Combining Texts

All the ideas for 'Remarks on the definition and nature of mathematics', 'Particulars in Particular Clothing' and 'works'

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8 ideas

5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
17807 To study formal systems, look at the whole thing, and not just how it is constructed in steps [Curry]
     Full Idea: In the study of formal systems we do not confine ourselves to the derivation of elementary propositions step by step. Rather we take the system, defined by its primitive frame, as datum, and then study it by any means at our command.
     From: Haskell B. Curry (Remarks on the definition and nature of mathematics [1954], 'The formalist')
     A reaction: This is what may potentially lead to an essentialist view of such things. Focusing on bricks gives formalism, focusing on buildings gives essentialism.
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
8717 Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) [Hilbert, by Friend]
     Full Idea: Hilbert wanted to derive ideal mathematics from the secure, paradox-free, finite mathematics (known as 'Hilbert's Programme'). ...Note that for the realist consistency is not something we need to prove; it is a precondition of thought.
     From: report of David Hilbert (works [1900], 6.7) by Michčle Friend - Introducing the Philosophy of Mathematics
     A reaction: I am an intuitive realist, though I am not so sure about that on cautious reflection. Compare the claims that there are reasons or causes for everything. Reality cannot contain contradicitions (can it?). Contradictions would be our fault.
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
17806 It is untenable that mathematics is general physical truths, because it needs infinity [Curry]
     Full Idea: According to realism, mathematical propositions express the most general properties of our physical environment. This is the primitive view of mathematics, yet on account of the essential role played by infinity in mathematics, it is untenable today.
     From: Haskell B. Curry (Remarks on the definition and nature of mathematics [1954], 'The problem')
     A reaction: I resist this view, because Curry's view seems to imply a mad metaphysics. Hilbert resisted the role of the infinite in essential mathematics. If the physical world includes its possibilities, that might do the job. Hellman on structuralism?
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
17808 Saying mathematics is logic is merely replacing one undefined term by another [Curry]
     Full Idea: To say that mathematics is logic is merely to replace one undefined term by another.
     From: Haskell B. Curry (Remarks on the definition and nature of mathematics [1954], 'Mathematics')
6. Mathematics / C. Sources of Mathematics / 7. Formalism
10113 The grounding of mathematics is 'in the beginning was the sign' [Hilbert]
     Full Idea: The solid philosophical attitude that I think is required for the grounding of pure mathematics is this: In the beginning was the sign.
     From: David Hilbert (works [1900]), quoted by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6
     A reaction: Why did people invent those particular signs? Presumably they were meant to designate something, in the world or in our experience.
10115 Hilbert substituted a syntactic for a semantic account of consistency [Hilbert, by George/Velleman]
     Full Idea: Hilbert replaced a semantic construal of inconsistency (that the theory entails a statement that is necessarily false) by a syntactic one (that the theory formally derives the statement (0 =1 ∧ 0 not-= 1).
     From: report of David Hilbert (works [1900]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6
     A reaction: Finding one particular clash will pinpoint the notion of inconsistency, but it doesn't seem to define what it means, since the concept has very wide application.
6. Mathematics / C. Sources of Mathematics / 8. Finitism
10116 Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions [Hilbert, by George/Velleman]
     Full Idea: Hilbert's project was to establish the consistency of classical mathematics using just finitary means, to convince all parties that no contradictions will follow from employing the infinitary notions and reasoning.
     From: report of David Hilbert (works [1900]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6
     A reaction: This is the project which was badly torpedoed by Gödel's Second Incompleteness Theorem.
8. Modes of Existence / B. Properties / 13. Tropes / a. Nature of tropes
18431 Internal relations combine some tropes into a nucleus, which bears the non-essential tropes [Simons, by Edwards]
     Full Idea: Simons's 'nuclear' option blends features of the substratum and bundle theories. First we have tropes collected by virtue of their internal relations, forming the essential kernel or nucleus. This nucleus then bears the non-essential tropes.
     From: report of Peter Simons (Particulars in Particular Clothing [1994], p.567) by Douglas Edwards - Properties 3.5
     A reaction: [compression of Edwards's summary] This strikes me as being a remarkably good theory. I am not sure of the ontological status of properties, such that they can (unaided) combine to make part of an object. What binds the non-essentials?