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Full Idea
Mathematics contains axioms (definitions) and conclusions from definitions. Its objects do not exist. The truth of its conclusions rests on the accuracy of logical thought.
Gist of Idea
Mathematics is just accurate inferences from definitions, and doesn't involve objects
Source
Friedrich Nietzsche (Unpublished Notebooks 1884-85 [1884], 25[307])
Book Ref
Nietzsche,Friedrich: 'Fragments from 1884-85 (v 15)', ed/tr. Loeb,P.S./Tinsley,D.F. [Stanford 2022], p.78
A Reaction
Not suprising to find Nietzsche defying platonism. This is evidence that he was a systematic philosopher, who knew mathematics could be a challenge to his naturalism.
| 10054 | Arithmetic and geometry achieve some certainty without worrying about existence [Descartes] |
| 10055 | Mathematical proofs work, irrespective of whether the objects exist [Locke] |
| 10056 | At bottom eternal truths are all conditional [Leibniz] |
| 14783 | Logic, unlike mathematics, is not hypothetical; it asserts categorical ends from hypothetical means [Peirce] |
| 21493 | Pure mathematics deals only with hypotheses, of which the reality does not matter [Peirce] |
| 24137 | Mathematics is just accurate inferences from definitions, and doesn't involve objects [Nietzsche] |
| 10053 | Geometrical axioms imply the propositions, but the former may not be true [Russell] |
| 10064 | Quine quickly dismisses If-thenism [Quine, by Musgrave] |
| 10066 | Putnam coined the term 'if-thenism' [Putnam, by Musgrave] |
| 10061 | The If-thenist view only seems to work for the axiomatised portions of mathematics [Musgrave] |
| 10065 | Perhaps If-thenism survives in mathematics if we stick to first-order logic [Musgrave] |
| 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
| 22291 | Deductivism can't explain how the world supports unconditional conclusions [Potter] |