# Talk:Euclidean space

From Encyclopedia of Mathematics

## 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)- OK, I see. Boris Tsirelson (talk) 22:52, 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

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Euclidean space.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Euclidean_space&oldid=42904