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"We Need Never Know Anything About the Actual World"

**2014 March 3**

Bertrand Russell in 1916.

THE PHILOSOPHICAL IMPORTANCE OF
MATHEMATICAL LOGIC

by Bertrand Russell.

[*Monist 22*, 481 (1913)]

"Lecture delivered in French at the School of Advanced Social Studies
(Ecole des Hautes Etudes Sociales) on March 22, 1911 ; translated by P. E. B.
Jourdain from the *Revue de Métaphysique
et de Morale,* Vol. XX, 1912, where
it appears under the title, *L'Importance philosophique de la Logistique.*
This translation has been most kindly revised by Mr. Russell."

Russell's words are in **bold.**

Weierstrass and his successors have "arithmetized" 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 Leibniz, 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*.

**
**[*In more modern language, any (proper) subset of
a (finite) set must have fewer elements than the set itself.*]**
**

**
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 generalization 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 α are members of a class β;
and if *x* is a
member of the class α, then *x* is a member
of the class β.* 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 characterize 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 generalization 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 α, then, if
all the members of α are members of β, it follows
that *x* is a β." 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 generalization, 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* are 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 hypo-
thetical form: *If* 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
generalize 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 about 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 generalization
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 emphasizes a distinction
of the first importance, that is to say the distinction be-
tween a premise and a hypothesis. When we say "Socrates
is a man, *therefore* Socrates 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 twofold 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 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 : 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 colored 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 color of the spectacles never
changes. Kant did not deign to tell us how he knew this.

(It is possible that the true interpretation of Kant is less psychological than I supposed here; but the historical question has only a secondary importance for us in the present discussion.)

As soon as we take into account the consequences of
Kant's hypotheses, 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.

B. RUSSELL.

TRINITY COLLEGE, CAMBRIDGE, ENGLAND.

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