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Climbing Mountains and Proving Theorems

February 2, 2010


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:

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

{\bullet } 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.

{\bullet } 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.

{\bullet} 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.

{\bullet } 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.

{\bullet } 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.

{\bullet } 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

{\bullet } 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.

{\bullet } 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.

{\bullet} 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.

{\bullet } 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 {\sqrt{d}} for

\displaystyle  d \in \{ -1,-2,-3,-7,-11,-19,-43,-67,-163 \} .

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.

{\bullet } 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.

{\bullet } 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.

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19 Comments leave one →
  1. February 2, 2010 1:49 pm

    I am reminded of a saying in rock climbing, “There are old climbers, and there are bold climbers.” Unfortunately, this too might apply to the scientific world — staying inside the system (getting jobs at the right time) requires quite a bit of strategic play.

  2. subruk permalink
    February 2, 2010 3:18 pm

    This reminds me of another analogy of problems to mountains, one that I had heard from Prof Craig Tovey at ISyE at a class.

    He had this insight, he used to think of problems/theorems/results as living in mountain peaks. He noted that different problems, live together in peaks. One example was that the following results — the simplex method, Separating Hyperplane theorem, Strong duality theorem (for LP), Hahn Banach theorem — all live together in the same peak. That is, if you prove one, it is easy to get to the
    other. Proving one from another is easy, but proving any one from scratch is hard. It is like they are all together in a peak. Once you get there it is easy to move to another in the same peak. But getting there is hard.

  3. February 2, 2010 3:29 pm

    Can they provide free 02 tanks at FOCS/STOC breaks?

  4. February 2, 2010 3:34 pm

    Can we say?:
    1) As one gets closer to solving a problem, the air gets rarefied and the oxygen level drops.
    2) If you try to move too fast, so that you start sweating, you may die of hypothermia.
    :)

  5. February 2, 2010 3:35 pm

    Dear Dick, I like the analogy a lot and the way you describe the different ingredient.

    Perhaps one more common item is: No matter how good you are you may fall and crash.

    (To offer more support for this analogy let me mention that I played with a similar analogy in chapter 31 of my Book “Gina Says” http://gilkalai.wordpress.com/gina-says/ )

    • February 2, 2010 5:47 pm

      Another common item is: If you want to reach the summit you cannot explore every beautiful flower on the way.

  6. Ernie Croot permalink
    February 2, 2010 4:25 pm

    That’s an interesting posting; and, in fact, I find all your postings interesting (your blog is one that I regularly skim).

    One way that I think math (and CS… and research in general) differs from mountain climbing is that there are many different ways to reach the “summit” and return — it is one of the things I like about math. Some people get there through sheer technical awesomeness; some are extremely creative; some have superior social skills are are good at networking (i.e. emailing the right people for help on annoying little sticking points in their research); and some have all or many of these qualities and more. Though probably you are right that the strongest are in this last category — they have most or many of the above.

    On the comment “You cannot reach the top if you do not climb”, I remember Carl Pomerance once telling me that, though maybe he was quoting Erdos. The quote was something like, “You’ll never solve a difficult problem unless you try to solve a difficult problem.” (though probably more eloquently than I put it).

    One last thing: I also read that book “Into thin Air”. For a while I was a fan of Jon Krakauer, after reading “Into the Wild” — what a tragic story. I have read some of his writings in various magazines. Haven’t read his book “Under the Banner of Heaven” yet, but it is somewhere on my reading list.

  7. Phd student permalink
    February 2, 2010 5:13 pm

    Great post. Nothing more to add

  8. February 2, 2010 7:07 pm

    Interesting analogies all. I’ve heard a couple of similar ones:

    When the great Russian mathematician Israel Gelfand passed away recently, a mathematician contrasted his style with his advisor Andrei Kolmogorov by stating that whenever the latter would encounter new mathematical territory, he would immediately try to climb the highest mountain, whereas Gelfand would immediately start building roads. Given the astonishing amount of technical machinery Gelfand has contributed over the years, I found that to be an interesting contrast.

    Another interesting analogy relates to Nash and his ingeniously roundabout problem-solving methodology: an observer stated that while most people would try to make sense of a peak by climbing it directly, Nash would scale another peak entirely and shine down a light on the first one one.

  9. David permalink
    February 2, 2010 7:26 pm

    Richard, are you applying for CS PhD programs right now? Because about half of the personal statements we get start by likening research to mountain climbing.

  10. eqnets permalink
    February 2, 2010 9:57 pm

    FYI I started a thread at MathOverflow discussing “success requires reaching the summit and getting down”; other examples are there: http://mathoverflow.net/questions/13896/what-are-some-famous-rejections-of-correct-mathematics

    Also re: the roads versus mountains comment on Gel’fand and Kolmogorov: Lee Smolin has made similar observations in the vein of physics.

  11. John Augustine permalink
    February 2, 2010 11:49 pm

    Here’s one more point that bolsters the analogy. Once a summit has been reached, awareness of the tricks/routes increases. The next person may scale the summit using an easier path. In fact, Mt. Everest has been scaled by many (probably with less effort) than Hillary and Norgay (BTW, I never understand why we still highlight Hillary more than Norgay). So as time goes by, the most “elegant” approach might emerge.

    Of course, we all know that first proofs are not always textbook proofs. It usually takes some time before the “book” proof emerges.

  12. breakingkeyboards permalink
    February 3, 2010 8:16 am

    …You will not reach the top if you don’t believe is possible to reach the top

  13. Liza permalink
    February 3, 2010 4:14 pm

    What is the altitude sickness equivalent in this analogy :)

  14. Prof permalink
    February 3, 2010 5:21 pm

    I’ve only climbed one peak of any note (not sure if it qualifies as a mountain), but I did make sure to select my Tenzing Norgays very carefully. The climb down was particularly difficult in the sense that we pretty much closed the area for some time, so while the result is universally accepted as true, it didn’t garnish as much attention as one would have expected. The paper was invited to the special issue of the big name conference in which it appeared, so it’s not like I’m complaining either.

  15. Claire permalink
    February 8, 2010 5:53 pm

    Interesting post for mountain fans such as myself (Kilimanjaro 2001).

    And reminds me of Oded Schramm :(

  16. proaonuiq permalink
    February 20, 2010 8:10 am

    A post about the same subject from a great mathematician, Peter Cameron (of permutation groups fame):

    http://cameroncounts.wordpress.com/2009/10/11/mathematics-and-mountain-climbing/

    Previuos post in his blog is about one of the greatest mountain climbers in history, Erdos. And the previous one about polyclimber projects in the pre-internet era.

  17. March 21, 2010 3:10 am

    This analogy reminds me of another analogy I had recently heard about insight.

    “It’s like climbing a mountain, as you climb the view gets more and more beautiful. And if you’re lucky you reach the top, and you have a breathtaking view. Suddenly you can see in all directions, and things make sense. But the moment you know something it raises more questions, in other words once you reach the top you’ll probably see even larger mountains to climb, maybe even a whole range. But there are endless amounts of mountain ranges that you can’t see.”

    -Greg Chaitin

    I find it interesting that there are various mountain climbing analogies with math, even though the thought is quite straightforward.

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