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Ideas for 'fragments/reports', 'The Human Body is a sort of Machine' and 'Philosophy of Mathematics'

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6 ideas

4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
8077 Stoic propositional logic is like chemistry - how atoms make molecules, not the innards of atoms [Chrysippus, by Devlin]
     Full Idea: In Stoic logic propositions are treated the way atoms are treated in present-day chemistry, where the focus is on the way atoms fit together to form molecules, rather than on the internal structure of the atoms.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Keith Devlin - Goodbye Descartes Ch.2
     A reaction: A nice analogy to explain the nature of Propositional Logic, which was invented by the Stoics (N.B. after Aristotle had invented predicate logic).
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
20791 Chrysippus has five obvious 'indemonstrables' of reasoning [Chrysippus, by Diog. Laertius]
     Full Idea: Chrysippus has five indemonstrables that do not need demonstration:1) If 1st the 2nd, but 1st, so 2nd; 2) If 1st the 2nd, but not 2nd, so not 1st; 3) Not 1st and 2nd, the 1st, so not 2nd; 4) 1st or 2nd, the 1st, so not 2nd; 5) 1st or 2nd, not 2nd, so 1st.
     From: report of Chrysippus (fragments/reports [c.240 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 07.80-81
     A reaction: [from his lost text 'Dialectics'; squashed to fit into one quote] 1) is Modus Ponens, 2) is Modus Tollens. 4) and 5) are Disjunctive Syllogisms. 3) seems a bit complex to be an indemonstrable.
4. Formal Logic / D. Modal Logic ML / 1. Modal Logic
10206 Modal operators are usually treated as quantifiers [Shapiro]
     Full Idea: It is common now, and throughout the history of philosophy, to interpret modal operators as quantifiers. This is an analysis of modality in terms of ontology.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10208 Axiom of Choice: some function has a value for every set in a given set [Shapiro]
     Full Idea: One version of the Axiom of Choice says that for every set A of nonempty sets, there is a function whose domain is A and whose value, for every a ∈ A, is a member of a.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 1)
10252 The Axiom of Choice seems to license an infinite amount of choosing [Shapiro]
     Full Idea: If the Axiom of Choice says we can choose one member from each of a set of non-empty sets and put the chosen elements together in a set, this licenses the constructor to do an infinite amount of choosing.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 6.3)
     A reaction: This is one reason why the Axiom was originally controversial, and still is for many philosophers.
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
10207 Anti-realists reject set theory [Shapiro]
     Full Idea: Anti-realists reject set theory.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
     A reaction: That is, anti-realists about mathematical objects. I would have thought that one could accept an account of sets as (say) fictions, which provided interesting models of mathematics etc.