Talk:Euclidean space

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Revision as of 22:05, 3 March 2018 by Richard Pinch (talk | contribs) (More or less generality: only categorical if you assume Dedekind's axiom)

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More or less generality

The phrase "In a more general sense, a Euclidean space is a finite-dimensional real vector space" does not sound right. Since a finite dimensional real vector space satisfies the abstract Euclidean axioms, it is less general, not more. The general sense should be that, on adding suitable axioms such as continuity and between-ness to those of Euclid, one obtains a system which more or less characterises the real vector space. Richard Pinch (talk) 20:02, 3 March 2018 (CET)

I guess, "more general" means here "not necessarily of dimension 2 or 3". Axioms of Euclid are too ancient... rather, one uses Hilbert system of axioms or another system that is equivalent to real vector space with inner product in dimensions 2 and 3. Boris Tsirelson (talk) 21:34, 3 March 2018 (CET)
Hilbert's axioms are only categorical if you assume Dedekind's axiom (IV.2 in that article), which has the effect of identifying a "line" with a copy of the real line. Without that, a finite-dimensional vector space over the field of real algebraic numbers would also satisfy the axioms. By the way, I think that describing Euclid's axioms as "too ancient" does not adequately capture their status with respect to Hilbert's system. As Hardy said Greek mathematics is the real thing. The Greeks first spoke a language which modern mathematicians can understand: as Littlewood said to me once, they are not clever schoolboys or ‘scholarship candidates’, but ‘Fellows of another college'. Richard Pinch (talk) 22:05, 3 March 2018 (CET)
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