Combining Texts

Ideas for 'Natural Kinds', 'Set Theory and Its Philosophy' and 'Idealism: a critical survey'

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3 ideas

6. Mathematics / A. Nature of Mathematics / 2. Geometry

16949	Klein summarised geometry as grouped together by transformations [Quine]
	Full Idea: Felix Klein's so-called 'Erlangerprogramm' in geometry involved characterizing the various branches of geometry by what transformations were irrelevant to each.
	From: Willard Quine (Natural Kinds [1969], p.137)

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

10712	If set theory didn't found mathematics, it is still needed to count infinite sets [Potter]
	Full Idea: Even if set theory's role as a foundation for mathematics turned out to be wholly illusory, it would earn its keep through the calculus it provides for counting infinite sets.
	From: Michael Potter (Set Theory and Its Philosophy [2004], 03.8)

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic

17882	It is remarkable that all natural number arithmetic derives from just the Peano Axioms [Potter]
	Full Idea: It is a remarkable fact that all the arithmetical properties of the natural numbers can be derived from such a small number of assumptions (as the Peano Axioms).
	From: Michael Potter (Set Theory and Its Philosophy [2004], 05.2)
	A reaction: If one were to defend essentialism about arithmetic, this would be grist to their mill. I'm just saying.