display all the ideas for this combination of philosophers
9 ideas
| 16918 | Mathematics cannot proceed just by the analysis of concepts [Kant] |
| Full Idea: Mathematics cannot proceed analytically, namely by analysis of concepts, but only synthetically. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284) | |
| A reaction: I'm with Kant insofar as I take mathematics to be about the world, no matter how rarefied and 'abstract' it may become. |
| 16930 | Geometry is not analytic, because a line's being 'straight' is a quality [Kant] |
| Full Idea: No principle of pure geometry is analytic. That the straight line beween two points is the shortest is a synthetic proposition. For my concept of straight contains nothing of quantity but only of quality. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 269) | |
| A reaction: I'm not sure what his authority is for calling straightness a quality rather than a quantity, given that it can be expressed quantitatively. It is a very nice example for focusing our questions about the nature of geometry. I can't decide. |
| 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. |
| 16919 | Geometry rests on our intuition of space [Kant] |
| Full Idea: Geometry is grounded on the pure intuition of space. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284) | |
| A reaction: I have the impression that recent thinkers are coming round to this idea, having attempted purely algebraic or logical accounts of geometry. |
| 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. |
| 16920 | Numbers are formed by addition of units in time [Kant] |
| Full Idea: Arithmetic forms its own concepts of numbers by successive addition of units in time. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284) | |
| A reaction: It is hard to imagine any modern philosopher of mathematics embracing this idea. It sounds as if Kant thinks counting is the foundation of arithmetic, which I quite like. |
| 16929 | 7+5 = 12 is not analytic, because no analysis of 7+5 will reveal the concept of 12 [Kant] |
| Full Idea: The concept of twelve is in no way already thought by merely thinking the unification of seven and five, and though I analyse my concept of such a possible sum as long as I please, I shall never find twelve in it. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 269) | |
| A reaction: It might be more plausible to claim that an analysis of 12 would reveal the concept of 7+5. Doesn't the concept of two collections of objects contain the concept of their combined cardinality? |
| 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. |