Climbing Mountains and Proving Theorems



The parallels between climbing great peaks and solving open problems



Edmund Hillary is not a theoretician, but is world renowned for the first successful ascent and descent of Everest. He did this with Tenzing Norgay in 1953.

Today I want to talk about the connection between mountain climbing and solving mathematical problems. By mountain climbing I mean climbing Everest or K2 or one of the dozen other 8,000 plus meter peaks; by solving mathematical problems I mean solving hard open problems.



I believe that there are some interesting connections between these two vastly different tasks. Let me explain.

A Disclaimer

I want to start by saying I could never climb anything higher than a step ladder. Even that might be a stretch for me. Equally, I have solved some nice problems over the years, but I certainly am not one who has knocked off many great, or even lesser, open problems. However, I still think that I can say something about the relationship between these two. I hope you will agree.

Climbing Mountains

I would like to start by listing some facts about climbing that are believe are true. One of my favorite types of books to read are true stories of mountain climbing. If you like this genre, some of my favorites are:

The Kid Who Climbed Everest, by Bear Grylls. Into Thin Air: A Personal Account of the Mount Everest Disaster, by Jon Krakauer. On the Ridge Between Life and Death: A Climbing Life Reexamined, by David Roberts. Forever on the Mountain: The Truth Behind One of Mountaineering’s Most Controversial and and Mysterious Disasters, by James M. Tabor. No shortcuts to the top: climbing the world’s 14 highest peaks, by Ed Viesturs, David Roberts. K2: Life and Death on the World’s Most Dangerous Mountain, by Ed Viesturs, David Roberts.

You cannot reach the top if you do not climb. This sounds trivial but it still is true. If you do not try to climb the summit of K2, then you certainly will not reach it.

Climbing requires a team. Everest is a perfect example, as are most of the 8,000 meters peaks. There are some who have climbed solo, but the rule is that a great deal of work has to be done by a team to get even one to the summit.

Great climbers have multiple skills. The best climbers need great technical skill, great mental toughness, careful planning, and luck. Ed Viesturs points out that mountains allow you to climb them, you cannot just attack and climb them if they do not wish to be climbed.

Success requires reaching the summit and getting down. George Mallory may have been the first to reach Everest’s summit of 29,029 feet—but he never made it down.

Know when to turn around and go down. Great climbers know when the summit is out of reach. They know when they need to climb back down the mountain. Perhaps they will reach the summit another time, but they know it will not happen now.

Never rely on another’s rope. The best climbers will check others’ ropes very carefully before they use them. Many will almost never rely on any rope they have not placed themselves or by a trusted colleague.

Solving Problems

You cannot reach the top if you do not climb. In solving this translates into: you cannot solve an open problem if you do not think about the problem. One of the many themes that I have repeated many times is, in today’s terms, get out there and try to climb a mountain.

Climbing requires a team. In solving this translates into: the team in mathematics is those that have worked already on the open problem. Just as in climbing there are “solo” climbs, but even these have often relied on past climbers who proved that an approach could work. In problem solving we need to use the work that has come before. Also we need to work in teams: today many papers seem to have more and more authors. Even when the final paper is written by one mathematician, there is almost always others who helped. This was true with Andrew Wiles solution of Fermat’s Last Problem; it was also true with Grigori Perelman’s solution of the Poincaré Conjecture.

Success has several components. In solving this translates into: clearly solving hard problems requires technical skill. They also require mental toughness. Imagine the resolve that Wiles had in working alone on his problem for years—literally in his attic. The same for almost anyone who has solved an open problem. Finally, just as in climbing there is, I believe, an element of “luck.” Some approaches that eventually worked on an open problem just make it: there may be a perfect cancellation of two terms, for example.

Success requires reaching the summit and getting down. In solving this translates into: the lesson here is to solve the problem, but to also get “down.” That means—in my view—to get the solution written up and made public. There are many open problems that have been solved more than once. So by “getting down” I mean that you need to write up, and make your result accessible to others. There are many famous examples of the failure to do this. One of my favorite concerns the following pretty theorem from number theory—that was conjectured previously by Carl Gauss:

Theorem: There are only a finite number of imaginary quadratic fields that have unique factorization. They are for

This was proved first by Kurt Heegner 1952, but the proof was thought to be flawed. The first accepted proof was by Harold Stark in 1967. Then, the Heegner proof was re-examined and discovered to be correct. Oh well.

Know when to turn around and go down. In solving this translates into: know when to quit on a problem. Or at least when to put the problem aside. I have talked about mathematical diseases before; a good solver knows when their ideas are not going to work. Staying forever working on the same problem—summit—is a bad idea. In climbing it will get you killed, while in solving it will waste a great deal of time and energy.

Never rely on another’s rope. In solving this translates into: if you are using another’s lemma or work, you need to understand their work. Of course if you are using Gauss’s Quadratic Reciprocity Theorem, then perhaps you could trust the result. But, if you are using a new result, or even a result that has been around “checking the rope” is probably a good idea. I have personally spent several months using a published result that was in a great journal, had been generalized by another, only to discovery eventually that their “ropes” were frayed. In the end multiple papers had to be retracted—several papers were wrong.

Open Problems

I hope you like the analogy to climbing. There are, of course, two big differences. First, when we fail, we do not get killed. Second, there are only fourteen mountains over 8,000 meters. In the theory we seem to have an unending supply of great peaks to climb. Somehow as soon a peak is scaled, another higher and harder one is unveiled.

Good luck, try and climb some high peaks, and be careful.