display all the ideas for this combination of texts
13 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. |
| 13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD] |
| Full Idea: We can show (using the axiom of choice) that the less-than relation, <, well-orders the ordinals, ...and that it partially orders the ordinals, ...and that it totally orders the ordinals. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) |
| 13491 | The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD] |
| Full Idea: The axiom of infinity with separation yields a least limit ordinal, which is called ω. | |
| From: William D. Hart (The Evolution of Logic [2010], 3) |
| 13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD] |
| Full Idea: Since we can map the transfinite ordinals one-one into the infinite cardinals, there are at least as many infinite cardinals as transfinite ordinals. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) |
| 13492 | Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD] |
| Full Idea: It is easier to generalize von Neumann's finite ordinals into the transfinite. All Zermelo's nonzero finite ordinals are singletons, but if ω were a singleton it is hard to see how if could fail to be the successor of its member and so not a limit. | |
| From: William D. Hart (The Evolution of Logic [2010], 3) |
| 13446 | 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD] |
| Full Idea: The real numbers were not isolated from geometry until the arithmetization of analysis during the nineteenth century. | |
| From: William D. Hart (The Evolution of Logic [2010], 1) |
| 13509 | We can establish truths about infinite numbers by means of induction [Hart,WD] |
| Full Idea: Mathematical induction is a way to establish truths about the infinity of natural numbers by a finite proof. | |
| From: William D. Hart (The Evolution of Logic [2010], 5) | |
| A reaction: If there are truths about infinities, it is very tempting to infer that the infinities must therefore 'exist'. A nice, and large, question in philosophy is whether there can be truths without corresponding implications of existence. |
| 13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD] |
| Full Idea: There is a familiar comparison between Euclid (unique parallel) and 'spherical' geometry (no parallel) and 'saddle' geometry (several parallels). | |
| From: William D. Hart (The Evolution of Logic [2010], 2) |
| 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. |
| 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
| Full Idea: The thesis that no existence proposition is analytic is one of the few constants in philosophical consciences, but there are many existence claims in mathematics, such as the infinity of primes, five regular solids, and certain undecidable propositions. | |
| From: William D. Hart (The Evolution of Logic [2010], 2) |