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Full Idea
Two opposite tendencies of thought, the idea of creative advance and of collection and completion (underlying the Kantian 'antinomies') find their symbolic representation and their symbolic reconciliation in the transfinite numbers based on well-ordering.
Gist of Idea
The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers
Source
Ernst Zermelo (On boundary numbers and domains of sets [1930], §5)
Book Ref
'From Kant to Hilbert: sourcebook Vol. 2', ed/tr. Ewald,William [OUP 1996], p.1233
A Reaction
[a bit compressed] It is this sort of idea, from one of the greatest set-theorists, that leads philosophers to think that the philosophy of mathematics may offer solutions to metaphysical problems. As an outsider, I am sceptical.
| 13986 | Plato found antinomies in ideas, Kant in space and time, and Bradley in relations [Plato, by Ryle] |
| 14150 | Plato's 'Parmenides' is perhaps the best collection of antinomies ever made [Russell on Plato] |
| 21454 | The battle of the antinomies is usually won by the attacker, and lost by any defender [Kant] |
| 15628 | The idea that contradiction is essential to rational understanding is a key modern idea [Hegel] |
| 15629 | Tenderness for the world solves the antinomies; contradiction is in our reason, not in the essence of the world [Hegel] |
| 15630 | Antinomies are not just in four objects, but in all objects, all representations, all objects and all ideas [Hegel] |
| 17626 | The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo] |
| 21691 | Antinomies contradict accepted ways of reasoning, and demand revisions [Quine] |
| 9125 | Denying problems, or being romantically defeated by them, won't make them go away [Sorensen] |