By Imre Lakatos

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**Additional resources for Problems in the Philosophy of Mathematics**

**Example text**

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.