Combining Philosophers

Ideas for Hermarchus, Michael Potter and Christine M. Korsgaard

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4 ideas

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.
6. Mathematics / C. Sources of Mathematics / 7. Formalism
22310 The formalist defence against Gödel is to reject his metalinguistic concept of truth [Potter]
     Full Idea: Gödel's theorem does not refute formalism outright, because the committed formalist need not recognise the metalinguistic notion of truth to which the theorem appeals.
     From: Michael Potter (The Rise of Analytic Philosophy 1879-1930 [2020], 45 'Log')
     A reaction: The theorem was prior to Tarski's account of truth. Potter says Gödel avoided explicit mention of truth because of this problem. In general Gödel showed that there are truths outside the formal system (which is all provable).
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
22298 Why is fictional arithmetic applicable to the real world? [Potter]
     Full Idea: Fictionalists struggle to explain why arithmetic is applicable to the real world in a way that other stories are not.
     From: Michael Potter (The Rise of Analytic Philosophy 1879-1930 [2020], 21 'Math')
     A reaction: We know why some novels are realistic and others just the opposite. If a novel aimed to 'model' the real world it would be even closer to it. Fictionalists must explain why some fictions are useful.