display all the ideas for this combination of texts
4 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. |