Great Proofs as Great Art



Can we view and appreciate great proofs as great art?



Leonardo da Vinci was not a complexity theorist, but according to our friends at Wikipedia he was just about everything else: a painter, a sculptor, an architect, a musician, a scientist, a mathematician, an engineer, an inventor, an anatomist, a geologist, a cartographer, a botanist and a writer. If he was alive now, I wonder what brilliant things he would be doing.

Today I want to discuss a question related to Leonardo as an artist: can we view great proofs as great art? I think we can, and I would like to explain why.



Connecting proofs and art reminds me of a legend I heard at Yale, many years ago, concerning the beginning programming class, CS 101. I do not believe the legend, but this will not stop me from repeating it. The legend goes there was once an exam given to a 101 class, containing the question:

Evaluate the following expression: where and .

The correct answer was , not , since the operator “ ” has higher precedence than the operator “ ”. But, the legend is one student “evaluated” the expression as follows:

The expression has a pleasing horizontal symmetry, and a near left-to-right central symmetry, and is also

You must remember Yale was filled with very bright students, but their main interests were often in the liberal arts, not in mathematics. The student was right, the expression does have some nice symmetries

Studying Great Art

How do we study great art? Consider Mona Lisa as an example—perhaps the best known painting in the entire world. We may, if in Paris, go to the Louvre and wait in a long line, and eventually get a glance at the famous painting. Or we may study reproductions of the Mona Lisa in art books, in other reproductions, or even on-line today.

Most of us look at the Mona Lisa and see different things. We may like it, we may love it, or we may prefer more modern paintings, but most are amazed at Leonardo’s achievement.

Our goal in looking at and studying a great work of art, like this, is not to reproduce it ourselves—that is impossible. Our goal is not to even try to understand how Leonardo achieved his great painting, how he captured her famous smile—that also is probably impossible.

Instead our goal is one of enjoyment, of pleasure, of the excitement in seeing the work of a master. We come away with a impression, a general feel, a holist view of what he did. We do not understand each brush stroke, we do not understand how he made his paints, we do not understand how he captured her look, we do not understand how he created the masterpiece. But, we do leave with something.

Studying Great Proofs

I have wondered if we could do something similar with the great proofs. Many read and study a great proof in an attempt to really understand them. If there is a masterpiece in your area of research you may need to understand the proof line for line—you may need to be able to reproduce the proof to be considered an expert in your area.

However, most of us do not even look at the masterpiece proofs, ever. We may know their statements, but most of us—I claim—have no idea of how they work. Nor do we need to understand them line for line. But my thesis is this:

We can learn from a study of the great proofs, even if we do not follow the proofs in detail.

We should study the great proofs, not to understand them completely, but in the same way people study the Mona Lisa. For enjoyment, for enrichment, for seeing a master at work, and for getting something out of the study. Not to be able to reproduce the Mona Lisa. What we learn may be intangible, but still invaluable. Again, we can leave with something.

Some Great Proofs

Here are some sample great proofs—there are many others, these are just a few to make our discussion concrete. The same comments will apply to many other of the great proofs.

One property of many of the great proofs—not all—is they are long. The opposite is not true: there are plenty of long proofs of relatively unimportant results. Length is no measure of greatness, just as the length of a book or the size of a painting is not a measure of its greatness. However, many of the great proofs are long, some are very long. One reason is: a great proof often requires several new ideas and it is reasonable to expect weaving these ideas together can be difficult to explain; hence, the great length of the great proofs.

Feit-Thompson’s Theorem: Walter Feit and John Thompson proved in 1962: Every group of odd order is solvable.

Cohen’s Theorem: Paul Cohen proved in 1962: The Axiom of Choice and the Continuum Hypothesis are independent from Zermelo–Fraenkel (ZF) set theory. Obviously, 1962 was a very good year for great proofs.

Szemerédi’s Theorem: Endre Szemerédi proved in 1975: Every dense enough set has arbitrary long arithmetic progressions.

Wiles’ Theorem: Andrew Wiles proved in 1994: Fermat’s Last Theorem.

What Can One Learn From This Study?

I would like to run a class on the study of great proofs as great art. I would have students “read” each of the great proofs, and we would have discussions about them. I would not expect, even the top students, to be able to understand the proofs fully, or even partially. I would not expect anyone to understand the proof at any deep level at all. They may, even should, understand parts of the proof completely—a lemma here, or an argument there.

I would expect students to learn some appreciation for what it takes to create a masterpiece—a great proof. I believe we can learn a great deal from reading great proofs, even without understanding them in the usual sense. Let’s look and see what we might learn from reading these proofs in this way.

Previous Work

Great proofs usually do not arise out of a complete vacuum. They are almost always based on previous work: they may extend an earlier argument, they may use techniques developed by others, they arise in a complex context.

Feit-Thompson: They used a great deal of machinery of group theory. One of the great previous results was due to William Burnside, who had proved every group of order with and both prime is solvable.

Cohen: He used Kurt Gödel famous proof of the relative consistency of the AC with ZF. Weave Many Ideas Together Great proofs often need to use much of the existing machinery. This is similar to the last point, but a bit different. Here the point is the creation of a great proof usually requires the mastery of many tools and techniques.

Feit-Thompson: They used almost all aspects of group theory: local analysis, character theory, and relation definitions of groups. They also needed to use non-trivial number theory. At the very end of their proof they could have saved a large amount of work if they could have proved this conjecture: there are no distinct primes and so that Since they could not, they needed to use even more group theory tricks to get the contradiction they needed.

Wiles: He used much of modern number theory, especially the theory of modular forms and Galois representation theory. The famous number theorist Enrico Bombieri said that he would need at least a full year to understand Wiles achievement—this is because of the diverse tools used in the proof. Golden Nuggets Great proofs often contain new ideas and methods, sometimes these can be more important than the theorem being proved. Note, I have talked all through this discussion about proofs not theorems. This is one reason for this: often the proof is much more important than the theorem itself.

Cohen: He created an entire new way of constructing models of set theory. This method, called forcing, is now a standard tool used by set theorists everyday. When Alfred Tarski heard about Cohen’s new method he said: They have a method, now they will get everything.

Szemerédi: He needed to prove a certain lemma to make his great proof work. This lemma is the now famous Regularity Lemma. Strategic Plan Great proofs almost always have a plan of attack. If someone tells you they have solved a long standing open problem by “just an induction”—it is pretty unlikely to be true. Usually, great proofs have a complex strategic plan for the attack of their proof.

Feit-Thompson: Their proof is divided into six chapters: each has its own abstract and they give a coherent overview of what each chapter does and how they all fit together.