Follow the Instructions!
One problem with the foundationalist take on proofs as showing off logical structuring of mathematical objects is that it doesn’t do so well with picture proofs. Now, it should be emphasized at the outset that some philosophers believe that diagrammatic proofs can’t truly prove anything, for various reasons. Of course, a major reason might be that picture proofs don’t work well with the foundationalist take on proofs! Let’s not go into this. Instead, you can judge for yourself through two simple examples.
The first is to show the following equality holds:
We can do this with the following diagram of an equilateral triangle:2
What!? How is that supposed to work? Well, imagine taking the whole triangle to have an area of 1. Then we can chop up the triangle into quarters as follows:
Well, if we take the top triangle and chop that into quarters again, each triangle is a quarter of that quarter, so will be (1/4) x (1/4) = (1/4)2. In fact, we can keep dividing the top quarter to get higher and higher powers of 1/4, leaving us with a diagram like this:
But then we can split this into three series of triangles: one along the left, one along the middle and one along the right, getting us our first diagram:
Each of the series (dark shaded, light shaded, and white) will now be of the form
But we also can see that the whole triangle is covered by these series and that they all have the exact same area, so each series must cover 1/3 of the triangle! As such it follows that
The foundationalist response to this demonstration could either be to deny that it shows what it appears to show, or to logicify it. However, it seems to me that no way of formalizing the idea in the above picture will maintain the elegance or intuitiveness that the original has.
Here is a second example, this time to prove the Pythagorean Theorem that a2*b2 = c2, where c is the length of hypotenuse of a triangle and a and b are the lengths of the other two sides. The proof is given by the following picture:
The idea is that we take the initial triangle, inflate it, and rotate it to get three different triangles. Furthermore, we can slot two of them together to get a new triangle (at the bottom left) which is identical to the third (at the bottom right). But for the two we slot together the length of the hypotenuse is a2+b2 while for the other it is c2. Since these two triangles are the same size (in technical terms, they are congruent), the two hypotenuse lengths are equal, so the Pythagorean Theorem is proved!
What I love about this proof is that it brings to the fore the active nature of the demonstration. You must take the original triangle and perform three different actions on it, as indicated by the arrows in the picture. By manipulating the triangle in various ways we can then come to see the truth of the theorem. Here the parallels with LEGO are far stronger than they were with the foundationalists and Platonists. In LEGO instructions, we are guided through a series of actions that we must perform. Of course, the diagrams are on the printed page and so obviously cannot move. But having a series of pictures showing what changes occur between one step and the next, with occasional uses of arrows, manages to nonetheless successfully communicate a series of actions for us to build our model. In general, we are not interested in the pictures in the instructions for their own sakes or for what they show (after all, it won’t be long before we have built the model itself so won’t need pictures of it anymore).
Likewise in diagrammatic proofs, we are not focused on the picture for its own sake, but instead we are interested in the series of actions that it tells us to perform in order to construct the proof. In our first example, the actions are those of carving up a triangle in particular ways in order to mirror the infinite series we were dealing with. In the second example, we are scaling up the triangle in various ways, then rotating and re-combining the results.
We might call the idea underlying all of this the LEGO account of diagrammatic proofs. A similar analogy for another book could have led us to an IKEA account of diagrammatic proofs, comparing the activities directed by picture proofs to IKEA instructions for building furniture. (Any Swedes reading this may take comfort in the fact that they have an alternative to the Danish LEGO domination.)