Combining Texts

All the ideas for 'works', 'On the Foundations of Logic and Arithmetic' and 'Phil of Mathematics: why nothing works'

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4 ideas

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism

3663	How can you contemplate Platonic entities without causal transactions with them? [Putnam]
	Full Idea: Platonism has the attendant problem of how we can succeed in thinking about and referring to entities we can have no causal transactions with.
	From: Hilary Putnam (Phil of Mathematics: why nothing works [1979], Modalism)

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism

17697	The existence of an arbitrarily large number refutes the idea that numbers come from experience [Hilbert]
	Full Idea: The standpoint of pure experience seems to me to be refuted by the objection that the existence, possible or actual, of an arbitrarily large number can never be derived through experience, that is, through experiment.
	From: David Hilbert (On the Foundations of Logic and Arithmetic [1904], p.130)
	A reaction: Alternatively, empiricism refutes infinite numbers! No modern mathematician will accept that, but you wonder in what sense the proposed entities qualify as 'numbers'.

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique

17698	Logic already contains some arithmetic, so the two must be developed together [Hilbert]
	Full Idea: In the traditional exposition of the laws of logic certain fundamental arithmetic notions are already used, for example in the notion of set, and to some extent also of number. Thus we turn in a circle, and a partly simultaneous development is required.
	From: David Hilbert (On the Foundations of Logic and Arithmetic [1904], p.131)
	A reaction: If the Axiom of Infinity is meant, it may be possible to purge the arithmetic from the logic. Then the challenge to derive arithmetic from it becomes rather tougher.

22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / e. Human nature

7667	There are two sides to men - the pleasantly social, and the violent and creative [Diderot, by Berlin]
	Full Idea: Diderot is among the first to preach that there are two men: the artificial man, who belongs in society and seeks to please, and the violent, bold, criminal instinct of a man who wishes to break out (and, if controlled, is responsible for works of genius.
	From: report of Denis Diderot (works [1769], Ch.3) by Isaiah Berlin - The Roots of Romanticism
	A reaction: This has an obvious ancestor in Plato's picture (esp. in 'Phaedrus') of the two conflicting sides to the psuché, which seem to be reason and emotion. In Diderot, though, the suppressed man has virtues, which Plato would deny.