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results from an improvement, not a rejection, of what has gone before.
(Russell, 1961, pp. 788 789)
LOGICAL EMPIRICISM
The Grand Theses of Logical Empiricism
Have Not Turned out to Be Correct
Not a single one of the great theses of Logical Empiricism (that Meaning
is Method of Verification; that metaphysical propositions are literally
without sense; that Mathematics is True by Convention) has turned out
to be correct. It detracts from the excitement of the fact that, by turning
140 LOGICAL POSITIVISM
philosophical theses into linguistic ones [as Carnap had tried to do] . . .
one can make philosophy more scientific and settle the truth value of
philosophical propositions by hard scientific research, if the results one
obtains are uniformly negative. (Putnam, 1975, p. 20)
LOGICAL POSITIVISM
The Question of the Validity and
Justification of Metaphysics
There have been many opponents of metaphysics from the Greek sceptics
to the empiricists of the nineteenth century. Criticisms of very diverse
kinds have been set forth. Many have declared that the doctrine of meta-
physics is false, since it contradicts our empirical knowledge. Others have
believed it to be uncertain, on the ground that its problems transcend the
limits of human knowledge. Many anti-metaphysicians have declared
that occupation with metaphysical questions is sterile. Whether or not
these questions can be answered, it is at any rate unnecessary to worry
about them; let us devote ourselves entirely to the practical tasks which
confront active men every day of their lives!
The development of modern logic has made it possible to give a new
and sharper answer to the question of the validity and justification of
metaphysics. The researchers of applied logic or the theory of knowl-
edge, which aim at clarifying the cognitive content of scientific state-
ments and thereby the meanings of the terms that occur in the
statements, by means of logical analysis, lead to a positive and to a neg-
ative result. The positive result is worked out in the domain of empirical
science; the various concepts of the various branches of science are clar-
ified; their formal, logical and epistemological connections are made ex-
plicit.
In the domain of metaphysics, including all philosophy of value and
normative theory, logical analysis yields the negative result that the al-
leged statements in this domain are entirely meaningless. Therewith a radical
elimination of metaphysics is attained, which was not yet possible from
the earlier anti-metaphysical standpoints. (Carnap, 1959, p. 60)
M
MACHINE
An Infallible Machine Cannot Also Be
Intelligent
In other words then, if a machine is expected to be infallible, it cannot
also be intelligent. There are several theorems which say almost exactly
that. But these theorems say nothing about how much intelligence may
be displayed if a machine makes no pretence at infallibility. (Turing,
1946, p. 124)
MACHINES
The Desire for the Creation of Machines
[T]he human desire to escape the flesh, which took one form in asceti-
cism, might take another form in the creation of machines. Thus, the
wish to rise above the bestial body manifested itself not only in angels
but in mechanical creatures. (Mazlish, 1993, p. 218)
142 MATHEMATICAL DISCOVERY
MATHEMATICAL DISCOVERY
Mathematical Discovery Consists of the
Discernment and Selection of Useful
Combinations
What, in fact, is mathematical discovery? It does not consist in making
new combinations with mathematical entities that are already known.
That can be done by anyone, and the combinations that could be so
formed would be infinite in number, and the greater part of them would
be absolutely devoid of interest. Discovery consists precisely in not con-
structing useless combinations, but in constructing those that are useful,
which are an infinitely small minority. Discovery is discernment, selec-
tion. (Poincaré, 1952, pp. 50 51)
MATHEMATICS
The World of Mathematics Is Really a
Beautiful World
The world of mathematics, which you contemn, is really a beautiful
world; it has nothing to do with life and death and human sordidness,
but is eternal, cold and passionless. To me pure mathematics is one of
the highest forms of art; it has a sublimity quite special to itself, and an
immense dignity derived from the fact that its world is exempt from
change and time. I am quite serious in this. . . . [M]athematics is the only
thing we know of that is capable of perfection; in thinking about it we
become Gods. (Russell [to Helen Thomas, 30 December 1901], 1992, Let-
ter No. 98, p. 224)
MATHEMATICS 143
MATHEMATICS
Why Mathematics Works
One of the deepest problems of nature is the success of mathematics as
a language for describing and discovering features of physical reality. In
short, why does mathematics work? . . .
We humans have stripped back the clouds that cloak our understand-
ing of our cosmic beginning and our current persistence to the stage that
exposes the mathematical structure of the world more clearly than it has
ever been observed before. . . . Furthermore, the attention of seriously
equipped thinkers, those thinkers we call scientists, is at last beginning
to turn to that other great conundrum of being: consciousness. . . . If we
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