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My training was in commutative algebra as a pure mathematician. I did dabble in combinatorial commutative algebra and went so far as purchase a book on the top
by yequalsx 10y ago
My training was in commutative algebra as a pure mathematician. I did dabble in combinatorial commutative algebra and went so far as purchase a book on the topic by Sturmfels and Miller. I doubt very much that you can find a published paper in mathematics in which the author guessed the answer to something and did not prove anything. Every paper I read (commutative algebra) proved things or discussed something heuristically with the goal of finding a method of proving.
I have the book by Stanley on combinatorics. There are not results of the form: I guessed A is the answer and it's right. Let's move on. This is does not happen. When one notices something is a solution the mathematician always wonders why. A mathematician wonders what underlying structure there is. Never is one satisfied by a guess.
If someone found a counterexample to the Riemann Hypothesis the first question would be, why is this number a counterexample? What caused the obstruction? You could problem publish a paper that just said, A is a counterexample to the Riemann Hypothesis. But you could not publish a paper that said, I guessed A is a solution to B and it turns out I was right.
- dxbydt 10y agoEven in commutative algebra, there is a fair amount of guesswork. If I tell you that 2+3=4 in an abelian group with {2,3,4}, then ask you what is 2+4, you will immediately guess 2+4=2. Its just what the millennials term "stupid obvious". Yeah you can do a lengthy proof on why 2+4=2. Proof: Its 2 because 4 must be the additive identity, and 4 is the additive identity because 2 & 3 are not, and they aren't because if 2 was then 2+3=3, and if 3 was then 2+3=2, but because I've told you 2+3=4, by closure 4 must be identity, which implies 2+4=2. That's the whole story. Literally no mathematician I know will sit down & write that lengthy proof I wrote. They'll just tell you 2+4=2, and if you pester them with "But why?" they'll say "Because" and excuse themselves :)
- yequalsx 10y agoIn an elementary group theory course this is the type of problem that would be given when groups are first introduced. It's a good homework/test question and just writing a*c = a as the answer would not suffice (depending on the level of the course and aim of the problem). The point at that level is for the student to learn how to justify their beliefs. This is especially so if one were in a basic mathematical logic course. Of course, in an algebraic topology course where this group showed up it would be assumed that everyone knows how to find the answer and why. No justification would be needed.
- 21 10y agohttp://www.openculture.com/2015/04/shortest-known-paper-in-a-serious-math-journal.html http://www.openculture.com/2015/04/shortest-known-paper-in-a...
- yequalsx 10y agoThe paper you cited supports what I've been saying. You can publish a paper that says, "Here is a counterexample." You can't publish a paper that says, "Here is a solution I guessed to be correct." Any method of finding a counterexample is accepted. Guessing a solution is not. EDIT: The paper linked to was published because it was a counterexample to a famous conjecture.