No More Learning

But the uninitiated may wonder how it is possible to deal with a
number which cannot be counted. It is impossible to count up _all_ the
numbers, one by one, because, however many we may count, there are
always more to follow. The fact is that counting is a very vulgar and
elementary way of finding out how many terms there are in a
collection. And in any case, counting gives us what mathematicians
call the _ordinal_ number of our terms; that is to say, it arranges
our terms in an order or series, and its result tells us what type of
series results from this arrangement. In other words, it is impossible
to count things without counting some first and others afterwards, so
that counting always has to do with order. Now when there are only a
finite number of terms, we can count them in any order we like; but
when there are an infinite number, what corresponds to counting will
give us quite different results according to the way in which we carry
out the operation. Thus the ordinal number, which results from what,
in a general sense may be called counting, depends not only upon how
many terms we have, but also (where the number of terms is infinite)
upon the way in which the terms are arranged.

The fundamental infinite numbers are not ordinal, but are what is
called _cardinal_. They are not obtained by putting our terms in order
and counting them, but by a different method, which tells us, to begin
with, whether two collections have the same number of terms, or, if
not, which is the greater. [14] It does not tell us, in the way in
which counting does, _what_ number of terms a collection has; but if
we define a number as the number of terms in such and such a
collection, then this method enables us to discover whether some other
collection that may be mentioned has more or fewer terms. An
illustration will show how this is done. If there existed some country
in which, for one reason or another, it was impossible to take a
census, but in which it was known that every man had a wife and every
woman a husband, then (provided polygamy was not a national
institution) we should know, without counting, that there were exactly
as many men as there were women in that country, neither more nor
less. This method can be applied generally. If there is some relation
which, like marriage, connects the things in one collection each with
one of the things in another collection, and vice versa, then the two
collections have the same number of terms. This was the way in which
we found that there are as many even numbers as there are numbers.
Every number can be doubled, and every even number can be halved, and
each process gives just one number corresponding to the one that is
doubled or halved. And in this way we can find any number of
collections each of which has just as many terms as there are finite
numbers. If every term of a collection can be hooked on to a number,
and all the finite numbers are used once, and only once, in the
process, then our collection must have just as many terms as there are
finite numbers. This is the general method by which the numbers of
infinite collections are defined.

But it must not be supposed that all infinite numbers are equal. On
the contrary, there are infinitely more infinite numbers than finite
ones. There are more ways of arranging the finite numbers in different
types of series than there are finite numbers. There are probably more
points in space and more moments in time than there are finite
numbers. There are exactly as many fractions as whole numbers,
although there are an infinite number of fractions between any two
whole numbers. But there are more irrational numbers than there are
whole numbers or fractions. There are probably exactly as many points
in space as there are irrational numbers, and exactly as many points
on a line a millionth of an inch long as in the whole of infinite
space. There is a greatest of all infinite numbers, which is the
number of things altogether, of every sort and kind. It is obvious
that there cannot be a greater number than this, because, if
everything has been taken, there is nothing left to add. Cantor has a
proof that there is no greatest number, and if this proof were valid,
the contradictions of infinity would reappear in a sublimated form.
But in this one point, the master has been guilty of a very subtle
fallacy, which I hope to explain in some future work. [15]

We can now understand why Zeno believed that Achilles cannot overtake
the tortoise and why as a matter of fact he can overtake it. We shall
see that all the people who disagreed with Zeno had no right to do so,
because they all accepted premises from which his conclusion followed.
The argument is this: Let Achilles and the tortoise start along a road
at the same time, the tortoise (as is only fair) being allowed a
handicap. Let Achilles go twice as fast as the tortoise, or ten times
or a hundred times as fast. Then he will never reach the tortoise. For
at every moment the tortoise is somewhere and Achilles is somewhere;
and neither is ever twice in the same place while the race is going
on. Thus the tortoise goes to just as many places as Achilles does,
because each is in one place at one moment, and in another at any
other moment. But if Achilles were to catch up with the tortoise, the
places where the tortoise would have been would be only part of the
places where Achilles would have been. Here, we must suppose, Zeno
appealed to the maxim that the whole has more terms than the part. [16]
Thus if Achilles were to overtake the tortoise, he would have been in
more places than the tortoise; but we saw that he must, in any period,
be in exactly as many places as the tortoise. Hence we infer that he
can never catch the tortoise. This argument is strictly correct, if we
allow the axiom that the whole has more terms than the part. As the
conclusion is absurd, the axiom must be rejected, and then all goes
well. But there is no good word to be said for the philosophers of the
past two thousand years and more, who have all allowed the axiom and
denied the conclusion.

The retention of this axiom leads to absolute contradictions, while
its rejection leads only to oddities. Some of these oddities, it must
be confessed, are very odd. One of them, which I call the paradox of
Tristram Shandy, is the converse of the Achilles, and shows that the
tortoise, if you give him time, will go just as far as Achilles.
Tristram Shandy, as we know, employed two years in chronicling the
first two days of his life, and lamented that, at this rate, material
would accumulate faster than he could deal with it, so that, as years
went by, he would be farther and farther from the end of his history.
Now I maintain that, if he had lived for ever, and had not wearied of
his task, then, even if his life had continued as event fully as it
began, no part of his biography would have remained unwritten. For
consider: the hundredth day will be described in the hundredth year,
the thousandth in the thousandth year, and so on. Whatever day we may
choose as so far on that he cannot hope to reach it, that day will be
described in the corresponding year. Thus any day that may be
mentioned will be written up sooner or later, and therefore no part of
the biography will remain permanently unwritten. This paradoxical but
perfectly true proposition depends upon the fact that the number of
days in all time is no greater than the number of years.

Thus on the subject of infinity it is impossible to avoid conclusions
which at first sight appear paradoxical, and this is the reason why so
many philosophers have supposed that there were inherent
contradictions in the infinite. But a little practice enables one to
grasp the true principles of Cantor's doctrine, and to acquire new and
better instincts as to the true and the false. The oddities then
become no odder than the people at the antipodes, who used to be
thought impossible because they would find it so inconvenient to stand
on their heads.

The solution of the problems concerning infinity has enabled Cantor to
solve also the problems of continuity. Of this, as of infinity, he has
given a perfectly precise definition, and has shown that there are no
contradictions in the notion so defined. But this subject is so
technical that it is impossible to give any account of it here.

The notion of continuity depends upon that of _order_, since
continuity is merely a particular type of order. Mathematics has, in
modern times, brought order into greater and greater prominence. In
former days, it was supposed (and philosophers are still apt to
suppose) that quantity was the fundamental notion of mathematics. But
nowadays, quantity is banished altogether, except from one little
corner of Geometry, while order more and more reigns supreme. The
investigation of different kinds of series and their relations is now
a very large part of mathematics, and it has been found that this
investigation can be conducted without any reference to quantity, and,
for the most part, without any reference to number. All types of
series are capable of formal definition, and their properties can be
deduced from the principles of symbolic logic by means of the Algebra
of Relatives. The notion of a limit, which is fundamental in the
greater part of higher mathematics, used to be defined by means of
quantity, as a term to which the terms of some series approximate as
nearly as we please. But nowadays the limit is defined quite
differently, and the series which it limits may not approximate to it
at all. This improvement also is due to Cantor, and it is one which
has revolutionised mathematics. Only order is now relevant to limits.
Thus, for instance, the smallest of the infinite integers is the limit
of the finite integers, though all finite integers are at an infinite
distance from it. The study of different types of series is a general
subject of which the study of ordinal numbers (mentioned above) is a
special and very interesting branch. But the unavoidable
technicalities of this subject render it impossible to explain to any
but professed mathematicians.

Geometry, like Arithmetic, has been subsumed, in recent times, under
the general study of order. It was formerly supposed that Geometry was
the study of the nature of the space in which we live, and accordingly
it was urged, by those who held that what exists can only be known
empirically, that Geometry should really be regarded as belonging to
applied mathematics. But it has gradually appeared, by the increase of
non-Euclidean systems, that Geometry throws no more light upon the
nature of space than Arithmetic throws upon the population of the
United States. Geometry is a whole collection of deductive sciences
based on a corresponding collection of sets of axioms. One set of
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
more evident with the progress of physics that large generalisations,
such as the conservation of energy or mass, are far from certain and
are very likely only approximate. Mass, which used to be regarded as
the most indubitable of physical quantities, is now generally believed
to vary according to velocity, and to be, in fact, a vector quantity
which at a given moment is different in different directions. The
detailed conclusions deduced from the supposed constancy of mass for
such motions as used to be studied in physics will remain very nearly
exact, and therefore over the field of the older investigations very
little modification of the older results is required. But as soon as
such a principle as the conservation of mass or of energy is erected
into a universal _a priori_ law, the slightest failure in absolute
exactness is fatal, and the whole philosophic structure raised upon
this foundation is necessarily ruined. The prudent philosopher,
therefore, though he may with advantage study the methods of physics,
will be very chary of basing anything upon what happen at the moment
to be the most general results apparently obtained by those methods.

(2) The philosophy of evolution, which was to be our second example,
illustrates the same tendency to hasty generalisation, and also
another sort, namely, the undue preoccupation with ethical notions.
There are two kinds of evolutionist philosophy, of which both Hegel
and Spencer represent the older and less radical kind, while
Pragmatism and Bergson represent the more modern and revolutionary
variety. But both these sorts of evolutionism have in common the
emphasis on _progress_, that is, upon a continual change from the
worse to the better, or from the simpler to the more complex. It
would be unfair to attribute to Hegel any scientific motive or
foundation, but all the other evolutionists, including Hegel's modern
disciples, have derived their impetus very largely from the history of
biological development. To a philosophy which derives a law of
universal progress from this history there are two objections. First,
that this history itself is concerned with a very small selection of
facts confined to an infinitesimal fragment of space and time, and
even on scientific grounds probably not an average sample of events in
the world at large. For we know that decay as well as growth is a
normal occurrence in the world. An extra-terrestrial philosopher, who
had watched a single youth up to the age of twenty-one and had never
come across any other human being, might conclude that it is the
nature of human beings to grow continually taller and wiser in an
indefinite progress towards perfection; and this generalisation would
be just as well founded as the generalisation which evolutionists base
upon the previous history of this planet. Apart, however, from this
scientific objection to evolutionism, there is another, derived from
the undue admixture of ethical notions in the very idea of progress
from which evolutionism derives its charm. Organic life, we are told,
has developed gradually from the protozoon to the philosopher, and
this development, we are assured, is indubitably an advance.
Unfortunately it is the philosopher, not the protozoon, who gives us
this assurance, and we can have no security that the impartial
outsider would agree with the philosopher's self-complacent
assumption. This point has been illustrated by the philosopher Chuang
Tz? in the following instructive anecdote:

"The Grand Augur, in his ceremonial robes, approached the shambles
and thus addressed the pigs: 'How can you object to die? I shall
fatten you for three months. I shall discipline myself for ten
days and fast for three. I shall strew fine grass, and place you
bodily upon a carved sacrificial dish. Does not this satisfy you? '

Then, speaking from the pigs' point of view, he continued: 'It is
better, perhaps, after all, to live on bran and escape the
shambles. . . . '

'But then,' added he, speaking from his own point of view, 'to
enjoy honour when alive one would readily die on a war-shield or
in the headsman's basket. '

So he rejected the pigs' point of view and adopted his own point
of view. In what sense, then, was he different from the pigs? "

I much fear that the evolutionists too often resemble the Grand Augur
and the pigs.

The ethical element which has been prominent in many of the most
famous systems of philosophy is, in my opinion, one of the most
serious obstacles to the victory of scientific method in the
investigation of philosophical questions. Human ethical notions, as
Chuang Tz? perceived, are essentially anthropocentric, and involve,
when used in metaphysics, an attempt, however veiled, to legislate for
the universe on the basis of the present desires of men. In this way
they interfere with that receptivity to fact which is the essence of
the scientific attitude towards the world. To regard ethical notions
as a key to the understanding of the world is essentially
pre-Copernican. It is to make man, with the hopes and ideals which he
happens to have at the present moment, the centre of the universe and
the interpreter of its supposed aims and purposes. Ethical metaphysics
is fundamentally an attempt, however disguised, to give legislative
force to our own wishes. This may, of course, be questioned, but I
think that it is confirmed by a consideration of the way in which
ethical notions arise. Ethics is essentially a product of the
gregarious instinct, that is to say, of the instinct to co-operate
with those who are to form our own group against those who belong to
other groups. Those who belong to our own group are good; those who
belong to hostile groups are wicked. The ends which are pursued by our
own group are desirable ends, the ends pursued by hostile groups are
nefarious. The subjectivity of this situation is not apparent to the
gregarious animal, which feels that the general principles of justice
are on the side of its own herd. When the animal has arrived at the
dignity of the metaphysician, it invents ethics as the embodiment of
its belief in the justice of its own herd. So the Grand Augur invokes
ethics as the justification of Augurs in their conflicts with pigs.
But, it may be said, this view of ethics takes no account of such
truly ethical notions as that of self-sacrifice. This, however, would
be a mistake. The success of gregarious animals in the struggle for
existence depends upon co-operation within the herd, and co-operation
requires sacrifice, to some extent, of what would otherwise be the
interest of the individual. Hence arises a conflict of desires and
instincts, since both self-preservation and the preservation of the
herd are biological ends to the individual. Ethics is in origin the
art of recommending to others the sacrifices required for co-operation
with oneself. Hence, by reflexion, it comes, through the operation of
social justice, to recommend sacrifices by oneself, but all ethics,
however refined, remains more or less subjective. Even vegetarians do
not hesitate, for example, to save the life of a man in a fever,
although in doing so they destroy the lives of many millions of
microbes. The view of the world taken by the philosophy derived from
ethical notions is thus never impartial and therefore never fully
scientific. As compared with science, it fails to achieve the
imaginative liberation from self which is necessary to such
understanding of the world as man can hope to achieve, and the
philosophy which it inspires is always more or less parochial, more or
less infected with the prejudices of a time and a place.

I do not deny the importance or value, within its own sphere, of the
kind of philosophy which is inspired by ethical notions. The ethical
work of Spinoza, for example, appears to me of the very highest
significance, but what is valuable in such work is not any
metaphysical theory as to the nature of the world to which it may give
rise, nor indeed anything which can be proved or disproved by
argument. What is valuable is the indication of some new way of
feeling towards life and the world, some way of feeling by which our
own existence can acquire more of the characteristics which we must
deeply desire. The value of such work, however immeasurable it is,
belongs with practice and not with theory. Such theoretic importance
as it may possess is only in relation to human nature, not in relation
to the world at large. The scientific philosophy, therefore, which
aims only at understanding the world and not directly at any other
improvement of human life, cannot take account of ethical notions
without being turned aside from that submission to fact which is the
essence of the scientific temper.


II

If the notion of the universe and the notion of good and evil are
extruded from scientific philosophy, it may be asked what specific
problems remain for the philosopher as opposed to the man of science?
It would be difficult to give a precise answer to this question, but
certain characteristics may be noted as distinguishing the province of
philosophy from that of the special sciences.

In the first place a philosophical proposition must be general. It
must not deal specially with things on the surface of the earth, or
with the solar system, or with any other portion of space and time. It
is this need of generality which has led to the belief that philosophy
deals with the universe as a whole. I do not believe that this belief
is justified, but I do believe that a philosophical proposition must
be applicable to everything that exists or may exist. It might be
supposed that this admission would be scarcely distinguishable from
the view which I wish to reject. This, however, would be an error, and
an important one. The traditional view would make the universe itself
the subject of various predicates which could not be applied to any
particular thing in the universe, and the ascription of such peculiar
predicates to the universe would be the special business of
philosophy. I maintain, on the contrary, that there are no
propositions of which the "universe" is the subject; in other words,
that there is no such thing as the "universe. " What I do maintain is
that there are general propositions which may be asserted of each
individual thing, such as the propositions of logic. This does not
involve that all the things there are form a whole which could be
regarded as another thing and be made the subject of predicates. It
involves only the assertion that there are properties which belong to
each separate thing, not that there are properties belonging to the
whole of things collectively. The philosophy which I wish to advocate
may be called logical atomism or absolute pluralism, because, while
maintaining that there are many things, it denies that there is a
whole composed of those things. We shall see, therefore, that
philosophical propositions, instead of being concerned with the whole
of things collectively, are concerned with all things distributively;
and not only must they be concerned with all things, but they must be
concerned with such properties of all things as do not depend upon the
accidental nature of the things that there happen to be, but are true
of any possible world, independently of such facts as can only be
discovered by our senses.

This brings us to a second characteristic of philosophical
propositions, namely, that they must be _a priori_. A philosophical
proposition must be such as can be neither proved nor disproved by
empirical evidence. Too often we find in philosophical books arguments
based upon the course of history, or the convolutions of the brain, or
the eyes of shell-fish. Special and accidental facts of this kind are
irrelevant to philosophy, which must make only such assertions as
would be equally true however the actual world were constituted.

We may sum up these two characteristics of philosophical propositions
by saying that _philosophy is the science of the possible_. But this
statement unexplained is liable to be misleading, since it may be
thought that the possible is something other than the general, whereas
in fact the two are indistinguishable.

Philosophy, if what has been said is correct, becomes indistinguishable
from logic as that word has now come to be used. The study of logic
consists, broadly speaking, of two not very sharply distinguished
portions. On the one hand it is concerned with those general statements
which can be made concerning everything without mentioning any one
thing or predicate or relation, such for example as "if _x_ is a member
of the class ? and every member of ? is a member of ? , then _x_ is a
member of the class ? , whatever _x_, ? , and ? may be. " On the other
hand, it is concerned with the analysis and enumeration of logical
_forms_, i. e. with the kinds of propositions that may occur, with the
various types of facts, and with the classification of the constituents
of facts. In this way logic provides an inventory of possibilities, a
repertory of abstractly tenable hypotheses.

It might be thought that such a study would be too vague and too
general to be of any very great importance, and that, if its problems
became at any point sufficiently definite, they would be merged in the
problems of some special science. It appears, however, that this is
not the case. In some problems, for example, the analysis of space and
time, the nature of perception, or the theory of judgment, the
discovery of the logical form of the facts involved is the hardest
part of the work and the part whose performance has been most lacking
hitherto. It is chiefly for want of the right logical hypothesis that
such problems have hitherto been treated in such an unsatisfactory
manner, and have given rise to those contradictions or antinomies in
which the enemies of reason among philosophers have at all times
delighted.

By concentrating attention upon the investigation of logical forms, it
becomes possible at last for philosophy to deal with its problems
piecemeal, and to obtain, as the sciences do, such partial and
probably not wholly correct results as subsequent investigation can
utilise even while it supplements and improves them. Most
philosophies hitherto have been constructed all in one block, in such
a way that, if they were not wholly correct, they were wholly
incorrect, and could not be used as a basis for further
investigations. It is chiefly owing to this fact that philosophy,
unlike science, has hitherto been unprogressive, because each original
philosopher has had to begin the work again from the beginning,
without being able to accept anything definite from the work of his
predecessors. A scientific philosophy such as I wish to recommend will
be piecemeal and tentative like other sciences; above all, it will be
able to invent hypotheses which, even if they are not wholly true,
will yet remain fruitful after the necessary corrections have been
made. This possibility of successive approximations to the truth is,
more than anything else, the source of the triumphs of science, and to
transfer this possibility to philosophy is to ensure a progress in
method whose importance it would be almost impossible to exaggerate.

The essence of philosophy as thus conceived is analysis, not
synthesis. To build up systems of the world, like Heine's German
professor who knit together fragments of life and made an intelligible
system out of them, is not, I believe, any more feasible than the
discovery of the philosopher's stone. What is feasible is the
understanding of general forms, and the division of traditional
problems into a number of separate and less baffling questions.
"Divide and conquer" is the maxim of success here as elsewhere.

Let us illustrate these somewhat general maxims by examining their
application to the philosophy of space, for it is only in application
that the meaning or importance of a method can be understood. Suppose
we are confronted with the problem of space as presented in Kant's
Transcendental AEsthetic, and suppose we wish to discover what are the
elements of the problem and what hope there is of obtaining a solution
of them. It will soon appear that three entirely distinct problems,
belonging to different studies, and requiring different methods for
their solution, have been confusedly combined in the supposed single
problem with which Kant is concerned. There is a problem of logic, a
problem of physics, and a problem of theory of knowledge. Of these
three, the problem of logic can be solved exactly and perfectly; the
problem of physics can probably be solved with as great a degree of
certainty and as great an approach to exactness as can be hoped in an
empirical region; the problem of theory of knowledge, however, remains
very obscure and very difficult to deal with. Let us see how these
three problems arise.

(1) The logical problem has arisen through the suggestions of
non-Euclidean geometry. Given a body of geometrical propositions, it
is not difficult to find a minimum statement of the axioms from which
this body of propositions can be deduced. It is also not difficult, by
dropping or altering some of these axioms, to obtain a more general or
a different geometry, having, from the point of view of pure
mathematics, the same logical coherence and the same title to respect
as the more familiar Euclidean geometry. The Euclidean geometry itself
is true perhaps of actual space (though this is doubtful), but
certainly of an infinite number of purely arithmetical systems, each
of which, from the point of view of abstract logic, has an equal and
indefeasible right to be called a Euclidean space. Thus space as an
object of logical or mathematical study loses its uniqueness; not only
are there many kinds of spaces, but there are an infinity of examples
of each kind, though it is difficult to find any kind of which the
space of physics may be an example, and it is impossible to find any
kind of which the space of physics is certainly an example. As an
illustration of one possible logical system of geometry we may
consider all relations of three terms which are analogous in certain
formal respects to the relation "between" as it appears to be in
actual space. A space is then defined by means of one such three-term
relation. The points of the space are all the terms which have this
relation to something or other, and their order in the space in
question is determined by this relation. The points of one space are
necessarily also points of other spaces, since there are necessarily
other three-term relations having those same points for their field.
The space in fact is not determined by the class of its points, but by
the ordering three-term relation. When enough abstract logical
properties of such relations have been enumerated to determine the
resulting kind of geometry, say, for example, Euclidean geometry, it
becomes unnecessary for the pure geometer in his abstract capacity to
distinguish between the various relations which have all these
properties. He considers the whole class of such relations, not any
single one among them. Thus in studying a given kind of geometry the
pure mathematician is studying a certain class of relations defined by
means of certain abstract logical properties which take the place of
what used to be called axioms. The nature of geometrical _reasoning_
therefore is purely deductive and purely logical; if any special
epistemological peculiarities are to be found in geometry, it must not
be in the reasoning, but in our knowledge concerning the axioms in
some given space.

(2) The physical problem of space is both more interesting and more
difficult than the logical problem. The physical problem may be
stated as follows: to find in the physical world, or to construct from
physical materials, a space of one of the kinds enumerated by the
logical treatment of geometry. This problem derives its difficulty
from the attempt to accommodate to the roughness and vagueness of the
real world some system possessing the logical clearness and exactitude
of pure mathematics. That this can be done with a certain degree of
approximation is fairly evident If I see three people _A_, _B_, and
_C_ sitting in a row, I become aware of the fact which may be
expressed by saying that _B_ is between _A_ and _C_ rather than that
_A_ is between _B_ and _C_, or _C_ is between _A_ and _B_. This
relation of "between" which is thus perceived to hold has some of the
abstract logical properties of those three-term relations which, we
saw, give rise to a geometry, but its properties fail to be exact, and
are not, as empirically given, amenable to the kind of treatment at
which geometry aims. In abstract geometry we deal with points,
straight lines, and planes; but the three people _A_, _B_, and _C_
whom I see sitting in a row are not exactly points, nor is the row
exactly a straight line. Nevertheless physics, which formally assumes
a space containing points, straight lines, and planes, is found
empirically to give results applicable to the sensible world. It must
therefore be possible to find an interpretation of the points,
straight lines, and planes of physics in terms of physical data, or at
any rate in terms of data together with such hypothetical additions as
seem least open to question. Since all data suffer from a lack of
mathematical precision through being of a certain size and somewhat
vague in outline, it is plain that if such a notion as that of a point
is to find any application to empirical material, the point must be
neither a datum nor a hypothetical addition to data, but a
_construction_ by means of data with their hypothetical additions. It
is obvious that any hypothetical filling out of data is less dubious
and unsatisfactory when the additions are closely analogous to data
than when they are of a radically different sort. To assume, for
example, that objects which we see continue, after we have turned away
our eyes, to be more or less analogous to what they were while we were
looking, is a less violent assumption than to assume that such objects
are composed of an infinite number of mathematical points. Hence in
the physical study of the geometry of physical space, points must not
be assumed _ab initio_ as they are in the logical treatment of
geometry, but must be constructed as systems composed of data and
hypothetical analogues of data. We are thus led naturally to define a
physical point as a certain class of those objects which are the
ultimate constituents of the physical world. It will be the class of
all those objects which, as one would naturally say, _contain_ the
point. To secure a definition giving this result, without previously
assuming that physical objects are composed of points, is an agreeable
problem in mathematical logic. The solution of this problem and the
perception of its importance are due to my friend Dr. Whitehead. The
oddity of regarding a point as a class of physical entities wears off
with familiarity, and ought in any case not to be felt by those who
maintain, as practically every one does, that points are mathematical
fictions. The word "fiction" is used glibly in such connexions by many
men who seem not to feel the necessity of explaining how it can come
about that a fiction can be so useful in the study of the actual world
as the points of mathematical physics have been found to be. By our
definition, which regards a point as a class of physical objects, it
is explained both how the use of points can lead to important
physical results, and how we can nevertheless avoid the assumption
that points are themselves entities in the physical world.

Many of the mathematically convenient properties of abstract logical
spaces cannot be either known to belong or known not to belong to the
space of physics. Such are all the properties connected with continuity.
For to know that actual space has these properties would require an
infinite exactness of sense-perception. If actual space is continuous,
there are nevertheless many possible non-continuous spaces which will be
empirically indistinguishable from it; and, conversely, actual space may
be non-continuous and yet empirically indistinguishable from a possible
continuous space. Continuity, therefore, though obtainable in the _a
priori_ region of arithmetic, is not with certainty obtainable in the
space or time of the physical world: whether these are continuous or not
would seem to be a question not only unanswered but for ever
unanswerable. From the point of view of philosophy, however, the
discovery that a question is unanswerable is as complete an answer as
any that could possibly be obtained. And from the point of view of
physics, where no empirical means of distinction can be found, there can
be no empirical objection to the mathematically simplest assumption,
which is that of continuity.

The subject of the physical theory of space is a very large one,
hitherto little explored. It is associated with a similar theory of
time, and both have been forced upon the attention of philosophically
minded physicists by the discussions which have raged concerning the
theory of relativity.

(3) The problem with which Kant is concerned in the Transcendental
AEsthetic is primarily the epistemological problem: "How do we come to
have knowledge of geometry _a priori_? " By the distinction between the
logical and physical problems of geometry, the bearing and scope of
this question are greatly altered. Our knowledge of pure geometry is
_a priori_ but is wholly logical. Our knowledge of physical geometry
is synthetic, but is not _a priori_. Our knowledge of pure geometry is
hypothetical, and does not enable us to assert, for example, that the
axiom of parallels is true in the physical world.