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Matheology § 156
Posted:
Nov 20, 2012 7:32 AM
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Matheology § 156
The object of this paper is to give a satisfactory account of the
Foundations of Mathematics in accordance with the general method of
Frege, Whitehead and Russell. Following these authorities, I hold that
mathematics is part of logic, and so belong to what may be called the
logical school as opposed to the formalist and intuitionist schools. I
have therefore taken Principia Mathematica as a basis for discussion
and amendment; and believe myself to have discovered how, by using the
work of Mr. Ludwig Wittgenstein, it can be rendered free from the
serious objections which have caused its rejection by the majority of
German authorities, who have deserted altogether its line of
approach.
In this chapter we shall be concerned with the general nature of pure
mathematics, and how it is distinguished from other sciences.
(Footnote: In the future by 'mathematics' will always be meant 'pure
mathematics'.) Here there are really two distinct categories of things
of which an account must be given -- the ideas or concepts of
mathematics, and the propositions of mathematics. This distinction is
neither artificial nor unnecessary, for the great majority of writers
on the subject have concentrated their attention on the explanation of
one or other of these categories, and erroneously supposed that a
satisfactory explanation of the other would immediately follow.
Thus the formalist school, of whom the most eminent representative
is now Hilbert, have concentrated on the propositions of mathematics,
such as '2 + 2 = 4'. They have pronounced these to be meaningless
formulae to be manipulated according to certain arbitrary rules, and
they hold that mathematical knowledge consists in knowing what
formulae can be derived from what others consistently with the rules.
Such being the propositions of mathematics, their account of its
concepts, for example the number 2, immediately follows. '2' is a
meaningless mark occurring in these meaningless formulae. But,
whatever may be thought of this as an account of mathematical
concepts, it is obviously hopeless as a theory of mathematical
concepts; for these occur not only in mathematical propositions, but
also in those of everyday life. Thus '2' occurs not merely in '2 + 2 =
4', but also in 'It is 2 miles to the station', which is not a
meaningless formula, but a significant proposition, in which '2'
cannot conceivably be a meaningless mark. Nor can there be any doubt
that '2' is used in the same sense in the two cases, for we can use '2
+ 2 = 4' to infer from 'It is two miles to the station and two miles
on to the Gogs' that 'It is four miles to the Gogs via the station',
so that these ordinary meanings of two and four are clearly involved
in '2 + 2 + 4'. So the hopelessly inadequate formalist theory is, to
some extent, the result of considering only the propositions of
mathematics and neglecting the analysis of its concepts, on which
additional light can be thrown by their occurrence outside mathematics
in the propositions of everyday life.
[F.P. Ramsey: "The Foundations of Mathematics", Proc. Lond. Math.
Soc., Vol.s2-25, Issue 1 (1926) 338-384]
http://www.hist-analytic.org/Ramsey.htm
http://www-history.mcs.st-and.ac.uk/Mathematicians/Ramsey.html
Regards, WM
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