Why we use $\mathbb{R}^{3}$as a model for space? More specifically why we don't use other number systems such as extensions of the real line (hyperreals, surreals, etc.)?
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$\begingroup$ my guess would be laziness - they probably do not offer enough benefit over the existing real formalisms to re-write all textbooks; the same could be said for rationals vs reals (we could do all our math with series of rationals to represent the reals), but they were introduced much earlier in the game and their benefit is probably more obvious... $\endgroup$– ChristophCommented Dec 6, 2011 at 19:18
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2$\begingroup$ We do it because it matches experiment ... why else? (And actually, strictly speaking, we don't---we use Minkowski coordinates for space-time.) If somebody can think of an experiment which will tell the difference between the reals and the hyperreals or surreals, we may eventually have to change the way we do things. $\endgroup$– Peter ShorCommented Dec 6, 2011 at 20:39
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3$\begingroup$ I'd actually be inclined to make this question just ask "why do we use $\mathbb{R}^3$ as a model for space?" In most cases, asking a question directly is preferable to asking for papers that answer the question, because a direct question invites both types of answers: links to references, as well as direct explanations. $\endgroup$– David ZCommented Dec 6, 2011 at 21:12
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$\begingroup$ @PeterShor: I am actually interested in why we don't just use the the rationals to model things. Although I understand that the Pythagorean theorem demands irrational distances, the fact is that no device ever measures irrational values. Does the completeness of the reals just make computation easier? And is it because the reals is the smallest complete space containing the rationals that the real number system is used so prominently? $\endgroup$– JudeCommented Dec 7, 2011 at 2:06
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1$\begingroup$ @Jude: your last comment is the answer. $\endgroup$– Ron MaimonCommented Dec 7, 2011 at 5:51
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3 Answers
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Superspace in supersymmetry has Grassmann numbers as coordinates. BPS compactifications in string theory have complex numbers as Calabi-Yau n-fold coordinates. Noncommutative geometry has noncommutative algebras as coordinates.
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1$\begingroup$ Unfortunately, I don't understand enough of what you mentioned to argue anything strongly. But presumably the ultimate goal of these theories is to describe phenomena we experience, which we ultimately use real numbers to describe. I'm guessing these coordinates you mentioned are simply mathematical tools. $\endgroup$– JudeCommented Dec 6, 2011 at 19:11
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1$\begingroup$ Grassman coordinates are not coordinates, and the complex number coordinates in Calabi Yaus are just pairs of reals, their complex structure is broken by perturbations like gravitons. The NC example is the only good one. $\endgroup$ Commented Dec 6, 2011 at 19:26
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1$\begingroup$ afaik grassmann are some algebraic abstract objects with no real representation whatsoever $\endgroup$– lurscherCommented Dec 6, 2011 at 20:22
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$\begingroup$ @Ron Maimon: It's a good example, in that a priori it doesn't have the continouity of the reals. But you can only physically measure a spectrum consisting of some real numbers, right? $\endgroup$ Commented Dec 6, 2011 at 20:30
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$\begingroup$ @Nikolaj: But the matrix model in string theory, for example, has dynamics which are purely noncommutative, and only reproduces space in the limit of large numbers of point particles and classical separations. The real numbers are inside, but the structure of space is not R^{10}, but M^10 where M is an N by N matrix space. The measurements do not correspond directly to coordinate measurements, the space is emergent from the dynamics. $\endgroup$ Commented Dec 7, 2011 at 5:43
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The two examples you gave, hyperreals and surreals, are exactly the same as the real numbers as far as physics is concerned. They are logically different from the real numbers, in that they both contain special infinitesimal elements $a>0$ which have the infinite list of properties $a<{1\over n}$ for each natural number n, while the real numbers do not. And this does make them different as a logical model of the axioms of the real number line, but this does not mean that they give a different as a model of space, as we see around us in nature.
A model of nature is always expressed by matching computations that you can perform to experiments which you can set up. The computations are done on a standard digital computer. In principle, you don't even need real numbers, because a computer runs on integers.
The real number models of space are always explicitly defined as a small $\epsilon$ limit of some discrete structure. The limit has to make sense, so that you can compute the small $\epsilon$ limit. In certain cases, the limit has been worked out already, and the theory is defined on the continuum from the start. In other cases, like in quantum fields, the limit still has to be taken by hand on a case-by-case basis.
The reason for using continuum to model space is ultimately because we are big. If you make a lattice model, the description at large distances is often by a continuum limit. Any model purporting to describe physics must use the continuum in such a way that if you take the spacing small, you recover the continuum answer. This is explicitly true of quantum fields, for example.
To show that this is not a completely empty statement: Two particles move through space, and when their distance is exactly rational $p/q$ in lowest terms, then they feel a mutual impulse in the direction of separation proportional to ${1\over q^3}$. This model is absurd as physics for obvious reasons--- it treats space-time points as resolved to arbitrary accuracy.
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Both the examples you gave are real number systems, they're just a different way to slice up and interpret values. As for why we don't use it, the better question is why would we.
