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For you, personally, the way you do maths, how much does the way forward seem like trial and error versus absolute?

I guess what I mean is that it seems to me that in maths there is kind of a goal that you are seeking, whether it be to solve a particular math problem or simply to direct the mind toward mathematical-type thinking. So the question is, to what extent does the path toward the goal feel kind of "preordained" versus... or maybe what I mean is if mathematics progresses in order (which i suppose is another question in itself but anyway), to what extent do some of the mathematical creatures that are observed turn out to be irrelevant to the goal / to what extent do particular constructions turn out to be unnecessary to the final construction? Do mathematicians allow walking down a certain path and then deciding the path is not important to the solution and turning around, walking back a ways before continuing down a new path, OR, does every path, all the ground that is tread on the way to the solution, contribute something to the solution?

Or as my dad says, it's a philosophical math question about dead-ends in math.

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closed as not constructive by Austin Mohr, no identity, Chris Eagle, J. M., Thomas Oct 7 '12 at 18:50

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4 Answers 4

This is a hard question to give a concrete answer to but I'll write a couple of my thoughts.

For me, hard problems require a lot of indirect exploring and feeling around before I try to understand the specifics - only easy problems will be attainable from a direct path from the solution right off the bat. I need to get a sense of the space I'm working in. So I'll construct a bunch of examples, as "weird" as possible, and tinker and explore until I feel I have a reasonable intuition. In this method nothing is really wasted time.

Of course, sometimes the space is too complicated and I will walk myself in circles doing this. In this case it usually helps to restrict the problem to something smaller, then try to understand that, and gradually lower the restrictions, waiting for understanding at each point before I move on, until I believe I can make another attempt at the general theorem.

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Steven McAdam once said to my abstract algebra class at the University of Texas, "Trial and Error is a legitimate problem-solving tool in mathematics." I agree 100%.

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I really learned some research method from Michael Barratt in the late 1950s. I recall saying to myself after a long session with Michael: "If Michael Barratt can try one damn fool thing after another, then so can I!" And I have followed this method ever since. (Actually, Michael's trials were not all that damn fool, but you get the idea.) One of my research students, Derek Waller, remarked that if 10% of your ideas are any good, and you have 100 ideas, then you have 10 good ideas; but if you have only 10 ideas, then you have problems!

How to get ideas? The composer Ravel advocated copying. He said: If you have some originality, then this will show. If not, never mind! In fact, you may need to copy several times before the rusty wheels of the brain start to turn.

Another method is to try and write up one topic in the style and using the methods of another subject. You might like to look at the article The methodology of mathematics. Part of the argument there is that any human activity needs a discussion of methodology.

It took me a long time to realise that trying to write good clear mathematics is very helpful to oneself! You may find areas where the current versions are not clear, or have seeming anomalies, so you wonder how to correct them. This was how I got led into investigating groupoids in the 1960s. It has been productive, and a lot of fun, even though, or perhaps because, at that time, and even later, big shots were often inclined to say "Groupoids are rubbish!". In fact that raised the stakes.

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I'll venture so far as to say that any problem that can be solved without trial-and-error has ceased to be a mathematical problem, but is merely one of handle-cranking.

It is possible to gain so much experience with a particular problem domain (say, elementary algebra) that the trial-and-error phase mostly happens unconsciously when you look at the problem statement, and then the path to the solution can seem almost preordained. But it still happens, and problems that can be solved that way remain mathematical (though "easy") ones.

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