display all the ideas for this combination of texts
11 ideas
| 8739 | Geometry studies the Euclidean space that dictates how we perceive things [Kant, by Shapiro] |
| Full Idea: For Kant, geometry studies the forms of perception in the sense that it describes the infinite space that conditions perceived objects. This Euclidean space provides the forms of perception, or, in Kantian terms, the a priori form of empirical intuition. | |
| From: report of Immanuel Kant (Critique of Pure Reason [1781]) by Stewart Shapiro - Thinking About Mathematics 4.2 | |
| A reaction: We shouldn't assume that the discovery of new geometries nullifies this view. We evolved in small areas of space, where it is pretty much Euclidean. We don't perceive the curvature of space. |
| 8740 | Geometry would just be an idle game without its connection to our intuition [Kant] |
| Full Idea: Were it not for the connection to intuition, geometry would have no objective validity whatever, but be mere play by the imagination or the understanding. | |
| From: Immanuel Kant (Critique of Pure Reason [1781], B298/A239), quoted by Stewart Shapiro - Thinking About Mathematics 4.2 | |
| A reaction: If we pursue the idealist reading of Kant (in which the noumenon is hopelessly inapprehensible), then mathematics still has not real application, despite connection to intuition. However, Kant would have been an intuitionist, and not a formalist. |
| 16899 | Geometrical truth comes from a general schema abstracted from a particular object [Kant, by Burge] |
| Full Idea: Kant explains the general validity of geometrical truths by maintaining that the particularity is genuine and ineliminable but is used as a schema. One abstracts from the particular elements of the objects of intuition in forming a general object. | |
| From: report of Immanuel Kant (Critique of Pure Reason [1781], B741/A713) by Tyler Burge - Frege on Apriority (with ps) 4 | |
| A reaction: A helpful summary by Burge of a rather wordy but very interesting section of Kant. I like the idea of being 'abstracted', but am not sure why that must be from one particular instance [certainty?]. The essence of triangles emerges from comparisons. |
| 9632 | Kant only accepts potential infinity, not actual infinity [Kant, by Brown,JR] |
| Full Idea: For Kant the only legitimate infinity is the so-called potential infinity, not the actual infinity. | |
| From: report of Immanuel Kant (Critique of Pure Reason [1781]) by James Robert Brown - Philosophy of Mathematics Ch.5 | |
| A reaction: This is part of what leads on the the Constructivist view of mathematics. There is a procedure for endlessly continuing, but no procedure for arriving. That seems to make good sense. |
| 3343 | Euclid's could be the only viable geometry, if rejection of the parallel line postulate doesn't lead to a contradiction [Benardete,JA on Kant] |
| Full Idea: The possible denial of the parallel lines postulate does not entail that Kant was wrong in considering Euclid's the only viable geometry. If the denial issued in a contradiction, then the postulate would be analytic, and Kant would be refuted. | |
| From: comment on Immanuel Kant (Critique of Pure Reason [1781]) by José A. Benardete - Metaphysics: the logical approach Ch.18 |
| 8737 | Kant suggested that arithmetic has no axioms [Kant, by Shapiro] |
| Full Idea: Kant suggested that arithmetic has no axioms. | |
| From: report of Immanuel Kant (Critique of Pure Reason [1781], B204-6/A164) by Stewart Shapiro - Thinking About Mathematics 4.2 | |
| A reaction: A hundred years later a queue was forming to spell out the axioms of arithmetic. The definitions of 0 and 1 always look to me more like logicians' tricks than profound truths. Some notions of successor and induction do, however, seem needed. |
| 5557 | Axioms ought to be synthetic a priori propositions [Kant] |
| Full Idea: Concerning magnitude ...there are no axioms in the proper sense. ....Axioms ought to be synthetic a priori propositions. | |
| From: Immanuel Kant (Critique of Pure Reason [1781], B205/A164) | |
| A reaction: This may be a hopeless dream, but it is (sort of) what all philosophers long for. Post-modern relativism may just be the claim that all axioms are analytic. Could a posteriori propositions every qualify as axioms? |
| 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. |
| 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? |
| 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 |