Topic: Diagrams are psychologically useful, but prove nothing; they can even be misleading.

I will argue against the claim. Diagrams are not only psychologically useful, but they can also serve as proofs for mathematical theorems. They are more than just heuristic devices.

If we create a good diagram that appropriately represents a mathematical statement, then we should have no problem relying on it to portray a mathematical proof. We can see that a good diagram can show certain patterns that are unchanging. When we take one step further in a proof and show that a mathematical statement holds for bigger n, for instance, we merely use the same pattern except that we expand the diagram further. As long as there is no change of the structure, then the diagram is just as valid as a rigid arithmetic proof, since the latter also relies on unchanging patterns and structures (as seen in mathematical induction proofs).

Although diagrams cannot expand infinitely, they are still capable of proving general results. All one has to do is to first imagine the case where the diagram is expanded far enough, and then form logical conclusions based on that expansion. It will not be difficult to tell, after using logical analysis, that the diagram can appropriately form a proof of a theorem.

It might be objected that there are diagrams that are misleading, as shown by the diagrams that attempted to prove the sum of the following series: “1 + 1/2 + 1/3 + …” and “1 + 1/2^2 + 1/3^2 + …”. It is too difficult to see that the former sum up to infinity while the latter sum up to a finite number since their corresponding diagrams look similar. However, I believe that, through more research and intellectual work, we can, one day, produce good diagrams that are not misleading for these series. Even if we cannot currently achieve it, the misleading part can be easily clarified by adding informal explanations, but we can still keep the diagrams for illustration purposes.