thanks mainly to the labour of mathematicians with philosophical minds
on the principles of their own science. If we admit that mathematics is
true--and it seems quite impossible to avoid the admission--we can now
see that neither the traditional Kant-Hegel doctrine nor the traditional
sensationalistic empiricism can be sound. Not to speak of inquiries
which have been actually created within our own life-time, it may fairly
be said that the whole of pure mathematics has been shown, or is on the
verge of being shown, to form a body of conclusions rigidly deduced from
a few unproved postulates which are of a purely logical character.
Descartes has proved to be right in his view that the exceptional
certainty men have always ascribed to mathematical knowledge is not due
to the supposed restriction of the science to relation of number and
magnitude--there is a good deal of pure mathematics which deals with
neither--but to the simplicity of its undefined notions and the high
plausibility of its unproved postulates. Bit by bit the bad logic has
been purged out of the Calculus and the Theory of Functions and these
branches of study have been made into patterns of accurate reasoning on
exactly stated premisses. It has appeared in the process that the
alleged contradictions in mathematics upon which the followers of Kant
and Hegel laid stress do not really exist at all, and only seemed to
exist because mathematicians in the past expressed their meaning so
awkwardly. Further, it has been established that the most fundamental
idea of all in mathematics is not that of number or magnitude but that
of _order_ in a series and that the whole doctrine of series is only a
branch of the logic of Relations. From the logical doctrine of serial
order we seem to be able to deduce the whole arithmetic of integers, and
from this it is easy to deduce further the arithmetic of fractions and
the arithmetic or algebra of the 'real' and 'complex' numbers. As the
logical principles of serial order enable us to deal with infinite as
well as with finite series, it further follows that the Calculus and the
Theory of Functions can now be built up without a single contradiction
or breach of logic. The puzzles about the infinitely great and
infinitely small, which used to throw a cloud of mystery over the
'higher' branches of Mathematics, have been finally dissipated by the
discovery that the 'infinite' is readily definable in purely ordinal
terms and that the 'infinitesimal' does not really enter into the
misnamed 'Infinitesimal Calculus' at all. Arithmetic and the theory of
serial order have been shown to be the sufficient basis of the whole
science which, as Plato long ago remarked, is 'very inappropriately
called geometry'. A résumé of the work which has been thus done may be
found in the stately volumes of the _Principia Mathematica_ of Whitehead
and Russell, or--to a large extent--in the _Formulario Matematico_ of
Professor Peano. Of other works dealing with the subject, the finest
from the strictly philosophical point of view is probably that of
Professor G. Frege on _The Fundamental Laws of Arithmetic_. The general
result of the whole development is that we are now at last definitely
freed from the haunting fear that there is some hidden contradiction in
the principles of the exact sciences which would vitiate all our
knowledge of universal truths. This removes the chief, if not the only
ground for the view that all the truths of Science are only 'partial'.
At the same time, the proof that pure mathematics is a strictly logical
development and that all its conclusions are of the hypothetical form,
'if _a b c_ ..., then _x_' definitely disproves the popular Kantian
doctrine that _sense_-data are a necessary constituent of scientific
knowledge. And with this dogma falls the _main_ ground for the denial
that knowledge about the soul and God is attainable. The recovery of a
sounder philosophical method has, as Mr. Russell himself says, disposed
of what was yesterday the accepted view that the function of Philosophy
is to narrow down the range of possible interpretations of facts until
only one is left. Philosophy rather opens doors than shuts them. It
multiplies the number of logically possible sets of premisses from which
consequences agreeing with empirical facts may be inferred. Mr.
Russell's unreasoned anti-theism seems to me to make him curiously blind
to an obvious application of this principle. On the other side, the
revived attention to the logical methods of the sciences is killing the
crude sensationalism of the days which saw the first publication of
Mach's _Science of Mechanics_ and Pearson's _Grammar of Science_. The
claims of 'induction' to be a method of establishing truths may be
fairly said to have been completely exposed. It is clearer now than it
was when Kant made the observation that each of the 'sciences' contains
just so much science as it contains mathematics, and that the Critical
Philosophy was fully justified in insisting that all science implies
universal _à priori_ postulates, though it went wrong in thinking that
these postulates are laws of the working of the human mind or are 'put
into' things by the human mind. How far Science has moved away from
crude sensationalistic empiricism may be estimated by a comparison of
the successive editions of the _Grammar of Science_. It must always have
been apparent to an attentive reader that the chapters of that
fascinating book which deal directly with the leading principles of
Physics and Biology are of very different quality from the earlier
chapters which expound, with many self-contradictions and much wrath
against metaphysicians and theologians whom the writer seems never to
have tried to understand, the fantastic 'metaphysics of the
telephone-exchange'. But the difference of quality is more marked in the
second edition than in the first, and in the (alas!) unfinished third
edition than in the second. So far, then, as the problem of the
unification of the sciences is concerned, the old prejudices which
divided the rationalist philosopher from the sensationalist scientific
man seem to have been, in the main, dissipated. We can see now that what
used to be called Philosophy and what used to be called Science are both
parts of one task, that they have a common method and presuppose a
common body of principles.
So far it may be said with truth that Philosophy is becoming more
faithful than Kant was himself to the leading ideas of 'Criticism', and
again that it is reverting once more, as it reverted in the days of
Galileo, to the positions of Plato. I do not mean that the whole
programme has been completely executed and that there is nothing for a
successor of Frege or Russell to do. It is instructive to observe that
at the very end of the great work on arithmetic to which I have referred
Frege found himself compelled by difficulties which had been overlooked
until Russell called attention to them to add an appendix confessing
that there was a single important flaw in his elaborate logical
construction of the principles of arithmetic. He had shown that if there
are certain things called 'integers', defined as he had defined them,
the whole of arithmetic follows. But he had not shown that there _is_
any object answering to his definition of an integer, and the logical
researches of Russell had thrown some doubt on the point. This proved
that some restatement of the initial assumptions of the theory was
needed. Since the date of Frege's appendix (1903), Mr. Russell and
others have done something towards the necessary rectification, and the
resulting 'Theory of Types' is pretty certainly one of the most
important contributions ever made to logical doctrine, but it may still
be reasonably doubted whether the 'Theory of Types', as expounded by
Whitehead and Russell in their _Principia Mathematica_, is the last word
required. At any rate, it seems clear that it is a great step on the
right road to the solution of a most difficult problem.
There still remains the greatest problem of all, the harmonization of
Science and Life. I cannot believe that this problem is an illegitimate
one, or that we must sit down content to accept the severance of 'fact'
and 'value' as final for our thought. Even the unification of the
sciences itself remains imperfect so long as we treat it as merely
something which 'happens to be the case' that there are many things and
many kinds of things in the universe and also a number of relations in
which they 'happen' to stand. It is significant that in his later
writings Mr. Russell has been driven to abandon the concept of personal
identity, which is so fundamental for practical life, and to assert that
each of us is not one man but an infinite series of men of whom each
only exists for a mathematical instant. I am sure that such a theory
requires the abandonment of the whole notion of value as an illusion,
and even more sure that it is ruinous to any practical rule of living,
and I cannot believe in the 'philosophy' of any man who is satisfied to
base his practice on what he regards as detected illusion. Hence I find
myself strongly in sympathy with my eminent Italian colleague Professor
Varisco, who has devoted his two chief works (_I Massimi Problemi_ and
_Conosci Te Stesso_) to an exceedingly subtle attempt to show that 'what
ought to be', in Platonic phrase 'the Good', is in the end the single
principle from which all things derive their existence as well as their
value. Mr. Russell's philosophy saves us half Plato, and that is much,
but I am convinced that it is whole and entire Plato whom a profounder
philosophy would preserve for us. I believe personally that such a
philosophy will be led, as Plato was in the end led, to a theistic
interpretation of life, that it is in the living God Who is over all,
blessed for ever, that it will find the common source of fact and value.
And again I believe that it will be led to its result very largely by
what is, after all, perhaps the profoundest thought of Kant, the
conviction that the most illuminating fact of all is the _fact_ of the
absolute and unconditional obligatoriness of the law of right. It is
precisely here that fact and value most obviously meet. For when we ask
ourselves what in fact we are, we shall assuredly find no true answer to
this question about what _is_ if we forget that we are first and
foremost beings who _ought_ to follow a certain way of life, and to
follow it for no other reason than that it is good. But I cannot, of
course, offer reasons here for this conviction, though I am sure that
adequate reasons can be given. Here I must be content to state this
ultimate conviction as a 'theological superstition', or, as I should
prefer to put it with a little more certainty, as a matter of faith. The
alternative is to treat the world as a stupid, and possibly malicious,
bad joke.
_Note_.--It may be thought that something should have been said about
the revolt against authority and tradition which has styled itself
variously 'Pragmatism' and 'Humanism', and also about the recent vogue
of Bergsonianism. I may in part excuse my silence by the plea that both
movements are, in my judgement, already spent forces. If I must say more
than this, I would only remark about Pragmatism that I could speak of it
with more confidence if its representatives themselves were more agreed
as to its precise principles. At present I can discern little agreement
among them about anything except that they all show a great impatience
with the business of thinking things quietly and steadily out, and that
none of them seems to appreciate the importance of the 'critical'
problem. 'Pragmatism' thus seems to me less a definite way of thinking
than a collective name for a series of 'guesses at truth'. Some of the
guesses may be very lucky ones, but I, at least, can hardly take the
claims of unmethodic guessing to be a philosophy very seriously. To
'give and receive argument' appears to me to be of the very essence of
Philosophy. As for M. Bergson, I yield to no one in admiration for his
brilliancy as a stylist and the happiness of many of his illustrations.
But I have always found it difficult to grasp his central idea--if he
really has one--because his whole doctrine has always seemed to me to be
based upon a couple of elementary blunders which will be found in the
opening chapter of his _Données Immédiates de la Conscience_. We are
there called on to reject the intellect in Philosophy on the grounds (1)
that, being originally developed in the services of practical needs, it
can at best tell us how to find our way about among the bodies around
us, and is thus debarred from knowing more than the _outsides_ of
things; (2) that its typical achievement is therefore geometry, and
geometry, _because it can measure only straight lines_, necessarily
misconceives the true character of 'real duration'. Now, as to the first
point, I should have thought it obvious that the establishment of a
_modus vivendi_ with one's fellows has always been as much of a
practical need as the avoidance of stones and pit-falls, and the alleged
conclusion about the defects of the intellect does not therefore seem to
me to follow from M. Bergson's premisses, even if we had any reason, as
I do not see that we have, to accept the premisses. And as to the second
point, I would ask whether M. Bergson possesses a clock or a watch, and
if he has, how he supposes time is measured on them? He seems to me to
have forgotten the elementary fact that angles can be measured as well
as straight lines. (I might add that he makes the further curious
assumption that all geometry is metrical.) It may be that something
would be left of the Bergsonian philosophy if one eliminated the
consequences of these initial blunders, but I do not know what the
remainder would be. At any rate, the anti-intellectualism which M.
Bergson and his disciple, Professor Carr, seem to regard as fundamental
will have to go, unless different and better grounds can be found for
it. I must leave it to others to judge of the adequacy of this apology.
FOR REFERENCE
Varisco, _The Great Problem_ (Macmillan).
Varisco, _Know Thyself_ (Macmillan).
Aliotta, _The Idealistic Reaction against Science_ (Macmillan).
Bertrand Russell, _Our Knowledge of the External World_ (Open Court
Publishing Co.).
Bertrand Russell, _The Problems of Philosophy_ (Home University
Library).
A.N. Whitehead, _The Principles of Natural Knowledge_ (Cambridge Press).
G.E. Moore, _Ethics_ (H.U.L.).
W. McDougall, _Philosophy_ (H.U.L.).
A.N. Whitehead, _Introduction to Mathematics_ (H.U.L.).
III
RECENT DEVELOPMENTS IN EUROPEAN THOUGHT ON THE EVOLUTION OF RELIGION
F.B. JEVONS
The living beings that exist or have existed upon the earth are of kinds
innumerable; and, in the opinion of man, the chief of them all is
mankind. Man, for the simple reason that he is man, is anthropomorphic
in all his judgements and not merely in his religious conceptions; he
holds himself to be the standard and measure in all things. If his right
so to regard himself were challenged, if he were called upon to justify
himself for having taken his foot as a unit of measurement, or his
fingers as the basis of his system of numbers, he might reply that
anything will serve as a standard for weights and measures, provided
that it never varies, but is always the same whenever referred to. But
the reply, valid though it is, does not do full justice to man: it
leaves room for the suspicion that a standard is something chosen by man
in a purely arbitrary manner and without reference to the facts of
nature. If that were really the case, then man's conception of himself
as superior to the other animals on earth might be but a prejudice of an
arbitrary kind. When, however, we consider, from the point of view of
evolution, the place of man among the other animals that occupy or have
occupied the earth, it is indubitable that the human organism is in
point of time the latest evolved and the human brain is in point of
complexity and efficiency the most highly developed. Further, the
evidence of embryology goes to show that the organism which has
eventually become human became so only by passing through successive
stages, each of which has its analogue in some of the existing forms of
animal life. Those forms of animal life exist side by side; and if we
conceive them to be represented diagrammatically by vertical lines,
differing in height according to their degree of evolution, the line
representing the human organism will be the tallest, and may be
considered to have become the tallest by successive increments or stages
corresponding to the height of the various other parallel vertical
lines.
When the conception of evolution, which had been employed to explain the
origin of species and the descent of man, and which had been gained by a
consideration of material organisms, came to be applied to the world of
man's thoughts, to the non-material and spiritual domain, and to be used
for the purpose of explaining the growth and the development of
religion, it was natural that the conception which had proved so
valuable in the one case should be applied without modification to the
other--as natural as that the first railway coach should be built on the
model of the stagecoach. The possibility that the theory of evolution
might itself evolve, and in evolving change, was one that was not, and
at that time could hardly be, present to the minds of those who were
extending the theory and in the process of extending it were developing
it. Yet the possibility was there, implicit in the very conception of
evolution, which involves continuous change--change in continuity and
continuity in change.
Any and every attempt to trace the evolution of religion seems at first
necessarily to involve the assumption that from the beginning religion
was there to be evolved. That was the position assumed by Robertson
Smith in _The Religion of the Semites_, which appeared in 1889. At that
date the aborigines of Australia were supposed to represent the human
race in its lowest and its earliest stage of development. In them,
therefore, if anywhere, we might expect to find what would be religion
in its lowest and earliest stage indeed but still religion. Reduced to
its lowest terms, religion, it was felt at first, must imply at least
belief in a god and communion with him. If, therefore, religion was to
be found amongst the representatives of the lowest and earliest stage in
the evolution of humanity, belief in a god and communion with him must
there be found. He who seeks finds. Robertson Smith found amongst the
Australians totem-gods and sacramental rites. Indeed, it was at that
time the belief universally held by students of the science of religion
that in Australia a totem was a god and a god might be a totem. It was
conjectured by Robertson Smith that in Australia the totem animal or
plant was eaten sacramentally. Since, then, the totem in Australia was
held to be both the god and the animal or plant in which the god
manifested himself, it followed that in Australia we had, preserved to
this day, the earliest form of sacrifice--that in which the totem animal
was itself the totem god to whom it was offered as a sacrifice, and was
itself--or rather himself--the sacramental meal furnished to his
worshippers. The totem was eaten, it was conjectured, with the object of
acquiring the qualities of the divine creature, or of absorbing them
into the person of the worshipper. That the totem was eaten
sacramentally rested, as has just been said, on the conjecture made by
Robertson Smith in 1889; but in 1899 Dr. (now Sir James) Frazer declared
that, thanks to the investigations of Messrs. Spencer and Gillen, 'here,
in the heart of Australia, among the most primitive savages known to
us, we find the actual observance of that totem sacrament which
Robertson Smith, with the intuition of genius, divined years ago, but of
which positive examples have hitherto been wanting.'
On the foundation thus laid by the intuition of Robertson Smith and
approved in 1899 by Sir James Frazer, a simple and complete theory of
the evolution of religion was possible. In any one tribe there were
several totem-kins. The totem of each kin was divine, but the
personality of a totem was so undeveloped in conception that, though it
might, and on the theory did, develop into a deity, it was originally
more of the nature of a spirit than a god, and totemism proper might
easily pass into polydaemonism, that is a system in which the beings
worshipped were conceived to possess a personality more clearly defined
than that attributed to totems but less developed than that assigned to
deities. From the beginning each tribe had worshipped a plurality of
totems; it was, therefore, readily intelligible that, as these totems
came to be credited with more and more definite and developed
personality, the plurality of totems became not only a polydaemonism,
but afterwards a plurality of deities, and a system of polytheism came
to be established. From polytheism then, amongst the Israelites,
monotheism was conceived to have been gradually developed.
On this theory the evolution of religion was, if we may so describe it,
linear or rectilinear: the process consisted in a series of successive
stages. In some cases, as for instance among the aborigines of
Australia, it never rose higher than its starting-point, totemism; in
others, as for instance in Samoa, it became polytheism without ceasing
to be totemism; in others again, as for instance amongst the Aryan
peoples, it became so completely polytheistic that even conjecture can
discover but few indications or 'survivals' of the totemism from which
it is supposed to have developed; and the polytheism of the Israelites
was so completely superseded by monotheism that the very existence of an
earlier stage of polytheism could be as strenuously denied in their case
as the pre-existence of totemism could be denied amongst the
polytheistic Aryans. Nevertheless the theory was that if we represent
the growth of the various religions of the world diagrammatically by
vertical lines parallel to one another but of various lengths, one line
standing for totemism, a longer line for polydaemonism, a still longer
one for polytheism, and the longest of all for monotheism, we should see
that the line of growth has been the same in all cases, and that it is
in their length, or (shall we say?) in their height alone that the
various lines differ, and that the longest line culminates in monotheism
only because it has been, so to speak, pulled out in the same way that a
telescope when closed, may be extended. The evolution of religion on
this view has been a process literally of 'evolution' or unfolding: the
idea of a god and of communion with him has been present from the
beginning; and, much though religion may have changed, it remains to the
end essentially the same thing. This view of the process of religious
evolution we may fairly term 'the pre-formation theory'; for on this
theory each stage is pre-formed in the stage immediately preceding it,
and in the earliest stage of all were all succeeding stages contained
pre-formed, though it depended on circumstances whether the seed should
spring up, and how many stages of its growth it should accomplish.
Robertson Smith's theory of the evolution of religion is in effect a
form of the pre-formation theory, and is liable to the same difficulties
as dog the theory of pre-formation in all its applications. On the
theory, if we cut open a seed we should find within it the plant
pre-formed; if we analyse totemism, the seed from which, in Robertson
Smith's view, all other forms of religion have grown in orderly
succession one after the other, we find in it religion in all its stages
pre-formed. In fact, however, if we cut open an acorn we do not find a
miniature oak-tree inside. The presumption, therefore, is that neither
in totemism, if we dissect it, shall we find religion pre-formed.
Indeed, there is the possibility that totemism, on dissection, will be
found to have no such content--that the hope or expectation of finding
anything in it is as vain, is as much doomed to disappointment, as is
the expectation of the child who cuts open his drum, thinking to find
inside it something which produces the sound.