Plato against the Geometers

It’s fairly obvious that for LEGO a large part of the fun is in the activities, but this idea is not as obvious when it comes to math. The blame for this attitude, I think, can be traced all the way back to Plato (427/8 bc-347/8 вс). In Book 7 of the Republic, Plato has Socrates (469/70 вс-399 вс) say the following:

Now, no one with even a little experience of geometry will dispute that [geometry] is entirely the opposite of what is said about it in the accounts of its practitioners. [...] They give ridiculous accounts of it, though they can’t help it, for they speak like practical men, and all their accounts refer to doing things. They talk of “squaring,” “applying,” “adding,” and the like, whereas the entire subject is pursued for the sake of [. ] knowing what always is, not what comes into being and 1

passes away.

The accusation here is that when we focus on the activities of mathematics, we lose sight of the very nature of the subject. Math is precisely about the mathsy objects like numbers and shapes; these mathsy objects are independently existing abstract objects. This means that they are real things that don’t exist anywhere concretely. This view, known as (surprise, surprise) Platonism, holds that objects like numbers are eternal and unchanging—and thus we can’t do anything to them. For a Platonist, the idea of focusing on mathematical practice might well be “ridiculous” because any activities we might engage in cannot have any effect on that mathematical stuff which math is about. In other words: in doing mathematics we want to find out about these eternal and unchanging mathematical objects themselves. Activities like adding up numbers and drawing circles are beside the point, because they can’t affect the objects we are trying to find out about. (This might even be a strong disanalogy with the LEGO case, because with LEGO we are directly playing or building with the bricks themselves.)

But in our quick discussion of Plato we find two key components for switching the negative attitude around entirely, toward the idea that a focus on the activities of math is not ridiculous at all. Rather, these activities are vital. The first reason is this: the geometers’ activities are about demonstrating the relations and truths of mathematics because these activities may be crucial for mathematical proofs. Indeed, Plato himself is concerned with knowledge, and one of the most important ways for us to come to know things about mathematics is through proving them to be the case. The second component is the fact that geometry in particular has caught our eye. Geometry makes far greater use of pictures and diagrams than tends to be the case for other areas of mathematics. So we’ll focus on diagrammatic proofs—that is, proofs which are wholly or primarily comprised of pictures—as a key case where proofs guide us through a series of actions. Such processes or activities, with pictures to guide us through, allow us to draw a very close parallel between mathematics and LEGO, and to highlight the role that following the instructions has both in learning mathematics via the use of picture proofs and in taking us from a box of LEGO bricks through to completed models.

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