Combining Texts

All the ideas for 'Symposium', 'On the Foundations of Logic and Arithmetic' and 'talk'

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4 ideas

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.

15. Nature of Minds / C. Capacities of Minds / 2. Imagination

6901	Understanding is needed for imagination, just as much as the other way around [Betteridge]
	Full Idea: Although it might be right to say that imagination is required in order to make reasoning and understanding possible, this also works the other way, as imagination cannot occur without some prior understanding.
	From: Alex Betteridge (talk [2005]), quoted by PG - Db (ideas)
	A reaction: This strikes me as a very illuminating remark, particularly for anyone who aspires to draw a simplified flowdiagram of the mind showing logical priority between its various parts. In fact, the parts are interdependent. Maybe imagination is understanding.

22. Metaethics / B. Value / 2. Values / h. Fine deeds

5845	Niceratus learnt the whole of Homer by heart, as a guide to goodness [Xenophon]
	Full Idea: Niceratus said that his father, because he was concerned to make him a good man, made him learn the whole works of Homer, and he could still repeat by heart the entire 'Iliad' and 'Odyssey'.
	From: Xenophon (Symposium [c.391 BCE], 3.5)
	A reaction: This clearly shows the status which Homer had in the teaching of morality in the time of Socrates, and it is precisely this acceptance of authority which he was challenging, in his attempts to analyse the true basis of virtue