No More Learning

[30] On this subject, compare _A Theory of Time and Space_, by Mr.
A. A. Robb (Camb. Univ. Press), which first suggested to me the views
advocated here, though I have, for present purposes, omitted what is
most interesting and novel in his theory. Mr. Robb has given a sketch
of his theory in a pamphlet with the same title (Heffer and Sons,
Cambridge, 1913).

[31] "Natural Realism and Present Tendencies in Philosophy," _Proc.
Arist. Soc. _, 1908-1909, p. 165.

[32] _Die Erfahrungsgrundlagen unseres Wissens_, p. 28.

[33] Cf. _Principia Mathematica_, Vol. I, * 14, and Introduction,
Chap. III. For the definition of _existence_, cf. * 14. 02.

[34] Cf. Edwin B. Holt, _The Place of Illusory Experience in a
Realistic World. _ "The New Realism," p. 303, both on this point and as
regards _seeing double_.




IX

ON THE NOTION OF CAUSE


In the following paper I wish, first, to maintain that the word
"cause" is so inextricably bound up with misleading associations as to
make its complete extrusion from the philosophical vocabulary
desirable; secondly, to inquire what principle, if any, is employed in
science in place of the supposed "law of causality" which philosophers
imagine to be employed; thirdly, to exhibit certain confusions,
especially in regard to teleology and determinism, which appear to me
to be connected with erroneous notions as to causality.

All philosophers, of every school, imagine that causation is one of
the fundamental axioms or postulates of science, yet, oddly enough, in
advanced sciences such as gravitational astronomy, the word "cause"
never occurs. Dr. James Ward, in his _Naturalism and Agnosticism_,
makes this a ground of complaint against physics: the business of
those who wish to ascertain the ultimate truth about the world, he
apparently thinks, should be the discovery of causes, yet physics
never even seeks them. To me it seems that philosophy ought not to
assume such legislative functions, and that the reason why physics has
ceased to look for causes is that, in fact, there are no such things.
The law of causality, I believe, like much that passes muster among
philosophers, is a relic of a bygone age, surviving, like the
monarchy, only because it is erroneously supposed to do no harm. In
order to find out what philosophers commonly understand by "cause," I
consulted Baldwin's _Dictionary_, and was rewarded beyond my
expectations, for I found the following three mutually incompatible
definitions:--

"CAUSALITY. (1) The necessary connection of events in the
time-series. . . .

"CAUSE (notion of). Whatever may be included in the thought or
perception of a process as taking place in consequence of
another process. . . .

"CAUSE AND EFFECT. (1) Cause and effect . . . are correlative terms
denoting any two distinguishable things, phases, or aspects of
reality, which are so related to each other that whenever the
first ceases to exist the second comes into existence
immediately after, and whenever the second comes into existence
the first has ceased to exist immediately before. "

Let us consider these three definitions in turn. The first, obviously,
is unintelligible without a definition of "necessary. " Under this
head, Baldwin's _Dictionary_ gives the following:--

"NECESSARY. That is necessary which not only is true, but would
be true under all circumstances. Something more than brute
compulsion is, therefore, involved in the conception; there is
a general law under which the thing takes place. "

The notion of cause is so intimately connected with that of necessity
that it will be no digression to linger over the above definition,
with a view to discovering, if possible, _some_ meaning of which it is
capable; for, as it stands, it is very far from having any definite
signification.

The first point to notice is that, if any meaning is to be given to
the phrase "would be true under all circumstances," the subject of it
must be a propositional function, not a proposition. [35] A
proposition is simply true or false, and that ends the matter: there
can be no question of "circumstances. " "Charles I's head was cut off"
is just as true in summer as in winter, on Sundays as on Mondays. Thus
when it is worth saying that something "would be true under all
circumstances," the something in question must be a propositional
function, i. e. an expression containing a variable, and becoming a
proposition when a value is assigned to the variable; the varying
"circumstances" alluded to are then the different values of which the
variable is capable. Thus if "necessary" means "what is true under all
circumstances," then "if _x_ is a man, _x_ is mortal" is necessary,
because it is true for any possible value of _x_. Thus we should be
led to the following definition:--

"NECESSARY is a predicate of a propositional function, meaning
that it is true for all possible values of its argument or
arguments. "

Unfortunately, however, the definition in Baldwin's _Dictionary_ says
that what is necessary is not only "true under all circumstances" but
is also "true. " Now these two are incompatible. Only propositions can
be "true," and only propositional functions can be "true under all
circumstances. " Hence the definition as it stands is nonsense. What is
meant seems to be this: "A proposition is necessary when it is a value
of a propositional function which is true under all circumstances,
i. e. for all values of its argument or arguments. " But if we adopt
this definition, the same proposition will be necessary or contingent
according as we choose one or other of its terms as the argument to
our propositional function. For example, "if Socrates is a man,
Socrates is mortal," is necessary if Socrates is chosen as argument,
but not if _man_ or _mortal_ is chosen. Again, "if Socrates is a man,
Plato is mortal," will be necessary if either Socrates or _man_ is
chosen as argument, but not if Plato or _mortal_ is chosen. However,
this difficulty can be overcome by specifying the constituent which is
to be regarded as argument, and we thus arrive at the following
definition:

"A proposition is _necessary_ with respect to a given constituent if
it remains true when that constituent is altered in any way compatible
with the proposition remaining significant. "

We may now apply this definition to the definition of causality quoted
above. It is obvious that the argument must be the time at which the
earlier event occurs. Thus an instance of causality will be such as:
"If the event [Math: e_{1}] occurs at the time [Math: t_{1}], it will
be followed by the event [Math: e_{2}]. " This proposition is intended
to be necessary with respect to [Math: t_{1}], i. e. to remain true
however [Math: t_{1}] may be varied. Causality, as a universal law,
will then be the following: "Given any event [Math: t_{1}], there is
an event [Math: e_{2}] such that, whenever [Math: t_{1}] occurs,
[Math: e_{2}] occurs later. " But before this can be considered
precise, we must specify how much later [Math: e_{2}] is to occur.
Thus the principle becomes:--

"Given any event [Math: e_{1}], there is an event [Math: e_{2}] and a
time-interval ? such that, whenever [Math: e_{1}] occurs, [Math:
e_{2}] follows after an interval ? . "

I am not concerned as yet to consider whether this law is true or
false. For the present, I am merely concerned to discover what the law
of causality is supposed to be. I pass, therefore, to the other
definitions quoted above.

The second definition need not detain us long, for two reasons. First,
because it is psychological: not the "thought or perception" of a
process, but the process itself, must be what concerns us in
considering causality. Secondly, because it is circular: in speaking
of a process as "taking place in consequence of" another process, it
introduces the very notion of cause which was to be defined.

The third definition is by far the most precise; indeed as regards
clearness it leaves nothing to be desired. But a great difficulty is
caused by the temporal contiguity of cause and effect which the
definition asserts. No two instants are contiguous, since the
time-series is compact; hence either the cause or the effect or both
must, if the definition is correct, endure for a finite time; indeed,
by the wording of the definition it is plain that both are assumed to
endure for a finite time. But then we are faced with a dilemma: if the
cause is a process involving change within itself, we shall require
(if causality is universal) causal relations between its earlier and
later parts; moreover, it would seem that only the later parts can be
relevant to the effect, since the earlier parts are not contiguous to
the effect, and therefore (by the definition) cannot influence the
effect. Thus we shall be led to diminish the duration of the cause
without limit, and however much we may diminish it, there will still
remain an earlier part which might be altered without altering the
effect, so that the true cause, as defined, will not have been
reached, for it will be observed that the definition excludes
plurality of causes. If, on the other hand, the cause is purely
static, involving no change within itself, then, in the first place,
no such cause is to be found in nature, and in the second place, it
seems strange--too strange to be accepted, in spite of bare logical
possibility--that the cause, after existing placidly for some time,
should suddenly explode into the effect, when it might just as well
have done so at any earlier time, or have gone on unchanged without
producing its effect. This dilemma, therefore, is fatal to the view
that cause and effect can be contiguous in time; if there are causes
and effects, they must be separated by a finite time-interval ? , as
was assumed in the above interpretation of the first definition.

What is essentially the same statement of the law of causality as the
one elicited above from the first of Baldwin's definitions is given by
other philosophers. Thus John Stuart Mill says:--

"The Law of Causation, the recognition of which is the main pillar of
inductive science, is but the familiar truth, that invariability of
succession is found by observation to obtain between every fact in
nature and some other fact which has preceded it. "[36]

And Bergson, who has rightly perceived that the law as stated by
philosophers is worthless, nevertheless continues to suppose that it
is used in science. Thus he says:--

"Now, it is argued, this law [the law of causality] means that every
phenomenon is determined by its conditions, or, in other words, that
the same causes produce the same effects. "[37]

And again:--

"We perceive physical phenomena, and these phenomena obey laws. This
means: (1) That phenomena _a_, _b_, _c_, _d_, previously perceived,
can occur again in the same shape; (2) that a certain phenomenon P,
which appeared after the conditions _a_, _b_, _c_, _d_, and after
these conditions only, will not fail to recur as soon as the same
conditions are again present. "[38]

A great part of Bergson's attack on science rests on the assumption
that it employs this principle. In fact, it employs no such principle,
but philosophers--even Bergson--are too apt to take their views on
science from each other, not from science. As to what the principle
is, there is a fair consensus among philosophers of different schools.
There are, however, a number of difficulties which at once arise. I
omit the question of plurality of causes for the present, since other
graver questions have to be considered. Two of these, which are forced
on our attention by the above statement of the law, are the
following:--

(1) What is meant by an "event"?

(2) How long may the time-interval be between cause and effect?

(1) An "event," in the statement of the law, is obviously intended to
be something that is likely to recur since otherwise the law becomes
trivial. It follows that an "event" is not a particular, but some
universal of which there may be many instances. It follows also that
an "event" must be something short of the whole state of the universe,
since it is highly improbable that this will recur. What is meant by
an "event" is something like striking a match, or dropping a penny
into the slot of an automatic machine. If such an event is to recur,
it must not be defined too narrowly: we must not state with what
degree of force the match is to be struck, nor what is to be the
temperature of the penny. For if such considerations were relevant,
our "event" would occur at most once, and the law would cease to give
information. An "event," then, is a universal defined sufficiently
widely to admit of many particular occurrences in time being instances
of it.

(2) The next question concerns the time-interval. Philosophers, no
doubt, think of cause and effect as contiguous in time, but this, for
reasons already given, is impossible. Hence, since there are no
infinitesimal time-intervals, there must be some finite lapse of time
? between cause and effect. This, however, at once raises insuperable
difficulties. However short we make the interval ? , something may
happen during this interval which prevents the expected result. I put
my penny in the slot, but before I can draw out my ticket there is an
earthquake which upsets the machine and my calculations. In order to
be sure of the expected effect, we must know that there is nothing in
the environment to interfere with it. But this means that the supposed
cause is not, by itself, adequate to insure the effect. And as soon as
we include the environment, the probability of repetition is
diminished, until at last, when the whole environment is included, the
probability of repetition becomes almost _nil_.

In spite of these difficulties, it must, of course, be admitted that
many fairly dependable regularities of sequence occur in daily life.
It is these regularities that have suggested the supposed law of
causality; where they are found to fail, it is thought that a better
formulation could have been found which would have never failed. I am
far from denying that there may be such sequences which in fact never
do fail. It may be that there will never be an exception to the rule
that when a stone of more than a certain mass, moving with more than a
certain velocity, comes in contact with a pane of glass of less than
a certain thickness, the glass breaks. I also do not deny that the
observation of such regularities, even when they are not without
exceptions, is useful in the infancy of a science: the observation
that unsupported bodies in air usually fall was a stage on the way to
the law of gravitation. What I deny is that science assumes the
existence of invariable uniformities of sequence of this kind, or that
it aims at discovering them. All such uniformities, as we saw, depend
upon a certain vagueness in the definition of the "events. " That
bodies fall is a vague qualitative statement; science wishes to know
how fast they fall. This depends upon the shape of the bodies and the
density of the air. It is true that there is more nearly uniformity
when they fall in a vacuum; so far as Galileo could observe, the
uniformity is then complete. But later it appeared that even there the
latitude made a difference, and the altitude. Theoretically, the
position of the sun and moon must make a difference. In short, every
advance in a science takes us farther away from the crude uniformities
which are first observed, into greater differentiation of antecedent
and consequent, and into a continually wider circle of antecedents
recognised as relevant.

The principle "same cause, same effect," which philosophers imagine to
be vital to science, is therefore utterly otiose. As soon as the
antecedents have been given sufficiently fully to enable the
consequent to be calculated with some exactitude, the antecedents have
become so complicated that it is very unlikely they will ever recur.
Hence, if this were the principle involved, science would remain
utterly sterile.

The importance of these considerations lies partly in the fact that
they lead to a more correct account of scientific procedure, partly in
the fact that they remove the analogy with human volition which makes
the conception of cause such a fruitful source of fallacies. The
latter point will become clearer by the help of some illustrations.
For this purpose I shall consider a few maxims which have played a
great part in the history of philosophy.

(1) "Cause and effect must more or less resemble each other. " This
principle was prominent in the philosophy of occasionalism, and is
still by no means extinct. It is still often thought, for example,
that mind could not have grown up in a universe which previously
contained nothing mental, and one ground for this belief is that
matter is too dissimilar from mind to have been able to cause it. Or,
more particularly, what are termed the nobler parts of our nature are
supposed to be inexplicable, unless the universe always contained
something at least equally noble which could cause them. All such
views seem to depend upon assuming some unduly simplified law of
causality; for, in any legitimate sense of "cause" and "effect,"
science seems to show that they are usually very widely dissimilar,
the "cause" being, in fact, two states of the whole universe, and the
"effect" some particular event.

(2) "Cause is analogous to volition, since there must be an
intelligible _nexus_ between cause and effect. " This maxim is, I
think, often unconsciously in the imaginations of philosophers who
would reject it when explicitly stated. It is probably operative in
the view we have just been considering, that mind could not have
resulted from a purely material world. I do not profess to know what
is meant by "intelligible"; it seems to mean "familiar to
imagination. " Nothing is less "intelligible," in any other sense, than
the connection between an act of will and its fulfilment. But
obviously the sort of nexus desired between cause and effect is such
as could only hold between the "events" which the supposed law of
causality contemplates; the laws which replace causality in such a
science as physics leave no room for any two events between which a
nexus could be sought.

(3) "The cause _compels_ the effect in some sense in which the effect
does not compel the cause. " This belief seems largely operative in the
dislike of determinism; but, as a matter of fact, it is connected with
our second maxim, and falls as soon as that is abandoned. We may
define "compulsion" as follows: "Any set of circumstances is said to
compel A when A desires to do something which the circumstances
prevent, or to abstain from something which the circumstances cause. "
This presupposes that some meaning has been found for the word
"cause"--a point to which I shall return later. What I want to make
clear at present is that compulsion is a very complex notion,
involving thwarted desire. So long as a person does what he wishes to
do, there is no compulsion, however much his wishes may be calculable
by the help of earlier events. And where desire does not come in,
there can be no question of compulsion. Hence it is, in general,
misleading to regard the cause as compelling the effect.

A vaguer form of the same maxim substitutes the word "determine" for
the word "compel"; we are told that the cause _determines_ the effect
in a sense in which the effect does not _determine_ the cause. It is
not quite clear what is meant by "determining"; the only precise
sense, so far as I know, is that of a function or one-many relation.
If we admit plurality of causes, but not of effects, that is, if we
suppose that, given the cause, the effect must be such and such, but,
given the effect, the cause may have been one of many alternatives,
then we may say that the cause determines the effect, but not the
effect the cause. Plurality of causes, however, results only from
conceiving the effect vaguely and narrowly and the cause precisely and
widely. Many antecedents may "cause" a man's death, because his death
is vague and narrow. But if we adopt the opposite course, taking as
the "cause" the drinking of a dose of arsenic, and as the "effect" the
whole state of the world five minutes later, we shall have plurality
of effects instead of plurality of causes. Thus the supposed lack of
symmetry between "cause" and "effect" is illusory.

(4) "A cause cannot operate when it has ceased to exist, because what
has ceased to exist is nothing. " This is a common maxim, and a still
more common unexpressed prejudice. It has, I fancy, a good deal to do
with the attractiveness of Bergson's "_duree_": since the past has
effects now, it must still exist in some sense. The mistake in this
maxim consists in the supposition that causes "operate" at all. A
volition "operates" when what it wills takes place; but nothing can
operate except a volition. The belief that causes "operate" results
from assimilating them, consciously or unconsciously, to volitions. We
have already seen that, if there are causes at all, they must be
separated by a finite interval of time from their effects, and thus
cause their effects after they have ceased to exist.

It may be objected to the above definition of a volition "operating"
that it only operates when it "causes" what it wills, not when it
merely happens to be followed by what it wills. This certainly
represents the usual view of what is meant by a volition "operating,"
but as it involves the very view of causation which we are engaged in
combating, it is not open to us as a definition. We may say that a
volition "operates" when there is some law in virtue of which a
similar volition in rather similar circumstances will usually be
followed by what it wills. But this is a vague conception, and
introduces ideas which we have not yet considered. What is chiefly
important to notice is that the usual notion of "operating" is not
open to us if we reject, as I contend that we should, the usual notion
of causation.

(5) "A cause cannot operate except where it is. " This maxim is very
widespread; it was urged against Newton, and has remained a source of
prejudice against "action at a distance. " In philosophy it has led to
a denial of transient action, and thence to monism or Leibnizian
monadism. Like the analogous maxim concerning temporal contiguity, it
rests upon the assumption that causes "operate," i. e. that they are in
some obscure way analogous to volitions. And, as in the case of
temporal contiguity, the inferences drawn from this maxim are wholly
groundless.

I return now to the question, What law or laws can be found to take
the place of the supposed law of causality?

First, without passing beyond such uniformities of sequence as are
contemplated by the traditional law, we may admit that, if any such
sequence has been observed in a great many cases, and has never been
found to fail, there is an inductive probability that it will be found
to hold in future cases. If stones have hitherto been found to break
windows, it is probable that they will continue to do so. This, of
course, assumes the inductive principle, of which the truth may
reasonably be questioned; but as this principle is not our present
concern, I shall in this discussion treat it as indubitable. We may
then say, in the case of any such frequently observed sequence, that
the earlier event is the _cause_ and the later event the _effect_.

Several considerations, however, make such special sequences very
different from the traditional relation of cause and effect. In the
first place, the sequence, in any hitherto unobserved instance, is no
more than probable, whereas the relation of cause and effect was
supposed to be necessary.
I do not mean by this merely that we are not
sure of having discovered a true case of cause and effect; I mean
that, even when we have a case of cause and effect in our present
sense, all that is meant is that on grounds of observation, it is
probable that when one occurs the other will also occur. Thus in our
present sense, A may be the cause of B even if there actually are
cases where B does not follow A. Striking a match will be the cause of
its igniting, in spite of the fact that some matches are damp and fail
to ignite.

In the second place, it will not be assumed that _every_ event has
some antecedent which is its cause in this sense; we shall only
believe in causal sequences where we find them, without any
presumption that they always are to be found.

In the third place, _any_ case of sufficiently frequent sequence will
be causal in our present sense; for example, we shall not refuse to
say that night is the cause of day. Our repugnance to saying this
arises from the ease with which we can imagine the sequence to fail,
but owing to the fact that cause and effect must be separated by a
finite interval of time, _any_ such sequence _might_ fail through the
interposition of other circumstances in the interval. Mill, discussing
this instance of night and day, says:--

"It is necessary to our using the word cause, that we should believe
not only that the antecedent always _has_ been followed by the
consequent, but that as long as the present constitution of things
endures, it always _will_ be so. "[39]

In this sense, we shall have to give up the hope of finding causal
laws such as Mill contemplated; any causal sequence which we have
observed may at any moment be falsified without a falsification of any
laws of the kind that the more advanced sciences aim at establishing.

In the fourth place, such laws of probable sequence, though useful in
daily life and in the infancy of a science, tend to be displaced by
quite different laws as soon as a science is successful. The law of
gravitation will illustrate what occurs in any advanced science. In
the motions of mutually gravitating bodies, there is nothing that can
be called a cause, and nothing that can be called an effect; there is
merely a formula. Certain differential equations can be found, which
hold at every instant for every particle of the system, and which,
given the configuration and velocities at one instant, or the
configurations at two instants, render the configuration at any other
earlier or later instant theoretically calculable. That is to say, the
configuration at any instant is a function of that instant and the
configurations at two given instants. This statement holds throughout
physics, and not only in the special case of gravitation. But there is
nothing that could be properly called "cause" and nothing that could
be properly called "effect" in such a system.

No doubt the reason why the old "law of causality" has so long
continued to pervade the books of philosophers is simply that the idea
of a function is unfamiliar to most of them, and therefore they seek
an unduly simplified statement. There is no question of repetitions of
the "same" cause producing the "same" effect; it is not in any
sameness of causes and effects that the constancy of scientific law
consists, but in sameness of relations. And even "sameness of
relations" is too simple a phrase; "sameness of differential
equations" is the only correct phrase. It is impossible to state this
accurately in non-mathematical language; the nearest approach would be
as follows: "There is a constant relation between the state of the
universe at any instant and the rate of change in the rate at which
any part of the universe is changing at that instant, and this
relation is many-one, i. e. such that the rate of change in the rate of
change is determinate when the state of the universe is given. " If the
"law of causality" is to be something actually discoverable in the
practice of science, the above proposition has a better right to the
name than any "law of causality" to be found in the books of
philosophers.

In regard to the above principle, several observations must be made--

(1) No one can pretend that the above principle is _a priori_ or
self-evident or a "necessity of thought. " Nor is it, in any sense, a
premiss of science: it is an empirical generalisation from a number of
laws which are themselves empirical generalisations.

(2) The law makes no difference between past and future: the future
"determines" the past in exactly the same sense in which the past
"determines" the future. The word "determine," here, has a purely
logical significance: a certain number of variables "determine"
another variable if that other variable is a function of them.

(3) The law will not be empirically verifiable unless the course of
events within some sufficiently small volume will be approximately
the same in any two states of the universe which only differ in regard
to what is at a considerable distance from the small volume in
question. For example, motions of planets in the solar system must be
approximately the same however the fixed stars may be distributed,
provided that all the fixed stars are very much farther from the sun
than the planets are. If gravitation varied directly as the distance,
so that the most remote stars made the most difference to the motions
of the planets, the world might be just as regular and just as much
subject to mathematical laws as it is at present, but we could never
discover the fact.

(4) Although the old "law of causality" is not assumed by science,
something which we may call the "uniformity of nature" is assumed, or
rather is accepted on inductive grounds. The uniformity of nature does
not assert the trivial principle "same cause, same effect," but the
principle of the permanence of laws. That is to say, when a law
exhibiting, e. g. an acceleration as a function of the configuration
has been found to hold throughout the observable past, it is expected
that it will continue to hold in the future, or that, if it does not
itself hold, there is some other law, agreeing with the supposed law
as regards the past, which will hold for the future. The ground of
this principle is simply the inductive ground that it has been found
to be true in very many instances; hence the principle cannot be
considered certain, but only probable to a degree which cannot be
accurately estimated.

The uniformity of nature, in the above sense, although it is assumed
in the practice of science, must not, in its generality, be regarded
as a kind of major premiss, without which all scientific reasoning
would be in error. The assumption that _all_ laws of nature are
permanent has, of course, less probability than the assumption that
this or that particular law is permanent; and the assumption that a
particular law is permanent for all time has less probability than the
assumption that it will be valid up to such and such a date. Science,
in any given case, will assume what the case requires, but no more. In
constructing the _Nautical Almanac_ for 1915 it will assume that the
law of gravitation will remain true up to the end of that year; but it
will make no assumption as to 1916 until it comes to the next volume
of the almanac. This procedure is, of course, dictated by the fact
that the uniformity of nature is not known _a priori_, but is an
empirical generalisation, like "all men are mortal. " In all such
cases, it is better to argue immediately from the given particular
instances to the new instance, than to argue by way of a major
premiss; the conclusion is only probable in either case, but acquires
a higher probability by the former method than by the latter.

In all science we have to distinguish two sorts of laws: first, those
that are empirically verifiable but probably only approximate;
secondly, those that are not verifiable, but may be exact. The law of
gravitation, for example, in its applications to the solar system, is
only empirically verifiable when it is assumed that matter outside the
solar system may be ignored for such purposes; we believe this to be
only approximately true, but we cannot empirically verify the law of
universal gravitation which we believe to be exact. This point is very
important in connection with what we may call "relatively isolated
systems. " These may be defined as follows:--

A system relatively isolated during a given period is one which,
within some assignable margin of error, will behave in the same way
throughout that period, however the rest of the universe may be
constituted.

A system may be called "practically isolated" during a given period
if, although there _might_ be states of the rest of the universe which
would produce more than the assigned margin of error, there is reason
to believe that such states do not in fact occur.

Strictly speaking, we ought to specify the respect in which the system
is relatively isolated. For example, the earth is relatively isolated
as regards falling bodies, but not as regards tides; it is
_practically_ isolated as regards economic phenomena, although, if
Jevons' sunspot theory of commercial crises had been true, it would
not have been even practically isolated in this respect.

It will be observed that we cannot prove in advance that a system is
isolated. This will be inferred from the observed fact that
approximate uniformities can be stated for this system alone. If the
complete laws for the whole universe were known, the isolation of a
system could be deduced from them; assuming, for example, the law of
universal gravitation, the practical isolation of the solar system in
this respect can be deduced by the help of the fact that there is very
little matter in its neighbourhood. But it should be observed that
isolated systems are only important as providing a possibility of
_discovering_ scientific laws; they have no theoretical importance in
the finished structure of a science.

The case where one event A is said to "cause" another event B, which
philosophers take as fundamental, is really only the most simplified
instance of a practically isolated system. It may happen that, as a
result of general scientific laws, whenever A occurs throughout a
certain period, it is followed by B; in that case, A and B form a
system which is practically isolated throughout that period. It is,
however, to be regarded as a piece of good fortune if this occurs; it
will always be due to special circumstances, and would not have been
true if the rest of the universe had been different though subject to
the same laws.

The essential function which causality has been supposed to perform is
the possibility of inferring the future from the past, or, more
generally, events at any time from events at certain assigned times.
Any system in which such inference is possible may be called a
"deterministic" system. We may define a deterministic system as
follows:--

A system is said to be "deterministic" when, given certain data,
[Math: e_{1}, e_{2}, . . . , e_{n}, at times t_{1}, t_{2}, . . . ,
t_{n}] respectively, concerning this system, if [Math: E_{t}] is
the state of the system at any time _t_, there is a functional
relation of the form

[Math: E_{t} = f (e_{1}, t_{1}, e_{2}, t_{2}, . . . , e_{n}, t_{n}, t)]. (A)

The system will be "deterministic throughout a given period" if
_t_, in the above formula, may be any time within that period,
though outside that period the formula may be no longer true. If
the universe, as a whole, is such a system, determinism is true of
the universe; if not, not. A system which is part of a
deterministic system I shall call "determined"; one which is not
part of any such system I shall call "capricious. "

The events [Math: e_{1}, e_{2}, . . . , e_{n}] I shall call "determinants"
of the system. It is to be observed that a system which has one set of
determinants will in general have many. In the case of the motions of
the planets, for example, the configurations of the solar system at any
two given times will be determinants.

We may take another illustration from the hypothesis of
psycho-physical parallelism. Let us assume, for the purposes of this
illustration, that to a given state of brain a given state of mind
always corresponds, and vice versa, i. e. that there is a one-one
relation between them, so that each is a function of the other. We may
also assume, what is practically certain, that to a given state of a
certain brain a given state of the whole material universe
corresponds, since it is highly improbable that a given brain is ever
twice in exactly the same state. Hence there will be a one-one
relation between the state of a given person's mind and the state of
the whole material universe. It follows that, if _n_ states of the
material universe are determinants of the material universe, then _n_
states of a given man's mind are determinants of the whole material
and mental universe--assuming, that is to say, that psycho-physical
parallelism is true.

The above illustration is important in connection with a certain
confusion which seems to have beset those who have philosophised on
the relation of mind and matter. It is often thought that, if the
state of the mind is determinate when the state of the brain is given,
and if the material world forms a deterministic system, then mind is
"subject" to matter in some sense in which matter is not "subject" to
mind. But if the state of the brain is also determinate when the state
of the mind is given, it must be exactly as true to regard matter as
subject to mind as it would be to regard mind as subject to matter. We
could, theoretically, work out the history of mind without ever
mentioning matter, and then, at the end, deduce that matter must
meanwhile have gone through the corresponding history. It is true that
if the relation of brain to mind were many-one, not one-one, there
would be a one-sided dependence of mind on brain, while conversely, if
the relation were one-many, as Bergson supposes, there would be a
one-aided dependence of brain on mind. But the dependence involved is,
in any case, only logical; it does not mean that we shall be
compelled to do things we desire not to do, which is what people
instinctively imagine it to mean.

As another illustration we may take the case of mechanism and
teleology. A system may be defined as "mechanical" when it has a set
of determinants that are purely material, such as the positions of
certain pieces of matter at certain times. It is an open question
whether the world of mind and matter, as we know it, is a mechanical
system or not; let us suppose, for the sake of argument, that it is a
mechanical system. This supposition--so I contend--throws no light
whatever on the question whether the universe is or is not a
"teleological" system. It is difficult to define accurately what is
meant by a "teleological" system, but the argument is not much
affected by the particular definition we adopt. Broadly, a
teleological system is one in which purposes are realised, i. e. in
which certain desires--those that are deeper or nobler or more
fundamental or more universal or what not--are followed by their
realisation. Now the fact--if it be a fact--that the universe is
mechanical has no bearing whatever on the question whether it is
teleological in the above sense. There might be a mechanical system in
which all wishes were realised, and there might be one in which all
wishes were thwarted. The question whether, or how far, our actual
world is teleological, cannot, therefore, be settled by proving that
it is mechanical, and the desire that it should be teleological is no
ground for wishing it to be not mechanical.

There is, in all these questions, a very great difficulty in avoiding
confusion between what we can infer and what is in fact determined.
Let us consider, for a moment, the various senses in which the future
may be "determined. " There is one sense--and a very important one--in
which it is determined quite independently of scientific laws, namely,
the sense that it will be what it will be. We all regard the past as
determined simply by the fact that it has happened; but for the
accident that memory works backward and not forward, we should regard
the future as equally determined by the fact that it will happen.
"But," we are told, "you cannot alter the past, while you can to some
extent alter the future. " This view seems to me to rest upon just
those errors in regard to causation which it has been my object to
remove. You cannot make the past other than it was--true, but this is
a mere application of the law of contradiction. If you already know
what the past was, obviously it is useless to wish it different. But
also you cannot make the future other than it will be; this again is
an application of the law of contradiction. And if you happen to know
the future--e. g. in the case of a forthcoming eclipse--it is just as
useless to wish it different as to wish the past different. "But," it
will be rejoined, "our wishes can _cause_ the future, sometimes, to be
different from what it would be if they did not exist, and they can
have no such effect upon the past. " This, again, is a mere tautology.
An effect being _defined_ as something subsequent to its cause,
obviously we can have no _effect_ upon the past. But that does not
mean that the past would not have been different if our present wishes
had been different. Obviously, our present wishes are conditioned by
the past, and therefore could not have been different unless the past
had been different; therefore, if our present wishes were different,
the past would be different. Of course, the past cannot be different
from what it was, but no more can our present wishes be different from
what they are; this again is merely the law of contradiction. The
facts seem to be merely (1) that wishing generally depends upon
ignorance, and is therefore commoner in regard to the future than in
regard to the past; (2) that where a wish concerns the future, it and
its realisation very often form a "practically independent system,"
i. e. many wishes regarding the future are realised. But there seems no
doubt that the main difference in our feelings arises from the
accidental fact that the past but not the future can be known by
memory.

Although the sense of "determined" in which the future is determined
by the mere fact that it will be what it will be is sufficient (at
least so it seems to me) to refute some opponents of determinism,
notably M. Bergson and the pragmatists, yet it is not what most people
have in mind when they speak of the future as determined. What they
have in mind is a formula by means of which the future can be
exhibited, and at least theoretically calculated, as a function of the
past. But at this point we meet with a great difficulty, which besets
what has been said above about deterministic systems, as well as what
is said by others.

If formulae of any degree of complexity, however great, are admitted,
it would seem that any system, whose state at a given moment is a
function of certain measurable quantities, must be a deterministic
system. Let us consider, in illustration, a single material particle,
whose co-ordinates at time _t_ are [Math: x_{t}, y_{t}, z_{t}]. Then,
however, the particle moves, there must be, theoretically, functions
[Math: f_{1}, f_{2}, f_{3}], such that

[Math: x_{t} = f_{t}(t), y_{t} = f_{2}(t), z_{t} = f_{3}(t). ]

It follows that, theoretically, the whole state of the material
universe at time _t_ must be capable of being exhibited as a function
of _t_. Hence our universe will be deterministic in the sense defined
above. But if this be true, no information is conveyed about the
universe in stating that it is deterministic. It is true that the
formulae involved may be of strictly infinite complexity, and therefore
not practically capable of being written down or apprehended. But
except from the point of view of our knowledge, this might seem to be
a detail: in itself, if the above considerations are sound, the
material universe _must_ be deterministic, _must_ be subject to laws.

This, however, is plainly not what was intended. The difference
between this view and the view intended may be seen as follows. Given
some formula which fits the facts hitherto--say the law of
gravitation--there will be an infinite number of other formulae, not
empirically distinguishable from it in the past, but diverging from it
more and more in the future. Hence, even assuming that there are
persistent laws, we shall have no reason for assuming that the law of
the inverse square will hold in future; it may be some other hitherto
indistinguishable law that will hold. We cannot say that _every_ law
which has held hitherto must hold in the future, because past facts
which obey one law will also obey others, hitherto indistinguishable
but diverging in future. Hence there must, at every moment, be laws
hitherto unbroken which are now broken for the first time. What
science does, in fact, is to select the _simplest_ formula that will
fit the facts. But this, quite obviously, is merely a methodological
precept, not a law of Nature. If the simplest formula ceases, after a
time, to be applicable, the simplest formula that remains applicable
is selected, and science has no sense that an axiom has been
falsified. We are thus left with the brute fact that, in many
departments of science, quite simple laws have hitherto been found to
hold. This fact cannot be regarded as having any _a priori_ ground,
nor can it be used to support inductively the opinion that the same
laws will continue; for at every moment laws hitherto true are being
falsified, though in the advanced sciences these laws are less simple
than those that have remained true. Moreover it would be fallacious to
argue inductively from the state of the advanced sciences to the
future state of the others, for it may well be that the advanced
sciences are advanced simply because, hitherto, their subject-matter
has obeyed simple and easily ascertainable laws, while the
subject-matter of other sciences has not done so.

The difficulty we have been considering seems to be met partly, if not
wholly, by the principle that the _time_ must not enter explicitly
into our formulae. All mechanical laws exhibit acceleration as a
function of configuration, not of configuration and time jointly; and
this principle of the irrelevance of the time may be extended to all
scientific laws. In fact we might interpret the "uniformity of nature"
as meaning just this, that no scientific law involves the time as an
argument, unless, of course, it is given in an integrated form, in
which case _lapse_ of time, though not absolute time, may appear in
our formulae. Whether this consideration suffices to overcome our
difficulty completely, I do not know; but in any case it does much to
diminish it.

It will serve to illustrate what has been said if we apply it to the
question of free will.

(1) Determinism in regard to the will is the doctrine that our
volitions belong to some deterministic system, i. e. are "determined"
in the sense defined above. Whether this doctrine is true or false, is
a mere question of fact; no _a priori_ considerations (if our previous
discussions have been correct) can exist on either side. On the one
hand, there is no _a priori_ category of causality, but merely certain
observed uniformities. As a matter of fact, there are observed
uniformities in regard to volitions; thus there is some empirical
evidence that volitions are determined. But it would be very rash to
maintain that the evidence is overwhelming, and it is quite possible
that some volitions, as well as some other things, are not determined,
except in the sense in which we found that everything must be
determined.

(2) But, on the other hand, the subjective sense of freedom, sometimes
alleged against determinism, has no bearing on the question whatever.
The view that it has a bearing rests upon the belief that causes
compel their effects, or that nature enforces obedience to its laws as
governments do. These are mere anthropomorphic superstitions, due to
assimilation of causes with volitions and of natural laws with human
edicts. We feel that our will is not compelled, but that only means
that it is not other than we choose it to be. It is one of the
demerits of the traditional theory of causality that it has created an
artificial opposition between determinism and the freedom of which we
are introspectively conscious.

(3) Besides the general question whether volitions are determined,
there is the further question whether they are _mechanically_
determined, i. e. whether they are part of what was above defined as a
mechanical system. This is the question whether they form part of a
system with purely material determinants, i. e. whether there are laws
which, given certain material data, make all volitions functions of
those data. Here again, there is empirical evidence up to a point, but
it is not conclusive in regard to all volitions. It is important to
observe, however that even if volitions are part of a mechanical
system, this by no means implies any supremacy of matter over mind. It
may well be that the same system which is susceptible of material
determinants is also susceptible of mental determinants; thus a
mechanical system may be determined by sets of volitions, as well as
by sets of material facts. It would seem, therefore, that the reasons
which make people dislike the view that volitions are mechanically
determined are fallacious.

(4) The notion of _necessity_, which is often associated with
determinism, is a confused notion not legitimately deducible from
determinism. Three meanings are commonly confounded when necessity is
spoken of:--

(? ) An _action_ is necessary when it will be performed however much
the agent may wish to do otherwise. Determinism does not imply that
actions are necessary in this sense.

(? ) A _propositional function_ is necessary when all its values are
true. This sense is not relevant to our present discussion.

(? ) A _proposition_ is necessary with respect to a given constituent
when it is the value, with that constituent as argument, of a
necessary propositional function, in other words, when it remains true
however that constituent may be varied. In this sense, in a
deterministic system, the connection of a volition with its
determinants is necessary, if the time at which the determinants occur
be taken as the constituent to be varied, the time-interval between
the determinants and the volition being kept constant. But this sense
of necessity is purely logical, and has no emotional importance.

We may now sum up our discussion of causality. We found first that the
law of causality, as usually stated by philosophers, is false, and is
not employed in science. We then considered the nature of scientific
laws, and found that, instead of stating that one event A is always
followed by another event B, they stated functional relations between
certain events at certain times, which we called determinants, and
other events at earlier or later times or at the same time. We were
unable to find any _a priori_ category involved: the existence of
scientific laws appeared as a purely empirical fact, not necessarily
universal, except in a trivial and scientifically useless form. We
found that a system with one set of determinants may very likely have
other sets of a quite different kind, that, for example, a
mechanically determined system may also be teleologically or
volitionally determined. Finally we considered the problem of free
will: here we found that the reasons for supposing volitions to be
determined are strong but not conclusive, and we decided that even if
volitions are mechanically determined, that is no reason for denying
freedom in the sense revealed by introspection, or for supposing that
mechanical events are not determined by volitions. The problem of free
will _versus_ determinism is therefore, if we were right, mainly
illusory, but in part not yet capable of being decisively solved.

FOOTNOTES:

[35] A propositional function is an expression containing a variable,
or undetermined constituent, and becoming a proposition as soon as a
definite value is assigned to the variable. Examples are: "A is A,"
"_x_ is a number. " The variable is called the _argument_ of the
function.

[36] _Logic_, Bk. III, Chap. V, ? 2.

[37] _Time and Free Will_, p. 199.

[38] _Time and Free Will.