By Imre Lakatos
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Additional resources for Problems in the Philosophy of Mathematics
Their methods were more or less geometrical. This is particularly true of Pascal who behaved idiosyncratically towards Cartesian methods. Leibniz’ starting point was a n integral transformation he found in Pascal’s work and stripped of its geometrical clothing. The gist of Leibniz’ efforts was the thorough algebraisation of calculus. The result was a n easy and prolific formalism, more practical than Newton’s, and rapidly accepted by most creative mathematicians. : Technique versus metaphysics in the calculus.
P’s . . S’s aren’t un-P’s . . Some S are P. P’s . . S’s aren’t P’s . . Some S are not P. All four categoricals are about (all) the S’s. I n A , P is affirmed, in E , P , its contrary, is affirmed. I shall therefore say that E contrafirms what A affirms. I n I , the cont,rary of P is denied of the S’s. I shall say that I contradenies A . I n 0, P is denied. The two fundamental modes of predication are affirmation and denial. A and E are affirmations, I and 0 are denials. But since either a term or its contrary may be affirmed or denied we get four logically distinct ways of predicating a term.
G. G. Leibniz touchant son sentiment sur le calcul diffBrentie1, Journal de Trdvoux, 1701, Mathematische Schriften (ed. C. I. Gerhardt) vol. 5, 1858, p. 350. [lo] A. ROBINSON, Non-standard Analysis, Studies in Logic and the Foundations of Mathematics, Amsterdam, 1966. Mathematical reasoning and its objects, George Berkeley [ I l l E. W. STRONG, lectures, University of California Publications in Philosophy, vol. 29, Berkeley and Los Angeles, 1957, pp. 65-88. DISCUSSION PETER GEACH: Infinity in scholastic philosophy.