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Full Idea
We must only assert of various geometries that the axioms imply the propositions, not that the axioms are true and therefore that the propositions are true.
Gist of Idea
Geometrical axioms imply the propositions, but the former may not be true
Source
Bertrand Russell (Foundations of Geometry [1897], Intro vii), quoted by Alan Musgrave - Logicism Revisited §4
Book Ref
-: 'British Soc for the Philosophy of Science' [-], p.109
A Reaction
Clearly the truth of the axioms can remain a separate issue from whether they actually imply the theorems. The truth of the axioms might be as much a metaphysical as an empirical question. Musgrave sees this as the birth of if-thenism.
| 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] |