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 "_durée_": 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.