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Full Idea
If-thenism denies that mathematics is in the business of discovering truths about abstracta. ...[their opponents] obviously don't regard any starting point, even a consistent one, as equally worthy of investigation.
Gist of Idea
Critics of if-thenism say that not all starting points, even consistent ones, are worth studying
Source
Penelope Maddy (Defending the Axioms [2011], 3.3)
Book Ref
Maddy,Penelope: 'Defending the Axioms' [OUP 2013], p.99
A Reaction
I have some sympathy with if-thenism, in that you can obviously study the implications of any 'if' you like, but deep down I agree with the critics.
| 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] |