axioms is Euclid's; other equally good sets of axioms lead to other
results. Whether Euclid's axioms are true, is a question as to which
the pure mathematician is indifferent; and, what is more, it is a
question which it is theoretically impossible to answer with certainty
in the affirmative. It might possibly be shown, by very careful
measurements, that Euclid's axioms are false; but no measurements
could ever assure us (owing to the errors of observation) that they
are exactly true. Thus the geometer leaves to the man of science to
decide, as best he may, what axioms are most nearly true in the actual
world. The geometer takes any set of axioms that seem interesting, and
deduces their consequences. What defines Geometry, in this sense, is
that the axioms must give rise to a series of more than one dimension.
And it is thus that Geometry becomes a department in the study of
order.
In Geometry, as in other parts of mathematics, Peano and his disciples
have done work of the very greatest merit as regards principles.
Formerly, it was held by philosophers and mathematicians alike that
the proofs in Geometry depended on the figure; nowadays, this is known
to be false. In the best books there are no figures at all. The
reasoning proceeds by the strict rules of formal logic from a set of
axioms laid down to begin with. If a figure is used, all sorts of
things seem obviously to follow, which no formal reasoning can prove
from the explicit axioms, and which, as a matter of fact, are only
accepted because they are obvious. By banishing the figure, it becomes
possible to discover _all_ the axioms that are needed; and in this way
all sorts of possibilities, which would have otherwise remained
undetected, are brought to light.
One great advance, from the point of view of correctness, has been
made by introducing points as they are required, and not starting, as
was formerly done, by assuming the whole of space. This method is due
partly to Peano, partly to another Italian named Fano.
To those
unaccustomed to it, it has an air of somewhat wilful pedantry. In this
way, we begin with the following axioms: (1) There is a class of
entities called _points_. (2) There is at least one point. (3) If _a_
be a point, there is at least one other point besides _a_. Then we
bring in the straight line joining two points, and begin again with
(4), namely, on the straight line joining _a_ and _b_, there is at
least one other point besides _a_ and _b_. (5) There is at least one
point not on the line _ab_. And so we go on, till we have the means of
obtaining as many points as we require. But the word _space_, as Peano
humorously remarks, is one for which Geometry has no use at all.
The rigid methods employed by modern geometers have deposed Euclid
from his pinnacle of correctness. It was thought, until recent times,
that, as Sir Henry Savile remarked in 1621, there were only two
blemishes in Euclid, the theory of parallels and the theory of
proportion. It is now known that these are almost the only points in
which Euclid is free from blemish. Countless errors are involved in
his first eight propositions. That is to say, not only is it doubtful
whether his axioms are true, which is a comparatively trivial matter,
but it is certain that his propositions do not follow from the axioms
which he enunciates. A vastly greater number of axioms, which Euclid
unconsciously employs, are required for the proof of his propositions.
Even in the first proposition of all, where he constructs an
equilateral triangle on a given base, he uses two circles which are
assumed to intersect. But no explicit axiom assures us that they do
so, and in some kinds of spaces they do not always intersect. It is
quite doubtful whether our space belongs to one of these kinds or not.
Thus Euclid fails entirely to prove his point in the very first
proposition. As he is certainly not an easy author, and is terribly
long-winded, he has no longer any but an historical interest. Under
these circumstances, it is nothing less than a scandal that he should
still be taught to boys in England.[17] A book should have either
intelligibility or correctness; to combine the two is impossible, but
to lack both is to be unworthy of such a place as Euclid has occupied
in education.
The most remarkable result of modern methods in mathematics is the
importance of symbolic logic and of rigid formalism.
Mathematicians,
under the influence of Weierstrass, have shown in modern times a care
for accuracy, and an aversion to slipshod reasoning, such as had not
been known among them previously since the time of the Greeks. The
great inventions of the seventeenth century--Analytical Geometry and
the Infinitesimal Calculus--were so fruitful in new results that
mathematicians had neither time nor inclination to examine their
foundations. Philosophers, who should have taken up the task, had too
little mathematical ability to invent the new branches of mathematics
which have now been found necessary for any adequate discussion. Thus
mathematicians were only awakened from their "dogmatic slumbers" when
Weierstrass and his followers showed that many of their most cherished
propositions are in general false. Macaulay, contrasting the certainty
of mathematics with the uncertainty of philosophy, asks who ever heard
of a reaction against Taylor's theorem? If he had lived now, he
himself might have heard of such a reaction, for this is precisely one
of the theorems which modern investigations have overthrown. Such rude
shocks to mathematical faith have produced that love of formalism
which appears, to those who are ignorant of its motive, to be mere
outrageous pedantry.
The proof that all pure mathematics, including Geometry, is nothing
but formal logic, is a fatal blow to the Kantian philosophy. Kant,
rightly perceiving that Euclid's propositions could not be deduced
from Euclid's axioms without the help of the figures, invented a
theory of knowledge to account for this fact; and it accounted so
successfully that, when the fact is shown to be a mere defect in
Euclid, and not a result of the nature of geometrical reasoning,
Kant's theory also has to be abandoned. The whole doctrine of _a
priori_ intuitions, by which Kant explained the possibility of pure
mathematics, is wholly inapplicable to mathematics in its present
form. The Aristotelian doctrines of the schoolmen come nearer in
spirit to the doctrines which modern mathematics inspire; but the
schoolmen were hampered by the fact that their formal logic was very
defective, and that the philosophical logic based upon the syllogism
showed a corresponding narrowness. What is now required is to give the
greatest possible development to mathematical logic, to allow to the
full the importance of relations, and then to found upon this secure
basis a new philosophical logic, which may hope to borrow some of the
exactitude and certainty of its mathematical foundation.
If this can
be successfully accomplished, there is every reason to hope that the
near future will be as great an epoch in pure philosophy as the
immediate past has been in the principles of mathematics. Great
triumphs inspire great hopes; and pure thought may achieve, within our
generation, such results as will place our time, in this respect, on a
level with the greatest age of Greece.[18]
FOOTNOTES:
[11] This subject is due in the main to Mr. C.S. Peirce.
[12] I ought to have added Frege, but his writings were unknown to me
when this article was written. [Note added in 1917.]
[13] Professor of Mathematics in the University of Berlin. He died in
1897.
[14] [Note added in 1917.] Although some infinite numbers are greater
than some others, it cannot be proved that of any two infinite numbers
one must be the greater.
[15] Cantor was not guilty of a fallacy on this point.
His proof that
there is no greatest number is valid. The solution of the puzzle is
complicated and depends upon the theory of types, which is explained
in _Principia Mathematica_, Vol. I (Camb. Univ. Press, 1910). [Note
added in 1917.]
[16] This must not be regarded as a historically correct account of
what Zeno actually had in mind. It is a new argument for his
conclusion, not the argument which influenced him. On this point, see
e.g. C.D. Broad, "Note on Achilles and the Tortoise,"
_Mind_, N.S.,
Vol. XXII, pp. 318-19. Much valuable work on the interpretation of
Zeno has been done since this article was written. [Note added in
1917.]
[17] Since the above was written, he has ceased to be used as a
textbook. But I fear many of the books now used are so bad that the
change is no great improvement. [Note added in 1917.]
[18] The greatest age of Greece was brought to an end by the
Peloponnesian War. [Note added in 1917.]
VI
ON SCIENTIFIC METHOD IN PHILOSOPHY
When we try to ascertain the motives which have led men to the
investigation of philosophical questions, we find that, broadly
speaking, they can be divided into two groups, often antagonistic, and
leading to very divergent systems. These two groups of motives are, on
the one hand, those derived from religion and ethics, and, on the
other hand, those derived from science. Plato, Spinoza, and Hegel may
be taken as typical of the philosophers whose interests are mainly
religious and ethical, while Leibniz, Locke, and Hume may be taken as
representatives of the scientific wing. In Aristotle, Descartes,
Berkeley, and Kant we find both groups of motives strongly present.
Herbert Spencer, in whose honour we are assembled to-day, would
naturally be classed among scientific philosophers: it was mainly from
science that he drew his data, his formulation of problems, and his
conception of method. But his strong religious sense is obvious in
much of his writing, and his ethical pre-occupations are what make him
value the conception of evolution--that conception in which, as a
whole generation has believed, science and morals are to be united in
fruitful and indissoluble marriage.
It is my belief that the ethical and religious motives in spite of
the splendidly imaginative systems to which they have given rise, have
been on the whole a hindrance to the progress of philosophy, and ought
now to be consciously thrust aside by those who wish to discover
philosophical truth. Science, originally, was entangled in similar
motives, and was thereby hindered in its advances. It is, I maintain,
from science, rather than from ethics and religion, that philosophy
should draw its inspiration.
But there are two different ways in which a philosophy may seek to
base itself upon science. It may emphasise the most general _results_
of science, and seek to give even greater generality and unity to
these results. Or it may study the _methods_ of science, and seek to
apply these methods, with the necessary adaptations, to its own
peculiar province. Much philosophy inspired by science has gone astray
through preoccupation with the _results_ momentarily supposed to have
been achieved. It is not results, but _methods_ that can be
transferred with profit from the sphere of the special sciences to the
sphere of philosophy. What I wish to bring to your notice is the
possibility and importance of applying to philosophical problems
certain broad principles of method which have been found successful in
the study of scientific questions.
The opposition between a philosophy guided by scientific method and a
philosophy dominated by religious and ethical ideas may be illustrated
by two notions which are very prevalent in the works of philosophers,
namely the notion of _the universe_, and the notion of _good and
evil_. A philosopher is expected to tell us something about the nature
of the universe as a whole, and to give grounds for either optimism or
pessimism. Both these expectations seem to me mistaken.
I believe the
conception of "the universe" to be, as its etymology indicates, a
mere relic of pre-Copernican astronomy: and I believe the question of
optimism and pessimism to be one which the philosopher will regard as
outside his scope, except, possibly, to the extent of maintaining that
it is insoluble.
In the days before Copernicus, the conception of the
"universe" was
defensible on scientific grounds: the diurnal revolution of the
heavenly bodies bound them together as all parts of one system, of
which the earth was the centre. Round this apparent scientific fact,
many human desires rallied: the wish to believe Man important in the
scheme of things, the theoretical desire for a comprehensive
understanding of the Whole, the hope that the course of nature might
be guided by some sympathy with our wishes. In this way, an ethically
inspired system of metaphysics grew up, whose anthropocentrism was
apparently warranted by the geocentrism of astronomy.
When Copernicus
swept away the astronomical basis of this system of thought, it had
grown so familiar, and had associated itself so intimately with men's
aspirations, that it survived with scarcely diminished force--survived
even Kant's "Copernican revolution," and is still now the unconscious
premiss of most metaphysical systems.
The oneness of the world is an almost undiscussed postulate of most
metaphysics. "Reality is not merely one and self-consistent, but is a
system of reciprocally determinate parts"[19]--such a statement would
pass almost unnoticed as a mere truism. Yet I believe that it embodies
a failure to effect thoroughly the "Copernican revolution," and that
the apparent oneness of the world is merely the oneness of what is
seen by a single spectator or apprehended by a single mind. The
Critical Philosophy, although it intended to emphasise the subjective
element in many apparent characteristics of the world, yet, by
regarding the world in itself as unknowable, so concentrated attention
upon the subjective representation that its subjectivity was soon
forgotten. Having recognised the categories as the work of the mind,
it was paralysed by its own recognition, and abandoned in despair the
attempt to undo the work of subjective falsification. In part, no
doubt, its despair was well founded, but not, I think, in any absolute
or ultimate sense. Still less was it a ground for rejoicing, or for
supposing that the nescience to which it ought to have given rise
could be legitimately exchanged for a metaphysical dogmatism.
I
As regards our present question, namely, the question of the unity of
the world, the right method, as I think, has been indicated by William
James.[20] "Let us now turn our backs upon ineffable or unintelligible
ways of accounting for the world's oneness, and inquire whether,
instead of being a principle, the 'oneness' affirmed may not merely be
a name like 'substance' descriptive of the fact that certain _specific
and verifiable connections_ are found among the parts of the
experiential flux.... We can easily conceive of things that shall have
no connection whatever with each other. We may assume them to inhabit
different times and spaces, as the dreams of different persons do even
now. They may be so unlike and incommensurable, and so inert towards
one another, as never to jostle or interfere. Even now there may
actually be whole universes so disparate from ours that we who know
ours have no means of perceiving that they exist. We conceive their
diversity, however; and by that fact the whole lot of them form what
is known in logic as 'a universe of discourse.' To form a universe of
discourse argues, as this example shows, no further kind of connexion.
The importance attached by certain monistic writers to the fact that
any chaos may become a universe by merely being named, is to me
incomprehensible." We are thus left with two kinds of unity in the
experienced world; the one what we may call the epistemological unity,
due merely to the fact that my experienced world is what _one_
experience selects from the sum total of existence: the other that
tentative and partial unity exhibited in the prevalence of scientific
laws in those portions of the world which science has hitherto
mastered. Now a generalisation based upon either of these kinds of
unity would be fallacious. That the things which we experience have
the common property of being experienced by us is a truism from which
obviously nothing of importance can be deducible: it is clearly
fallacious to draw from the fact that whatever we experience is
experienced the conclusion that therefore everything must be
experienced. The generalisation of the second kind of unity, namely,
that derived from scientific laws, would be equally fallacious, though
the fallacy is a trifle less elementary. In order to explain it let us
consider for a moment what is called the reign of law.
People often
speak as though it were a remarkable fact that the physical world is
subject to invariable laws. In fact, however, it is not easy to see
how such a world could fail to obey general laws. Taking any arbitrary
set of points in space, there is a function of the time corresponding
to these points, i.e. expressing the motion of a particle which
traverses these points: this function may be regarded as a general law
to which the behaviour of such a particle is subject.
Taking all such
functions for all the particles in the universe, there will be
theoretically some one formula embracing them all, and this formula
may be regarded as the single and supreme law of the spatio-temporal
world. Thus what is surprising in physics is not the existence of
general laws, but their extreme simplicity. It is not the uniformity
of nature that should surprise us, for, by sufficient analytic
ingenuity, any conceivable course of nature might be shown to exhibit
uniformity. What should surprise us is the fact that the uniformity is
simple enough for us to be able to discover it. But it is just this
characteristic of simplicity in the laws of nature hitherto discovered
which it would be fallacious to generalise, for it is obvious that
simplicity has been a part cause of their discovery, and can,
therefore, give no ground for the supposition that other undiscovered
laws are equally simple.
The fallacies to which these two kinds of unity have given rise
suggest a caution as regards all use in philosophy of general
_results_ that science is supposed to have achieved. In the first
place, in generalising these results beyond past experience, it is
necessary to examine very carefully whether there is not some reason
making it more probable that these results should hold of all that has
been experienced than that they should hold of things universally. The
sum total of what is experienced by mankind is a selection from the
sum total of what exists, and any general character exhibited by this
selection may be due to the manner of selecting rather than to the
general character of that from which experience selects.
In the second
place, the most general results of science are the least certain and
the most liable to be upset by subsequent research. In utilizing these
results as the basis of a philosophy, we sacrifice the most valuable
and remarkable characteristic of scientific method, namely, that,
although almost everything in science is found sooner or later to
require some correction, yet this correction is almost always such as
to leave untouched, or only slightly modified, the greater part of the
results which have been deduced from the premiss subsequently
discovered to be faulty. The prudent man of science acquires a certain
instinct as to the kind of uses which may be made of present
scientific beliefs without incurring the danger of complete and utter
refutation from the modifications likely to be introduced by
subsequent discoveries. Unfortunately the use of scientific
generalisations of a sweeping kind as the basis of philosophy is just
that kind of use which an instinct of scientific caution would avoid,
since, as a rule, it would only lead to true results if the
generalisation upon which it is based stood in _no_ need of
correction.
We may illustrate these general considerations by means of two
examples, namely, the conservation of energy and the principle of
evolution.
(1) Let us begin with the conservation of energy, or, as Herbert
Spencer used to call it, the persistence of force. He says:[21]
"Before taking a first step in the rational interpretation of
Evolution, it is needful to recognise, not only the facts that
Matter is indestructible and Motion continuous, but also the fact
that Force persists. An attempt to assign the _causes_
of
Evolution would manifestly be absurd if that agency to which the
metamorphosis in general and in detail is due, could either come
into existence or cease to exist. The succession of phenomena
would in such case be altogether arbitrary, and deductive Science
impossible."
This paragraph illustrates the kind of way in which the philosopher is
tempted to give an air of absoluteness and necessity to empirical
generalisations, of which only the approximate truth in the regions
hitherto investigated can be guaranteed by the unaided methods of
science. It is very often said that the persistence of something or
other is a necessary presupposition of all scientific investigation,
and this presupposition is then thought to be exemplified in some
quantity which physics declares to be constant. There are here, as it
seems to me, three distinct errors. First, the detailed scientific
investigation of nature does not _presuppose_ any such general laws as
its results are found to verify. Apart from particular observations,
science need presuppose nothing except the general principles of
logic, and these principles are not laws of nature, for they are
merely hypothetical, and apply not only to the actual world but to
whatever is _possible_. The second error consists in the identification of a constant quantity with a persistent entity. Energy
is a certain function of a physical system, but is not a thing or
substance persisting throughout the changes of the system. The same is
true of mass, in spite of the fact that mass has often been defined as
_quantity of matter_. The whole conception of quantity, involving, as
it does, numerical measurement based largely upon conventions, is far
more artificial, far more an embodiment of mathematical convenience,
than is commonly believed by those who philosophise on physics. Thus
even if (which I cannot for a moment admit) the persistence of some
entity were among the necessary postulates of science, it would be a
sheer error to infer from this the constancy of any physical quantity,
or the _a priori_ necessity of any such constancy which may be
empirically discovered. In the third place, it has become more and