Cathy’s post touched off a lot of discussion of math contests, and whether they do or do not, in her formulation, suck. My thoughts on this are pretty simple.

**Big good thing about math contests:** They reveal that math is more than what’s taught in school, and that there’s a whole community of kids around the world who are passionate about math.

**Big bad thing about math contests:** They help promote the idea that the most important thing about math is whether you’re the best at it.

Of course you can design your contest to provide more of the big good and less of the big bad. The question isn’t so much whether the good minus the bad is positive; it’s whether there are other ways of getting at the good that avoid the bad. I think programs like Hampshire and MathCamp and PROMYS and Ross are like this. But they clearly don’t scale to the size of an AMC. Mary O’Keeffe‘s comments on Cathy’s blog were particularly interesting, since they give a good sense of what math contests are like in 2011 for those of us whose direct experience is substantially less recent.

Some people, I think, don’t think my big bad is so bad. I disagree. Somebody on my Facebook feed recently linked to this letter from algebraist Donald Weidman to *Science*, headed “Emotional Perils of Mathematics.” Weidman numbers among these perils the following frustration:

“The history of mathematics makes plain that all the general outlines and most of the major results have been obtained by a few geniuses who are not the ordinary run of mathematicians. These few big men make the long strides forward, then the lesser lights come scurrying in to fill in the chinks, make generalizations, and find some new applications; meanwhile the giants are making further strides.”

This is so profoundly wrong it makes my teeth hurt. Mathematics is like Earthball. The weight of our ignorance is tremendous and all of us push together to move it a bit to one side over the course of our lifetimes. We may vary in strength but what we have in common is that there’s very little we can do alone.

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A math “competition” I greatly enjoyed as a youngster was the USAMTS ( http://www.usamts.org/ ), where you’re given a month to work out solutions to several problems, and you earn prizes if you get enough correct over the course of the year. Somehow that system (as opposed to a first, second, third place sort of thing) made me feel that my participation was less of a competition against others but really against the problems / myself. I also liked having lots of time to really get absorbed in the problems, which I think has helped me to this day.

I’d forgotten about the prizes! I loved USAMTS, the prizes, however, didn’t go up my valley. Not that I don’t like the idea of the prizes, but I always wonder why I kept getting books on number theory and cryptography…

It is remarkable that the culture of math contests is so similar between very different countries (communist Romania and USA, 20+ years ago). I side with Cathy: in retrospect I realize I really hated the math contests I went to, due to their highly competitive nature. In the words of a popular song, “don’t read beauty magazines, they will only make you feel ugly.” [For full disclosure: I started out as the beauty queen in 7th grade (won 1st prize at the national level), and turned ugly later. Boy, were the grapes sour.]

On the plus side, I met many of my friends (who later also became mathematicians) at the training camps. This is to my mind the only benefit I got from these contests, the feeling of belonging to a select group of like-minded people. But I also always felt second rate because I never made the IMO team, the supreme prize. Luckily I was able to forge myself a new identity as a computer geek (and go to the less desirable CS equivalent of the IMO). Otherwise I probably would have quit both maths and CS. Only after college was I mature enough to realize I liked math enough to do it without having to compare myself to others.

I am not sure I was exposed to so much non-traditional math at the training camps. It was more like learning the myriad tricks used to solve inequalities. I learned a whole lot more from Martin Gardner’s books, or later from Donald Knuth’s.

In Romania there was also an equivalent of the USAMTS, where the prize was an invitation to participate in a free, extremely dingy, camp. (We’d wander the hills outside the town where the camp took place to steal plums to supplement our nutrition.) But the idea of spending a week with kids interested in math, with no contest at the end, was a huge winner and almost all my friends looked forward to these camps.

I found a quote somewhere else that made me look up this great little post of yours.

Over at teleskopos </a? Rebekah Higgitt quoted Augustus de Morgan:

That’s much more like it, I think.

You’re big bad is an exaggeration. Competitions are wonderful. Competitions provoke people–especially men–in ways that nothing else can. Competitions separate the mathematical wheat from the chaff. If you’re uncomfortable with inequality–with loser and winners–that’s too bad. Because it’s an ineliminable feature of life.

If some kids come away from competitions thinking being the best at mathematics is the most important thing, oh well. Really, who cares?

It’s a heck of a lot closer to the truth than believing “not being the best mathematician is the most important thing.” Or” not being the best is just as good as being the best”. Excellence is something to be desired. Being the best is better than not being the best.

If some neurotic freak wishes to take these obvious truths, and transmogrify it into your “the most important thing about math is being the best at it”, just maybe the problem is his personality. Maybe he doesn’t belong in math competitions. In any event, it’s not moving the needle: I’m unaware of any serious, durable, deleterious effects of mathematical competitions. Neither do I see any evidence of it harming the field or being a cause of a chronic undersupply of mathematicians.

This type competition is no materially different than any other competitive endeavor boys engage in. There will of course be excesses. But it’s obviously a net positive. Don’t try to be the pathetic mommy who changes the rules of a game so that nobody comes out a loser.

Secondly, Weidman is spot on. There is glaring variablity in mathematical ability even amongst professional mathematicians. Given your own extreme precocity you should recognize this rather obvious fact and its equally obvious corollary: some mathematical insights are beyond the ken of certain individuals. And there’s no guarantee that if you put together a room full of inferiors, they’ll necessarily be equal in insight to a single superior intellect. There may be certain things that only those at above that superior intellect level are able to fully fathom.

And it’s bloody obvious that giants like Poincare, Gauss, Euler, and Laplace contribute leagues more than the innumerable minor mathematicians consigned to historical oblivion. Likewise for physics, where giants like Newton, Einstein, Schrodinger and Feynman punch above their weight.

Your Earthball analogy maybe a more apt description of contemporary mathematics–or physics. Sure, there’s little we can do alone once the aforementioned intellectual giants have picked through all the low-hanging fruit. Now we often employ teams of scientists and computers funded by huge budgets. But *historically*, as Weidman stipulated, that wasn’t the case. Furthermore, Weidman’s claim is not that each genius came into the world as some tabula rasa and proved everything de novo from first principles. Newton himself famously stood on the shoulders of giants. But there’s no denying his work was unique, pathbreaking, and the result of a profound intelligence–perhaps only rivaled by one contemporary, Leibniz.

Even if you wish to view mathematics as some vast collective effort to conquer MATH, that doesn’t change the fact that some are more able than others, and that the state of mathematics would be far more harmed by the death of Grigori Perelman or Terence Tao (and minds like theirs) than by the early demise of Prof. Schuyler at Union County Community College (and minds like his). And it is the former brains –the Gauss’, Euler’s and Ramanujan’s–that make the huge breakthroughs, Jordan. The breakthroughs too often require outsized mathematical intelligence that is the province of a *very* rarefied few.

How any one could take exception to this statement: “The history of mathematics makes plain that all the general outlines and most of the major results have been obtained by a few geniuses who are not the ordinary run of mathematicians.” ?!?!

It’s just beyond me. Isn’t it obvious to you that “most of the major results” have been obtained by a few geniuses who are not the ordinary run of mathematicians”.

(Jeez, speaking of which: another wonderful confirmation of Pareto’s 80/20 heuristic.)

And if competition is what it takes to identify, incentivize, and hone that skill. Then bring it, baby!

I’m sorry, you seem to have mistaken me for an unsympathetic minor character in an Ayn Rand novel.

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That article by Weidman appeared in 1965 when math and the attitudes towards math were very different from today. Math is a much huger enterprise with more room for contributions now. Also with so much collaboration it’s much easier for a lesser player to participate more than before.

But most of all, this idea that that somehow secondary contributions are of little value is what bothers me. If it’s a nice argument and you enjoyed working on it, does it matter if it revolutionizes the field or not? Sounds to me like the problem with Weidman was his attitude towards research… he apparently was trying to make breakthroughs through working feverishly on big problems. Just because that led him to frustration doesn’t mean there’s no satisfying way to be a mathematician who isn’t a leader.

The date of the letter (1965) certainly shows: “The mathematician must risk frustration. Most of the time, in fact, he finds himself…” ; “… he will almost certainly…”, “… these few big men…”, “… no run-of-the-mill mathematician expects in his heart to prove a major theorem himself”.

Obviously, the missing half (or so) of humanity did not even have a starting chance! (And “Naturally, his family and his friends have no feeling for the significance of his accomplishments”).

(And by the way, concerning comment #5, from what I’ve read, Schrödinger himself was rather a run-of-the-mill physicist before he more or less stumbled on an equation that was much in the air, badly misunderstood its physical meaning, and later quite openly claimed to be disgusted by its probabilistic interpretation…)

I actually looked on Mathscinet out of curiosity. As it happens, D. Weidman has a single 6 page publication (taken from his 1964 PhD), so it seems pretty clear that that he had basically no insight whatsoever about what goes on in any other mathematician’s mind than his.

It’s actually funny to see that this letter was published in “Science”; it looks like something which is of interest mostly as self-analysis than anything else.

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