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Recently I have been perplexed by a non-mathematical problem.

One of my Indian student-friend is preparing for the next year's IMO exam but he says that he doesn't like simply doing hard and challenging problems. To him it is kind of boring to only solve problems. For example he likes to read the Logical Foundations of Geometry rather than solving geometry problems. He has recently appeared for a number of M. C. Q. exams but did not qualify in any of them.

He couldn't answer even the half of the questions because he didn't want to answer any question which he couldn't prove. I don't think in this way he can crack the IMO exam because in IMO the skill of problem solving is acquired by solving harder and harder problems and its the thing that is required most.

The only thing that he has to his credit is that he has given an elementary proof of Sylvester-Gallai Theorem (a modification of my wrong proof that I asked for verification in Mathematics Stack Exchange), some theorems on Elementary Number Theory (among which one deals with a non-trial method of finding the order of an integer modulo some integer) which are perhaps new (I don't know precisely because I am not an expert in this field). He wants to be judged by his works, not by passing any exam (even the IMO!). He even tried to send his works to some journal and to some mathematicians also but either it was rejected or he got no reply from the corresponding mathematicians.

Basing upon the incidents, I think that the best way to motivate him would be to have recommendation for his work from some renowned mathematician of his interested fields which are only Number Theory, Logic and Euclidean Geometry but I don't think that it will be logical for any mathematician to waste time on his work without knowing that it is something significant. I can tell this to him but probably it will do much harm to him rather than doing good.

Is there any other way to tell him the truth or is there some other way to motivate him to do mathematics rather than trying to seek its logical consistency?

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    $\begingroup$ I don't understand the question, but your post raises lots of red flags. I'm skeptical of most people who claim not to solve problems simply because they're boring. Add on poor past performance (on recent exams) and rejections/non-replies when he has sent out his actual work, and the issue sounds bigger than boredom. Why even take the IMO if he doesn't want to solve problems? And as far as the S-G Theorem is concerned: Was his proof the one in Engel's Problem Solving Strategies? (mualphatheta.org/problem_corner/Mathematical_Log/Issues/0402/…) $\endgroup$ – Benjamin Dickman Apr 25 '14 at 16:09
  • $\begingroup$ it's exactly 100 years after the first encounter between Ramanujan and Hardy :-) $\endgroup$ – mau Apr 25 '14 at 16:48
  • $\begingroup$ @Benjamin Dickman: No, the proof is not the one Engel's Problem Solving Strategies. He proved it using Mathematical Induction and the axiom of Euclidean Geometry that two straight lines intersect in at most two points. I have read his proof. Probably it is not wrong. And by the way what is meant by "it's exactly 100 years after the first encounter between Ramanujan and Hardy"? $\endgroup$ – William Hilbert Apr 25 '14 at 16:55
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    $\begingroup$ I suggest improving the title of your question; it doesn't give any indication of what the question actually is. $\endgroup$ – Mike Shulman Apr 25 '14 at 18:43
  • $\begingroup$ @BenjaminDickman: I forgot to mention in my previous comment that he bothers to take the IMO exam because in India there is a institute called Indian Statistical Institute in which he wants to study and if he can crack the IMO, it would warrant his admission there. But if he couldn't then he has to take the ISI Entrance Exam which consists of two sets of questions, the 1st part being M. C. Q and the 2nd part being elaborate type answers. The problem is that unless you score a minimum (determined by the authority) in the M. C. Q questions, your subjective answers will not be checked. $\endgroup$ – William Hilbert Apr 26 '14 at 6:57
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There's an important distinction to make. It's not clear to me if your friend is failing to really learn at all (reading about abstract topics, feeling satisfied and not doing any problems at all, but not really grasping the subject---I know I'm prone to this mistake), or if your friend is in fact learning in his preferred way, but is struggling with obtaining the recognition needed to proceed to the next academic step.

Since you only raise the latter topic, I'm only going to try to address that. It sounds like your friend is in a difficult position because in order to advance academically, he needs to demonstrate a different set of skills than the ones he's actually good at (even though his actual skills may justify advancement).

The IMO is prestigious, but it also focuses on a relatively narrow kind of mathematical problem---the sorts of problems that require a great deal of cleverness but not a lot of deep theory. So while doing well on the IMO is a good indicator of mathematical ability, not doing well doesn't mean much. From what you say of your friend, it's not his sort of math, so it's unsurprising he's not motivated by it.

Similarly, your friend's desire to only solve problems when he can prove the answer sounds fairly ordinary to me, and many professional mathematicians share that attitude. (I'd much rather read about the logical foundations of geometry than solve geometry problems!)

I imagine focusing on the suggestion that he's doing math wrong is counterproductive and causing him to dig in his heels. It might be better to acknowledge that these questions aren't interesting or mathematically important to him, and that the source of motivation has to be doing something boring to get into ISI.

(I should add that the attitudes he's showing can be taken too far---some amount of problem solving is necessary for understanding. The details you gave don't reflect that, but perhaps he's doing that as well.)

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  • $\begingroup$ I am sorry for not being so elaborate and precise. But the guesses you made are mostly true. Actually, he has a typical nature. In fact, he has asked me to deduce a Generating Function of Goldbach Partition of Even Integer (I did that but he told me that he already deduced that and asked me to post the problem to MSE) and thereby showed that the number of GPEI is bounded. This is the kind of mathematics he likes to do. Anyway, your suggestion seems to me to work on him. Thanks. $\endgroup$ – William Hilbert Apr 26 '14 at 16:43
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    $\begingroup$ I agree, but as a friend says (rather rudely): "Each profession has it's 1/4 of whore." There are certain things you have to do, like it or not. $\endgroup$ – vonbrand Apr 26 '14 at 17:40
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    $\begingroup$ @WilliamHilbert I think that vonbrand didn't stress it enough: $$\Large\text{There are certain things you have to do, like it or not.}$$ The sooner he learns it, the easier for him it would be. If he's a genius, maybe he will have to do only a part of it, but there is no way to avoid it all. $\endgroup$ – dtldarek Apr 26 '14 at 23:42
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If

He wants to be judged by his works

, perhaps point out that being able to replicate other people's proofs isn't intrinsically any more original than just using the results of the proofs.

Most true craftsmen start off by using existing tools before they design their own, and they're probably more competent for all that as well.

So there should be no shame in becoming proficient in the existing toolset.

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  • $\begingroup$ The thing isn't that he doesn't want to become proficient in the 'existing toolset' the point is that he doesn't want to be only 'proficient in the existing toolset', rather he wants to 'enrich the existing toolset by some tools of his own.' $\endgroup$ – William Hilbert Apr 25 '14 at 17:16
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    $\begingroup$ It's a bit immature, though, to be so unwilling to get to grips with the existing toolset, that he can't even pass basic exams. If he's as good as he imagines, he needs to either put up or shut up. Otherwise, he'll be left by the wayside. $\endgroup$ – ChrisA Apr 25 '14 at 20:43
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Perhaps he should hang out in MSE a bit, and ask for confirmation/refutation/"this is well known, check ..." on results.

It is a fact that the standard "tell people of some results/have them do exercises/test them" cycle isn't for everyone. But it is the only accepted way to select the few whith the abilities to go farther. Yes, there have been exceptions, but

In any case, remind them the oft-quoted standing on shoulders of giants; if you want to check everything out strictly by yourself, you won't get very far.

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