Combining Texts

All the ideas for 'On the Senses', 'The Foundations of Mathematics' and 'Concepts and Counting'

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4 ideas

5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle

18844	You would cripple mathematics if you denied Excluded Middle [Hilbert]
	Full Idea: Taking the principle of Excluded Middle away from the mathematician would be the same, say, as prohibiting the astronomer from using the telescope or the boxer from using his fists.
	From: David Hilbert (The Foundations of Mathematics [1927], p.476), quoted by Ian Rumfitt - The Boundary Stones of Thought 9.4
	A reaction: [p.476 in Van Heijenoort]

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure

17462	A single object must not be counted twice, which needs knowledge of distinctness (negative identity) [Rumfitt]
	Full Idea: One requirement for a successful count is that double counting should be avoided: a single object should not be counted twice. ...but that is to make a knowledgeable judgement of distinctness - to resolve a question of identity in the negative.
	From: Ian Rumfitt (Concepts and Counting [2002], III)
	A reaction: He also notes later (p.65) that you must count all and only the right things.

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers

17461	Some 'how many?' answers are not predications of a concept, like 'how many gallons?' [Rumfitt]
	Full Idea: We hit trouble if we hear answers to some 'How many?' questions as predications about concepts. The correct answer to 'how many gallons of water are in the tank?' may be 'ten', but that doesn''t mean ten things instantiate 'gallon of water in the tank'.
	From: Ian Rumfitt (Concepts and Counting [2002], I)
	A reaction: Rumfitt makes the point that a huge number of things instantiate that concept in a ten gallon tank of water. No problem, says Rumfitt, because Frege wouldn't have counted that as a statement of number.

13. Knowledge Criteria / E. Relativism / 6. Relativism Critique

20921	How can we state relativism of sweet and sour, if they have no determinate nature? [Theophrastus]
	Full Idea: How could what is bitter for us be sweet and sour for others, if there is not some determinate nature for them?
	From: Theophrastus (On the Senses [c.321 BCE], 70)
	A reaction: The remark is aimed at Democritus. This is part of the general question of how you can even talk about relativism, without attaching stable meanings to the concepts employed.