display all the ideas for this combination of philosophers
10 ideas
| 8726 | Geometry can lead the mind upwards to truth and philosophy [Plato] |
| Full Idea: Geometry can attract the mind towards truth. It can produce philosophical thought, in the sense that it can reverse the midguided downwards tendencies we currently have. | |
| From: Plato (The Republic [c.374 BCE], 527b) | |
| A reaction: Hence the Academy gate bore the inscription "Let no one enter here who is ignorant of geometry". He's not necessarily wrong. Something in early education must straighten out some of the kinks in the messy human mind. |
| 9867 | It is absurd to define a circle, but not be able to recognise a real one [Plato] |
| Full Idea: It will be ridiculous if our student knows the definition of the circle and of the divine sphere itself, but cannot recognize the human sphere and these our circles, used in housebuilding. | |
| From: Plato (Philebus [c.353 BCE], 62a) | |
| A reaction: This is the equivalent of being able to recite numbers, but not to count objects. It also resembles Molyneux's question (to Locke), of whether recognition by one sense entails recognition by others. Nice (and a bit anti-platonist!). |
| 13155 | If you add one to one, which one becomes two, or do they both become two? [Plato] |
| Full Idea: I cannot convince myself that when you add one to one either the first or the second one becomes two, or they both become two by the addition of the one to the other, ...or that when you divide one, the cause of becoming two is now the division. | |
| From: Plato (Phaedo [c.382 BCE], 097d) | |
| A reaction: Lovely questions, all leading to the conclusion that two consists of partaking in duality, to which you can come by several different routes. |
| 9865 | Daily arithmetic counts unequal things, but pure arithmetic equalises them [Plato] |
| Full Idea: The arithmetic of the many computes sums of unequal units, such as two armies, or two herds, ..but philosopher's arithmetic computes when it is guaranteed that none of those infinitely many units differed in the least from any of the others. | |
| From: Plato (Philebus [c.353 BCE], 56d) | |
| A reaction: But of course 'the many' are ironing out the differences too, when they say there are 'three armies'. Shocking snob, Plato. Even philosophers are interested in the difference between three armies and three platoons. |
| 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. |
| 10216 | We master arithmetic by knowing all the numbers in our soul [Plato] |
| Full Idea: It must surely be true that a man who has completely mastered arithmetic knows all numbers? Because there are pieces of knowledge covering all numbers in his soul. | |
| From: Plato (Theaetetus [c.368 BCE], 198b) | |
| A reaction: This clearly views numbers as objects. Expectation of knowing them all is a bit startling! They also appear to be innate in us, and hence they appear to be Forms. See Aristotle's comment in Idea 645. |
| 16150 | One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato] |
| Full Idea: If one is, there must also necessarily be number - Necessarily - But if there is number, there would be many, and an unlimited multitude of beings. ..So if all partakes of being, each part of number would also partake of it. | |
| From: Plato (Parmenides [c.364 BCE], 144a) | |
| A reaction: This seems to commit to numbers having being, then to too many numbers, and hence to too much being - but without backing down and wondering whether numbers had being after all. Aristotle disagreed. |
| 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. |