| 10054 | Arithmetic and geometry achieve some certainty without worrying about existence [Descartes] |
| Full Idea: Arithmetic, geometry and sciences of that kind only treat of things without taking any great trouble to ascertain whether they are actually existent or not, and contain some measure of certainty. | |
| From: René Descartes (Meditations [1641], §1), quoted by Alan Musgrave - Logicism Revisited §4 | |
| A reaction: This is Musgrave's earliest quotation which seems to take the if-thenist view. |
| 10055 | Mathematical proofs work, irrespective of whether the objects exist [Locke] |
| Full Idea: All the demonstrations of mathematicians are the same, whether there be any square or circle existing in the world or no. | |
| From: John Locke (Essay Conc Human Understanding (2nd Ed) [1694], 4.04.08) | |
| A reaction: Musgrave gives this as an early indication of the if-thenist view of mathematics. |
| 10056 | At bottom eternal truths are all conditional [Leibniz] |
| Full Idea: At bottom eternal truths are all conditional, saying 'granted such a thing, such another thing is'. | |
| From: Gottfried Leibniz (New Essays on Human Understanding [1704], 4.11.14), quoted by Alan Musgrave - Logicism Revisited §4 | |
| A reaction: Thus showing Leibniz to have sympathy with the if-thenist view. He cites geometry as his illustration. |
| 14783 | Logic, unlike mathematics, is not hypothetical; it asserts categorical ends from hypothetical means [Peirce] |
| Full Idea: Mathematics is purely hypothetical: it produces nothing but conditional propositions. Logic, on the contrary, is categorical in its assertions. True, it is a normative science, and not a mere discovery of what really is. It discovers ends from means. | |
| From: Charles Sanders Peirce (The Nature of Mathematics [1898], II) |
| 21493 | Pure mathematics deals only with hypotheses, of which the reality does not matter [Peirce] |
| Full Idea: The pure mathematician deals exclusively with hypotheses. Whether or not there is any corresponding real thing, he does not care. | |
| From: Charles Sanders Peirce (works [1892], CP5.567), quoted by Albert Atkin - Peirce 3 'separation' | |
| A reaction: [Dated 1902] Maybe we should identify a huge branch of human learning as Hyptheticals. Professor of Hypotheticals at Cambridge University. The trouble is it would have to include computer games. So why does maths matter more than games? |
| 24137 | Mathematics is just accurate inferences from definitions, and doesn't involve objects [Nietzsche] |
| 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. | |
| From: Friedrich Nietzsche (Unpublished Notebooks 1884-85 [1884], 25[307]) | |
| 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. |
| 10053 | Geometrical axioms imply the propositions, but the former may not be true [Russell] |
| 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. | |
| From: Bertrand Russell (Foundations of Geometry [1897], Intro vii), quoted by Alan Musgrave - Logicism Revisited §4 | |
| 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. |
| 10064 | Quine quickly dismisses If-thenism [Quine, by Musgrave] |
| Full Idea: Quine quickly dismisses If-thenism. | |
| From: report of Willard Quine (Truth by Convention [1935], p.327) by Alan Musgrave - Logicism Revisited §5 | |
| A reaction: [Musgrave quotes a long chunk of Quine which is hard to compress!] Effectively, he says If-thenism is cheating, or begs the question, by eliminating whole sections of perfectly good mathematics, because they cannot be derived from axioms. |
| 10066 | Putnam coined the term 'if-thenism' [Putnam, by Musgrave] |
| Full Idea: Putnam coined the term 'if-thenism'. | |
| From: report of Hilary Putnam (The Thesis that Mathematics is Logic [1967]) by Alan Musgrave - Logicism Revisited §5 n |
| 10061 | The If-thenist view only seems to work for the axiomatised portions of mathematics [Musgrave] |
| Full Idea: The If-thenist view seems to apply straightforwardly only to the axiomatised portions of mathematics. | |
| From: Alan Musgrave (Logicism Revisited [1977], §5) | |
| A reaction: He cites Lakatos to show that cutting-edge mathematics is never axiomatised. One might reply that if the new mathematics is any good then it ought to be axiomatis-able (barring Gödelian problems). |
| 10065 | Perhaps If-thenism survives in mathematics if we stick to first-order logic [Musgrave] |
| Full Idea: If we identify logic with first-order logic, and mathematics with the collection of first-order theories, then maybe we can continue to maintain the If-thenist position. | |
| From: Alan Musgrave (Logicism Revisited [1977], §5) | |
| A reaction: The problem is that If-thenism must rely on rules of inference. That seems to mean that what is needed is Soundness, rather than Completeness. That is, inference by the rules must work properly. |
| 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
| 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. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.3) | |
| 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. |
| 22291 | Deductivism can't explain how the world supports unconditional conclusions [Potter] |
| Full Idea: Deductivism is a good account of large parts of mathematics, but stumbles where mathematics is directly applicable to the world. It fails to explain how we detach the antecedent so as to arrive at unconditional conclusions. | |
| From: Michael Potter (The Rise of Analytic Philosophy 1879-1930 [2020], 12 'Deduc') | |
| A reaction: I suppose the reply would be that we have designed deductive structures which fit our understanding of reality - so it is all deductive, but selected pragmatically. |