MATHEMATICS AS AN ELEMENT IN
THE HISTORY OF THOUGHT
The science of Pure Mathematics, in its modern developments, may claim
to be the most original creation of the human spirit. Another claimant
for this position is music. But we will put aside all rivals, and
consider the ground on which such a claim can be made for mathematics.
The originality of mathematics consists in the fact that in mathematical
science connections between things are exhibited which, apart from the
agency of human reason, are extremely unobvious. Thus the ideas, now in
the minds of contemporary mathematicians, lie very remote from any
notions which can be immediately derived by perception through the
senses; unless indeed it be perception stimulated and guided by
antecedent mathematical knowledge. This is the thesis which I proceed to
exemplify.
Suppose we project our imaginations backwards through many thousands of
years, and endeavour to realise the simple-mindedness of even the
greatest intellects in those early societies. Abstract ideas which to us
are immediately obvious must have been, for them, matters only of the
most dim apprehension. For example take the question of number. We think
of the number ‘five’ as applying to appropriate groups of any entities
whatsoever—to five fishes, five children, five apples, five days. Thus
in considering the relations of the number ‘five’ to the number ‘three,’
we are thinking of two groups of things, one with five members and the
other with three members. But we are entirely abstracting from any
consideration of any particular entities, or even of any particular
sorts of entities, which go to make up the membership of either of the
two groups. We are merely thinking of those relationships between those
two groups which are entirely independent of the individual essences of
any of the members of either group. This is a very remarkable feat of
abstraction; and it must have taken ages for the human race to rise to
it. During a long period, groups of fishes will have been compared to
each other in respect to their multiplicity, and groups of days to each
other. But the first man who noticed the analogy between a group of
seven fishes and a group of seven days made a notable advance in the
history of thought. He was the first man who entertained a concept
belonging to the science of pure mathematics. At that moment it must
have been impossible for him to divine the complexity and subtlety of
these abstract mathematical ideas which were waiting for discovery. Nor
could he have guessed that these notions would exert a widespread
fascination in each succeeding generation. There is an erroneous
literary tradition which represents the love of mathematics as a
monomania confined to a few eccentrics in each generation. But be this
as it may, it would have been impossible to anticipate the pleasure
derivable from a type of abstract thinking which had no counterpart in
the then-existing society. Thirdly, the tremendous future effect of
mathematical knowledge on the lives of men, on their daily avocations,
on their habitual thoughts, on the organization of society, must have
been even more completely shrouded from the foresight of those early
thinkers. Even now there is a very wavering grasp of the true position
of mathematics as an element in the history of thought. I will not go so
far as to say that to construct a history of thought without profound
study of the mathematical ideas of successive epochs is like omitting
Hamlet from the play which is named after him. That would be claiming
too much. But it is certainly analogous to cutting out the part of
Ophelia. This simile is singularly exact. For Ophelia is quite essential
to the play, she is very charming,—and a little mad. Let us grant that
the pursuit of mathematics is a divine madness of the human spirit, a
refuge from the goading urgency of contingent happenings.
When we think of mathematics, we have in our mind a science devoted to
the exploration of number, quantity, geometry, and in modern times also
including investigation into yet more abstract concepts of order, and
into analogous types of purely logical relations. The point of
mathematics is that in it we have always got rid of the particular
instance, and even of any particular sorts of entities. So that for
example, no mathematical truths apply merely to fish, or merely to
stones, or merely to colours. So long as you are dealing with pure
mathematics, you are in the realm of complete and absolute abstraction.
All you assert is, that reason insists on the admission that, if any
entities whatever have any relations which satisfy such-and-such purely
abstract conditions, then they must have other relations which satisfy
other purely abstract conditions.
Mathematics is thought moving in the sphere of complete abstraction from
any particular instance of what it is talking about. So far is this view
of mathematics from being obvious, that we can easily assure ourselves
that it is not, even now, generally understood. For example, it is
habitually thought that the certainty of mathematics is a reason for the
certainty of our geometrical knowledge of the space of the physical
universe. This is a delusion which has vitiated much philosophy in the
past, and some philosophy in the present. This question of geometry is a
test case of some urgency. There are certain alternative sets of purely
abstract conditions possible for the relationships of groups of
unspecified entities, which I will call _geometrical conditions_. I give
them this name because of their general analogy to those conditions,
which we believe to hold respecting the particular geometrical relations
of things observed by us in our direct perception of nature. So far as
our observations are concerned, we are not quite accurate enough to be
certain of the exact conditions regulating the things we come across in
nature. But we can by a slight stretch of hypothesis identify these
observed conditions with some one set of the purely abstract geometrical
conditions. In doing so, we make a particular determination of the group
of unspecified entities which are the _relata_ in the abstract science.
In the pure mathematics of geometrical relationships, we say that, if
_any_ group of entities enjoy _any_ relationships among its members
satisfying _this_ set of abstract geometrical conditions, then
such-and-such additional abstract conditions must also hold for such
relationships. But when we come to physical space, we say that some
definitely observed group of physical entities enjoys some definitely
observed relationships among its members which do satisfy this
above-mentioned set of abstract geometrical conditions. We thence
conclude that the additional relationships which we concluded to hold in
_any_ such case, must therefore hold in _this particular_ case.
The certainty of mathematics depends upon its complete abstract
generality. But we can have no _à priori_ certainty that we are right in
believing that the observed entities in the concrete universe form a
particular instance of what falls under our general reasoning. To take
another example from arithmetic. It is a general abstract truth of pure
mathematics that any group of forty entities can be subdivided into two
groups of twenty entities. We are therefore justified in concluding that
a particular group of apples which we believe to contain forty members
can be subdivided into two groups of apples of which each contains
twenty members. But there always remains the possibility that we have
miscounted the big group; so that, when we come in practice to subdivide
it, we shall find that one of the two heaps has an apple too few or an
apple too many.
Accordingly, in criticising an argument based upon the application of
mathematics to particular matters of fact, there are always three
processes to be kept perfectly distinct in our minds. We must first scan
the purely mathematical reasoning to make sure that there are no mere
slips in it—no casual illogicalities due to mental failure. Any
mathematician knows from bitter experience that, in first elaborating a
train of reasoning, it is very easy to commit a slight error which yet
makes all the difference. But when a piece of mathematics has been
revised, and has been before the expert world for some time, the chance
of a casual error is almost negligible. The next process is to make
quite certain of all the abstract conditions which have been presupposed
to hold. This is the determination of the abstract premises from which
the mathematical reasoning proceeds. This is a matter of considerable
difficulty. In the past quite remarkable oversights have been made, and
have been accepted by generations of the greatest mathematicians. The
chief danger is that of oversight, namely, tacitly to introduce some
condition, which it is natural for us to presuppose, but which in fact
need not always be holding. There is another opposite oversight in this
connection which does not lead to error, but only to lack of
simplification. It is very easy to think that more postulated conditions
are required than is in fact the case. In other words, we may think that
some abstract postulate is necessary which is in fact capable of being
proved from the other postulates that we have already on hand. The only
effects of this excess of abstract postulates are to diminish our
aesthetic pleasure in the mathematical reasoning, and to give us more
trouble when we come to the third process of criticism.
This third process of criticism is that of verifying that our abstract
postulates hold for the particular case in question. It is in respect to
this process of verification for the particular case that all the
trouble arises. In some simple instances, such as the counting of forty
apples, we can with a little care arrive at practical certainty. But in
general, with more complex instances, complete certainty is
unattainable. Volumes, libraries of volumes, have been written on the
subject. It is the battle ground of rival philosophers. There are two
distinct questions involved. There are particular definite things
observed, and we have to make sure that the relations between these
things really do obey certain definite exact abstract conditions. There
is great room for error here. The exact observational methods of science
are all contrivances for limiting these erroneous conclusions as to
direct matters of fact. But another question arises. The things directly
observed are, almost always, only samples. We want to conclude that the
abstract conditions, which hold for the samples, also hold for all other
entities which, for some reason or other, appear to us to be of the same
sort. This process of reasoning from the sample to the whole species is
Induction. The theory of Induction is the despair of philosophy—and yet
all our activities are based upon it. Anyhow, in criticising a
mathematical conclusion as to a particular matter of fact, the real
difficulties consist in finding out the abstract assumptions involved,
and in estimating the evidence for their applicability to the particular
case in hand.
It often happens, therefore, that in criticising a learned book of
applied mathematics, or a memoir, one’s whole trouble is with the first
chapter, or even with the first page. For it is there, at the very
outset, where the author will probably be found to slip in his
assumptions. Farther, the trouble is not with what the author does say,
but with what he does not say. Also it is not with what he knows he has
assumed, but with what he has unconsciously assumed. We do not doubt the
author’s honesty. It is his perspicacity which we are criticising. Each
generation criticises the unconscious assumptions made by its parents.
It may assent to them, but it brings them out in the open.
The history of the development of language illustrates this point. It is
a history of the progressive analysis of ideas. Latin and Greek were
inflected languages. This means that they express an unanalyzed complex
of ideas by the mere modification of a word; whereas in English, for
example, we use prepositions and auxiliary verbs to drag into the open
the whole bundle of ideas involved. For certain forms of literary
art,—though not always—the compact absorption of auxiliary ideas into
the main word may be an advantage. But in a language such as English
there is the overwhelming gain in explicitness. This increased
explicitness is a more complete exhibition of the various abstractions
involved in the complex idea which is the meaning of the sentence.
By comparison with language, we can now see what is the function in
thought which is performed by pure mathematics. It is a resolute attempt
to go the whole way in the direction of complete analysis, so as to
separate the elements of mere matter of fact from the purely abstract
conditions which they exemplify.
The habit of such analysis enlightens every act of the functioning of
the human mind. It first (by isolating it) emphasizes the direct
aesthetic appreciation of the content of experience. This direct
appreciation means an apprehension of what this experience is in itself
in its own particular essence, including its immediate concrete values.
This is a question of direct experience, dependent upon sensitive
subtlety. There is then the abstraction of the particular entities
involved, viewed in themselves, and as apart from that particular
occasion of experience in which we are then apprehending them. Lastly
there is the further apprehension of the absolutely general conditions
satisfied by the particular relations of those entities as in that
experience. These conditions gain their generality from the fact that
they are expressible without reference to those particular relations or
to those particular relata which occur in that particular occasion of
experience. They are conditions which might hold for an indefinite
variety of other occasions, involving other entities and other relations
between them. Thus these conditions are perfectly general because they
refer to no particular occasion, and to no particular entities (such as
green, or blue, or trees) which enter into a variety of occasions, and
to no particular relationships between such entities.
There is, however, a limitation to be made to the generality of
mathematics; it is a qualification which applies equally to all general
statements. No statement, except one, can be made respecting any remote
occasion which enters into no relationship with the immediate occasion
so as to form a constitutive element of the essence of that immediate
occasion. By the ‘immediate occasion’ I mean that occasion which
involves as an ingredient the individual act of judgment in question.
The one excepted statement is,—If anything out of relationship, then
complete ignorance as to it. Here by ‘ignorance,’ I mean _ignorance_;
accordingly no advice can be given as to how to expect it, or to treat
it, in ‘practice’ or in any other way. Either we know something of the
remote occasion by the cognition which is itself an element of the
immediate occasion, or we know nothing. Accordingly the full universe,
disclosed for every variety of experience, is a universe in which every
detail enters into its proper relationship with the immediate occasion.
The generality of mathematics is the most complete generality consistent
with the community of occasions which constitutes our metaphysical
situation.
It is further to be noticed that the particular entities require these
general conditions for their ingression into any occasions; but the same
general conditions may be required by many types of particular entities.
This fact, that the general conditions transcend any one set of
particular entities, is the ground for the entry into mathematics, and
into mathematical logic, of the notion of the ‘variable.’ It is by the
employment of this notion that general conditions are investigated
without any specification of particular entities. This irrelevance of
the particular entities has not been generally understood: for example,
the shape-iness of shapes, _e.g._, circularity and sphericity and
cubicality as in actual experience, do not enter into the geometrical
reasoning.
The exercise of logical reason is always concerned with these absolutely
general conditions. In its broadest sense, the discovery of mathematics
is the discovery that the totality of these general abstract conditions,
which are concurrently applicable to the relationships among the
entities of any one concrete occasion, are themselves inter-connected in
the manner of a pattern with a key to it. This pattern of relationships
among general abstract conditions is imposed alike on external reality,
and on our abstract representations of it, by the general necessity that
every thing must be just its own individual self, with its own
individual way of differing from everything else. This is nothing else
than the necessity of abstract logic, which is the presupposition
involved in the very fact of interrelated existence as disclosed in each
immediate occasion of experience.
The key to the pattern means this fact:—that from a select set of those
general conditions, exemplified in any one and the same occasion, a
pattern involving an infinite variety of other such conditions, also
exemplified in the same occasion, can be developed by the pure exercise
of abstract logic. Any such select set is called the set of postulates,
or premises, from which the reasoning proceeds. The reasoning is nothing
else than the exhibition of the whole pattern of general conditions
involved in the pattern derived from the selected postulates.
The harmony of the logical reason, which divines the complete pattern as
involved in the postulates, is the most general aesthetic property
arising from the mere fact of concurrent existence in the unity of one
occasion. Wherever there is a unity of occasion there is thereby
established an aesthetic relationship between the general conditions
involved in that occasion. This aesthetic relationship is that which is
divined in the exercise of rationality. Whatever falls within that
relationship is thereby exemplified in that occasion; whatever falls
without that relationship is thereby excluded from exemplification in
that occasion. The complete pattern of general conditions, thus
exemplified, is determined by any one of many select sets of these
conditions. These key sets are sets of equivalent postulates. This
reasonable harmony of being, which is required for the unity of a
complex occasion, together with the completeness of the realisation (in
that occasion) of all that is involved in its logical harmony, is the
primary article of metaphysical doctrine. It means that for things to be
together involves that they are reasonably together. This means that
thought can penetrate into every occasion of fact, so that by
comprehending its key conditions, the whole complex of its pattern of
conditions lies open before it. It comes to this:—provided we know
something which is perfectly general about the elements in any occasion,
we can then know an indefinite number of other equally general concepts
which must also be exemplified in that same occasion. The logical
harmony involved in the unity of an occasion is both exclusive and
inclusive. The occasion must exclude the inharmonious, and it must
include the harmonious.
Pythagoras was the first man who had any grasp of the full sweep of this
general principle. He lived in the sixth century before Christ. Our
knowledge of him is fragmentary. But we know some points which establish
his greatness in the history of thought. He insisted on the importance
of the utmost generality in reasoning, and he divined the importance of
number as an aid to the construction of any representation of the
conditions involved in the order of nature. We know also that he studied
geometry, and discovered the general proof of the remarkable theorem
about right-angled triangles. The formation of the Pythagorean
Brotherhood, and the mysterious rumours as to its rites and its
influence, afford some evidence that Pythagoras divined, however dimly,
the possible importance of mathematics in the formation of science. On
the side of philosophy he started a discussion which has agitated
thinkers ever since. He asked, ‘What is the status of mathematical
entities, such as numbers for example, in the realm of things?’ The
number ‘two,’ for example, is in some sense exempt from the flux of time
and the necessity of position in space. Yet it is involved in the real
world. The same considerations apply to geometrical notions—to circular
shape, for example. Pythagoras is said to have taught that the
mathematical entities, such as numbers and shapes, were the ultimate
stuff out of which the real entities of our perceptual experience are
constructed. As thus boldly stated, the idea seems crude, and indeed
silly. But undoubtedly, he had hit upon a philosophical notion of
considerable importance; a notion which has a long history, and which
has moved the minds of men, and has even entered into Christian
theology. About a thousand years separate the Athanasian Creed from
Pythagoras, and about two thousand four hundred years separate
Pythagoras from Hegel. Yet for all these distances in time, the
importance of definite number in the constitution of the Divine Nature,
and the concept of the real world as exhibiting the evolution of an
idea, can both be traced back to the train of thought set going by
Pythagoras.
The importance of an individual thinker owes something to chance. For it
depends upon the fate of his ideas in the minds of his successors. In
this respect Pythagoras was fortunate. His philosophical speculations
reach us through the mind of Plato. The Platonic world of ideas is the
refined, revised form of the Pythagorean doctrine that number lies at
the base of the real world. Owing to the Greek mode of representing
numbers by patterns of dots, the notions of number and of geometrical
configuration are less separated than with us. Also Pythagoras, without
doubt, included the shape-iness of shape, which is an impure
mathematical entity. So to-day, when Einstein and his followers proclaim
that physical facts, such as gravitation, are to be construed as
exhibitions of local peculiarities of spatio-temporal properties, they
are following the pure Pythagorean tradition. In a sense, Plato and
Pythagoras stand nearer to modern physical science than does Aristotle.
The two former were mathematicians, whereas Aristotle was the son of a
doctor, though of course he was not thereby ignorant of mathematics. The
practical counsel to be derived from Pythagoras, is to measure, and thus
to express quality in terms of numerically determined quantity. But the
biological sciences, then and till our own time, have been
overwhelmingly classificatory. Accordingly, Aristotle by his Logic
throws the emphasis on classification. The popularity of Aristotelian
Logic retarded the advance of physical science throughout the Middle
Ages. If only the schoolmen had measured instead of classifying, how
much they might have learnt!
Classification is a halfway house between the immediate concreteness of
the individual thing and the complete abstraction of mathematical
notions. The species take account of the specific character, and the
genera of the generic character. But in the procedure of relating
mathematical notions to the facts of nature, by counting, by
measurement, and by geometrical relations, and by types of order, the
rational contemplation is lifted from the incomplete abstractions
involved in definite species and genera, to the complete, abstractions
of mathematics. Classification is necessary. But unless you can progress
from classification to mathematics, your reasoning will not take you
very far.
Between the epoch which stretches from Pythagoras to Plato and the epoch
comprised in the seventeenth century of the modern world nearly two
thousand years elapsed. In this long interval mathematics had made
immense strides. Geometry had gained the study of conic sections and
trigonometry; the method of exhaustion had almost anticipated the
integral calculus; and above all the Arabic arithmetical notation and
algebra had been contributed by Asiatic thought. But the progress was on
technical lines. Mathematics, as a formative element in the development
of philosophy, never, during this long period, recovered from its
deposition at the hands of Aristotle. Some of the old ideas derived from
the Pythagorean-Platonic epoch lingered on, and can be traced among the
Platonic influences which shaped the first period of evolution of
Christian theology. But philosophy received no fresh inspiration from
the steady advance of mathematical science. In the seventeenth century
the influence of Aristotle was at its lowest, and mathematics recovered
the importance of its earlier period. It was an age of great physicists
and great philosophers; and the physicists and philosophers were alike
mathematicians. The exception of John Locke should be made; although he
was greatly influenced by the Newtonian circle of the Royal Society. In
the age of Galileo, Descartes, Spinoza, Newton, and Leibniz, mathematics
was an influence of the first magnitude in the formation of philosophic
ideas. But the mathematics, which now emerged into prominence, was a
very different science from the mathematics of the earlier epoch. It had
gained in generality, and had started upon its almost incredible modern
career of piling subtlety of generalization upon subtlety of
generalization; and of finding, with each growth of complexity, some new
application, either to physical science, or to philosophic thought. The
Arabic notation had equipped the science with almost perfect technical
efficiency in the manipulation of numbers. This relief from a struggle
with arithmetical details (as instanced, for example, in the Egyptian
arithmetic of B. C. 1600) gave room for a development which had already
been faintly anticipated in later Greek mathematics. Algebra now came
upon the scene, and algebra is a generalisation of arithmetic. In the
same way as the notion of number abstracted from reference to any one
particular set of entities, so in algebra abstraction is made from the
notion of any particular numbers. Just as the number ‘5’ refers
impartially to any group of five entities, so in algebra the letters are
used to refer impartially to any number, with the proviso that each
letter is to refer to the same number throughout the same context of its
employment.
This usage was first employed in equations, which are methods of asking
complicated arithmetical questions. In this connection, the letters
representing numbers were termed ‘unknowns.’ But equations soon
suggested a new idea, that, namely, of a function of one or more general
symbols, these symbols being letters representing any numbers. In this
employment the algebraic letters are called the ‘arguments’ of the
function, or sometimes they are called the ‘variables.’ Then, for
instance, if an angle is represented by an algebraical letter, as
standing for its numerical measure in terms of a given unit,
Trigonometry is absorbed into this new algebra. Algebra thus develops
into the general science of analysis in which we consider the properties
of various functions of undetermined arguments. Finally the particular
functions, such as the trigonometrical functions, and the logarithmic
functions, and the algebraic functions, are generalised into the idea of
‘any function.’ Too large a generalisation leads to mere barrenness. It
is the large generalisation, limited by a happy particularity, which is
the fruitful conception. For instance the idea of any _continuous_
function, whereby the limitation of continuity is introduced, is the
fruitful idea which has led to most of the important applications. This
rise of algebraic analysis was concurrent with Descartes’ discovery of
analytical geometry, and then with the invention of the infinitesimal
calculus by Newton and Leibniz. Truly, Pythagoras, if he could have
foreseen the issue of the train of thought which he had set going would
have felt himself fully justified in his brotherhood with its excitement
of mysterious rites.
The point which I now want to make is that this dominance of the idea of
functionality in the abstract sphere of mathematics found itself
reflected in the order of nature under the guise of mathematically
expressed laws of nature. Apart from this progress of mathematics, the
seventeenth century developments of science would have been impossible.
Mathematics supplied the background of imaginative thought with which
the men of science approached the observation of nature. Galileo
produced formulae, Descartes produced formulae, Huyghens produced
formulae, Newton produced formulae.
As a particular example of the effect of the abstract development of
mathematics upon the science of those times, consider the notion of
periodicity. The general recurrences of things are very obvious in our
ordinary experience. Days recur, lunar phases recur, the seasons of the
year recur, rotating bodies recur to their old positions, beats of the
heart recur, breathing recurs. On every side, we are met by recurrence.
Apart from recurrence, knowledge would be impossible; for nothing could
be referred to our past experience. Also, apart from some regularity of
recurrence, measurement would be impossible. In our experience, as we
gain the idea of exactness, recurrence is fundamental.
In the sixteenth and seventeenth centuries, the theory of periodicity
took a fundamental place in science. Kepler divined a law connecting the
major axes of the planetary orbits with the periods in which the planets
respectively described their orbits: Galileo observed the periodic
vibrations of pendulums: Newton explained sound as being due to the
disturbance of air by the passage through it of periodic waves of
condensation and rarefaction: Huyghens explained light as being due to
the transverse waves of vibration of a subtle ether: Mersenne connected
the period of the vibration of a violin string with its density,
tension, and length. The birth of modern physics depended upon the
application of the abstract idea of periodicity to a variety of concrete
instances. But this would have been impossible, unless mathematicians
had already worked out in the abstract the various abstract ideas which
cluster round the notions of periodicity. The science of trigonometry
arose from that of the relations of the angles of a right-angled
triangle, to the ratios between the sides and hypotenuse of the
triangle. Then, under the influence of the newly discovered mathematical
science of the analysis of functions, it broadened out into the study of
the simple abstract periodic functions which these ratios exemplify.
Thus trigonometry became completely abstract; and in thus becoming
abstract, it became useful. It illuminated the underlying analogy
between sets of utterly diverse physical phenomena; and at the same time
it supplied the weapons by which any one such set could have its various
features analysed and related to each other.[2]
Nothing is more impressive than the fact that, as mathematics withdrew
increasingly into the upper regions of ever greater extremes of abstract
thought, it returned back to earth with a corresponding growth of
importance for the analysis of concrete fact. The history of the
seventeenth century science reads as though it were some vivid dream of
Plato or Pythagoras. In this characteristic the seventeenth century was
only the forerunner of its successors.
Footnote 2:
For a more detailed consideration of the nature and function of pure
mathematics _cf._ my _Introduction to Mathematics_, Home University
Library, Williams and Norgate, London.
The paradox is now fully established that the utmost abstractions are
the true weapons with which to control our thought of concrete fact. As
the result of the prominence of mathematicians in the seventeenth
century, the eighteenth century was mathematically minded, more
especially where French influence predominated. An exception must be
made of the English empiricism derived from Locke. Outside France,
Newton’s direct influence on philosophy is best seen in Kant, and not in
Hume.
In the nineteenth century, the general influence of mathematics waned.
The romantic movement in literature, and the idealistic movement in
philosophy were not the products of mathematical minds. Also, even in
science, the growth of geology, of zoology, and of the biological
sciences generally, was in each case entirely disconnected from any
reference to mathematics. The chief scientific excitement of the century
was the Darwinian theory of evolution. Accordingly, mathematicians were
in the background, so far as the general thought of that age was
concerned. But this does not mean that mathematics was being neglected,
or even that it was uninfluential. During the nineteenth century pure
mathematics made almost as much progress as during all the preceding
centuries from Pythagoras onwards. Of course progress was easier,
because the technique had been perfected. But allowing for that, the
change in mathematics between the years 1800 and 1900 is very
remarkable. If we add in the previous hundred years, and take the two
centuries preceding the present time, one is almost tempted to date the
foundation of mathematics somewhere in the last quarter of the
seventeenth century. The period of the discovery of the elements
stretches from Pythagoras to Descartes, Newton, and Leibniz, and the
developed science has been created during the last two hundred and fifty
years. This is not a boast as to the superior genius of the modern
world; for it is harder to discover the elements than to develop the
science.
Throughout the nineteenth century, the influence of the science was its
influence on dynamics and physics, and thence derivatively on
engineering and chemistry. It is difficult to overrate its indirect
influence on human life through the medium of these sciences. But there
was no direct influence of mathematics upon the general thought of the
age.
In reviewing this rapid sketch of the influence of mathematics
throughout European history, we see that it had two great periods of
direct influence upon general thought, both periods lasting for about
two hundred years. The first period was that stretching from Pythagoras
to Plato, when the possibility of the science, and its general
character, first dawned upon the Grecian thinkers. The second period
comprised the seventeenth and eighteenth centuries of our modern epoch.
Both periods had certain common characteristics. In the earlier, as in
the later period, the general categories of thought in many spheres of
human interest, were in a state of disintegration. In the age of
Pythagoras, the unconscious Paganism, with its traditional clothing of
beautiful ritual and of magical rites, was passing into a new phase
under two influences. There were waves of religious enthusiasm, seeking
direct enlightenment into the secret depths of being; and at the
opposite pole, there was the awakening of critical analytical thought,
probing with cool dispassionateness into ultimate meanings. In both
influences, so diverse in their outcome, there was one common element—an
awakened curiosity, and a movement towards the reconstruction of
traditional ways. The pagan mysteries may be compared to the Puritan
reaction and to the Catholic reaction; critical scientific interest was
alike in both epochs, though with minor differences of substantial
importance.
In each age, the earlier stages were placed in periods of rising
prosperity, and of new opportunities. In this respect, they differed
from the period of gradual declension in the second and third centuries
when Christianity was advancing to the conquest of the Roman world. It
is only in a period, fortunate both in its opportunities for
disengagement from the immediate pressure of circumstances, and in its
eager curiosity, that the Age-Spirit can undertake any direct revision
of those final abstractions which lie hidden in the more concrete
concepts from which the serious thought of an age takes its start. In
the rare periods when this task can be undertaken, mathematics becomes
relevant to philosophy. For mathematics is the science of the most
complete abstractions to which the human mind can attain.
The parallel between the two epochs must not be pressed too far. The
modern world is larger and more complex than the ancient civilization
round the shores of the Mediterranean, or even than that of the Europe
which sent Columbus and the Pilgrim Fathers across the ocean. We cannot
now explain our age by some simple formula which becomes dominant and
will then be laid to rest for a thousand years. Thus the temporary
submergence of the mathematical mentality from the time of Rousseau
onwards appears already to be at an end. We are entering upon an age of
reconstruction, in religion, in science, and in political thought. Such
ages, if they are to avoid mere ignorant oscillation between extremes,
must seek truth in its ultimate depths. There can be no vision of this
depth of truth apart from a philosophy which takes full account of those
ultimate abstractions, whose interconnections it is the business of
mathematics to explore.
In order to explain exactly how mathematics is gaining in general
importance at the present time, let us start from a particular
scientific perplexity and consider the notions to which we are naturally
led by some attempt to unravel its difficulties. At present physics is
troubled by the quantum theory. I need not now explain[3] what this
theory is, to those who are not already familiar with it. But the point
is that one of the most hopeful lines of explanation is to assume that
an electron does not continuously traverse its path in space. The
alternative notion as to its mode of existence is that it appears at a
series of discrete positions in space which it occupies for successive
durations of time. It is as though an automobile moving at the average
rate of thirty miles an hour along a road, did not traverse the road
continuously; but appeared successively at the successive milestones,
remaining for two minutes at each milestone.
Footnote 3:
_Cf._ Chapter VIII.
In the first place there is required the purely technical use of
mathematics to determine whether this conception does in fact explain
the many perplexing characteristics of the quantum theory. If the notion
survives this test, undoubtedly physics will adopt it. So far the
question is purely one for mathematics and physical science to settle
between them, on the basis of mathematical calculations and physical
observations.
But now a problem is handed over to the philosophers. This discontinuous
existence in space, thus assigned to electrons, is very unlike the
continuous existence of material entities which we habitually assume as
obvious. The electron seems to be borrowing the character which some
people have assigned to the Mahatmas of Tibet. These electrons, with the
correlative protons, are now conceived as being the fundamental entities
out of which the material bodies of ordinary experience are composed.
Accordingly, if this explanation is allowed, we have to revise all our
notions of the ultimate character of material existence. For when we
penetrate to these final entities, this startling discontinuity of
spatial existence discloses itself.
There is no difficulty in explaining the paradox, if we consent to apply
to the apparently steady undifferentiated endurance of matter the same
principles as those now accepted for sound and light. A steadily
sounding note is explained as the outcome of vibrations in the air: a
steady colour is explained as the outcome of vibrations in ether. If we
explain the steady endurance of matter on the same principle, we shall
conceive each primordial element as a vibratory ebb and flow of an
underlying energy, or activity. Suppose we keep to the physical idea of
energy: then each primordial element will be an organized system of
vibratory streaming of energy. Accordingly there will be a definite
period associated with each element; and within that period the
stream-system will sway from one stationary maximum to another
stationary maximum,—or, taking a metaphor from the ocean tides, the
system will sway from one high tide to another high tide. This system,
forming the primordial element, is nothing at any instant. It requires
its whole period in which to manifest itself. In an analogous way, a
note of music is nothing at an instant, but it also requires its whole
period in which to manifest itself.
Accordingly, in asking where the primordial element is, we must settle
on its average position at the centre of each period. If we divide time
into smaller elements, the vibratory system as one electronic entity has
no existence. The path in space of such a vibratory entity—where the
entity is _constituted by_ the vibrations—must be represented by a
series of detached positions in space, analogously to the automobile
which is found at successive milestones and at nowhere between.
We first must ask whether there is any evidence to associate the quantum
theory with vibration. This question is immediately answered in the
affirmative. The whole theory centres round the radiant energy from an
atom, and is intimately associated with the periods of the radiant
wave-systems. It seems, therefore, that the hypothesis of essentially
vibratory existence is the most hopeful way of explaining the paradox of
the discontinuous orbit.
In the second place, a new problem is now placed before philosophers and
physicists, if we entertain the hypothesis that the ultimate elements of
matter are in their essence vibratory. By this I mean that apart from
being a periodic system, such an element would have no existence. With
this hypothesis we have to ask, what are the ingredients which form the
vibratory organism. We have already got rid of the matter with its
appearance of undifferentiated endurance. Apart from some metaphysical
compulsion, there is no reason to provide another more subtle stuff to
take the place of the matter which has just been explained away. The
field is now open for the introduction of some new doctrine of organism
which may take the place of the materialism with which, since the
seventeenth century, science has saddled philosophy. It must be
remembered that the physicists’ energy is obviously an abstraction. The
concrete fact, which is the organism, must be a complete expression of
the character of a real occurrence. Such a displacement of scientific
materialism, if it ever takes place, cannot fail to have important
consequences in every field of thought.
Finally, our last reflection must be, that we have in the end come back
to a version of the doctrine of old Pythagoras, from whom mathematics,
and mathematical physics, took their rise. He discovered the importance
of dealing with abstractions; and in particular directed attention to
number as characterizing the periodicities of notes of music. The
importance of the abstract idea of periodicity was thus present at the
very beginning both of mathematics and of European philosophy.
In the seventeenth century, the birth of modern science required a new
mathematics, more fully equipped for the purpose of analysing the
characteristics of vibratory existence. And now in the twentieth century
we find physicists largely engaged in analysing the periodicities of
atoms. Truly, Pythagoras in founding European philosophy and European
mathematics, endowed them with the luckiest of lucky guesses—or, was it
a flash of divine genius, penetrating to the inmost nature of things?