The Geometers Strike Back
Thinking back to where we began, if we accept the LEGO account of diagrammatic proofs then we have strongly sided with the geometers against Plato. Where Plato thought that the geometers were ridiculous for using active, practice-based terminology for their demonstrations, on our LEGO account of proofs this is the most important aspect of those proofs. The main principle of departure from Plato is to not think so much in terms of mathematical objects but instead focus on mathematical activities. The same thought is meant to apply to LEGO: the particular bricks are only important insofar as they facilitate the things we can practically do with them.
It might even be possible to extend the LEGO account of diagrammatic proofs to a full account of proofs. For example, we might take all proofs to be describing a series of activities for us to undertake to come to appreciate the truth of mathematical theorems. The philosopher Ludwig Wittgenstein (1889-1951) puts the idea as follows: “The mathematical proposition says to me: Proceed like this!”3
On this view, mathematics isn’t about the statements of truths, or arranging them into a neat logical structure as the foundationalist might think, but is instead about dynamic activities. The ultimate philosophy of mathematics we may end up with from here is one that values the practice of doing mathematics and according to which proofs are like a list of instructions to follow. Like the instruction booklets included with LEGO models, they must be seen not as the ends in themselves, but rather as a set of imperatives or guiding moves to follow in order to come to understand the wonderful realm of mathematics.
A final thought: we started out with parallel questions about LEGO and math. Why are they interesting? Why do we encourage kids to learn math and to play with LEGO? Well, with the LEGO account of proofs in hand we can tentatively suggest some answers. Though there are some aspects of both LEGO and mathematics that are about the objects (for example, we want to collect the awesome new Star Wars® LEGO sets and learn about the properties of triangles), we also want to concentrate on the activities that go along with them, like building, playing, proving, and discovering. These activities are interesting and useful precisely because they teach us how to solve puzzles effectively and play creatively. We can even solve problems with LEGO and play with mathematics, so the two are not so far apart after all. Maybe, with this in mind, we can throw away the old stereotype of mathematics as abstractly detached from reality, and see it as both important for doing things and fun in its own right.