Euclidean geometry, nominalism, and drawings

[Posted on july 9, 2017.]

In middle and high school, geometry is a very visual affair, always illustrated by drawings. Indeed, reasoning on a problem is usually made much easier by glancing at a sketch, even one not to scale.

Of course, problems can arise if the sketch is not able to capture the essence of the problem, as in the well-known missing square puzzle, which is a good opportunity to tell pupils about the difference between abstract mathematical thinking, and visual “proofs”.

Picture of the puzzle by wikimedia user Krauss

Please note that I am not saying “difference between ideal mathematical objects and their visual approximation” : that would be a Platonistic view, which is not at all compulsory. For instance, a nominalist take on this issue is completely possible, and preferable as far as I’m concerned. See Jody Azzouni’s take in Mathematics, Substance and Surmise: Views on the Meaning and Ontology of Mathematics, in particular the beginning of its section 5:

Now,  in Euclidean geometry we have precise metric statements (a length of , an angle of ), and also some that seem more topological in nature (points belonging to some part of the plane as a consequence of this or that, even though they could also be recast entirely in terms of an angle or length not having some exact value).

So what I’m wondering now is whether there is a framework for approximate geometry, saying that “any set of pseudo-lines [“topological lines”, something not really straight, and of varying thickness, just like on a drawing] and pseudo-points [some “fat dots”], when they are in such and such configuration, imply that another pseudo-line or pseudo-point has a certain property exactly”. In other words, a setting for which every actual drawing on a piece of paper is exactly capturing the essence of the question.

I would be very interested by any relevant comments or references on that topic. Has this been done already?