(1) Keep trying to prove it.
You might never get an answer, but your attempts can help to narrow
down the possibilities, and close in on the quarry. Mathematics has
always been a communal enterprise. Honourable failed attempts on the Big
Conjectures have nearly always led to insights into other things, and even
proved significant side results along the way. The story of
thousands of years of attempts to solve the great construction problems of
antiquity (ultimately shown impossible) is the story of large tracts of
modern mathematics spawned in the process. Some beautiful mathematics has
arisen out of ``failed'' endeavours. The recent success of Andrew Wiles in
cracking Fermat's Last Theorem at last (see Issue 8.1 page 7) has actually
encouraged many mathematicians to have a go at some of the Big Ones, like
the Riemann Conjecture - the Holy Grail of mathematics (Issue 8.1 page 8).
(2) Try to disprove it.
Try to show that assuming it leads to something absurd, or try to find
exceptions/counterexamples. The mental activity demanded here is often
quite different to that in (1), but the two can serve each other. R. H.
Bing said of the Poincaré conjecture that he would alternate, spending 2
weeks trying to prove it, then switching to 2 weeks trying to construct a
counterexample. (See the article following this.)
(3) Prove that it would be implied by some other things you don't yet know
are true.
The story of Fermat's Last Theorem (like that of the Four Colour Theorem)
is full of heroes who succeeded in reducing the problem to verifying one
thing or the next thing. Seek for a good forward position, a staging
post, half-way house, a series
of base camps for your assault on the summit. Think of a simpler problem
whose solution would help with the harder problem. It would be OK if only
... It reduces to showing.... It will be sufficient to show... This
strategy often surprises - suddenly you are nearly all the way there!
(4) Prove that it implies other things you would like to know.
If this then what? It would be wonderful if this were so, because then....
Pointing out the gains can load the problem with significance - make its
solution more worthwhile and provide motivation for solving it. The
conjectures in mathematics which have most prestige are those which not
only have defeated the best for the longest, but whose solution would open
the gates to other treasures.
(5) Generalize the problem.
Sometimes mathematicians do this to cover their embarrassment, and make
the problem sound harder and more grand! But sometimes, indeed, the more
general expression touches on some other field where progress is being
made, and makes fruitful connections.
(6) Ignore it.
Consider first: maybe there is no great advantage in knowing it, and your
isolated effort in proving it may lead to no great acclaim or new
insights. On the other hand, it may be important for you symbolically -
cracking it may give you great encouragement. It may be crucial for your
self-esteem not to pass by, or give up.
It may be an important step forward, a
significant block along the way, and be worth substantial sacrifice of
time and effort. Perhaps, like the really Big Conjectures, it lurks
at the very heart of mathematics and its importance cannot be
overestimated: it lures you on irresistably. The
trouble is that taking on something too hard can seriously discourage
you, dissipate your energies - even kill you. George
Polya once had a
young mathematician confide to him that he was working on the great
Riemann hypothesis. `` I think about it every day when I wake up in the
morning,'' he said. Polya sent him a reprint of a faulty proof that had
been once been submitted by a mathematician who was convinced he'd solved
it, together with a note: ``If you want to climb the Matterhorn1 you might
first wish to go to Zermat where those who have tried are buried.''
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``My student Luke Woodward came in and showed this example that disproved my conjecture about palindromic symmetry. He was a bit worried that I was going to be angry or something, but he was obviously really pleased with his work and he did this lovely build-up. He showed me some of the things he had been doing, and then a few more, and he kept this example to the last minute and then said: `Oh, look, I've found this!' And you know, you just have to accept what nature gives you. There is always a silver lining to every discovery, so I'm not too disappointed: there is clearly something going on there. We can't have calculated all these examples and seen all these beautiful symmetries without there being something going on.''
1A
mountain in Switzerland.
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