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Difference between revisions of "Talk:Euclidean space"

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(More or less generality: not necessarily of dimension 2 or 3)
(More or less generality: OK, I see)
 
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The phrase "<u>In a more general sense</u>, 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 <u>less</u> 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.  [[User:Richard Pinch|Richard Pinch]] ([[User talk:Richard Pinch|talk]]) 20:02, 3 March 2018 (CET)
 
The phrase "<u>In a more general sense</u>, 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 <u>less</u> 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.  [[User:Richard Pinch|Richard Pinch]] ([[User talk: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. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 21:34, 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. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 21:34, 3 March 2018 (CET)
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::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'.''  [[User:Richard Pinch|Richard Pinch]] ([[User talk:Richard Pinch|talk]]) 22:05, 3 March 2018 (CET)
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:::OK, I see. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 22:52, 3 March 2018 (CET)

Latest revision as of 22:52, 3 March 2018

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)
How to Cite This Entry:
Euclidean space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Euclidean_space&oldid=42902