6 ideas
| 17832 | Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M] |
| Full Idea: Zermelo's paper sets out to show that the standard set-theoretic axioms (what he calls the 'constitutive axioms', thus the ZF axioms minus the axiom of infinity) have an unending sequence of different models, thus that they are non-categorical. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Michael Hallett - Introduction to Zermelo's 1930 paper p.1209 | |
| A reaction: Hallett says later that Zermelo is working with second-order set theory. The addition of an Axiom of Infinity seems to have aimed at addressing the problem, and the complexities of that were pursued by Gödel. |
| 13028 | Replacement was added when some advanced theorems seemed to need it [Zermelo, by Maddy] |
| Full Idea: Zermelo included Replacement in 1930, after it was noticed that the sequence of power sets was needed, and Replacement gave the ordinal form of the well-ordering theorem, and justification for transfinite recursion. | |
| From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Penelope Maddy - Believing the Axioms I §1.8 | |
| A reaction: Maddy says that this axiom suits the 'limitation of size' theorists very well, but is not so good for the 'iterative conception'. |
| 17626 | The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo] |
| 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. | |
| From: Ernst Zermelo (On boundary numbers and domains of sets [1930], §5) | |
| 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. |
| 12733 | Because of the definitions of cause, effect and power, cause and effect have the same power [Leibniz] |
| Full Idea: The primary mechanical axiom is that the whole cause and the entire effect have the same power [potentia]. ..This depends on the definition of cause, effect and power. | |
| From: Gottfried Leibniz (De arcanus motus [1676], 203), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 6 | |
| A reaction: This is a useful reminder that if one is going to build a metaphysics on powers (which I intend to do), then the conservation laws in physics are highly relevant. |
| 12734 | Every necessary proposition is demonstrable to someone who understands [Leibniz] |
| Full Idea: Every necessary proposition is demonstrable, at least by someone who understands it. | |
| From: Gottfried Leibniz (De arcanus motus [1676], 203), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 6 | |
| A reaction: This kind of optimism leads to the crisis of the Hilbert Programme in the 1930s. Gödel seems to have conclusively proved that Leibniz was wrong. What would Leibniz have made of Gödel? |
| 12729 | The cause of a change is not the real influence, but whatever gives a reason for the change [Leibniz] |
| Full Idea: That thing from whose state a reason for the changes is most readily provided is adjudged to be the cause. ...Causes are not derived from a real influence, but from the providing of a reason. | |
| From: Gottfried Leibniz (Specimen inventorum [1689], A6.4.1620), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 5 | |
| A reaction: Leibniz is not denying that there are real influences. He seems to be offering the thesis which I am pursuing, that the need for explanation is the crucial factor in shaping the structure of our metaphysics. |