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Full Idea
One of the culminating achievements of Euclidean geometry was categorical axiomatisations, that describe the geometric structure so completely that any two models of the axioms are isomorphic. The axioms are second-order.
Gist of Idea
The culmination of Euclidean geometry was axioms that made all models isomorphic
Source
Vann McGee (Logical Consequence [2014], 7)
Book Ref
'Bloomsbury Companion to Philosophical Logic', ed/tr. Horsten,L/Pettigrew,R [Bloomsbury 2014], p.45
A Reaction
[He cites Veblen 1904 and Hilbert 1903] For most mathematicians, categorical axiomatisation is the best you can ever dream of (rather than a single true axiomatisation).
Related Idea
Idea 10246 The limit of science is isomorphism of theories, with essences a matter of indifference [Weyl]
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| 18754 | Logically valid sentences are analytic truths which are just true because of their logical words [McGee] |
| 18755 | Validity is explained as truth in all models, because that relies on the logical terms [McGee] |
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| 18760 | The culmination of Euclidean geometry was axioms that made all models isomorphic [McGee] |
| 18761 | Second-order variables need to range over more than collections of first-order objects [McGee] |
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