Talk:Space (mathematics)

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The differences section in this article is unaccountably poor. There is no evidence or citations for it either. It sounds like the musings of a semi-informed dilettante. —Preceding unsigned comment added by (talk) 20:41, 12 November 2009 (UTC)

I agree that I am a semi-informed dilettante in the history of mathematics. However, I do not agree that "there is no evidence or citations for it either". Just the opposite! Each item of the table corresponds to some phrase of the "History" section. And each of these phrases has a reference. Boris Tsirelson (talk) 07:14, 13 November 2009 (UTC)
In particular, the first two lines of the table, "axioms are obvious implications of definitions" – "axioms are conventional" and "theorems are absolute objective truth" – "theorems are implications of the corresponding axioms", correspond to the last phrase of the first paragraph of "History": "At that time geometric theorems were treated as an absolute objective truth knowable through intuition and reason, similarly to objects of natural science;[2] and axioms were treated as obvious implications of definitions.[3]", with two refs, and paragraph 6: "This discovery forced the abandonment of the pretensions to the absolute truth of Euclidean geometry. It showed that axioms are not "obvious", nor "implications of definitions". Rather, they are hypotheses. To what extent do they correspond to an experimental reality? This important physical problem has nothing anymore to do with mathematics. Even if a "geometry" does not correspond to an experimental reality, its theorems remain no less "mathematical truths".[3]"
Line 3 of the table, "relationships between points, lines etc. are determined by their nature" – "relationships between points, lines etc. are essential; their nature is not" corresponds to paragraph 7: "It shows that relations between objects are essential in mathematics, while the nature of the objects is not."
Line 4 of the table, "mathematical objects are given to us with their structure" – "each mathematical theory describes its objects by some of their properties" corresponds to paragraph 3: "The relation between the two geometries, Euclidean and projective,[4] shows that mathematical objects are not given to us with their structure.[5] Rather, each mathematical theory describes its objects by some of their properties, precisely those that are put as axioms at the foundations of the theory.[6]"
Line 5 of the table, "geometry corresponds to an experimental reality" – "geometry is a mathematical truth" corresponds to paragraph 6 again (see above).
Line 6 of the table, "all geometric properties of the space follow from the axioms" – "axioms of a space need not determine all geometric properties" corresponds to the last paragraph of Section 3.2 "Two relations...": "Euclidean axioms leave no freedom, they determine uniquely all geometric properties of the space. More exactly: all three-dimensional Euclidean spaces are mutually isomorphic. In this sense we have "the" three-dimensional Euclidean space. In terms of Bourbaki, the corresponding theory is univalent. In contrast, topological spaces are generally non-isomorphic, their theory is multivalent. A similar idea occurs in mathematical logic: a theory is called categorical if all its models are mutually isomorphic. According to Bourbaki,[15] the study of multivalent theories is the most striking feature which distinguishes modern mathematics from classical mathematics."
Line 7 of the table, "geometry is an autonomous and living science" – "classical geometry is a universal language of mathematics" corresponds to paragraph 2 of Section 2.2 "The golden age...": "According to Bourbaki,[11] "passed over in its role as an autonomous and living science, classical geometry is thus transfigured into a universal language of contemporary mathematics"."
Lines 8 and 9 of the table are probably uncontroversial.
Anyway, I'll be happy of corrections/improvements made by an expert in the history of mathematics. Boris Tsirelson (talk) 07:26, 13 November 2009 (UTC)
I appreciate the fact that you took the time to respond in such depth. I believe the main problem is that the table doesn't really add anything to the article. As it stands, it's a rather subjective and incomplete summary of the text. I think statements like "geometry is an autonomous and living science" and "classical geometry is a universal language of mathematics" are too vague to be useful, are inherently subjective, and don't really add anything to the article. While it's true that you can find instances of mathematicians say those things, ripped out of context like that, they become meaningless. I don't really know what it ACTUALLY means to be an "autonomous science" or a "universal language" of mathematics. Are the two things opposed to one another? I think it would be remiss to characterise the many different philosophical views mathematicians past and present have into such a cut and dry binary classification as "modern" vs. "classic".

I'd be interested in hearing from other readers on this. (talk) 02:11, 17 November 2009 (UTC)

Well, then, really, I do not know. It seems to me the summary is useful, but maybe it is not. Also I'd be interested in hearing from other readers on this. Boris Tsirelson (talk) 06:57, 17 November 2009 (UTC)
I just think it looks slightly unprofessional. I wouldn't expect to find a table like this in an encyclopaedia or textbook. I think the content in the table is too nuanced and subtle to be put into a table. The ideas in that table need to be explained in complete sentences, with qualifications, explanations, citations and references. (by the way, I am the original anon posting from a different computer) (talk) 12:04, 19 November 2009 (UTC)

Hierarchy Image[edit]

Are the image and associated information saying that a space having an inner product has a norm, ∴ a metric, ∴ a topology? If so, the image is an overcomplicated representation for such a chain reaction. If not, it should be clarified. Regardless of what's being communicated, the image does the job much less efficiently and effectively than could a simple line of text with some arrows or other symbols. See containment hierarchy LokiClock (talk) 17:26, 20 December 2009 (UTC)

The image is a part of a stub to which I have added a lot. Maybe it really is an overcomplicated representation for the chain. Delete it if you wish. I did not delete it mostly because it seems to be desirable to have at least one image, and I did not invent a better one. Boris Tsirelson (talk) 20:06, 22 December 2009 (UTC)

I think it is a good image, but it would be interesting to add Uniform spaces between Metric spaces and Topological spaces. Bruno321 (talk) 17:01, 16 February 2010 (UTC)

Taxonomy of spaces[edit]

Zoo of spaces[edit]

Linear and topological spaces[edit]

Part of paragraph 2 reads: "Every complex linear space is also a real linear space (the latter underlies the former), since each real number is also a complex number." Isn't a real linear space also a complex linear space, not the other way around?
Surely not. For example, the real line is a (one-dimensional) real linear space. But it is not a complex linear space. On the other hand, the complex plane is a one-dim complex linear space, but can also be treated as a two-dim real linear space. Boris Tsirelson (talk) 07:08, 11 March 2010 (UTC)
Someone was too bold to implement the wrong idea expressed above (see version of 00:01, 10 April 2010). Thus I've added an explanatory note to the article. Hope it will help. Boris Tsirelson (talk) 16:24, 10 April 2010 (UTC)

Category theory[edit]

This article should mention category theory: in a way, it provides a unified framework that treats most (all?) of the enumerated types of spaces at once. —Preceding unsigned comment added by (talk) 02:38, 3 April 2010 (UTC)

Difference between Space and Algebraic structure[edit]

I think it would be useful to note the difference between a Space and an Algebraic structure. For example, a Vector space is both a space and an algebraic structure, so what is the difference? Can there be spaces that are not algebraic structures, and reverse? And so on. --Aqwis (talk) 06:56, 23 July 2012 (UTC)

Why only algebraic structure? Topological structure is not. Boris Tsirelson (talk) 13:47, 23 July 2012 (UTC)


The article should have a section explaining the concept of subspace. Currently, subspace (disambiguation) makes an appearance that there are merely several terminological coincidences. Incnis Mrsi (talk) 11:05, 27 April 2013 (UTC)

Or maybe a separate article "subspace (mathematics)"? Boris Tsirelson (talk) 16:00, 27 April 2013 (UTC)