What role do mistakes play in the everyday work of a mathematician?
Types of mistakes:
- Logical mistakes, wrong calculations – easy to check (?!)
- Wrong tactics – choosing manipulations which don’t deliver intermediate solutions
- Wrong strategies – although intermediate steps might work, the chosen path doesn’t lead to the intended result
Good and bad mistakes:
- Are there good and bad mistakes?
- Obviously there are good mistakes, as remarks by G. Shimura about Y. Taniyama and remarks by G. H. Hardy about S. Ramanujan suggest.
- How can a good mistake be recognized?
- Good mathematics does not necessarily need to be correct mathematics. Obviously good mistakes are good mathematics.
Communication of mistakes:
- The work on a mathematical problem contains mistakes.
- This holds for every mathematical problem and for every mathematician.
- The proficiency in coping with mistakes constitutes a crucial part of the expertise of a mathematician.
- Mistakes are banned from published papers
- Mistakes have no place in dead mathematics
- Mistakes are seldom reported, especially in modern mathematical practice
- There is maybe a change of paradigm on the way, related to the polymath projects.
- standards should be developed for the form of a paper which presents unsuccessful work but also the rhetorical and stylistic tools will have to be different from those for “correct” papers
- standards for the evaluation of such papers would be needed (how should such a paper be reviewed)
- the rhetorical (and stylistic) tools will have to be different from those used to present successful research
How can the communication of a strategic mistake look like? “I tried to relate field A to field B. Don’t try it again, because of C.”
- C should contain important wrong mathematics
- The communication of C can result in
- Others finding successful paths
- Other but different wrong attempts
- Questioning of some intuitions
- “declaring” this relation as non-existing (or at least as non-productive)
Concerning my own interest in a problem from number theory, an expert (E. Hlawka) told me not to work on it: “Don’t burn your fingers with this!”. Obviously he had some reasons for this which he did not or could not make explicit.
What are the reasons for a certain problem to be considered as “hard”? Is anything like this communicated formally, thus published? Can the notion of “hardness” be formalized?
Maybe from refereeing each other’s paper submissions which were eventually rejected. But these are only those attempts where the author could not find the mistake himself. I would suspect that a mathematician will find most of his mistakes by himself (compare to chess) and will not submit them for publishing.
Can a mathematician judge the quality of his own mistakes? “I made this mistake but it was a good mistake and so I will write a paper about it.” “This other mistake was a “bad” one. Forget it!”
Are there “typical” mistakes in research mathematics like there are in school mathematics?
- Mistakes and understanding
Wrong vs. correct understanding
- Inconsistent conventions (important??)
- Goro Shimura (documentary ‘Fermat’s Last Theorem’, 12:48 to 13:03 ):
Tanyiama was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction and so eventually he got right answers. I tried to imitate him but I found out that it is very difficult to make good mistakes.
I have in the past tried to say things like “his failure was more wonderful than any of his triumphs”, but that is absurd. It is no use trying to pretend that failure is something else. All we can say is that his failures give us additional, surprising evidence of his imagination and versatility. And we can respect him as one who let his mind run free, instead of keeping it under saddle and blinkers like so many others do.
- V.I. Arnold (“Polymathematics: Is mathematics a single science or a set of arts?“):
Mistakes are an important and instructive part of mathematics, perhaps as important a part as the proofs. [… in the text Arnold gives examples of mistakes by Leibniz, Cauchy, Lagrange, Poincaré, Hilbert, Burnside, Petrovsky, Komogorov, Pontrjagin and Rokhlin]