The Philosophy of Logical Analysis
(Chapter XXXI of "A History of Western Philosophy", 1945)
In philosophy ever since the time of Pythagoras there has been an opposition
between the men whose thought was mainly inspired by mathematics and those who
were more influenced by the empirical sciences. Plato, Thomas Aquinas, Spinoza,
and Kant belong to what may be called the mathematical party; Democritus,
Aristotle, and the modern empiricist from Locke onwards, belong to the opposite
party. In our day a school of philosophy has arisen which sets to work to
eliminate Pythagoreanism from the principles of mathematics, and to combine
empiricism with an interest in the deductive parts of human knowledge. The aims
of this school are less spectacular than those of most philosophers in the
past, but some of its achievements are as solid as those of the men of science.
The origin of this philosophy is in the achievements of mathematicians who set
to work to purge their subject of fallacies and slipshod reasoning. The great
mathematicians of the seventeenth century were optimistic and anxious for quick
results; consequently they left the foundations of analytical geometry and the
infinitesimal calculus insecure. Leibnitz believed in actual infinitesimals,
but although this belief suited his metaphysics it has no sound basis in
mathematics. Weierstrass, soon after the middle of the nineteenth century,
showed how to establish the calculus without infinitezimals, and thus at last
made it logically secure. Next came Georg Cantor, who developed the theory of
continuity and infinite number. "Continuity" had been, until he
defined it, a vague word, convenient for philosophers like Hegel, who wished to
introduce metaphysical muddles into mathematics. Cantor gave a precise
significance to the word, and showed that continuity, as he defined it, was the
concept needed by mathematicians and physicist. By this means a great deal of
mysticism, such as that of Bergson, was rendered antiquated.
Cantor also overcame the long-standing logical puzzles about infinite number.
Take the series of whole numbers from 1 onwards; how many of them are there?
Clearly the number is not finite. Up to a thousand, there are a thousand
numbers; up to million, a million. Whatever finite number you mention, there
are evidently more numbers than that, because from 1 up to the number in
question there are just that number of numbers, and then there are others that
are greater. The number of finite whole numbers must, therefore, be an infinite
number. But now comes a curious fact: The number of even numbers must be the
same as the number of all whole numbers. Consider two rows:
1, 2, 3, 4, 5, 6, ....
2, 4, 6, 8, 10, 12, ....
There is one entry in the lower row for every one in the top row; therefore the
number of terms in the two rows must be the same, although the lower row
consists of only half the terms in the top row. Leibnitz, who noticed this,
thought it a contradiction, and concluded, though there are infinite
collections, there are no infinite numbers. Georg Cantor, on the contrary,
boldly denied that it is a contradiction. he was right; it is only an oddity.
Georg Cantor defined an "infinite" collection as one which has parts
containing as many terms as the whole collection contains. On this basis he was
able to built up a most interesting mathematical theory of infinite numbers,
thereby taking into realm of exact logic a whole region formerly given over to
mysticism and confusion.
The next man of importance was Frege, who published his first work in 1879, and
his definition of "number" in 1884; but in spite of the epoch-making
nature of his discoveries, he remained wholly without recognition until I drew
attention to him in 1903. It is remarkable that, before Frege, every definition
of number that had been suggested contained elementary logical blunders. It was
customary to identify "number" with "plurality". But an
instance of number is a particular number, say 3, and an instance of 3 is a
particular triad. The triad is a plurality, but the class of all triads - which
Frege identified with the number 3 - is a plurality of pluralities, and number
in general, of which 3 is an instance, is a plurality of pluralities of
pluralities. The elementary grammatical mistake of confounding this with the
simple plurality of a given triad made the whole philosophy of number, before
Frege, a tissue of nonsense in the strictest sense of the term
From Frege's work it followed that arithmetic, and pure mathematics generally,
is nothing but a prolongations of deductive logic. This disproved Kant's theory
that arithmetical propositions are "synthetic" and involve a
reference to time. The development of pure mathematics from logic was set forth
in detail in Principia Mathematica, by Whitehead and myself.
It gradually became clear that a great part of philosophy can be reduced to
something that may be called "syntax", though the word has to be used
in a somewhat wider sense than has hitherto been customary. Some men, notably
Carnap, have advanced the theory that all philosophical problems are really
syntactical, and that, when errors in syntax are avoided, a philosophical
problem is thereby either solved or shown to be insoluble. I think this is an
overstatment, but there can be no doubt that the utility of philosophical
syntax in relation to traditional problems is very great.
I will illustrate its utility by a brief explanation of what is called the
theory of descriptions. By a "description" I mean a phrase such as
"The present President of the United States", in which a person or
thing is designated, not by name, but by some property which is supposed or
known to be peculiar to him or it. Such phrases had given a lot of trouble.
Suppose I say "The golden mountain does not exist" and suppose you
ask "What is it that does not exist?" It would seem that, if I say
"It is the golden mountain", I am attributing some sort of existence
to it. Obviously I am not making the same statement as if I said, "The
round square does not exist". This seemed to imply that the golden
mountain is one thing and the round square is another, although neither exists.
The theory of descriptions was designed to meet this and other difficulties.
According to this theory, when a statement containing a phrase of the form
"the so-and-so" is rightly analyzed, the phrase "the
so-and-so" disappears. For example, take the statement "Scott was the
author of "Waverley". The theory interprets this statement as
"One and the only one man wrote Waverley, and that man was
Scott". Or, more fully:
"There is an entity c such that the statement 'x wrote
Waverley' is true if x is c and false otherwise; moreover
c is Scott".
The first part of this, before the word "moreover", is defined as
meaning: "The author of Waverley exist (or existed or will
exist)". Thus "The golden mountain does not exist" means:
"There is no entity c such that 'x is golden and
mountainous' is true when x is c, but not otherwise".
With this definition the puzzle as to what is meant when we say "The
golden mountain does not exist" disappears.
"Existence", according to this theory, can only be asserted of
descriptions. We can say "The author of Waverley exists", but
to say "Scott exists" is bad grammar, or rather bad syntax. This
clears up two millennia of muddle-headless about "existence",
beginning with Plato's Theateus.
One result of the work we have been considering is to dethrone mathematics from
the lofty place that it has occupied since Phythagoras and Plato, and to
destroy the presumption against empiricism which has been derived from it.
Mathematical knowledge, it is true, is not obtained by induction from
experience; our reason for believing that 2 and 2 are 4 is not that we have so
often found, by observation, that one couple and another couple together make a
quartet. In this sense, mathematical knowledge is still not empirical. But it
is also not a priori knowledge about the world. It is, in fact, merely
verbal knowledge. "3" means "2 + 1" and "4" means
"3 + 1". Hence it follows (though the proof is long) that
"4" means the same as "2 + 2". Thus mathematical knowledge
ceases to be mysterious. It is all of the same nature as the "great
truth" that there are three feet in a yard.
Physics, as well as pure mathematics, has supplied material for the philosophy
of logical analysis. This has occurred especially through the theory of
relativity and quantum mechanics.
What is important to the philosopher in the theory of relativity is the
substitution of space-time for space and time. Common sense thinks of the
physical world as composed of "things" which persist through a
certain period of time and move in space. Philosophy and physics developed the
notion of "thing" into that of "material substance", and
thought of material substance as consisting of particles, each very small, and
each persisting throughout all time. Einstein substituted events for particles;
each event had to each other a relation called "interval", which
could be analyzed in various ways into a time-element and space-element. The
choice between these various ways was arbitrary and no one of them was
theoretically preferable to any other. Given two elements A and B, in different
regions, it might happen that according to one convention they were
simultaneous, according to another A was earlier than B, and according to yet
another B was earlier than A. No physical facts correspond to these different
From all this it seems to follow that events, not particles, must be the
"stuff" of physics. What has been thought of as a particle will have
to be thought of as a series of events. The series of events that replaces a
particle has certain important physical properties, and therefore demands our
attention; but it has no more substantiality than any other series of events
that we might arbitrary single out. Thus "matter" is not part of the
ultimate material of the world, but merely a convenient way of collecting
events into bundles.
Quantum theory reinforces this conclusion, but its chief philosophical
importance is that it regards physical phenomena as possibly discontinuous. It
suggests that, in an atom (interpreted as above), a certain state of affairs
persists for a certain time, and then suddenly is replaced by a finitely
different state of affairs. Continuity of motion, which had always been
assumed, appears to have been a mere prejudice. The philosophy appropriate to
quantum theory, however, has not yet been adequately developed. I suspect that
it will demand even more radical departures from the traditional doctrine of
space and time than those demanded by the theory of relativity.
While physics has been making matter less material, psychology has been making
mind less mental. We had occasion in a former chapter to compare the
association of ideas with the conditioned reflex. The latter, which has
replaced the former, is much more physiological. (This is only one
illustration; I do not wish to exaggerate the scope of the conditioned reflex).
Thus from both ends physics and psychology have been approaching each other,
and making more possible the doctrine of "neutral monism" suggested
by William James's criticism of "consciousness". The distinction of
mind and matter came into philosophy form religion, although, for a long time,
it seemed to have valid grounds. I think that both mind and matter are merely
convenient ways of grouping events. Some single events, I should admit, belong
only to material groups, but others belong to both kinds of groups, and are
therefore at once mental and material. This doctrine effects a great
simplification in our picture of the structure of the world.
Modern Physics and physiology throw a new light upon the ancient problem of
perception. If there is to be anything that can be called
"perception", it must be in some degree an effect of the object
perceived, and must more or less resemble the object if it is to be a source of
knowledge of the object. The first requisite can only be fulfilled if there are
causal chains which are, to a greater or less extent, independent of the rest
of the world. According to physics, this is the case. Light-waves travel from
the sun to the earth, and in doing so obey their own laws. This is only roughly
true. Einstein has shown that light-rays are affected by gravitation. When they
reach our atmosphere, they suffer refraction, and some are more scattered than
others. When they reach a human eye, all sorts of things happen which would not
happen elsewhere, ending up with what we call "seeing the sun". But
although the sun of our visual experience is very different from the sun of the
astronomer, it is still a source of knowledge as to the latter, because
"seeing the sun" differs from "seeing the moon" in ways
that are causally connected with the difference between the astronomer's sun
and the astronomer's moon. What we can know of physical objects in this way,
however, is only certain abstract properties of structure. We can know that the
sun is round in a sense, though not quite the sense in which what we see is
round; but we have no reason to suppose that it is bright or warm, because
physics can account for its seeming so without supposing that it is so. Our
knowledge of the physical world, therefore, is only abstract and mathematical.
Modern analytical empiricism, of which I have been giving an outline, differs
from that of Locke, Berkeley, and Hume by its incorporation of mathematics and
its development of powerful logical technique. It is thus able, in regard to
certain problems, to achieve definite answers, which have the quality of
science rather that of philosophy. It has the advantage, as compared with the
philosophies of the system-builders, of being able to tackle its problems one
at a time, instead of having to invent at one stroke a block theory of the
whole universe. Its methods, in this respect, resemble those of science. I have
no doubt that, in so far as philosophical knowledge is possible, it is by such
methods, many ancient problems are completely soluble.
There remains, however, a vast field, traditionally included in philosophy,
where scientific methods are inadequate. This field includes ultimate questions
of value; science alone, for example, cannot prove that it is bad to enjoy the
infliction of cruelty. Whatever can be known, can be known by means of science;
but things which are legitimately matters of feeling lie outside its province.
Philosophy, throughout its history, has consisted of two parts inharmoniously
blended; on the one hand a theory as to the nature of the world, on the other
an ethical or political doctrine as to the best way of living. The failure to
separate these two with sufficient clarity has been a source of much confused
thinking. Philosophers, from Plato to William James, have allowed their
opinions as to the constitution of the universe to be influenced by the desire
for edificacion: knowing, as they supposed, what beliefs would make men
virtuous, they have invented arguments, often very sophistical, to prove that
these beliefs are true. For my part I reprobate this kind of bias, both on
moral and on intellectual grounds. Morally, a philosopher who uses his
professional competency for anything except a disinterested search for truth is
guilty of a kind of treachery. And when he assumes, in advance of inquiry, that
certain beliefs, whether true or false, are such as to promote good behavior,
he is so limiting the scope of philosophical speculation as to make philosophy
trivial; the true philosopher is prepared to examine all preconceptions.
When any limits are placed, consciously or unconsciously, upon the pursuit of
truth, philosophy becomes paralyzed by fear, and the ground is prepared for a
government censorship punishing those who utter "dangerous thoughts"
- in fact, the philosopher has already places such a censorship over his own
Intellectually, the effect of mistaken moral considerations upon philosophy has
been to impede progress to an extraordinary extent. I do not myself believe
that philosophy can either prove or disprove the truth of religious dogmas, but
ever since Plato most philosophers have considered it part of their business to
produce "proofs" of immortality and the existence of God. They have
found fault with the proofs of their predecessors - Saint Thomas rejected Saint
Anselm's proofs, and Kant rejected Descartes' - but they have supplied new ones
of their own. In order to make their proofs seem valid, they have had to
falsify logic, to make mathematics mystical, and to pretend that deep-seated
prejudices were heaven-sent intuitions.
All this is rejected by the philosophers who make logical analysis the main
business of philosophy. They confess frankly that the human intellect is unable
to find conclusive answers to many questions of profound importance to mankind,
but they refuse to believe that there are some "higher" way of
knowing, by which we can discover truths hidden from science and the intellect.
For this renunciation they have been rewarded by the discovery that many
questions, formerly obscured by the fog of metaphysics, can be answered with
precision, and by objective methods which introduce nothing of the
philosopher's temperament except the desire to understand. Take such questions
as: What is number? What are space and time? What is mind, and what is matter?
I do not say that a method has been discovered by which, as in science, we can
make successive approximations to the truth, in which each new stage results
from an improvment, not a rejection, of what has gone before.
In the welter of conflicting fanaticisms, one of the few unifying forces is
scientific truthfulness, by which i mean the habit of basing our beliefs upon
observations and inferences as impersonal, and as much diverse of local and
temperamental bias, as is possible for human beings. To have insisted upon the
introduction of his virtue into philosophy, and to have invented a powerful
method by which it can be rendered fruitful, are the chief merits of the
philosophical school of which I am a member. The habit of careful veracity
acquired in the practice of this philosophical method can be extended to the
whole sphere of human activity, producing, wherever it exists, a lessening of
fanaticism with an increasing capacity of sympathy and mutual understanding. In
abandoning a part of its dogmatic pretensions, philosophy does not cease to
suggest and inspire a way of life