display all the ideas for this combination of texts
4 ideas
| 9863 | We aim for elevated discussion of pure numbers, not attaching them to physical objects [Plato] |
| Full Idea: Our discussion of numbers leads the soul forcibly upward and compels it to discuss the numbers themselves, never permitting anyone to propose for discussion numbers attached to visible or tangible bodies. | |
| From: Plato (The Republic [c.374 BCE], 525d) | |
| A reaction: This strikes me as very important, because it shows that the platonist view of numbers places little or no importance on counting, inviting the question of whether they could be understood in complete ignorance of the process of counting. |
| 9864 | In pure numbers, all ones are equal, with no internal parts [Plato] |
| Full Idea: With those numbers that can be grasped only in thought, ..each one is equal to every other, without the least difference and containing no internal parts. | |
| From: Plato (The Republic [c.374 BCE], 526a) | |
| A reaction: [Two voices in the conversation are elided] Intriguing and tantalising. Does 13 have internal parts, in the platonist view? If so, is it more than the sum of its parts? Is Plato committed to numbers being built from indistinguishable abstract units/ |
| 8727 | Geometry is not an activity, but the study of unchanging knowledge [Plato] |
| Full Idea: Geometers talk as if they were actually doing something, and the point of their theorems is to have some effect (like 'squaring'). ...But the sole purpose is knowledge, of things which exist forever, not coming into existence and passing away. | |
| From: Plato (The Republic [c.374 BCE], 527a) | |
| A reaction: Modern Constructivism defends the view which Plato is attacking. The existence of real infinities can be doubted simply because we have not got enough time to construct them. |
| 9861 | The same thing is both one and an unlimited number at the same time [Plato] |
| Full Idea: We see the same thing to be both one and an unlimited number at the same time. | |
| From: Plato (The Republic [c.374 BCE], 525a) | |
| A reaction: Frege makes the same point, that a pair of boots is both two and one. The point is at its strongest in opposition to empirical accounts of arithmetic. However, Mill observes that pebbles can be both 5 and 3+2, without contradiction. |