u_{1} + S). }
Here we remember that (_u_{1}, _u_{2}, _u_{3}) are
the velocities of any particle according to A's "space and clock"
system, and that (U_{1}, U_{2}, U_{3}) are the velocities of the same
point according to B's "space and clock" system. We have obtained the
most general relations consistent with the facts that (1) they both
employ Euclidean systems, related as described above, and (2) they
agree in their judgments on the uniformity of velocity.
We now compare their judgments on the magnitudes of velocities.
Let the magnitude of the velocity of P be V according to A's judgment,
and V′ according to B's’ judgment.
Then
V^2 = _u_{1}^2 + _u_{2}^2 + _u_{3}^2,
V′^2 = U_{1}^2 + U_{2}^2 + U_{3}^2.
Also we can put
_u_{1} = _l_V, _u_{2} = _m_V, _u_{3} = _n_V,
where (_l_, _m_, _n_) have nothing to do with the magnitude V, but
simply depend on the direction of motion. In fact (_l_, _m_, _n_) are
the "direction cosines" of the velocity according to A's judgment. By
substituting in the above equation for V^2 we see that
_l_^2 + _m_^2 + _n_^2 = 1.
Now, substituting for (_u_{1}, _u_{2}, _u_{3}) in
the equations (I) above, and squaring and adding, and eliminating
_m_^2 + _n_^2 by the relation just found, we at once find
V′^2 = ((β^2 - 1)V^2_l_^2 - 2β^2V_vl_ + β^2_v_^2 + V^2)/(PV_l_ + S)^2.
It is thus seen that in general the relation of V′ to V depends on the
direction cosine _l_. Now _l_ is the cosine of the angle
which the direction of the velocity V makes with O_x_, according
to A's judgment.
The meaning of this relation is, that if A discharges, from guns at
the point P, shells with a given muzzle velocity V according to his
judgment, B will consider that their muzzle velocities are different
from each other, except in the case of pairs of guns equally inclined
to the axis OO′. Instances of this type of diversity of judgment can
be noted any day by any one who looks out of the window of a railway
carriage, and forgets that he is travelling.
Now, suppose the velocity V′ bears a relation to the velocity V, which
is independent of _l_. Then _l_ must disappear from the above
formula. There are two conditions to be satisfied
One condition is
V^2 = β^2_v_^2/(β^2 - 1),
or in a more convenient form
β^2 = 1/(1 - _v_^2/V^2).
The meaning of this condition is, that there is one, and only one,
muzzle velocity V (according to A's judgment), namely, the muzzle
velocity given by the above formula, which can have the property that
B will judge that all the guns are firing in their diverse directions
with one common muzzle velocity.
Let us now suppose that V has this peculiar value: that is, if we look
on this value V as known, we must suppose that β is given by the second
of the above formulæ.
The other condition allows P and S to be put in the forms
P = -β_v_/λV^2, S = β/λ,
where
V′ = λV.
Thus we have the bundle of formulæ
β^2 = 1/(1 - _v_^2/V^2),
T = β{_t_ - _vx_/V^2}/λ,
V′ = λV.
The value which we give to λ is purely a matter for the adjustment of
units. If we want A and B to agree in their judgments of the magnitude
of this peculiar muzzle-velocity, we put λ = 1.
We then get the formulæ usually adopted, namely
β^2 = 1/(1 - _v_^2/V^2), }
T = β{_t_ - _vx_/V^2}, } (II)
V′ = V. }
But if we prefer that A and B should reckon (according to A's judgment)
in the same units of time, we put λ = β, and obtain
β^2 = 1/(1 - _v_^2/V^2),
T = (_t_ - _vx_)/V^2,
V′ = βV.
But A and B are in any case in such hopeless difficulties over their
comparisons of time-judgments that the detail of using the same units
does not help them much. Accordingly the formulæ marked (II) are those
used. Thus A and B agree in their judgments as to the magnitude of one
special velocity V, whatever may be the direction in which the entity
possessing it is moving.
In order to reach this measure of agreement, they have to disagree as
to their space judgments and their time judgments. The root cause of
their disagreement is their diverse judgment as to which axis system is
to be taken at rest for the purpose of measuring velocities.
Before discussing the nature of the disagreement disclosed in formula
(II), let us ask why we should bring these difficulties on our heads by
supposing that two people in relative motion, who both (for the purpose
of measuring velocities) assume that they are at rest, should agree in
their judgments in respect to this special velocity V.
Such an agreement has no counterpart in any of our obvious judgments
made from railway carriages. Surely we can wait till the contingency
occurs before discussing the confusion which it creates.
But the contingency has occurred. It occurs when we consider the
velocity of light. Perhaps I may venture to remind a philosophical
society that light moves so very quickly that it is difficult to
consider its velocity at all. So we need not be surprised that this
peculiar fact concerning its velocity is not more obvious.
Now V being the velocity of light, unless _v_ is large, _v_/V
(and still more _v_^2/V^2) will be quite inappreciable. The only
velocity ready to hand which is big enough to give _v_/V an
appreciable value is the velocity of the earth in its orbit.
Many diverse experiments have been made, and they all agree in
concluding that a man who assumes the earth to be at rest will find by
measurement that the velocity of light is the same in all directions.
Furthermore, when the same man turns his attention to interstellar or
interplanetary phenomena, and assumes the sun to be at rest, he will
again find the velocity of light to be the same in all directions.
These are well-attested experiments made at long intervals of time.
This is the exact contingency contemplated above.
Again the velocity of light _in vacuo_ has recently taken on a
new dignity. It used to be one among other wave velocities such as
the velocity of sound in air, or in water, or the velocity of surface
waves in water. But Clerk Maxwell discovered that all electromagnetic
influences are propagated with the velocity of light, and now modern
physical science half suspects that electromagnetic influences are
the only physical influences which relate the changes in the physical
world. Accordingly the velocity of light becomes the fundamental
natural velocity, and experiment shows that our judgment of its
magnitude is not affected by our choice of the framework at rest, so
long as we keep to a set of dynamical axes. These experiments on light
have been confirmed by other electromagnetic experiments not involving
light.
Thus we are driven to equations (II), where V is the velocity of light.
The first conclusion to be drawn from equations (II) is that two people
who make different choices of bodies at rest will disagree as to
their measuring rods in the way described above. There is no peculiar
difficulty about that. The only wonder is that all people agree so well
in their judgments as to metrical systems. A mathematical angel would
naturally expect incarnate men to be in violent disagreement on this
subject.
But the case of time is different. For simplicity of statement we speak
of A as at O, and B as at O′. We remember that O′ is moving relatively
to O with velocity _v_ in direction OO′. Suppose A and B are
looking in this direction; and they both measure their time from the
instant when they met, as O′ passed over O. Then we have
T = ((_t_ - _vx_)/V^2)/((1 - _v_^2)/V^2).
Now, suppose we consider all the events all over space which A
considers to have happened simultaneously at the time _t_. The
events of this set which occurred anywhere on a plane perpendicular to
OO′ at a distance _x_ in front of O (according to A's reckoning),
will have occurred according to B's reckoning at the time T as given
above. Let us fix our attention on the fact that B does not consider
all these events to be simultaneous. For let T_{1}, and T_{2} be B's
times for such events on planes _x__{1}, and _x__{2}. Then
T_{1} - T_{2} = _v_(_x__{2} - _x__{1})/(V^2 - _v_^2).
Thus if _x__{2} be greater than _x__{1}, T_{2} is less than
T_{1}. Thus B judges the more distant events in front of him to have
happened earlier than the nearer events in front of him, and _vice
versa_ for the events behind. This disturbance of the judgment of
simultaneity is the fundamental fact. Obviously the measurement of time
intervals is a detail compared to simultaneity. A may think a sermon
long, and B may think it short, but at least they should both agree
that it stopped when the clock hand pointed at the hour. The worst of
the matter is that so far as any test can be applied there is no method
of discriminating between the validities of their judgments.
Thus we are confronted with two distinct concepts of the common world,
A's space-time concept, and B's space-time concept. Who is right? It is
no use staying for an answer. We must follow the example of the wise
old Roman, and pass on to other things.
Thus estimates of quantity in space and time, and, to some extent, even
estimates of order, depend on the individual observer. But what are
the crude deliverances of sensible experience, apart from that world
of imaginative reconstruction which for each of us has the best claim
to be called our real world? Here the experimental psychologist steps
in. We cannot get away from him. I wish we could, for he is frightfully
difficult to understand. Also, sometimes his knowledge of the
principles of mathematics is rather weak, and I sometimes suspect----
No, I will not say what I sometimes think: probably he, with equal
reason, is thinking the same sort of thing of us.
I will, however, venture to summarise conclusions, which are, I
believe, in harmony with the experimental evidence, both physical and
psychological, and which are certainly suggested by the materials for
that unwritten chapter in mathematical logic which I have already
commended to your notice. The concepts of space and time and of
quantity are capable of analysis into bundles of simpler concepts. In
any given sensible experience it is not necessary, or even usual, that
the whole complete bundle of such concepts apply. For example, the
concept of externality may apply without that of linear order, and the
concept of linear order may apply without that of linear distance.
Again, the abstract mathematical concept of a space-relation
may confuse together distinct concepts which apply to the given
perceptions. For example, linear order in the sense of a linear
projection from the observer is distinct from linear order in the sense
of a row of objects stretching across the line of sight.
Mathematical physics assumes a given world of definitely related
objects, and the various space-time systems are alternative ways of
expressing those relations as concepts in a form which also applies to
the immediate experience of observers.
Yet there must be one way of expressing the relations between objects
in a common external world. Alternative methods can only arise as the
result of alternative standpoints; that is to say, as the result of
leaving something added by the observer sticking (as it were) in the
universe.
But this way of conceiving the world of physical science, as composed
of hypothetical objects, leaves it as a mere fairy tale. What is really
actual are the immediate experiences. The task of deductive science
is to consider the concepts which apply to these data of experience,
and then to consider the concepts relating to these concepts, and
so on to any necessary degree of refinement. As our concepts become
more abstract, their logical relations become more general, and less
liable to exception. By this logical construction we finally arrive
at conceptions, (i) which have determinate exemplifications in the
experience of the individuals, and (ii) whose logical relations have
a peculiar smoothness. For example, conceptions of mathematical time,
of mathematical space, are such smooth conceptions. No one lives in
"an infinite given whole," but in a set of fragmentary experiences.
The problem is to exhibit the concepts of mathematical space and time
as the necessary outcome of these fragments by a process of logical
building up. Similarly for the other physical concepts. This process
builds a common world of conceptions out of fragmentary worlds of
experience. The material pyramids of Egypt are a conception, what is
actual are the fragmentary experiences of the races who have gazed on
them.
So far as science seeks to rid itself of hypothesis, it cannot go
beyond these general logical constructions. For science, as thus
conceived, the divergent time orders considered above present no
difficulty. The different time systems simply register the different
relations of the mathematical construct to those individual
experiences (actual or hypothetical) which could exist as the crude
material from which the construct is elaborated.
But after all it should be possible so to elaborate the mathematical
construct so as to eliminate specific reference to particular
experiences. Whatever be the data of experience, there must be
something which can be said of them as a whole, and that something is
a statement of the general properties of the common world. It is hard
to believe that with proper generalisation time and space will not be
found among such properties.
_Commentary added on reading the Paper before the Aristotelian
Society_
The first six pages of the paper consist of a summary of ideas which
ought to be in our minds while considering problems of time and space.
The ideas are mostly philosophical, and the summary has been made by
an amateur in that science; so there is no reason to ascribe to it any
importance except that of a modest reminder. There are only two points
in this summary to which I would draw attention.
On pp. 192 and 193 there occurs--
"Wherever mankind ... unending uniform succession."
If I understand Kant rightly--which I admit to be very
problematical--he holds that in the act of experience we are aware
of space and time as ingredients necessary for the occurrence of
experience. I would suggest--rather timidly--that this doctrine should
be given a different twist, which in fact turns it in the opposite
direction--namely, that in the act of experience we perceive a whole
formed of related differentiated parts. The relations between these
parts possess certain characteristics, and time and space are the
expressions of some of the characteristics of these relations. Then the
generality and uniformity which are ascribed to time and space express
what may be termed the uniformity of the texture of experience.
The success of mankind--modest though it is--in deducing uniform laws
of nature is, so far as it goes, a testimony that this uniformity of
texture goes beyond those characteristics of the data of experience
which are expressed as time and space. Time and space are necessary
to experience in the sense that they are characteristics of our
experience; and, of course, no one can have our experience without
running into them. I cannot see that Kant's deduction amounts to much
more than saying that "what is, is"--true enough, but not very helpful.
But I admit that what I have termed the "uniformity of the texture of
experience" is a most curious and arresting fact. I am quite ready to
believe that it is a mere illusion; and later on in the paper I suggest
that this uniformity does not belong to the immediate relations of the
crude data of experience, but is the result of substituting for them
more refined logical entities, such as relations between relations, or
classes of relations, or classes of classes of relations. By this means
it can be demonstrated--I think--that the uniformity which must be
ascribed to experience is of a much more abstract attenuated character
than is usually allowed. This process of lifting the uniform time and
space of the physical world into the status of logical abstractions has
also the advantage of recognising another fact, namely, the extremely
fragmentary nature of all direct individual experience.
My point in this respect is that fragmentary individual experiences
are all that we know, and that all speculation must start from these
_disjecta membra_ as its sole datum. It is not true that we are
directly aware of a smooth running world, which in our speculations we
are to conceive as given. In my view the creation of the world is the
first unconscious act of speculative thought; and the first task of a
self-conscious philosophy is to explain how it has been done.
There are roughly two rival explanations. One is to assert the world
as a postulate. The other way is to obtain it as a deduction, not a
deduction through a chain of reasoning, but a deduction through a chain
of definitions which, in fact, lifts thought on to a more abstract
level in which the logical ideas are more complex, and their relations
are more universal. In this way the broken limited experiences sustain
that connected infinite world in which in our thoughts we live. There
are three more remarks while on this point I wish to make--
(i) The fact that immediate experience is capable of this deductive
superstructive must mean that it itself has a certain uniformity of
texture. So this great fact still remains.
(ii) I do not wish to deny the world as a postulate. Speaking without
prejudice, I do not see how in our present elementary state of
philosophical advance we can get on without middle axioms, which, in
fact, we habitually assume.
My position is, that by careful scrutiny we should extrude such
postulates from every part of our organised knowledge in which it is
possible to do without them.
Now, physical science organises our knowledge of the relations between
the deliverances of our various senses. I hold that in this department
of knowledge such postulates, though not entirely to be extruded, can
be reduced to a minimum in the way which I have described.
I have not the slightest knowledge of theories respecting our emotions,
affections, and moral sentiments, and I can well believe that in
dealing with them further postulates are required. And in practice I
recognise that we all make such postulates, uncritically.
(iii) The next paragraphs on pp. 193 and 194 are as follows--
"The opposed philosophical method ... physical science."
It will be noted that, in the light of what has just been stated, the
first of these paragraphs (which, I hope, faithfully expresses the
experimental way of approaching the problem) really obscures the point
which I have been endeavouring to make. The phrase, "If we form the
exact concepts of points, etc.," is fatally ambiguous as between the
method of postulating entities with assigned relations, and the method
of forming logical constructions, and thus reaching points, etc., as
the result of a chain of definitions.
Turning now to pp. 194-195, we come to the following paragraphs--
"The other theory ... eighteenth century."
We note again that the relational theory of space from another point
of view brings us back to the idea of the fundamental space-entities
as being logical constructs from the relations between things. The
difference is, that this paragraph is written from a more developed
point of view, as it implicitly assumes the things in space, and
conceives space as an expression of certain of their relations.
Combining this paragraph with what has gone before, we see that the
suggested procedure is first to define "things" in terms of the data of
experience, and then to define space in terms of the relations between
things.
This procedure is explicitly assumed in the next short paragraph: "In
this connection ... from the events."
The gist of the remaining paragraphs of this section is contained in
the paragraph at the bottom of p. 196: "The point in time ... new
importance."
The sentence, "We live in durations, and not in points," can be
amplified by the addition, "We live in space-extensions and not in
space-points."
It must be noted that "whole and part" as applied to extensions in
space or time must be different from the "all and some" of logic,
unless we admit points to be the fundamental entities. For "spatial
whole and spatial part" can only mean "all and some" if they really
mean "all the points and some of the points." But if extensions and
their relations are more fundamental than points, this interpretation
is precluded. I suggest that "spatial whole and spatial part" is
intimately connected with the fundamental relation between things from
which our space ideas spring.
The relation of space whole to space part has many formal properties
which are identical with the properties of "all and some." Also when
points have been defined, we can replace it by the conception of "all
the points and some of the points." But the confusion between the two
relations is fatal to sound views on the subject.
_Diverse Euclidean Measure Systems_
The next section deals with the measure systems applicable to space.
A measure system is a group of congruent transformations of space
into itself. Consider a rigid body occupying all space. Let this body
be moved in any way so that the particles of the body which occupied
points P_{1}, P_{2}, P_{3}, etc., now occupy points Q_{1}, Q_{2},
Q_{3}, etc. Then any point P_{1} in space is uniquely related to the
corresponding point Q_{1} in space by a one-to-one transformation with
certain characteristics. By the aid of these transformations we can
achieve the definition of distance in a way which definitely determines
the distance between any two points, provided that we can define what
we mean by a congruent transformation without introducing the idea of
distance. If we introduced the idea of distance, we should simply say
that a congruent transformation is one which leaves all distances
unchanged, _i. e._, if P_{1}, P_{2} are transformed into Q_{1},
Q_{2} then the distance P_{1}P_{2} is equal to the distance Q_{1}Q_{2}.
But mathematicians have succeeded in defining congruent transformations
without any reference to distance.
There are alternative groups of such congruent transformations, and
each group gives a different measure system for space. The distance
P_{1}P_{2} may equal the distance Q_{1}Q_{2} for one measure system,
and will not equal it for another measure system. All these different
measure systems are on the same level, equally applicable. A being
with a strong enough head could think of them all at once as applying
to space. The result so far as it interests us in respect to the
theory of relativity is explained on pp. 197-200, ending with "The
most extraordinary fact ... same metrical system." This final sentence
bears on Poincaré's assertion that the measure system adopted is
purely "conventional." I presume that by "conventional" a certain
arbitrariness of choice is meant; and in that case, I must express
entire dissent. It is true that within the circle of geometrical ideas
there is no means of giving any preference to any one measure system,
and any one is as good as any other. But it is not true that if we
look at a normal carriage wheel, and at an oval curve one foot broad
and ten feet long, we experience any arbitrariness of judgment in
deciding which has approximately the form of a circle. Accordingly to
Poincaré the choice between them, as representing a circle, is entirely
conventional.
Again, we equally form immediate judgments as to whether a body is
approximately rigid. We know that a paving stone is rigid, and that a
concertina is not rigid. This again necessitates a determinate measure
system, selected from among the others.
Accordingly we conclude that (i) each being does, in fact, employ a
determinate measure system, which remains the same, except possibly
for very small variations, and (ii) the measure systems of different
human beings agree, to within the limits of our observations. These
conclusions are not the less extraordinary because no plain man has
ever doubted them.
It is an interesting subject to investigate exactly what are the
fundamental uniformities of experience which necessitate this
conclusion. It is not so easy as it looks, since we have to divest
ourselves of all aid of scientific hypothesis if our conclusions are to
be demonstrative.
_Relativity in Modern Physics_
Pp. 201-202, "Owing to the fact ... which B so regards."
The fundamental formulæ for the theory of relativity are the relations
between diverse co-ordinate systems given on p. 203, and formulæ II
at the bottom of p. 207. The general explanation of one method in
which these formulæ arise--namely, Einstein's method--is given on
pp. 201-211. Namely, we seek the condition that for all dynamical
axes the velocity of light should be the same, and the same in all
directions. It should be noted that the experiments which, so far as
they go, confirm these formulæ, can also be explained in another way
which makes the theory of relativity unnecessary. We need only ascribe
to the ether a certain property of contraction in the direction of
motion, and the thing is done. So no one need be bludgeoned into
accepting the rather bizarre doctrine of relativity, nor indeed any
other scientific generalisation. The good old homely ether, which we
all know, can in this case serve the purpose. Just as an author of
genius, if he lives long enough, survives the inevitable accusation
of immorality, so the ether by dint of persistence has outlived all
reputation of extravagance. But if we detach ourselves from the glib
phraseology concerning it, the scientific ether is uncommonly like the
primitive explanation of the soul, as a little man inside us, which can
sometimes be caught escaping in the form of a butterfly. As soon as the
ether has to be patched up with special properties to explain special
experiments, its scientific use is problematical, and its philosophic
use is _nil_.
Philosophically the ether seems to me to be an ambitious attempt
to give a complete explanation of the physical universe by making
an elephant stand on a tortoise. Scientifically it has a perfectly
adequate use by veiling the extremely abstract character of scientific
generalisations under a myth, which enables our imaginations to work
more freely. I am not advocating the extrusion of ether from our
scientific phraseology, even though at special points we have to
abandon it.
But the key to the reasons why it is worth while to consider seriously
the doctrine of relativity is to be found on pp. 209, 210: "Again the
velocity of light ... not involving light." Namely, we have begun to
suspect that all physical influences require time for their propagation
in space. This generalisation is a long way from being proved.
Gravitation stands like a lion in the path. But if it be the case, then
all idea of an immediate presentation to us of an aspect of the world
as it in fact is, must necessarily be abandoned. What we perceive at
any instant is already ancient history, with the dates of the various
parts hopelessly mixed.
We must add to this the difficulty of determining what is at rest
and what is in motion, and the further difficulty of determining a
definite uniform flow of time. It is no use discussing this matter
as though, but for the silly extravagant doctrine of relativity,
everything would be plain sailing. It isn't. You may be quite sure
that when, after prolonged study, you endeavour to give the simplest
explanation of a grave difficulty, you will be accused of extravagance.
I have no responsibility for the doctrine of relativity, and hold no
brief for it, but it has some claim to be considered as a comparatively
simple way out of a scientific maze.
In the first place, we use the Newtonian dynamical sets of axes,
and the Newtonian clock to extricate ourselves partially from the
difficulties of rest, motion, and time. These have proved capable
of scientific determination within the limits of our experimental
accuracy. Thus the only thing left over is the choice of the axes
at rest, which is a completely indeterminate problem on Newtonian
principles.
Again, so far as we can at present guess by adopting the theory
that all metrical influence is electromagnetic, all influences
are propagated with the velocity of light _in vacuo_. This
electromagnetic hypothesis is by no means established, but it gives
the simplest of all possible results in respect to the propagation of
influence, which we therefore adopt.
But what dynamical axes are we taking as at rest? Now our practical
choice gives a range of relative velocities small compared to that of
light. So except for certain refined experiments it does not matter.
There are two possibilities--
(i) We may assume that one set of axes are at rest, and that the others
will show traces of motion in respect to the velocity of light; or--
(ii) That the velocity of light is the same in all directions whichever
be the dynamical axes assumed.
The first supposition is negatived by experiment, and hence we are
driven to the second supposition; which immediately lands us in the
whole theory of relativity.
But if we will not have this theory we must reject the earlier
supposition that the velocity of light _in vacuo_ is the same in
all directions. This we do, in fact, by assuming an ether, and assuming
a certain law for its modification. Then we, in fact, adopt the first
supposition so far as to hold that there are dynamical axes specially
at rest, namely, at rest relatively to the undisturbed ether. Then an
assumed law for the modification of the ether so alters the velocity of
light that we explain why no dynamical axes show traces of motion.
I wish now to go back to the point which I made a few minutes ago,
that what we perceive at any instant is ancient history with its dates
hopelessly mixed. In the earlier part of my comments I emphasised the
point that our only data as to the physical world are our sensible
perceptions. We must not slip into the fallacy of assuming that we are
comparing a given world with given perceptions of it. The physical
world is in some general sense of the term a deduced concept.
Our problem is, in fact, to fit the world to our perceptions, and not
our perceptions to the world.
PRINTED IN GREAT BRITAIN BY RICHARD CLAY & SONS, LIMITED,
BRUNSWICK ST., STAMFORD ST., S.E., AND BUNGAY, SUFFOLK.
From WILLIAMS & NORGATE'S LIST
=A SPIRITUAL PILGRIMAGE.= By the Rev. R. J. CAMPBELL.
(The Rev. R. J. Campbell's own Story of his Religious Life.) 2nd
Impression. Demy 8vo. =7s. 6d.= net.
"This is eminently a book for which to be thankful--simple,
straightforward, kindly. Neither does it lack the saving grace of
humour."--_Guardian._
"Mr. Campbell devotes a chapter to his re-ordination, and it must be
admitted that his defence of this act is effective."--_Methodist
Times._
=EDINBURGH.= By the Right Hon. Sir HERBERT MAXWELL,
Bart., D.C.L., Author of "Life of the Duke of Wellington," "Scottish
Gardens," etc. President of the Society of Antiquaries of Scotland,
1910-13, and Chairman of the Royal Commission on Scottish Historical
Monuments. With a Coloured Frontispiece and 64 pages of Illustrations
of the past and present city. Medium 8vo. =10s. 6d.= net.
A History of the Scottish Capital from the earliest times to the
nineteenth century.
=FURTHER PAGES OF MY LIFE.= By the Right Rev. W. BOYD
CARPENTER, K.C.V.O., D.D., D.C.L., Formerly Bishop of Ripon,
Author of "Some Pages of My Life," "The Witness of Religious
Experience," etc. With Illustrations. Medium 8vo. =10s. 6d.= net.
Reflections mingled with intimate reminiscences and recollections of
Royal personages and of men eminent in many spheres.
=SOME PAGES OF MY LIFE.= By the Right Rev. W. BOYD
CARPENTER. New and Cheaper Edition. Large post 8vo. =5s.=
net.
All who have listened to his eloquent preaching will read with delight
his musing on what life has brought the author.
=RAPHAEL MELDOLA=, Hon. D.Sc. (Oxon.), Hon. LL.D. (St. And.),
F.R.S., Professor of Chemistry in the City and Guilds of London
Technical College. Reminiscences of his worth and work, by those who
knew him, together with a Chronological List of his Publications
(1868-1915). Edited by JAMES MARCHANT. With a Preface by the
Right Hon. LORD MOULTON, K.C.B., F.R.S. With a Portrait.
Crown 8vo. =5s.= net.
"The book is one of the most delightful biographies of scientific
men that have been published during recent years."--_Chemist and
Druggist._
=THE MANUFACTURE OF HISTORICAL MATERIAL.= An Elementary Study in
the Sources of Story. By J. W. JEUDWINE, LL.B. (Camb.), of
Lincoln's Inn, Barrister-at-Law, Author of "The First Twelve Centuries
of British Story," and other works. Crown 8vo. =6s.= net.
A consideration of the successive phases of Historical Research; it
explains necessary processes through which all the material has to
pass before it is placed before us as history.
LONDON: WILLIAMS AND NORGATE
SOME BOOKS ON RUSSIA AND POLAND
With a Coloured Frontispiece, 12 Photogravure Plates, 28 Illustrations
in the text, and 8 Maps. Demy 8vo. =7s. 6d.= net (postage
=6d.= extra).
=A Thousand Years of Russian History=
By SONIA E. HOWE.
"We can recommend the volume as an excellent, careful, and
well-written history of a great nation."--_Daily Telegraph._
Demy 8vo. Cloth, =6s.= net.
=The False Dmitri=
A Russian Romance and Tragedy. Described by British Eye-witnesses,
1604-1612.
Edited with a Preface by SONIA E. HOWE.
Illustrated by reproduction of contemporary portraits.
"With more than a whiff of the atmosphere Hakluyt got into his famous
volume."--_Northern Whig._
"Of special interest at the present time."--_Oxford Chronicle._
With 16 Plates and 28 Illustrations in the text. Demy 8vo. =7s.
6d.= net.
=Some Russian Heroes, Saints and Sinners=
A portrait gallery of outstanding figures in Russian history who were
typical of their times and their country.
By SONIA E. HOWE.
"It gives a vivid study of Russia ... and interesting book."--_Land
and Water._
Fcap. 8vo. Cloth, =1s. 3d.= net. In full leather presentation
binding, =2s. 6d.= net.
=An Outline of Russian Literature=
By the Hon. MAURICE BARING,
Author of "With the Russians in Manchuria," "A Year in Russia," "The
Russian People," etc.
∵ Gives a clear and interesting account of the growth of Russian
literature.
Fcap. 8vo. Cloth, =1s. 3d.= net. In full leather binding, =2s.
6d.= net.
=Poland=
By W. ALISON PHILLIPS, M.A.,
Lecky Professor of Modern History in the University of Dublin.
∵ A concise survey of the History of Poland up to 1915.
LONDON: WILLIAMS AND NORGATE
14 HENRIETTA STREET, COVENT GARDEN, W.C.
=TRANSCRIBER’S NOTES=
Simple typographical errors have been silently corrected; unbalanced
quotation marks were remedied when the change was obvious, and
otherwise left unbalanced.
Punctuation, hyphenation, and spelling were made consistent when a
predominant preference was found in the original book; otherwise they
were not changed.
*** END OF THE PROJECT GUTENBERG EBOOK THE ORGANISATION OF THOUGHT ***
Updated editions will replace the previous one—the old editions will
be renamed.
Creating the works from print editions not protected by U.S. copyright
law means that no one owns a United States copyright in these works,
so the Foundation (and you!) can copy and distribute it in the United
States without permission and without paying copyright
royalties. Special rules, set forth in the General Terms of Use part
of this license, apply to copying and distributing Project
Gutenberg™ electronic works to protect the PROJECT GUTENBERG™
concept and trademark. Project Gutenberg is a registered trademark,
and may not be used if you charge for an eBook, except by following
the terms of the trademark license, including paying royalties for use
of the Project Gutenberg trademark. If you do not charge anything for
copies of this eBook, complying with the trademark license is very
easy. You may use this eBook for nearly any purpose such as creation
of derivative works, reports, performances and research. Project
Gutenberg eBooks may be modified and printed and given away—you may
do practically ANYTHING in the United States with eBooks not protected
by U.S. copyright law. Redistribution is subject to the trademark
license, especially commercial redistribution.
START: FULL LICENSE
THE FULL PROJECT GUTENBERG LICENSE
PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK
To protect the Project Gutenberg™ mission of promoting the free
distribution of electronic works, by using or distributing this work
(or any other work associated in any way with the phrase “Project
Gutenberg”), you agree to comply with all the terms of the Full
Project Gutenberg™ License available with this file or online at
www.gutenberg.org/license.
Section 1. General Terms of Use and Redistributing Project Gutenberg™
electronic works
1.A. By reading or using any part of this Project Gutenberg™
electronic work, you indicate that you have read, understand, agree to
and accept all the terms of this license and intellectual property
(trademark/copyright) agreement. If you do not agree to abide by all
the terms of this agreement, you must cease using and return or
destroy all copies of Project Gutenberg™ electronic works in your
possession. If you paid a fee for obtaining a copy of or access to a
Project Gutenberg™ electronic work and you do not agree to be bound
by the terms of this agreement, you may obtain a refund from the person
or entity to whom you paid the fee as set forth in paragraph 1.E.8.
1.B. “Project Gutenberg” is a registered trademark. It may only be
used on or associated in any way with an electronic work by people who
agree to be bound by the terms of this agreement. There are a few
things that you can do with most Project Gutenberg™ electronic works
even without complying with the full terms of this agreement. See
paragraph 1.C below. There are a lot of things you can do with Project
Gutenberg™ electronic works if you follow the terms of this
agreement and help preserve free future access to Project Gutenberg™
electronic works. See paragraph 1.E below.
1.C. The Project Gutenberg Literary Archive Foundation (“the
Foundation” or PGLAF), owns a compilation copyright in the collection
of Project Gutenberg™ electronic works. Nearly all the individual
works in the collection are in the public domain in the United
States. If an individual work is unprotected by copyright law in the
United States and you are located in the United States, we do not
claim a right to prevent you from copying, distributing, performing,
displaying or creating derivative works based on the work as long as
all references to Project Gutenberg are removed. Of course, we hope
that you will support the Project Gutenberg™ mission of promoting
free access to electronic works by freely sharing Project Gutenberg™
works in compliance with the terms of this agreement for keeping the
Project Gutenberg™ name associated with the work. You can easily
comply with the terms of this agreement by keeping this work in the
same format with its attached full Project Gutenberg™ License when
you share it without charge with others.
1.D. The copyright laws of the place where you are located also govern
what you can do with this work. Copyright laws in most countries are
in a constant state of change. If you are outside the United States,
check the laws of your country in addition to the terms of this
agreement before downloading, copying, displaying, performing,
distributing or creating derivative works based on this work or any
other Project Gutenberg™ work. The Foundation makes no
representations concerning the copyright status of any work in any
country other than the United States.
1.E. Unless you have removed all references to Project Gutenberg:
1.E.1. The following sentence, with active links to, or other
immediate access to, the full Project Gutenberg™ License must appear
prominently whenever any copy of a Project Gutenberg™ work (any work
on which the phrase “Project Gutenberg” appears, or with which the
phrase “Project Gutenberg” is associated) is accessed, displayed,
performed, viewed, copied or distributed:
This eBook is for the use of anyone anywhere in the United States and most
other parts of the world at no cost and with almost no restrictions
whatsoever. You may copy it, give it away or re-use it under the terms
of the Project Gutenberg License included with this eBook or online
at www.gutenberg.org. If you
are not located in the United States, you will have to check the laws
of the country where you are located before using this eBook.
1.E.2. If an individual Project Gutenberg™ electronic work is
derived from texts not protected by U.S. copyright law (does not
contain a notice indicating that it is posted with permission of the
copyright holder), the work can be copied and distributed to anyone in
the United States without paying any fees or charges. If you are
redistributing or providing access to a work with the phrase “Project
Gutenberg” associated with or appearing on the work, you must comply
either with the requirements of paragraphs 1.E.1 through 1.E.7 or
obtain permission for the use of the work and the Project Gutenberg™
trademark as set forth in paragraphs 1.E.8 or 1.E.9.
1.E.3. If an individual Project Gutenberg™ electronic work is posted
with the permission of the copyright holder, your use and distribution
must comply with both paragraphs 1.E.1 through 1.E.7 and any
additional terms imposed by the copyright holder. Additional terms
will be linked to the Project Gutenberg™ License for all works
posted with the permission of the copyright holder found at the
beginning of this work.
1.E.4. Do not unlink or detach or remove the full Project Gutenberg™
License terms from this work, or any files containing a part of this
work or any other work associated with Project Gutenberg™.
1.E.5. Do not copy, display, perform, distribute or redistribute this
electronic work, or any part of this electronic work, without
prominently displaying the sentence set forth in paragraph 1.E.1 with
active links or immediate access to the full terms of the Project
Gutenberg™ License.
1.E.6. You may convert to and distribute this work in any binary,
compressed, marked up, nonproprietary or proprietary form, including
any word processing or hypertext form. However, if you provide access
to or distribute copies of a Project Gutenberg™ work in a format
other than “Plain Vanilla ASCII” or other format used in the official
version posted on the official Project Gutenberg™ website
(www.gutenberg.org), you must, at no additional cost, fee or expense
to the user, provide a copy, a means of exporting a copy, or a means
of obtaining a copy upon request, of the work in its original “Plain
Vanilla ASCII” or other form. Any alternate format must include the
full Project Gutenberg™ License as specified in paragraph 1.E.1.
1.E.7. Do not charge a fee for access to, viewing, displaying,
performing, copying or distributing any Project Gutenberg™ works
unless you comply with paragraph 1.E.8 or 1.E.9.
1.E.8. You may charge a reasonable fee for copies of or providing
access to or distributing Project Gutenberg™ electronic works
provided that:
• You pay a royalty fee of 20% of the gross profits you derive from
the use of Project Gutenberg™ works calculated using the method
you already use to calculate your applicable taxes. The fee is owed
to the owner of the Project Gutenberg™ trademark, but he has
agreed to donate royalties under this paragraph to the Project
Gutenberg Literary Archive Foundation. Royalty payments must be paid
within 60 days following each date on which you prepare (or are
legally required to prepare) your periodic tax returns. Royalty
payments should be clearly marked as such and sent to the Project
Gutenberg Literary Archive Foundation at the address specified in
Section 4, “Information about donations to the Project Gutenberg
Literary Archive Foundation.”
• You provide a full refund of any money paid by a user who notifies
you in writing (or by e-mail) within 30 days of receipt that s/he
does not agree to the terms of the full Project Gutenberg™
License. You must require such a user to return or destroy all
copies of the works possessed in a physical medium and discontinue
all use of and all access to other copies of Project Gutenberg™
works.
• You provide, in accordance with paragraph 1.F.3, a full refund of
any money paid for a work or a replacement copy, if a defect in the
electronic work is discovered and reported to you within 90 days of
receipt of the work.
• You comply with all other terms of this agreement for free
distribution of Project Gutenberg™ works.
1.E.9. If you wish to charge a fee or distribute a Project
Gutenberg™ electronic work or group of works on different terms than
are set forth in this agreement, you must obtain permission in writing
from the Project Gutenberg Literary Archive Foundation, the manager of
the Project Gutenberg™ trademark. Contact the Foundation as set
forth in Section 3 below.
1.F.
1.F.1. Project Gutenberg volunteers and employees expend considerable
effort to identify, do copyright research on, transcribe and proofread
works not protected by U.S. copyright law in creating the Project
Gutenberg™ collection. Despite these efforts, Project Gutenberg™
electronic works, and the medium on which they may be stored, may
contain “Defects,” such as, but not limited to, incomplete, inaccurate
or corrupt data, transcription errors, a copyright or other
intellectual property infringement, a defective or damaged disk or
other medium, a computer virus, or computer codes that damage or
cannot be read by your equipment.
1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the “Right
of Replacement or Refund” described in paragraph 1.F.3, the Project
Gutenberg Literary Archive Foundation, the owner of the Project
Gutenberg™ trademark, and any other party distributing a Project
Gutenberg™ electronic work under this agreement, disclaim all
liability to you for damages, costs and expenses, including legal
fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT
LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE
PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE FOUNDATION, THE
TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE
LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR
INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH
DAMAGE.
1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a
defect in this electronic work within 90 days of receiving it, you can
receive a refund of the money (if any) you paid for it by sending a
written explanation to the person you received the work from. If you
received the work on a physical medium, you must return the medium
with your written explanation. The person or entity that provided you
with the defective work may elect to provide a replacement copy in
lieu of a refund. If you received the work electronically, the person
or entity providing it to you may choose to give you a second
opportunity to receive the work electronically in lieu of a refund. If
the second copy is also defective, you may demand a refund in writing
without further opportunities to fix the problem.
1.F.4. Except for the limited right of replacement or refund set forth
in paragraph 1.F.3, this work is provided to you ‘AS-IS’, WITH NO
OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT
LIMITED TO WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE.
1.F.5. Some states do not allow disclaimers of certain implied
warranties or the exclusion or limitation of certain types of
damages. If any disclaimer or limitation set forth in this agreement
violates the law of the state applicable to this agreement, the
agreement shall be interpreted to make the maximum disclaimer or
limitation permitted by the applicable state law. The invalidity or
unenforceability of any provision of this agreement shall not void the
remaining provisions.
1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the
trademark owner, any agent or employee of the Foundation, anyone
providing copies of Project Gutenberg™ electronic works in
accordance with this agreement, and any volunteers associated with the
production, promotion and distribution of Project Gutenberg™
electronic works, harmless from all liability, costs and expenses,
including legal fees, that arise directly or indirectly from any of
the following which you do or cause to occur: (a) distribution of this
or any Project Gutenberg™ work, (b) alteration, modification, or
additions or deletions to any Project Gutenberg™ work, and (c) any
Defect you cause.
Section 2. Information about the Mission of Project Gutenberg™
Project Gutenberg™ is synonymous with the free distribution of
electronic works in formats readable by the widest variety of
computers including obsolete, old, middle-aged and new computers. It
exists because of the efforts of hundreds of volunteers and donations
from people in all walks of life.
Volunteers and financial support to provide volunteers with the
assistance they need are critical to reaching Project Gutenberg™’s
goals and ensuring that the Project Gutenberg™ collection will
remain freely available for generations to come. In 2001, the Project
Gutenberg Literary Archive Foundation was created to provide a secure
and permanent future for Project Gutenberg™ and future
generations. To learn more about the Project Gutenberg Literary
Archive Foundation and how your efforts and donations can help, see
Sections 3 and 4 and the Foundation information page at www.gutenberg.org.
Section 3. Information about the Project Gutenberg Literary Archive Foundation
The Project Gutenberg Literary Archive Foundation is a non-profit
501(c)(3) educational corporation organized under the laws of the
state of Mississippi and granted tax exempt status by the Internal
Revenue Service. The Foundation’s EIN or federal tax identification
number is 64-6221541. Contributions to the Project Gutenberg Literary
Archive Foundation are tax deductible to the full extent permitted by
U.S. federal laws and your state’s laws.
The Foundation’s business office is located at 809 North 1500 West,
Salt Lake City, UT 84116, (801) 596-1887. Email contact links and up
to date contact information can be found at the Foundation’s website
and official page at www.gutenberg.org/contact
Section 4. Information about Donations to the Project Gutenberg
Literary Archive Foundation
Project Gutenberg™ depends upon and cannot survive without widespread
public support and donations to carry out its mission of
increasing the number of public domain and licensed works that can be
freely distributed in machine-readable form accessible by the widest
array of equipment including outdated equipment. Many small donations
($1 to $5,000) are particularly important to maintaining tax exempt
status with the IRS.
The Foundation is committed to complying with the laws regulating
charities and charitable donations in all 50 states of the United
States. Compliance requirements are not uniform and it takes a
considerable effort, much paperwork and many fees to meet and keep up
with these requirements. We do not solicit donations in locations
where we have not received written confirmation of compliance. To SEND
DONATIONS or determine the status of compliance for any particular state
visit www.gutenberg.org/donate.
While we cannot and do not solicit contributions from states where we
have not met the solicitation requirements, we know of no prohibition
against accepting unsolicited donations from donors in such states who
approach us with offers to donate.
International donations are gratefully accepted, but we cannot make
any statements concerning tax treatment of donations received from
outside the United States. U.S. laws alone swamp our small staff.
Please check the Project Gutenberg web pages for current donation
methods and addresses. Donations are accepted in a number of other
ways including checks, online payments and credit card donations. To
donate, please visit: www.gutenberg.org/donate.
Section 5. General Information About Project Gutenberg™ electronic works
Professor Michael S. Hart was the originator of the Project
Gutenberg™ concept of a library of electronic works that could be
freely shared with anyone. For forty years, he produced and
distributed Project Gutenberg™ eBooks with only a loose network of
volunteer support.
Project Gutenberg™ eBooks are often created from several printed
editions, all of which are confirmed as not protected by copyright in
the U.S. unless a copyright notice is included. Thus, we do not
necessarily keep eBooks in compliance with any particular paper
edition.
Most people start at our website which has the main PG search
facility: www.gutenberg.org.
This website includes information about Project Gutenberg™,
including how to make donations to the Project Gutenberg Literary
Archive Foundation, how to help produce our new eBooks, and how to
subscribe to our email newsletter to hear about new eBooks.
