mathematical demonstrations which distinguishes modern mathematicians
from their predecessors, and points out that this must lead to a
critical investigation of the definition of number. He proceeds to show
the inadequacy of previous philosophical theories, especially of the
"synthetic _a priori_" theory of Kant and the empirical theory of Mill.
This brings him to the question: What kind of object is it that number
can properly be ascribed to? He points out that physical things may be
regarded as one or many: for example, if a tree has a thousand leaves,
they may be taken altogether as constituting its foliage, which would
count as one, not as a thousand; and _one_ pair of boots is the same
object as _two_ boots. It follows that physical things are not the
subjects of which number is properly predicated; for when we have
discovered the proper subjects, the number to be ascribed must be
unambiguous. This leads to a discussion of the very prevalent view that
number is really something psychological and subjective, a view which
Frege emphatically rejects. "Number," he says, "is as little an object
of psychology or an outcome of psychical processes as the North Sea....
The botanist wishes to state something which is just as much a fact when
he gives the number of petals in a flower as when he gives its colour.
The one depends as little as the other upon our caprice. There is
therefore a certain similarity between number and colour; but this does
not consist in the fact that both are sensibly perceptible in external
things, but in the fact that both are objective" (p. 34).
"I distinguish the objective," he continues, "from the palpable, the
spatial, the actual. The earth's axis, the centre of mass of the solar
system, are objective, but I should not call them actual, like the earth
itself" (p. 35). He concludes that number is neither spatial and
physical, nor subjective, but non-sensible and objective. This
conclusion is important, since it applies to all the subject-matter of
mathematics and logic. Most philosophers have thought that the physical
and the mental between them exhausted the world of being. Some have
argued that the objects of mathematics were obviously not subjective,
and therefore must be physical and empirical; others have argued that
they were obviously not physical, and therefore must be subjective and
mental. Both sides were right in what they denied, and wrong in what
they asserted; Frege has the merit of accepting both denials, and
finding a third assertion by recognising the world of logic, which is
neither mental nor physical.
The fact is, as Frege points out, that no number, not even 1, is
applicable to physical things, but only to general terms or
descriptions, such as "man," "satellite of the earth," "satellite of
Venus." The general term "man" is applicable to a certain number of
objects: there are in the world so and so many men. The unity which
philosophers rightly feel to be necessary for the assertion of a number
is the unity of the general term, and it is the general term which is
the proper subject of number. And this applies equally when there is one
object or none which falls under the general term. "Satellite of the
earth" is a term only applicable to one object, namely, the moon. But
"one" is not a property of the moon itself, which may equally well be
regarded as many molecules: it is a property of the general term
"earth's satellite." Similarly, 0 is a property of the general term
"satellite of Venus," because Venus has no satellite. Here at last we
have an intelligible theory of the number 0. This was impossible if
numbers applied to physical objects, because obviously no physical
object could have the number 0. Thus, in seeking our definition of
number we have arrived so far at the result that numbers are properties
of general terms or general descriptions, not of physical things or of
mental occurrences.
Instead of speaking of a general term, such as "man," as the subject of
which a number can be asserted, we may, without making any serious
change, take the subject as the class or collection of objects--_i.e._
"mankind" in the above instance--to which the general term in question
is applicable. Two general terms, such as "man" and "featherless biped,"
which are applicable to the same collection of objects, will obviously
have the same number of instances; thus the number depends upon the
class, not upon the selection of this or that general term to describe
it, provided several general terms can be found to describe the same
class. But some general term is always necessary in order to describe a
class. Even when the terms are enumerated, as "this and that and the
other," the collection is constituted by the general property of being
either this, or that, or the other, and only so acquires the unity which
enables us to speak of it as _one_ collection. And in the case of an
infinite class, enumeration is impossible, so that description by a
general characteristic common and peculiar to the members of the class
is the only possible description. Here, as we see, the theory of number
to which Frege was led by purely logical considerations becomes of use
in showing how infinite classes can be amenable to number in spite of
being incapable of enumeration.
Frege next asks the question: When do two collections have the same
number of terms? In ordinary life, we decide this question by counting;
but counting, as we saw, is impossible in the case of infinite
collections, and is not logically fundamental with finite collections.
We want, therefore, a different method of answering our question. An
illustration may help to make the method clear. I do not know how many
married men there are in England, but I do know that the number is the
same as the number of married women. The reason I know this is that the
relation of husband and wife relates one man to one woman and one woman
to one man. A relation of this sort is called a one-one relation. The
relation of father to son is called a one-many relation, because a man
can have only one father but may have many sons; conversely, the
relation of son to father is called a many-one relation. But the
relation of husband to wife (in Christian countries) is called one-one,
because a man cannot have more than one wife, or a woman more than one
husband. Now, whenever there is a one-one relation between all the terms
of one collection and all the terms of another severally, as in the case
of English husbands and English wives, the number of terms in the one
collection is the same as the number in the other; but when there is not
such a relation, the number is different. This is the answer to the
question: When do two collections have the same number of terms?
We can now at last answer the question: What is meant by the number of
terms in a given collection? When there is a one-one relation between
all the terms of one collection and all the terms of another severally,
we shall say that the two collections are "similar." We have just seen
that two similar collections have the same number of terms. This leads
us to define the number of a given collection as the class of all
collections that are similar to it; that is to say, we set up the
following formal definition:
"The number of terms in a given class" is defined as meaning "the class
of all classes that are similar to the given class."
This definition, as Frege (expressing it in slightly different terms)
showed, yields the usual arithmetical properties of numbers. It is
applicable equally to finite and infinite numbers, and it does not
require the admission of some new and mysterious set of metaphysical
entities. It shows that it is not physical objects, but classes or the
general terms by which they are defined, of which numbers can be
asserted; and it applies to 0 and 1 without any of the difficulties
which other theories find in dealing with these two special cases.
The above definition is sure to produce, at first sight, a feeling of
oddity, which is liable to cause a certain dissatisfaction. It defines
the number 2, for instance, as the class of all couples, and the number
3 as the class of all triads. This does not _seem_ to be what we have
hitherto been meaning when we spoke of 2 and 3, though it would be
difficult to say _what_ we had been meaning. The answer to a feeling
cannot be a logical argument, but nevertheless the answer in this case
is not without importance. In the first place, it will be found that
when an idea which has grown familiar as an unanalysed whole is first
resolved accurately into its component parts--which is what we do when
we define it--there is almost always a feeling of unfamiliarity produced
by the analysis, which tends to cause a protest against the definition.
In the second place, it may be admitted that the definition, like all
definitions, is to a certain extent arbitrary. In the case of the small
finite numbers, such as 2 and 3, it would be possible to frame
definitions more nearly in accordance with our unanalysed feeling of
what we mean; but the method of such definitions would lack uniformity,
and would be found to fail sooner or later--at latest when we reached
infinite numbers.
In the third place, the real desideratum about such a definition as that
of number is not that it should represent as nearly as possible the
ideas of those who have not gone through the analysis required in order
to reach a definition, but that it should give us objects having the
requisite properties. Numbers, in fact, must satisfy the formulae of
arithmetic; any indubitable set of objects fulfilling this requirement
may be called numbers. So far, the simplest set known to fulfil this
requirement is the set introduced by the above definition. In comparison
with this merit, the question whether the objects to which the
definition applies are like or unlike the vague ideas of numbers
entertained by those who cannot give a definition, is one of very little
importance. All the important requirements are fulfilled by the above
definition, and the sense of oddity which is at first unavoidable will
be found to wear off very quickly with the growth of familiarity.
There is, however, a certain logical doctrine which may be thought to
form an objection to the above definition of numbers as classes of
classes--I mean the doctrine that there are no such objects as classes
at all. It might be thought that this doctrine would make havoc of a
theory which reduces numbers to classes, and of the many other theories
in which we have made use of classes. This, however, would be a mistake:
none of these theories are any the worse for the doctrine that classes
are fictions. What the doctrine is, and why it is not destructive, I
will try briefly to explain.
On account of certain rather complicated difficulties, culminating in
definite contradictions, I was led to the view that nothing that can be
said significantly about things, _i.e._ particulars, can be said
significantly (_i.e._ either truly or falsely) about classes of things.
That is to say, if, in any sentence in which a thing is mentioned, you
substitute a class for the thing, you no longer have a sentence that has
any meaning: the sentence is no longer either true or false, but a
meaningless collection of words. Appearances to the contrary can be
dispelled by a moment's reflection. For example, in the sentence, "Adam
is fond of apples," you may substitute _mankind_, and say, "Mankind is
fond of apples." But obviously you do not mean that there is one
individual, called "mankind," which munches apples: you mean that the
separate individuals who compose mankind are each severally fond of
apples.
Now, if nothing that can be said significantly about a thing can be said
significantly about a class of things, it follows that classes of things
cannot have the same kind of reality as things have; for if they had, a
class could be substituted for a thing in a proposition predicating the
kind of reality which would be common to both. This view is really
consonant to common sense. In the third or fourth century B.C. there
lived a Chinese philosopher named Hui Tzu, who maintained that "a bay
horse and a dun cow are three; because taken separately they are two,
and taken together they are one: two and one make three."[53] The author
from whom I quote says that Hui Tzu "was particularly fond of the
quibbles which so delighted the sophists or unsound reasoners of ancient
Greece," and this no doubt represents the judgment of common sense upon
such arguments. Yet if collections of things were things, his contention
would be irrefragable. It is only because the bay horse and the dun cow
taken together are not a new thing that we can escape the conclusion
that there are three things wherever there are two.
[53] Giles, _The Civilisation of China_ (Home University Library),
p. 147.
When it is admitted that classes are not things, the question arises:
What do we mean by statements which are nominally about classes? Take
such a statement as, "The class of people interested in mathematical
logic is not very numerous." Obviously this reduces itself to, "Not very
many people are interested in mathematical logic." For the sake of
definiteness, let us substitute some particular number, say 3, for "very
many." Then our statement is, "Not three people are interested in
mathematical logic." This may be expressed in the form: "If _x_ is
interested in mathematical logic, and also _y_ is interested, and also
_z_ is interested, then _x_ is identical with _y_, or _x_ is identical
with _z_, or _y_ is identical with _z_." Here there is no longer any
reference at all to a "class." In some such way, all statements
nominally about a class can be reduced to statements about what follows
from the hypothesis of anything's having the defining property of the
class. All that is wanted, therefore, in order to render the _verbal_
use of classes legitimate, is a uniform method of interpreting
propositions in which such a use occurs, so as to obtain propositions in
which there is no longer any such use. The definition of such a method
is a technical matter, which Dr Whitehead and I have dealt with
elsewhere, and which we need not enter into on this occasion.[54]
[54] Cf. _Principia Mathematica_, Sec. 20, and Introduction, chapter iii.
If the theory that classes are merely symbolic is accepted, it follows
that numbers are not actual entities, but that propositions in which
numbers verbally occur have not really any constituents corresponding to
numbers, but only a certain logical form which is not a part of
propositions having this form. This is in fact the case with all the
apparent objects of logic and mathematics. Such words as _or_, _not_,
_if_, _there is_, _identity_, _greater_, _plus_, _nothing_,
_everything_, _function_, and so on, are not names of definite objects,
like "John" or "Jones," but are words which require a context in order
to have meaning. All of them are _formal_, that is to say, their
occurrence indicates a certain form of proposition, not a certain
constituent. "Logical constants," in short, are not entities; the words
expressing them are not names, and cannot significantly be made into
logical subjects except when it is the words themselves, as opposed to
their meanings, that are being discussed.[55] This fact has a very
important bearing on all logic and philosophy, since it shows how they
differ from the special sciences. But the questions raised are so large
and so difficult that it is impossible to pursue them further on this
occasion.
[55] In the above remarks I am making use of unpublished work by my
friend Ludwig Wittgenstein.
LECTURE VIII
ON THE NOTION OF CAUSE, WITH APPLICATIONS TO THE FREE-WILL PROBLEM
The nature of philosophic analysis, as illustrated in our previous
lectures, can now be stated in general terms. We start from a body of
common knowledge, which constitutes our data. On examination, the data
are found to be complex, rather vague, and largely interdependent
logically. By analysis we reduce them to propositions which are as
nearly as possible simple and precise, and we arrange them in deductive
chains, in which a certain number of initial propositions form a logical
guarantee for all the rest. These initial propositions are _premisses_
for the body of knowledge in question. Premisses are thus quite
different from data--they are simpler, more precise, and less infected
with logical redundancy. If the work of analysis has been performed
completely, they will be wholly free from logical redundancy, wholly
precise, and as simple as is logically compatible with their leading to
the given body of knowledge. The discovery of these premisses belongs to
philosophy; but the work of deducing the body of common knowledge from
them belongs to mathematics, if "mathematics" is interpreted in a
somewhat liberal sense.
But besides the logical analysis of the common knowledge which forms our
data, there is the consideration of its degree of certainty. When we
have arrived at its premisses, we may find that some of them seem open
to doubt, and we may find further that this doubt extends to those of
our original data which depend upon these doubtful premisses. In our
third lecture, for example, we saw that the part of physics which
depends upon testimony, and thus upon the existence of other minds than
our own, does not seem so certain as the part which depends exclusively
upon our own sense-data and the laws of logic. Similarly, it used to be
felt that the parts of geometry which depend upon the axiom of parallels
have less certainty than the parts which are independent of this
premiss. We may say, generally, that what commonly passes as knowledge
is not all equally certain, and that, when analysis into premisses has
been effected, the degree of certainty of any consequence of the
premisses will depend upon that of the most doubtful premiss employed in
proving this consequence. Thus analysis into premisses serves not only a
logical purpose, but also the purpose of facilitating an estimate as to
the degree of certainty to be attached to this or that derivative
belief. In view of the fallibility of all human beliefs, this service
seems at least as important as the purely logical services rendered by
philosophical analysis.
In the present lecture, I wish to apply the analytic method to the
notion of "cause," and to illustrate the discussion by applying it to
the problem of free will. For this purpose I shall inquire: I., what is
meant by a causal law; II., what is the evidence that causal laws have
held hitherto; III., what is the evidence that they will continue to
hold in the future; IV., how the causality which is used in science
differs from that of common sense and traditional philosophy; V., what
new light is thrown on the question of free will by our analysis of the
notion of "cause."
I. By a "causal law" I mean any general proposition in virtue of which
it is possible to infer the existence of one thing or event from the
existence of another or of a number of others. If you hear thunder
without having seen lightning, you infer that there nevertheless was a
flash, because of the general proposition, "All thunder is preceded by
lightning." When Robinson Crusoe sees a footprint, he infers a human
being, and he might justify his inference by the general proposition,
"All marks in the ground shaped like a human foot are subsequent to a
human being's standing where the marks are." When we see the sun set, we
expect that it will rise again the next day. When we hear a man
speaking, we infer that he has certain thoughts. All these inferences
are due to causal laws.
A causal law, we said, allows us to infer the existence of one _thing_
(or _event_) from the existence of one or more others. The word "thing"
here is to be understood as only applying to particulars, _i.e._ as
excluding such logical objects as numbers or classes or abstract
properties and relations, and including sense-data, with whatever is
logically of the same type as sense-data.[56] In so far as a causal law
is directly verifiable, the thing inferred and the thing from which it
is inferred must both be data, though they need not both be data at the
same time. In fact, a causal law which is being used to extend our
knowledge of existence must be applied to what, at the moment, is not a
datum; it is in the possibility of such application that the practical
utility of a causal law consists. The important point, for our present
purpose, however, is that what is inferred is a "thing," a "particular,"
an object having the kind of reality that belongs to objects of sense,
not an abstract object such as virtue or the square root of two.
[56] Thus we are not using "thing" here in the sense of a class of
correlated "aspects," as we did in Lecture III. Each "aspect" will
count separately in stating causal laws.
But we cannot become acquainted with a particular except by its being
actually given. Hence the particular inferred by a causal law must be
only _described_ with more or less exactness; it cannot be _named_ until
the inference is verified. Moreover, since the causal law is _general_,
and capable of applying to many cases, the given particular from which
we infer must allow the inference in virtue of some general
characteristic, not in virtue of its being just the particular that it
is. This is obvious in all our previous instances: we infer the
unperceived lightning from the thunder, not in virtue of any peculiarity
of the thunder, but in virtue of its resemblance to other claps of
thunder. Thus a causal law must state that the existence of a thing of a
certain sort (or of a number of things of a number of assigned sorts)
implies the existence of another thing having a relation to the first
which remains invariable so long as the first is of the kind in
question.
It is to be observed that what is constant in a causal law is not the
object or objects given, nor yet the object inferred, both of which may
vary within wide limits, but the _relation_ between what is given and
what is inferred. The principle, "same cause, same effect," which is
sometimes said to be the principle of causality, is much narrower in its
scope than the principle which really occurs in science; indeed, if
strictly interpreted, it has no scope at all, since the "same" cause
never recurs exactly. We shall return to this point at a later stage of
the discussion.
The particular which is inferred may be uniquely determined by the
causal law, or may be only described in such general terms that many
different particulars might satisfy the description. This depends upon
whether the constant relation affirmed by the causal law is one which
only one term can have to the data, or one which many terms may have. If
many terms may have the relation in question, science will not be
satisfied until it has found some more stringent law, which will enable
us to determine the inferred things uniquely.
Since all known things are in time, a causal law must take account of
temporal relations. It will be part of the causal law to state a
relation of succession or coexistence between the thing given and the
thing inferred. When we hear thunder and infer that there was lightning,
the law states that the thing inferred is earlier than the thing given.
Conversely, when we see lightning and wait expectantly for the thunder,
the law states that the thing given is earlier than the thing inferred.
When we infer a man's thoughts from his words, the law states that the
two are (at least approximately) simultaneous.
If a causal law is to achieve the precision at which science aims, it
must not be content with a vague _earlier_ or _later_, but must state
how much earlier or how much later. That is to say, the time-relation
between the thing given and the thing inferred ought to be capable of
exact statement; and usually the inference to be drawn is different
according to the length and direction of the interval. "A quarter of an
hour ago this man was alive; an hour hence he will be cold." Such a
statement involves two causal laws, one inferring from a datum something
which existed a quarter of an hour ago, the other inferring from the
same datum something which will exist an hour hence.
Often a causal law involves not one datum, but many, which need not be
all simultaneous with each other, though their time-relations must be
given. The general scheme of a causal law will be as follows:
"Whenever things occur in certain relations to each other (among which
their time-relations must be included), then a thing having a fixed
relation to these things will occur at a date fixed relatively to their
dates."
The things given will not, in practice, be things that only exist for an
instant, for such things, if there are any, can never be data. The
things given will each occupy some finite time. They may be not static
things, but processes, especially motions. We have considered in an
earlier lecture the sense in which a motion may be a datum, and need not
now recur to this topic.
It is not essential to a causal law that the object inferred should be
later than some or all of the data. It may equally well be earlier or at
the same time. The only thing essential is that the law should be such
as to enable us to infer the existence of an object which we can more or
less accurately describe in terms of the data.