Why do we prove things? (Or, what is a "good" proof?)


It's a fact that's taken for granted that mathematicians like to prove things. Advanced math courses in college are proof-based, and students are expected to start caring about proofs and writing proofs of their own. Professors in introductory proof-based courses might sometimes (though certainly not always) offer students guidance on how to write proofs and on what constitutes a formal mathematical proof, but rarely do they talk about *why* we prove things. One reason is that it's fairly self-evident. But I'd like to analyze this question, because it leads us to a much more interesting question: What determines whether a proof is a "good" proof? It seems natural to judge the quality of a proof by how well it fulfills the underlying purpose behind writing the proof in the first place.


So what is the *purpose* of a proof? In my mind, a proof has two basic purposes, which are related but not the same: (A) to *convince* someone that the theorem is true, and (B) to *explain why* the theorem is true. (I conjecture that these were the motivating factors behind the original development of Greek-style proof-based mathematics.) I believe that the quality of a proof should be evaluated on how well it achieves purposes (A) and (B). Typically, a convincing proof is precisely one that explains why the theorem is true, but (A) and (B) do not always go hand-in-hand.


An example of an "A without B" proof is when each step can easily be checked without the reader understanding why the overall argument works or where it came from. This is a familiar phenomenon in math textbooks. (It is also the complaint I discuss in my mini-essay on the irrationality of pi.) The most extreme version of this would be a two-column proof, using only logical rules of deduction and a list of given already-proven theorems. This is basically the gold standard for mathematical certainty and roughly corresponds to a technical mathematical definition of the word "proof." Each step can be trivially checked by anyone who understands the rules of logic, and therefore can be checked for correctness by a computer. Any sane person would agree that such a proof is basically useless for any sufficiently complicated theorem, because it obscures what the proof is all about, and it would also be extremely long and tedious. This is an extreme example of how (B) can be sacrificed in order to obtain more (A). A more realistic standard for what constitutes a super-rigorous proof is that the proof should be explicit enough that someone *could* theoretically convert the proof into a purely logical two-column proof (of the sort described above) in a straightforward manner, given enough time and patience. (Of course, this is a subjective standard.) Most proofs in undergraduate textbooks live up to this standard. Most proofs in research papers do not. It is nearly impossible to describe an appropriate standard of rigor for research papers, but I contend it doesn't really matter. The important question is, after reading the proof, how convinced are you that the theorem is true? Or in other words, I don't much care how rigorous the proof is in any technical sense. I just care how well the proof achieved purpose (A).


It is also possible to have a "B without A" proof. An example of this might be a "sketch" of a proof, in which the main ideas are presented, but the details are all left out. A deeper example would be when an argument is extremely compelling and obviously important, but it is not so clear that the argument proves what it says it does.  Most of 18th century mathematics fits this description since many important mathematical ideas and "truths" were discovered, despite the fact that there was no logical foundation to stand on.  More recent examples might include Thurston's "proof" of the Orbifold Theorem and Perelman's "proof" of the Poincare conjecture. In both cases, the missing details had to be filled in by other mathematicians before the results became accepted as true. An extreme example of a "B without A" argument is the physical explanation of mirror symmetry, in which heuristic (and totally non-rigorous) physical arguments are used to explain a mathematical conjecture.


Here's another reason why we shouldn't fetishize the formal, rigorous proof. A rigorous proof is the standard way to achieve purpose (A), but there might be other (less-convincing) ways to convince someone that's something is true. Consider the formula


1 + 2 + 3 + 4+ . . . + n = n (n + 1) / 2


You can check that this works for n from 1 up to about 20. By then, you're pretty darn convinced that this fact is true for all n. (That is, unless you've been polluted by formal mathematical training.) We have a tendency to indoctrinate students with the idea that you must prove this rigorously because maybe there's some huge number out there for which the formula fails. While I do agree with this reasoning, I'd say that that the more pressing need for a proof is to fulfill purpose (B); we want to know *why* this formula works.


In a good textbook, every proof can fulfill purposes (A) and (B) so that all the proofs are pretty decent, but in a typical research paper, things are not so clean. The motivation for this post came from my own desire to evaluate the merits of computer-assisted proofs, such as the proofs of the Four Color Theorem and the Kepler Theorem. (See the Appendix below for explanations of these theorems.) My conclusion is that these proofs have the potential to do pretty well at (A) but are destined to do poorly at (B). It's possible that this is unavoidable, because there are some theorems that will inevitably boil down to checking a huge number of cases, and there might be no better explanation than, "Hey, the cases check out."


Appendix:


Four Color Theorem: If I give you a map of the United States and four different-colored crayons, it is possible to color-in each state so that any two states that share a border have different colors. (States such as Utah and New Mexico, which meet at a single point, are allowed to have the same color). Similarly, given a map of Europe and four crayons, you could color-in each country so that any two countries that share a border have different colors. The Four Color Theorem says that no matter how the countries and borders might change in Europe, four colors will always be enough to get the job done. (However, you have to assume that each country is contiguous.)


Kepler Theorem: Suppose I give you a bunch of equal-sized cannonballs to stack neatly (or if you are a pacificist, then I give you equal-sized oranges), and you do it the way any ordinary, rational person would do it. The Kepler Theorem states that this is the densest possible way to arrange your cannonballs. Or in other words, if you wanted to fit the most possible oranges inside a huge cargo container, you should stack them neatly in the "usual" way. There is no bizarre or clever way that will ever do any better.