**In** speaking of "Mathematical logic", I use this word in a very
broad sense. By it I understand the works of Cantor on transfinite numbers as
well as the logical work of Frege and Peano. Weierstrass and his successors have
"arithmetised" mathematics; that is to say, they have reduced the whole of
analysis to the study of integer numbers. The accomplishment of this reduction
indicated the completion of a very important stage, at the end of which the
spirit of dissection might well be allowed a short rest. However, the theory of
integer numbers cannot be constituted in an autonomous manner, especially when
we take into account the likeness in properties of the finite and infinite
numbers. It was, then, necessary to go farther and reduce arithmetic, and above
all the definition of numbers, to logic. By the name "mathematical logic", then,
I will denote any logical theory whose object is the analysis and deduction of
arithmetic and geometry by means of concepts which belong evidently to logic. It
is this modern tendency that I intend to discuss here.

In an examination of the work done by mathematical logic, we may consider either the mathematical results, the method of mathematical reasoning as revealed by modern work, or the intrinsic nature of mathematical propositions according to the analysis which mathematical logic makes of them. It is impossible to distinguish exactly these three aspects of the subject, but there is enough of a distinction to serve the purpose of a framework for discussion. It might be thought that the inverse order would be the best; that we ought first to consider what a mathematical proposition is, then the method by which such propositions are demonstrated, and finally the results to which this method leads us. But the problem which we have to resolve, like every truly philosophical problem, is a problem of analysis; and in problems of analysis the best method is that which sets out from results and arrives at the premises. In mathematical logic it is the conclusions which have the greatest degree of certainty: the nearer we get to the ultimate premises the more uncertainty and difficulty do we find.

From the philosophical point of view, the most brilliant results of the new
method are the exact theories which we have been able to form about infinity and
continuity. We know that when we have to do with infinite collections, for
example the collection of finite integer numbers, it is possible to establish a
one-to-one correspondence between the whole collection and a part of itself. For
example, there is such a correspondence between the finite integers and the even
numbers, since the relation of a finite number to its double is one-to-one. Thus
it is evident that the number of an infinite collection is equal to the number
of a part of this collection. It was formerly believed that this was a
contradiction; even Leibnitz, although he was a partisan of the actual infinite,
denied infinite number because of this supposed contradiction. But to
demonstrate that there is a contradiction we must suppose that all numbers obey
mathematical induction. To explain mathematical induction, let us call by the
name "hereditary property" of a number a property which belongs to *n* *+
*1 whenever it belongs to *n*. Such is, for example, the property of
being greater than 100. If a number is greater than 100, the next number after
it is greater than 100. Let us call by the name "inductive property" of a number
a hereditary property which is possessed by the number zero. Such a property
must belong to 1, since it is hereditary and belongs to 0; in the same way, it
must belong to 2, since it belongs to 1; and so on. Consequently the numbers of
daily life possess every inductive property. Now, amongst the inductive
properties of numbers is found the following. If any collection has the number
n, no part of this collection can have the same number *n*. Consequently,
if all numbers possess all inductive properties, there is a contradiction with
the result that there are collections which have the same number as a part of
themselves. This contradiction, however, ceases to subsist as soon as we admit
that there are numbers which do not possess all inductive properties. And then
it appears that there is no contradiction in infinite number. Cantor has even
created a whole arithmetic of infinite numbers, and by means of this arithmetic
he has completely resolved the former problems on the nature of the infinite
which have disturbed philosophy since ancient times.

The problems of the *continuum *are closely connected with the problems
of the infinite and their solution is effected by the same means. The paradoxes
of Zeno the Eleatic and the difficulties in the analysis of space, of time, and
of motion, are all completely explained by means of the modern theory of
continuity. This is because a non-contradictory theory has been found, according
to which the continuum is composed of an infinity of distinct elements; and this
formerly appeared impossible. The elements cannot all be reached by continual
dichotomy; but it does not follow that these elements do not exist.

From this follows a complete revolution in the philosophy of space and time. The realist theories which were believed to be contradictory are so no longer, and the idealist theories have lost any excuse there might have been for their existence. The flux, which was believed to be incapable of analysis into indivisible elements, shows itself to be capable of mathematical analysis, and our reason shows itself to be capable of giving an explanation of the physical world and of the sensible world without supposing jumps where there is continuity, and also without giving up the analysis into separate and indivisible elements.

The mathematical theory of motion and other continuous changes uses, besides
the theories of infinite number and of the nature of the continuum, two
correlative notions, that of a *function *and that of a *variable. *
The importance of these ideas may be shown by an example. We still find in books
of philosophy a statement of the law of causality in the form: "When the same
cause happens again, the same effect will also happen." But it might be very
justly remarked that the same cause never happens again. What actually takes
place is that there is a constant relation between causes of a certain kind and
the effects which result from them. Wherever there is such a constant relation,
the effect is a function of the cause. By means of the constant relation we sum
up in a single formula an infinity of causes and effects, and we avoid the
worn-out hypothesis of the *repetition *of the same cause. It is the idea
of functionality, that is to say the idea of constant relation, which gives the
secret of the power of mathematics to deal simultaneously with an infinity of
data.

To understand the part played by the idea of a function in mathematics, we
must first of all understand the method of mathematical deduction. It will be
admitted that mathematical demonstrations, even those which are performed by
what is called mathematical induction, are always deductive. Now, in a deduction
it almost always happens that the validity of the deduction does not depend on
the subject spoken about, but only on the form of what is said about it. Take
for example the classical argument: All men are mortal, Socrates is a man,
therefore Socrates is mortal. Here it is evident that what is said remains true
if Plato or Aristotle or anybody else is substituted for Socrates. We can, then,
say: If all men are mortal, and if* x *is a man, then* x *is mortal.
This is a first generalisation of the proposition from which we set out. But it
is easy to go farther. In the deduction which has been stated, nothing depends
on the fact that it is men and mortals which occupy our attention. If all the
members of any class a are members of a class *s*, and if* x *is a
member of the class a, then* x *is a member of the class *s*. In this
statement, we have the pure logical form which underlies all the deductions of
the same form as that which proves that Socrates is mortal. To obtain a
proposition of pure mathematics (or of mathematical logic, which is the same
thing), we must submit a deduction of any kind to a process analogous to that
which we have just performed, that is to say, when an argument remains valid if
one of its terms is changed, this term must be replaced by a variable, i.e. by
an indeterminate object. In this way we finally reach a proposition of pure
logic, that is to say a proposition which does not contain any other constant
than logical constants. The definition of the *logical constants *is not
easy, but this much may be said: A *constant is logical *if the
propositions in which it is found still contain it when we try to replace it by
a variable. More exactly, we may perhaps characterise the logical constants in
the following manner: If we take any deduction and replace its terms by
variables, it will happen, after a certain number of stages, that the constants
which still remain in the deduction belong to a certain group, and, if we try to
push generalisation still farther, there will always remain constants which
belong to this same group. 'This group is the group of logical constants. The
logical constants are those which constitute pure form; a formal proposition is
a proposition which does not contain any other constants than logical constants.
We have just reduced the deduction which proves that Socrates is mortal to the
following form: "If *x* is an *a*, then, if all the members of *a*
are members of *b, *it follows that* x *is a *b*." The constants
here are: is-*a, all, and if-then. *These are logical constants and
evidently they are purely formal concepts.

Now, the validity of any valid deduction depends on its form, and its form is obtained by replacing the terms of the deduction by variables, until there do not remain any other constants than those of- logic. And conversely: every valid deduction can be obtained by starting from a deduction which operates on variables by means of logical constants, by attributing to variables definite values with which the hypothesis becomes true.

By means of this operation of generalisation, we separate the strictly
deductive element in an argument from the element which depends on the
particularity of what is spoken about. Pure mathematics concerns itself
exclusively with the deductive element. We obtain propositions of pure
mathematics by a process of *purification*. If I say: "Here are two things,
and here are two other things, therefore here arc four things in all", I do not
state a proposition of pure mathematics because here particular data come into
question. The proposition that I have stated is an *application* of the
general proposition: "Given any two things and also any two other things, there
are four things in all." 'The latter proposition is a proposition of pure
mathematics, while the former is a proposition of applied mathematics.

It is obvious that what depends on the particularity of the subject is the
verification of the hypothesis, and this permits us to assert, not merely that
the hypothesis implies the thesis, but that, since the hypothesis is true, the
thesis is true also. This assertion is not made in pure mathematics. Here we
content ourselves with the hypothetical form: It- any subject satisfies such and
such a hypothesis, it will also satisfy such and such a thesis. It is thus that
pure mathematics becomes entirely hypothetical, and concerns itself exclusively
with any indeterminate subject, that is to say with a *variable. *Any valid
deduction finds its form in a hypothetical proposition belonging to pure
mathematics; but in pure mathematics itself we affirm neither the hypothesis nor
the thesis, unless both can be expressed in terms of logical constants.

If it is asked why it is worth while to reduce deductions to such a form, I
reply that there are two associated reasons for this. In the first place, it is
a good thing to generalise any truth as much as possible; and, in the second
place, an economy of work is brought about by making the deduction with an
indeterminate x. When we reason al-out Socrates, we obtain results which apply
only to Socrates, so that, if we wish to know something about Plato, we have to
perform the reasoning all over again. But when we operate on x, we obtain
results which we know to be valid for every* x *which satisfies the
hypothesis. The usual scientific motives of economy and generalisation lead us,
then, to the theory of mathematical method which has just been sketched.

After what has just been said it is easy to see what must be thought about
the intrinsic nature of propositions of pure mathematics. In pure mathematics we
have never to discuss facts that are applicable to such and such an individual
object; we need never know anything about the actual world. We are concerned
exclusively with variables, that is to say, with any subject, about which
hypotheses are made which may be fulfilled sometimes, but whose verification for
such and such an object is only necessary for the *importance *of the
deductions, and not for their truth. At first sight it might appear that
everything would be arbitrary in such a science. But this is not so. It is
necessary that the hypothesis truly implies the thesis. If we make the
hypothesis that the hypothesis implies the thesis, we can only make deductions
in the case when this new hypothesis truly implies the new thesis. Implication
is a logical constant and cannot be dispensed with. Consequently we need true
propositions about implication. If we took as premises propositions on
implication which were not true, the consequences which would appear to flow
from them would not be truly implied by the premises, so that we would not
obtain even a hypothetical proof. This necessity for true premises emphasises a
distinction of the first importance, that is to say the distinction between a
premise and a hypothesis. When we say "Socrates is a man, *therefore Soc*rates
is mortal", the proposition "Socrates is a man" is a premise; but when we say: "*If*
Socrates is a man, then Socrates is mortal", the proposition "Socrates is a man"
is only a hypothesis. Similarly when I say: "If from *p* we deduce *q*
and from *q* we deduce r, then from *p* we deduce *r*", the
proposition "From *p* we deduce *q* and from *q* we deduce *r*"
is a hypothesis, but the whole proposition is not a hypothesis, since I affirm
it, and, in fact, it is true. This proposition is a rule of deduction, and the
rules of deduction have a two-fold use in mathematics: both as premises and as a
method of obtaining consequences of the premises. Now, if the rules of deduction
were not true, the consequences that would be obtained by using them would not
truly be consequences, so that we should not have even a correct deduction
setting out from a false premise. It is this twofold use of the rules of
deduction which differentiates the foundations of mathematics from the later
parts. In the later parts, we use the same rules of deduction to deduce, but we
no longer use them immediately as premises. Consequently, in the later parts,
the immediate premises may be false without the deductions being logically
incorrect, but, in the foundations, the deductions will be incorrect if the
premises are not true. It is necessary to be clear about this point, for
otherwise the part of arbitrariness and of hypothesis might appear greater than
it is in reality.

Mathematics, therefore, is wholly composed of propositions which only contain variables and logical constants, that is to say, purely formal propositions-for the logical constants are those which constitute form. It is remarkable that we have the power of knowing such propositions. The consequences of the analysis of mathematical knowledge are not without interest for the theory of knowledge. In the first place it is to be remarked, in opposition to empirical theories, that mathematical knowledge needs premises which are not based on the data of sense. Every general proposition goes beyond the limits of knowledge obtained through the senses, which is wholly restricted to what is individual. If we say that the extension of the given case to the general is effected by means of induction, we are forced to admit that induction itself is not proved by means of experience. Whatever may be the exact formulation of the fundamental principle of induction, it is evident that in the first place this principle is general, and in the second place that it cannot, without a vicious circle, be itself demonstrated by induction.

It is to be supposed that the principle of induction can be formulated more
or less in the following way. If we are given the fact that any two properties
occur together in a certain number of cases, it is more probable that a new case
which possesses one of these properties will possess the other than it would be
if we had not such a datum. I do not say that this is a satisfactory formulation
of the principle of induction; I only say that the principle of induction must
be like this in so far as it must be an absolutely general principle which
contains the notion of probability. Now it is evident that sense-experience
cannot demonstrate such a principle, and cannot even make it probable; for it is
only in virtue of the principle itself that the fact that it has often been
successful gives grounds for the belief that it will probably be successful in
the future. Hence inductive knowledge, like all knowledge which is obtained by
reasoning, needs logical principles which are *a priori* and universal. By
formulating the principle of induction, we transform every induction into a
deduction; induction is nothing else than a deduction which uses a certain
premise, namely the principle of induction.

In so far as it is primitive and undemonstrated, human knowledge is thus divided into two kinds: knowledge of particular facts, which alone allows us to affirm existence, and knowledge of logical truth, which alone allows us to reason about data. In science and in daily life the two kinds of knowledge are intermixed: the propositions which are affirmed are obtained from particular premises by means of logical principles. In pure perception we only find knowledge of particular facts: in pure mathematics, we only find knowledge of logical truths. In order that such a knowledge be possible, it is necessary that there should be self-evident logical truths, that is to say, truths which are known without demonstration. These are the truths which are the premises of pure mathematics as well as of the deductive elements in every demonstration on any subject whatever.

It is, then, possible to make assertions, not only about cases which we have
been able to observe, but about all actual or possible cases. The existence of
assertions of this kind and their necessity for almost all pieces of knowledge
which are said to be founded on experience shows that traditional empiricism is
in error and that there is *a priori* and universal knowledge.

In spite of the fact that traditional empiricism is mistaken in its theory of
knowledge, it must not be supposed that idealism is right. Idealism at least
every theory of knowledge which is derived from Kant-assumes that the
universality of *a priori* truths comes from their property of expressing
properties of the mind: I things appear to be thus because the nature of the
appearance depends on the subject in the same way that, if we have blue
spectacles, everything appears to be blue. The categories of Kant are the
coloured spectacles of the mind; truths *a priori* are the false
appearances produced by those spectacles. Besides, we must know that everybody
has spectacles of the same kind and that the colour of the spectacles never
changes. Kant did not deign to tell us how he knew this.

As soon as we take into account the consequences of Kant's hypothesis, it
becomes evident that general and *a priori* truths must have the same
objectivity, the same independence of the mind, that the particular facts of the
physical world possess. In fact, if general truths only express psychological
facts, we could not know that they would be constant from moment to moment or
from person to person, and we could never use them legitimately to deduce a fact
from another fact, since they would not connect facts but our ideas about the
facts. Logic and mathematics force us, then, to admit a kind of realism in the
scholastic sense, that is to say, to admit that there is a world of universals
and of truths which do not bear directly on such and such a particular
existence. This world of universals must *subsist, *although it cannot *
exist *in the same sense as that in which particular data exist. We have
immediate knowledge of an indefinite number of propositions about universals:
this is an ultimate fact, as ultimate as sensation is. Pure mathematics-which is
usually called "logic" in its elementary parts-is the sum of everything that we
can know, whether directly or by demonstration, about certain universals.

On the subject of self-evident truths it is necessary to avoid a misunderstanding. Self-evidence is a psychological property and is therefore subjective and variable. It is essential to knowledge, since all knowledge must be either self-evident or deduced from self-evident knowledge. But the order of knowledge which is obtained by starting from what is self-evident is not the same thing as the order of logical deduction, and we must not suppose that when we give such and such premises for a deductive system, we are of opinion that these premises constitute what is self-evident in the system. In the first place self-evidence has degrees: It is quite possible that the consequences are more evident than the premises. In the second place it may happen that we are certain of the truth of many of the consequences, but that the premises only appear probable, and that their probability is due to the fact that true consequences flow from them. In such a case, what we can be certain of is that the premises imply all the true consequences that it was wished to place in the deductive system. This remark has an application to the foundations of mathematics, since many of the ultimate premises are intrinsically less evident than many of the consequences which are deduced from them. Besides, if we lay too much stress on the self-evidence of the premises of a deductive system, we may be led to mistake the part played by intuition (not spatial but logical) in mathematics. The question of the part of logical intuition is a psychological question and it is not necessary, when constructing a deductive system, to have an opinion on it.

To sum up, we have seen, in the first place, that mathematical logic has
resolved the problems of infinity and continuity, and that it has made possible
a solid philosophy of space, time, and motion. In the second place, we have seen
that pure mathematics can be defined as the class of propositions which are
expressed exclusively in terms of variables and logical constants, that is to
say as the class of purely formal propositions. In the third place, we have seen
that the possibility of mathematical knowledge refutes both empiricism and
idealism, since it shows that human knowledge is not wholly deduced from facts
of sense, but that *a priori* knowledge can by no means be explained in a
subjective or psychological manner.