The influence of mathematics on philosophy and vice versâ can be
inferred from the historical progress of both studies, though it has
not been possible till about within the last fifteen years to give
a logical explanation for the relations between them. As long as it
was believed, according to the Kantian view, that the science of
mathematics was based on intuitions of time and space, the alliance
between philosophy and mathematics could not be proved to be closer
than that between philosophy and experimental science, although the
historical fact remained that philosophy and mathematics exercised a
mutual stimulus, and developed at the same periods of history.
But mathematics, as now defined, is independent of intuitions of
space and time, and also of axioms and hypotheses.[1] Mathematics,
as now understood, is based, like formal logic, on the prerequisites
of thought, not on the notions of space and time. Here there is no
definition of number or space, but the conception of number and
space,[2] which is more complicated, can be derived from them. All
other complicated mind processes can, in the same way, be reduced to
the simple elements of the prerequisites of thought.
Such a science might exist out of conditions of time and space as we
know them. It is a science of relations rather than of mere number.
Founded, then, on the laws of symbolic logic, it is a valuable aid
and illustration to philosophy; philosophy, on the other hand, can
imagine lines for the exercise of the constructive power involved in
mathematics. It is the object of this paper to show that the close
though apparently accidental union of philosophy and mathematics
throughout the history of thought can now be explained, and that the
problems with which pure mathematics is now concerned are those which
lie at the core of philosophic thought and speculation. (Symbolic
logic has developed to meet the new demands made upon it. It does
not now reduce itself to the syllogism, as Aristotle thought it did;
the prerequisites of thought are shown to be manifold instead of
single.[3])
The use of the word philosophic in this connection suggests a
necessity for further definition. Philosophy is held to include at
least two great branches--Metaphysics and Ethics. The influence of
mathematics is most evident on the metaphysical side of philosophy;
in fact, the grouping of mathematics and metaphysics as allied
sciences tends to bring out the essential distinction between
metaphysics and ethics, and--though not by any means to imply a break
in their real relation--to show where this has been misunderstood.
No philosophy has been equally strong on both sides; they represent
different forms of activity of the human mind; but it is still true,
and from the conditions always must be, that an ethical system grows
out of metaphysics as practice follows precept and conduct implies
belief. The new definition of mathematics does not touch these
consequences; it merely marks the limits within which philosophy on
the metaphysical side can submit to, or rest upon, the conclusions of
mathematics.
As to the historical relation between metaphysics and mathematics,
the subject is so vast that we shall only attempt a very rapid
generalisation of its results on the growth of the conception of
the Finite and the Infinite. (Of course, there are many other sides
of the relation which might be studied.) The general result of the
inquiry has been, as far as we can judge, that metaphysics has
exercised an inspiring force on mathematics, and mathematics has
defined and strengthened the conceptions of metaphysics at every
critical stage in the history of philosophy. But where metaphysics
has been treated as the proof of science, where it has been laid
down as the foundation for exact knowledge, the results have not
corresponded with the truth of experience, and the quality of thought
has become degenerate. Progress depends on the right perception of
the relations between the sciences and parts of philosophy.
Such progress is especially evident in the early Greek and in the
modern periods, while the large period from the Christian era to the
Renaissance gives examples of the unfortunate reversal of the parts
of metaphysics and science and consequent confusion of thought.[4]
The problems of the metaphysician are no doubt in a sense always the
same; but this is equally true of the problems of any other science.
The methods by which the problems are attacked and the adequacy
of the solutions they receive vary, from age to age, in close
correspondence with the general development of science. Every great
metaphysical conception has exercised its influence on the general
history of science, and in return every important movement in science
has affected the development of metaphysics. The metaphysician could
not if he would, and would not if he could, escape the duty of
estimating the bearing of the great scientific theories of his time
upon our ultimate conceptions of the nature of the world as a whole.
Every fundamental advance in science thus calls for a restatement and
reconsideration of the old metaphysical problems in the light of the
new discovery.
* * * * *
During the Greek period mathematics was the only branch of science
which was at all developed, and its development coincided with
the age of the philosophers. Thus when Plato spoke of science he
always meant mathematics. And even later, when the physical sciences
had begun to develop, Aristotle put mathematical ideas into close
connection with metaphysical ones when he stated that they occupied
the middle term between the ideal and the sensible. Both Plato and
Aristotle referred to and depended upon mathematical proofs and
illustrations of philosophical questions. During this Greek period
the conception of Infinity took shape. The pre-Platonic notion,
reproduced again later in the decline of Socratic theory by the
Stoics, was that the Infinite was the aggregate of the Finite; the
Platonic and Aristotelian theory, that, namely, of the most vigorous
moment of Greek thought, was that the Infinite was _more_ than the
aggregate of the Finite; that it had a self-determined existence from
which the Finite had been derived. Existence, as known to man, was
treated as a compromise between the Finite and the Infinite.
Neo-Platonism altogether separated the Infinite from the Finite. In
the Alexandrine metaphysics, which represented a decadent stage of
philosophy and its deviation from the sciences, the conception of
the Infinite became less clear and logical; it diverged from the
view which had been affected by mathematical thought, and tended
to assimilate to itself the ideas of perfection and universality,
which, philosophically speaking, are conceptions distinct from that
of Infinity--universality referring to a common principle of unity,
and perfection involving the moral ideal. Real progress was deferred
by the too rapid coherence of ideas only partially analysed and
understood. Thinkers passed quickly from the exclusive contemplation
of subject to that of object and back again,[5] each new period
negativing all previous experience, till the result was the exclusion
of an imperfectly analysed Relative and Finite from an insufficiently
apprehended Absolute and Infinite.
After the Christian era Greek philosophy drifted off into
scholasticism and lost touch of reality, the grammar of Aristotelian
logic replacing the vital connection of ideas. St. Anselm, it is
true, attempted to find a rational proof of the existence of God, and
identified Him with the Infinite of Greek thought; but St. Thomas
Aquinas led away the argument to a discussion as to how far form and
matter, separately considered, shared in the quality of Infinity. (He
thought form did, but not matter.) An overpowering sense of mystery,
joined to a premature desire for definition without scientific
analysis, sapped the vigour of mediæval thought.
Throughout the middle ages, then, we see the conditions of the Greek
period reversed: philosophy during the second period is not, as
in the first, engaged in giving a stimulus to the efforts of pure
reason; rather the intuitions of philosophy are treated as axiomatic,
and a false superstructure of knowledge, alien to experience and
reality, is erected upon these foundations. Philosophy, in fact, is
used as a general basis for science. The parts of philosophy and
mathematics, correctly though imperfectly seen by the Greeks, are in
the second period exchanged, and the result is confusion of idea.
The notion of the Infinite, as in the Alexandrine metaphysics, is
held to include perfection and universality, and does not exist as a
conception apart from these.
After the Renaissance, the scholastic philosophy falling into disuse,
the attempt to find an explanation of the Cosmos, a synthesis of the
universe, was abandoned, and replaced by the Cartesian idea--the
inference of existence from thought, and the limitation of the sphere
of inquiry to that which could be known by the ego. New scientific
and mathematical discoveries kept pace with this new analysis and
development of thought,[6] and the surer ground in philosophy was
definitely allied with the work of the mathematical mind. The
philosophical thesis developed from “The Infinite is the negation
of the Finite,” to “The Infinite presupposes the Finite and does
not exclude it.” The problem of the Finite and the Infinite became
the great idea of the age, and there was a reversion to the Greek
notion of existence as a compromise between the two, and almost the
hint of a coming explanation of them. In the decline of Cartesian
philosophy, when it drifted off into Pantheism, there was only a
vague conception of the Infinite, and we trace a tendency to identify
the notion of Infinity with that of the Cosmos. In mediæval thought
the idea of the Infinite had become confused with that of the Perfect
and Universal; in the modern period the effort to give a concrete
expression to the notions of Infinity, Perfection, and Universality
diverted the ideas from their relation to the Creator and applied
them to the Creation.
Kant, who gave a new impulse to some parts of the Cartesian idea,
neglected both mathematical proofs and the search for a metaphysical
Absolute. In avoiding the subject he helped to perpetuate the
vague descriptions of the Finite and the Infinite, uncorrected by
mathematical thought, which had been the currency of the philosophy
of his age, and which corrupted the philosophy of the succeeding
century. The nineteenth century produced nothing more than guesses
at truth, which were, perhaps, not very far wrong, and which the
present century is engaged in correcting and substantiating. The
same vagueness afflicted both mathematics (theory of functions)
and philosophy. Fichte, Schelling, and Hegel, particularly Hegel,
identify the metaphysical Absolute with reality, infinity, and the
universal. The ideas of continuity and infinity are not separated by
them from those of perfection and universality, nor from one another,
and their nature is not understood.
Leaving aside the French neo-critical School (Renouvier) and the
English School (Spencer)--the first of whom deny the Infinite,
thus acting in opposition to mathematical reasoning, while the
second perpetuate Kant’s error of considering the Infinite, though
thinkable, as unknowable (Dr. Caird has pointed out that this
position is illogical)--we arrive at a moment in history which is
more fruitful in result on the mathematical side, and will, no doubt,
have an effect on metaphysics. For, owing to recent discoveries
in Germany and England, mathematics is now in a position to give
greater support than before to the intuitions of philosophy.
Hitherto, philosophers have been reluctant to allow full value to
the mathematical conceptions of Infinity, and with some justice, as
the notion had not been sufficiently analysed. Philosophers, who
never attempted the analysis, have been inclined to accept certain
contradictions in their conception as inherent in the nature of
Infinity. Within the last twenty-five years Cantor and Dedekind have
cleared up the notion of continuity, and Russell has given greater
precision to the idea, and has applied this reasoning to philosophy.
Present-day metaphysicians seem to be divided into two groups; on
the one side, those who consider in philosophy the value of a theory
of being, and, on the other, those who chiefly consider the value
of a theory of knowledge, _i.e._ the Epistemologists. The first
group, devoting themselves to psychology, evolution, and history,
have no _necessary_ belief in the Infinite. The Epistemologists,
whose work is founded on Kant, discuss the theory of knowledge and
enumerate the conditions of knowledge. Their argument may not touch,
but does not exclude, the notion of the Infinite. The position of
the Epistemologist has been made infinitely more secure by recent
mathematical work. That of the psychologist remains almost untouched.
It is necessary now to examine more closely the mathematical results
to which reference has been made.
In general terms it may be said that mathematics has, as a study,
led immediately from the nature of the subject to the perception
of the Infinite, and to a knowledge of the connection between the
Infinite and the Finite. The simplest form in which the idea can be
put is stated by St. Augustine, who said that numbers considered
individually were finite, but considered as an aggregate were
infinite.[7] Before St. Augustine, and after him down the long
stream of philosophic thought, the theologian and the philosopher
have turned to mathematics for illustrations of the infinitely great
and infinitely little, as developed from the concrete processes of
arithmetic and geometry. The recurring decimal in arithmetic, the
properties of the circle and ellipse in geometry, of the cone in
conic sections, and of the surd in algebra, all touch the problem of
number and space on the side of Infinity.
In higher mathematics it is possible to start from the idea of the
Finite and reach the conception of the Infinite; or to reverse the
process, and from the Infinite to deduce the Finite. Thus in the
familiar puzzle of the subdivision of the parts of a straight line
by halving the remainder, there will be a crowding and a coalescing
of the points of division towards one end of the line, the points of
division getting infinitely nearer, but the steps will never meet.
Here in the centre of a straight line--a limited straight line--we
are confronted with the problem of Infinity.
Again, from a series of finite numbers we can gain the notion of an
infinite series. Take two series which have a correspondence with one
another. If for every element of the one we can choose an element of
the other, and of the other there is an element for the one, when at
any point we cut off its progress to infinity, this happens:--
One series, if summed up, will give a larger numerical result than
the other, and therefore can be said to be greater than the second.
Let us call the first series A, and the second B. Let us now imagine
the two series, though starting at a definite point, are never cut
at the further end. Then to all infinity series B is without certain
numbers which series A possesses, _and as an infinite series_ is
smaller than series A. But, on the other hand, when neither series
is cut, series B retains its correspondence with series A. Thus we
attain a definition of an infinite series. It is such that the part,
while being less than the whole, has yet a complete correspondence
with the whole. The whole is greater than the part, but take away
the part from the whole and that which remains corresponds to it _in
infinity_, because the test of summing the series (which would give
a contrary result) involves limitation, and thus cannot be applied.
Subtraction can take place in Infinity without loss.
By reversing this process, and by starting from the theory of the
Infinite, we may gain some idea of the discovery of the Finite. So
Dedekind and Russell define finite numbers not only in the usual way
as those which can be reached by mathematical induction, starting
from 0 and increasing by 1 at each step, but also as those of classes
which are _not_ similar to the parts of themselves obtained by taking
away single terms. That is the reversal of the process applied just
now. Dedekind also has deduced the Finite from the Infinite by a
novel process. He predicates a world of thought which we each and all
possess, filled with thoughts and things, to each thing corresponding
a thought. There are thus two “trans-finite” series in the minds of
each and all of us; we cannot say when the series of thoughts and
things will end; but they have number, though it is infinite number.
(Number exists wherever there is a correspondence, one to one,
between two aggregates.) But in this _Gedankenwelt_, says Dedekind,
there is one thing to which there is no corresponding thought: that
is the ego. Each man is part of his own world of thought, but
there is no thought of himself in his mind corresponding exactly to
himself, as a thought in his mind corresponds to another object.[8]
Two important results follow from Dedekind’s theory: first, the
existence of a finite number one, the number of the ego, as deduced
from the _Gedankenwelt_ of two infinite systems; second, by putting
together all the _Gedankenwelts_ there are or may be, we get the
notion of series of series, which seems to transcend Infinity, and
it gives us the conditions which are possibly gathered up in the
Absolute. Now the argument from the Finite to the Infinite and the
converse process may both be employed in mathematics (or both may
be neglected, as in the elementary methods of calculation used in
arithmetic). A discussion has taken place in the _Hibbert Journal_ on
the relative value of the two methods. Keyser, in an article called
the Axiom of Infinity, argued that one method, that of Dedekind,
should be exclusively developed. Russell answered him, stating that
it was not necessary to hold exclusively to either. If the Finite
and the Infinite can in turn be deduced from one another, neither
conception can be truly called an axiom. The real axiom is existence,
which includes both, and which is defined by mathematicians as that
which is _not self-contradictory_.
Now the problem of Infinity includes also that of continuity; in
other words, the problem of number includes that of _cardinal_ and
_ordinal_ number. It is time to get to the mathematical definition of
number, which we have found as a conception can be attached both to
the Finite and to the Infinite. What is number in mathematics?
Take any collection of things--we call that an _aggregate_. If an
aggregate corresponds one to one with another aggregate, they are
both said to have a number, and the same number. Subtract from the
idea of an aggregate the idea of quality or kind, and order or
arrangement, what is left is its _cardinal number_. If you subtract
quality and not order, the result is an _ordinal number_. This
reasoning applies both to finite and infinite aggregates; in fact,
the Infinite may be said to possess most of the properties which
we attach to the Finite. Two infinite aggregates, for example, can
have an ordinal correspondence, and infinite aggregates submit, like
finite ones, to arithmetical processes.
The mathematician analyses still more closely the relation between
the Finite and the Infinite, as follows:--
He starts from the aggregate, which he analyses into the Finite
and the Infinite, and the latter he analyses into the Transfinite
and the Absolute. Of these two elements, one only has till just
lately been the subject of mathematical treatment--the one called
the Transfinite. It is the transfinite subdivision of the Infinite
to which the idea of number is applicable, and which is, therefore,
in a sense inseparable from the Finite. Infinite numbers or series
ought then to be more correctly described as _Transfinite_. But the
processes of mathematics do not end here; they reach up to the idea
of the _Absolute Infinite_, the conception of which has been attained
in recent years by mathematical work. The results of this work may
now be briefly summarised.
I. The Absolute appears to have the same relation to the Transfinite
as the Transfinite to the Finite. If the Finite deals with numbers,
and the Transfinite with series of numbers, the Absolute deals
with series of series. Thus there are at least two examples of the
Infinite within our grasp which lead up to the idea of the Absolute.
One is the class of all classes of propositions; the other is the
series of all worlds of thought, in Dedekind’s sense.
II. The Finite, Transfinite, and Absolute can be further defined in
this way. There is no greatest finite number, but there is a least
transfinite number, which has been called Aleph 0, and which can
be proved to be greater than any possible finite number, however
large, because if there were a last number it must be smaller than
the sum of the whole series. There are unending series of Alephs or
infinite numbers, which are as distinct from one another in idea
as 1 is from 0, and which can no more be derived from one another
by a mathematical process than 1 can be derived from 0, but can
be reached in the same way by induction. Beyond the Transfinite we
cannot discover in the Absolute the idea of least or of greatest.
III. The relation of cardinal and ordinal number also throws some
light on the Finite, Transfinite, and Absolute. In the Finite,
cardinals and ordinals are parallel to one another; in the
Transfinite they strikingly diverge; in the Absolute we cannot trace
any connection between cardinals and ordinals, _i.e._, it is possible
to have an ordinal series to which there can be no corresponding
cardinal number or type.[9]
IV. If arithmetical processes are applied to the Finite, Transfinite,
or Absolute, we get interesting results. We know the effect of
addition, multiplication, and raising to a power, on the Finite.
The first two processes have been applied to the Alephs; the last
has been formulated, but the mathematical results have not yet been
brought to a satisfactory conclusion. Broadly speaking, we may say
that the raising of an Aleph to a power may make it transcend the
Finite and the Transfinite and melt into the Absolute. Thus all
mathematical processes which find their goal in the Absolute would
find their annihilation there. No finite mathematical conception
would be applicable to it.
Now the conception of this Absolute Infinite, of which the aggregate
of all ordinal numbers is perhaps a symbol,[10] has been subjected to
criticism. Some mathematicians[11] think that it exists, but has no
number. It is discovered by a logical process, but defies analysis
and the application to it of the notion of number. All mathematical
conceptions find in it their aim and conclusion. The importance of
this theory, its practical importance, lies in the very much simpler
mathematical formulæ that can be produced now that the logical
process is shown to extend from the Finite to the Absolute Infinite
(in the same way that the labour of summing a series arithmetically
by statement and addition is shortened by the application of
algebraical principles which depend on larger knowledge). Its
philosophical importance is great: the Absolute is here, as
elsewhere, the goal of human thought, and is the mathematician’s name
for the highest power discoverable by human reason.
It would be very interesting to discuss the probable attitude of a
Pascal or a Hegel to these mathematical conceptions, if they had been
aware of them. Take Pascal’s puzzle of the Finite and the Infinite.
He thought that if the Finite could be subtracted from the Infinite,
the Infinite would thereby lose some of its quality of infinity. How
differently would it have appeared to him had he realised that an
aggregate infinite cardinal can have subtracted from it either finite
or transfinite terms: if transfinite terms, many different answers
result, giving different degrees of transfinity: if only finite terms
are taken away, the Infinite remains in its entirety.
How, again, would Hegel have rejoiced in a definition of thought and
existence which would bridge over the logical gulf in his system!
Hegel asserted that thought and existence were one. He is objected
to by many philosophers, who ask where is the _tertium quid_ which
makes it possible to reach from one to the other, or predicate their
essential unity? But the mathematician defines existence as something
which is not self-contradictory. Thought, then, to him is a form
of existence, for thought is not self-contradictory; but the two,
thought and existence, are not necessarily conterminous.[12] Hence,
to say that non-contradiction is a fundamental condition of true
thinking is as much as to say that it is a fundamental characteristic
of real existence, and he identifies thought with reality.
Dr. Caird remarks that the secular conscience conceives of the
Infinite as opposed to the Finite; the religious conscience treats
the Infinite as real, presupposed by the illusory Finite. Where does
the truth lie? Mathematics does not admit the necessity of adopting
either view at the expense of the other.
Metaphysics standing alone produces results that may be disproved,
but cannot be proved. Mathematics standing alone produces results
that are susceptible of proof. Both are based on logic, and rest on
the prerequisites of thought. Together they are a field for the best
powers of human reason: metaphysics supplies insight, intuition,
imagination; mathematics offers the indubitable proof and translates
the ideal into the actual.
But the element in philosophical thought which, employing the
psychological method, tends to the discussion of a theory of being
rather than that of knowledge, and thus to the realisation of an
ethical system rather than to metaphysical discovery, is averse from
accepting these conclusions. It remains, therefore, for us to examine
the criticism offered by the psychological school on what they call
the mathematising of philosophy; and it will be found that the attack
deals both with the ground of the alliance and its results.
A typical exponent of this school is Moisant, who, in the _Revue
Philosophique_ for January 1905, attacked what he considered to be
the characteristic of modern philosophy and also its vice. It will
be observed that at the outset he reverses the _rôles_ of philosophy
and mathematics as we have apprehended them. Philosophy, he says,
should expect to be inspired by mathematics, but should avoid its
method. Next, he connects the modern movement with the theories of
Leibniz, who aimed at substituting general formulæ for elementary
forms of reason and calculation. These short cuts, which seem to the
mathematician to liberate the mind from a burden which prevents it
from employing its full activity, seem to the psychologist to tend to
a mechanical method, in which the thinker is only aware of premises
and results, and in which the mathematical concept tends to replace
the real idea. Then he attacks the new definition of mathematics as
the science of relations, asserting that it still contains notions of
space.[13]
Finally, he comes to the real question at issue, and enters into
the comparison of a metaphysical and a mathematical problem. He
takes as his subject the argument from the known to the unknown.
Descartes had said that argument should lead from the known to
the unknown, simple to complex, and had defined the first as that
which could be known without the help of the second. This logical
order of reasoning has been attributed to mathematics, but has been
considered to be inapplicable to philosophy. Mathematics, in its
recent development, by the argument from the Finite to the Infinite
and back again, starts from two propositions, neither of which can
be said to be axiomatic, because each in turn can be proved from
the other, but in the course of argument from either mathematics
makes use of the logical process. The real axiom, as has been shown,
is that of _existence_ or _being_. A metaphysical argument has the
same root--that of existence--but a metaphysical problem deals with
paradoxes, with questions which are sometimes defined as having two
answers, each equally correct, and sometimes as yielding no answer
at all. The method of thesis, antithesis, and synthesis is in the
Hegelian logic applied to their solution.
A mathematical and a metaphysical problem are not, then, problems of
the same kind to be solved by the same method; nor is the conception
of the mathematical Absolute reached in the same way as that of
the metaphysical Absolute. We are even unable to say how far they
correspond except in respect of their absoluteness.[14] But the
contention of the mathematician to-day and of the epistemologist
school of philosophy is not the identity of methods and results in
the two sciences. It is the axiom of existence on which they both
depend: the law of thought by which all methods are developed, and,
above all, the _correlative value of each science to the other_,
which allows us, in developing our knowledge from the standpoint
of the two sciences, to recognise something of the greatness of the
Absolute principle to which they both reach up, and in which their
being consists.