I am at the Prague Midsummer Combinatorics Workshop, a delightful meeting where there are not too many talks. We have talks in the morning; in the afternoon we can work, sightsee in this beautiful city, or simply talk to friends over a coffee.
On the first morning of the workshop, Geoff Whittle talked about real representations of matroids. Geoff intended to become a philosopher and got into mathematics more or less by accident; now he is a leader in the attempt to understand representability of matroids over finite fields, but had turned his attention to representability over the real numbers; he found this brought him back to his roots.
“Real representations of matroids” sounds very technical, but the rank 3 case is simply about what pictures you can draw on the blackboard with points and lines. We are not concerned with the distances or angles, or even the ordering of points on lines; simply the incidence. The question in this case is, given an abstract incidence structure satisfying the obvious conditions (two points on a unique line), what extra properties guarantee that it can be drawn in the plane? No satisfactory answer to this simple question is known!
Geoff reminded us of Kant’s notion of “synthetic a priori” truths. Crudely put, a proposition is analytic if it can be proved (e.g. statements of mathematics), and synthetic if an experiment is required to determine its truth; it is a priori if you know it will be true before it happens, a posteriori if you cannot be sure until afterwards. These two partitions might appear to be the same, but Kant claimed that there exist synthetic a priori propositions, and cited propositions about time and Euclidean geometry.
Many people believe that Kant’s claim that Euclidean geometry is a priori is refuted by the existence of non-Euclidean geometry and the possibility that the universe in which we lie may be non-Euclidean. According to Geoff, this refutation does not work. Can you picture a non-Euclidean plane in which the lines are “straight”? Geoff claimed not.
This seems to be true for the elliptic plane, which is just the surface of the sphere with antipodal points identified. There is no doubt that the lines are curved there. Indeed, it is hard to imagine a geometry in which lines are straight and any two lines meet.
The hyperbolic plane is less clear. Certainly, when I try to picture it, I think of the Poincaré unit disc model, where the plane is the interior of a disc, and the lines are the Euclidean circular arcs or straight line segments which meet the boundary at right angles. Probably this springs to mind because Escher’s “Circle Limit” pictures make everything so clear (the lines bend, and things become smaller as you move towards the boundary). But imagine yourself actually in the model; can you see the lines as straight?
I suggested to Geoff the Beltrami model of the hyperbolic plane as an alternative. The points are again the interior of a disc, but this time the lines are all the Euclidean line segments. Although the lines are “straight”, you pay the price: the model is not conformal, so that angles are not preserved, and so shapes are distorted.
It would be an interesting exercise, for someone with expertise in geometry software, to take Escher’s “Circle limit” pictures and transform them to the Beltrami model. (This can be done, if I remember correctly, by building a hemisphere on the disc, projecting onto the hemisphere from the south pole of the sphere, and then projecting back vertically.) The point of the experiment would be to decide which is more unnatural to our perception, bent lines or distorted shapes.
Geoff claimed that properties of real-representable matroids are other candidates for a priori synthetic truths. Exactly what makes a matroid real representable is a difficult question; recent results of Geoff, with Dillon Mayhew and Mike Newman, hint at its difficulty.
I realised, and Geoff confirmed, that representability in Euclidean, elliptic and hyperbolic space are equivalent conditions. So an experimental test of a condition for real representability would probe something deeper about the universe than merely whether its geometry is Euclidean or not. Geoff says that a physicist colleague of his has assured him that no physicist anywhere is thinking about this issue.
Finally, like so many of the big questions, this one turns on the relation between the universe and our perception of it, and in particular, whether we have in-built modules for perceiving the universe in certain ways (such as equating “geodesic” with “straight line”). Kant may have thought that we could throw light on such questions by introspection; I am not so sure. If not inborn, these facilities are presmably acquired at a very early age; for some reason, we lose contact with our thought processes at such an age. (Certainly true in my case; I have no very early childhood memories at all.)