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The original title, "has the bases of geometry been reconsidered in 20th century" of this question refers to Riemann's paper "On the Hypotheses which lie at the Bases of Geometry", an English version can be found here.

When we talk about the physical "space", we model it in terms of manifold, and usually to do some geometry one starts with definitions of Riemannian manifolds, on which one can talk about length, angle, area, volume, etc. This, of course, models our intuition of the space we are in pretty well and it seems rather reasonable to believe the space we are in carries a Riemannian metric. However, Riemann predates all the physics break through in the 20th century, which includes general relativity and quantum mechanics. The former is like a major triumph of Riemannian geometry but I don't know if the latter has anything to do with it.

My question is the following: with these physics breakthrough,

has the hypotheses which lie at the bases of geometry been reconsidered?

In other words, Riemannian geometry seems to model geometry on the scale of daily-life objects (i.e. not too small or too large) pretty well, and it seems to model the space pretty well on a large scale, but what about the microscopic level? Is it still a good model of the physical "space" on that scale?

In case people want to label this as "not a real question", let me quote Riemann's own question at the end of his paper ("plan of inquiry", copied from the English version):

III. Application to Space.

§ 1. System of facts which suffice to determine the measure-relations of space assumed in geometry.

§ 2. How far is the validity of these empirical determinations probable beyond the limits of observation towards the infinitely great?

§ 3. How far towards the infinitely small? Connection of this question with the interpretation of nature.

Question 2 and 3 on his list is the thing I want to ask here. (I don't understand what 1 means here.) .

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Why are there still 2 votes to close? At least consider this as a math-history question OK? –  36min Nov 17 '12 at 2:07
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The philosophical landscape on the relationship between mathematics and physics changed greatly over the 20th century. The generation of Hilbert and Riemann thought that reality IS mathematical, whether they interpreted the word 'reality' in a realist or Kantian idealist sense. Nowadays, most philosophers, and even most mathematicians and physicists, only accept the weaker claim that reality can be described mathematically, or perhaps that reality is best described mathematically. In our context, questions like Riemann's or Hilbert's 6th problem (axiomatize physics) seem quite nonsensical. –  Alexander Woo Nov 17 '12 at 16:18

2 Answers 2

It seems easier to answer the (current) title than the body of the question, so here goes: The notion of "space" has been vastly enlarged in the 20th century, even including some notions that Riemann would probably not recognize as sufficiently like classical spaces to deserve the name. Topological spaces (and their relatives like uniform spaces and proximity spaces) arrived early in the century. Metrizable ones are fairly close to Riemann's picture, but general topological spaces can be far from metrizable. We also have "pointless" topology, where a space is identified with a lattice, which would be the lattice of open sets in the case of a topological space but can be more general. Although most authors use the words "frame" for the lattice and "locale" for the associated space (the same object but in the opposite category) others have used "locale" for the lattice and "space" for the dual object. Grothendieck gave us two generalizations of the notion of space, in two different directions, namely schemes and topoi. Schemes have been further generalized to things like stacks. The basic idea underlying both locales and schemes --- namely to take some algebraic entities (frames in the one case, commutative rings in the other), to regard the dual entities formally as spaces, and to discover that some spatial concepts and intuitions are still useful in this generality --- has been extended further to give us non-commutative spaces.

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It seems to me (from the body of the question) that the OP did actually want to hear about alternatives, not generalizations of his 19th century "notion of space". I don't think that non-metrizable spaces, topoi, etc, could be seen as viable alternatives in this context (unless you convince us that they might well have something to do with quantum mechanics). Perhaps Cohen reals or ZF+AD could be better entitled to amount to reconsidering "hypotheses which lie at the bases of geometry"? If not, there's the Hilbert space (of operators having the meaning, say, of location of a particle) ... –  Sergey Melikhov Nov 17 '12 at 9:35
    
... as well as more recent developments, arxiv.org/pdf/gr-qc/9905087v1.pdf –  Sergey Melikhov Nov 17 '12 at 9:36
    
@Sergey: I agree with you about what the OP probably wanted. That's why I began my answer by saying that I'm answering the title, not the body of the question. –  Andreas Blass Nov 17 '12 at 14:24

This question is ambigous because the word "geometry" can have several different meanings. Rieman in his talk outlined the foundations of what is called Riemannian geometry now. This is the theory of spaces (manifolds) equipped with a Riemannian metric. These spaces studied in Riemannian geometry may have trivial isometry groups.

But there is a different approach and different point of view on geometry, outlined by Klein in his Erlangen program. Here the main object is certain group of transformations acting on the space. The space does not have to be equpped with a metric, and when it is equipped, it does not have to be a Riemannian one.

This point of view is on my opinion closer to the geometry of Euclid, and to classical projective geometry. It is in this sense that the word "geometry" is used in the expression "geometrization program".

And when we talk about algebraic geometry, we mean something very different again.

So I think the question is meaningless as stated. All these various "Geometries" have their own foundations.

By the way, the translation of Riemann you refer to is not a good one. For a better Engish translation of this work I recommend "A comprehensive introduction yo differential Geometry" by M. Spivak, where this lecture is reproduced in Spivak's ranslation. And it is followed by the chapter called "What did Riemann say?"

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OK, what I really care here is the notion of "space", instead of all different kind of geometry that has an axiomatic approach available. –  36min Nov 17 '12 at 2:50
    
I edited the title, maybe now it is better. –  36min Nov 17 '12 at 2:53
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The notion of "space" has even more different meanings than geometry:-) Do you include all infinite dmensional spaces of functional analysis? A 'space" nowadays is almost a synonim of a "set with some structure". –  Alexandre Eremenko Nov 17 '12 at 4:22
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I just wanted to remark that Riemannian geometry and Klein geometry have been subsumed by the more general notion of Cartan geometry, as it is outlined in the book by R.W.Sharpe Differential geometry, Cartan's generalization of Klein's Erlangen program. –  Qfwfq Nov 17 '12 at 16:11

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