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3 ideas
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| Full Idea: Cantor calls mathematics an empirical science in so far as it begins with consideration of things in the external world; on his view, number originates only by abstraction from objects. | |
| From: report of George Cantor (works [1880]) by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §21 | |
| A reaction: Frege utterly opposed this view, and he seems to have won the day, but I am rather thrilled to find the great Cantor endorsing my own intuitions on the subject. The difficulty is to explain 'abstraction'. |
| 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? |
| 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') |