For comparison, actual solutions to a major problem tend to be arrived at by a process more like the following (often involving several mathematicians over a period of years or decades, with many of the intermediate steps described here being significant publishable papers in their own right):
- Isolate a toy model case x of major problem X.
- Solve model case x using method A.
- Try using method A to solve the full problem X.
- This does not succeed, but method A can be extended to handle a few more model cases of X, such as x’ and x”.
- Eventually, it is realised that method A relies crucially on a property P being true; this property is known for x, x’, and x”, thus explaining the current progress so far.
- Conjecture that P is true for all instances of problem X.
- Discover a family of counterexamples y, y’, y”, … to this conjecture. This shows that either method A has to be adapted to avoid reliance on P, or that a new method is needed.
- Take the simplest counterexample y in this family, and try to prove X for this special case. Meanwhile, try to see whether method A can work in the absence of P.
- Discover several counterexamples in which method A fails, in which the cause of failure can be definitively traced back to P. Abandon efforts to modify method A.
- Realise that special case y is related to (or at least analogous to) a problem z in another field of mathematics. Look up the literature on z, and ask experts in that field for the latest perspectives on that problem.
- Learn that z has been successfully attacked in that field by use of method B. Attempt to adapt method B to solve y.
- After much effort, an adapted method B’ is developed to solve y.
- Repeat the above steps 1-12 with A replaced by B’ (the outcome will of course probably be a little different from the sample storyline presented above). Continue doing this for a few years, until all model special cases can be solved by one method or another.
- Eventually, one possesses an array of methods that can give partial results on X, each of having their strengths and weaknesses. Considerable intuition is gained as to the circumstances in which a given method is likely to yield something non-trivial or not.
- Begin combining the methods together, simplifying the execution of these methods, locating new model problems, and/or finding a unified and clarifying framework in which many previous methods, insights, results, etc. become special cases.
- Eventually, one realises that there is a family of methods A^ (of which A was the first to be discovered) which, roughly speaking, can handle all cases in which property P^ (a modern generalisation of property P) occurs. There is also a rather different family of methods B^ which can handle all cases in which Q^ occurs.
- From all the prior work on this problem, all known model examples are known to obey either P^ or Q^. Formulate Conjecture C: all cases of problem X obey either P^ or Q^.
- Verify that Conjecture C in fact implies the problem. This is a major reduction!
- Repeat steps 1-18, but with problem X replaced by Conjecture C. (Again, the storyline may be different from that presented above.) This procedure itself may iterate a few times.
- Finally, the problem has been boiled down to its most purified essence: a key conjecture K which (morally, at least) provides the decisive input into the known methods A^, B^, etc. which will settle conjecture C and hence problem X.
- A breakthrough: a new method Z is introduced to solve an important special case of K.
- The endgame: method Z is rapidly developed and extended, using the full power of all the intuition, experience, and past results, to fully settle K, then C, and then at last X.
- The technology developed to solve major problem X is adapted to solve other related problems in the field. But now a natural successor question X’ to X arises, which lies just outside of the reach of the newly developed tools… and we go back to Step 1.
作者：dhchen (虽然原作者肯定不是他)
链接：https://www.zhihu.com/question/51219380/answer/235795482
来源：知乎