display all the ideas for this combination of philosophers
7 ideas
| 12421 | Kant's intuitions struggle to judge relevance, impossibility and exactness [Kitcher on Kant] |
| Full Idea: Kant's intuitions have the Irrelevance problem (which structures of the mind are just accidental?), the Practical Impossibility problem (how to show impossible-in-principle?), and the Exactness problem (are entities exactly as they seem?). | |
| From: comment on Immanuel Kant (Critique of Pure Reason [1781]) by Philip Kitcher - The Nature of Mathematical Knowledge 03.1 | |
| A reaction: [see Kitcher for an examination of these] Presumably the answer to all three must be that we have meta-intuitions about our intuitions, or else intuitions come with built-in criteria to deal with the three problems. We must intuit something specific. |
| 16910 | Mathematics can only start from an a priori intuition which is not empirical but pure [Kant] |
| Full Idea: We find that all mathematical knowledge has this peculiarity, that it must first exhibit its concept in intuition, and do so a priori, in an intuition that is not empirical but pure. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 281) | |
| A reaction: Later thinkers had grave doubts about this Kantian 'intuition', even if they though maths was known a priori. Personally I am increasing fan of rational intuition, even if I am not sure how to discern whether it is rational on any occasion. |
| 16917 | All necessary mathematical judgements are based on intuitions of space and time [Kant] |
| Full Idea: Space and time are the two intuitions on which pure mathematics grounds all its cognitions and judgements that present themselves as at once apodictic and necessary. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284) | |
| A reaction: This unlikely proposal seems to be based on the idea that mathematics must arise from the basic categories of our intuition, and these two are the best candidates he can find. I would say that high-level generality is the basis of mathematics. |
| 17617 | Maths is a priori, but without its relation to empirical objects it is meaningless [Kant] |
| Full Idea: Although all these principles .....are generated in the mind completely a priori, they would still not signify anything at all if we could not always exhibit their significance in appearances (empirical objects). | |
| From: Immanuel Kant (Critique of Pure Reason [1781], B299/A240) | |
| A reaction: This is the subtle Kantian move that we all have to take seriously when we try to assert 'realism' about anything. Our drive for meaning creates our world for us? |
| 16928 | Mathematics cannot be empirical because it is necessary, and that has to be a priori [Kant] |
| Full Idea: Mathematical propositions are always judgements a priori, and not empirical, because they carry with them necessity, which cannot be taken from experience. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 268) | |
| A reaction: Presumably there are necessities in the physical world, and we might discern them by generalising about that world, so that mathematics is (by a tortuous abstract route) a posteriori necessary? Just a thought… |
| 12458 | Kant taught that mathematics is independent of logic, and cannot be grounded in it [Kant, by Hilbert] |
| Full Idea: Kant taught - and it is an integral part of his doctrine - that mathematics treats a subject matter which is given independently of logic. Mathematics, therefore, can never be grounded solely in logic. | |
| From: report of Immanuel Kant (Critique of Pure Reason [1781]) by David Hilbert - On the Infinite p.192 | |
| A reaction: Presumably Gödel's Incompleteness Theorems endorse the Kantian view, that arithmetic is sui generis, and beyond logic. |
| 2795 | If 7+5=12 is analytic, then an infinity of other ways to reach 12 have to be analytic [Kant, by Dancy,J] |
| Full Idea: Kant claimed that 7+5=12 is synthetic a priori. If the concept of 12 analytically involves knowing 7+5, it also involves an infinity of other arithmetical ways to reach 12, which is inadmissible. | |
| From: report of Immanuel Kant (Critique of Pure Reason [1781], B205/A164) by Jonathan Dancy - Intro to Contemporary Epistemology 14.3 |