The natural numbers can all be finitely represented, as can the rational numbers. The real numbers, however, cannot be so represented and require some notion of “infinity” to define. This makes it both computationally and philosophically interesting to determine for what purposes you need the real numbers, and for what purposes you need only the rationals.
It’s pretty clear that spatial concepts having to do with distances and rotation require the real numbers. For example, if we took as our model of the plane, the distance from to would not be rational, and we would not be able to rotate the point about the point by most angles.
But I always implicitly thought that spatial notions not depending on distances or angles required only the rationals. It turns out that I was wrong: there are spatial notions not depending on distances or angles which differ depending on whether you take space to be or . The fact that I was wrong follows from a theorem of Micha Perles which is very famous in combinatorics, but which I only found out about recently.
Actually, the fact that I was wrong follows just from a lemma in the proof of Perles’s result, which I will state before telling you what Perles’s main result was.
Consider the following system of points and lines (the image is stolen from Pak’s manuscript):
The lemma is then that while there is a collinearity- and noncollinearity-preserving embedding of this diagram into , there is not one into . Note that the question of a collinearity- and noncollinearity-preserving embedding of the diagram says nothing about angles or distances. The proof is simply to assume that there is a rational embedding, then to find a rational transformation of the configuration to one where you know that one of the points has an irrational coordinate. This proof appears on page 108 of Pak’s book.
Perles’s main theorem is the following, and I think it’s quite striking: A polytope is the convex hull of a finite set of points in some , where we consider two polytopes equivalent if they are combinatorially equivalent: i.e., if there is a bijection between the two sets of vertices such that if one pair of vertices has an edge between them, the corresponding pair does as well, etc. Then for all dimensions greater than 3, there is an -dimensional polytope which is not equivalent to one which is the convex hull of a set of points with only rational coordinates.
The discussion of this in Pak’s book is in Part I, Section 12.5.
Edit: I removed a paragraph on planar graphs because it didn’t really fit the article, and I took out the phrase “purely combinatorial property,” which was misleading and probably incorrect.