Since I was born on Oct 1, 1960 (that's not true---if I posted my real birthday I might get my identity stolen), I will do a nature vs nurture post based on my life, which seems less likely to offend then doing it on someone else's life. I'll just rattle off some random points on Nature vs Nurture.
1) Is it plausible that I was born with some math talent? Is plausible that I was born with some talent to understand the polynomial van der Warden theorem? What is the granularity of nature-given or nurture-given abilities?
2) My dad was a HS English teacher and later Vice-Principal. My mom taught English at a Community college. Readers of the blog might think, given my spelling and grammar, that I was switched at birth. My mom says (jokingly?) that I was switched at birth since she thinks I am good at math.
a) I am not THAT good at math. Also see next point.
b) While there are some math families, there are not many. See my post here.
c) I think being raised in an intellectual atmosphere by two EDUCATORS who had the money to send me to college and allowed me the freedom to study what I wanted to is far more important than the rather incidental matter of what field I studied.
d) Since my parents never went into math or the sciences it is very hard to tell if they were `good at math' or even what that means.
3) There were early signs I was INTERESTED in math, though not that I was good at it.
a) In fourth grade I wanted to know how many SECONDS were in a century so I spend some time figuring it out on paper. Did I get the right answer? I forgot about leap years.
b) I was either a beneficiary of, or a victim of, THE NEW MATH. So I learned about comm. and assoc. operations in 8th grade. We were asked to come up with our own operations. I wanted to come up with an operation that was comm. but not assoc. I did! Today I would write it as f(x,y) = |x-y|. This is the earliest I can think of where I made up a nice math problem. Might have been the last time I made up a nice math problem AND solved it without help.
c) In 10th grade I took some Martin Gardner books out of the library. The first theorem I learned not-in-school was that a graph is Eulerian iff every vertex has even degree. I read the chapter on Soma cubes and bought a set. (Soma cubes are explained here.)
d) I had a talent (nature?) at Soma Cubes. I did every puzzle in the book in a week, diagrammed them, and even understood (on some level) the proofs that some could not be done. Oddly I am NOT good at 3-dim geom. Or even 2-dim geom. For 1-dim I hold my own!
e) Throughout my childhood I noticed odd logic and odd math things that were said:
``Here at WCOZ (a radio station) we have an AXIOM, that's like a saying man, that weekends should be SEVEN DAYS LONG'' (Unless that axiom resolves CH, I don't think it should be assumed.)
``To PROVE we have the lowest prices in town we will give you a free camera!'' (how does that prove anything?)
``This margarine tastes JUST LIKE BUTTER'' (Okay-- so why not just buy butter?)
e) In 9th grade when I learned the quadratic formula I re-derived it once-a-month since I though it was important that one can prove such things. I heard (not sure from where) that there was no 5th degree equation. At that very moment I told my parents:
I am going to major in math so I can find out why there is no 5th degree equation.
There are worse things for parents to hear from their children. See here for dad's reaction.
f) When I learned that the earth's orbit around the sun is an ellipse and that the earth was one of the foci, I wondered where the other foci is and if its important. I still wonder about this one. Google has not helped me here, though perhaps I have not phrased the question properly. If you know the answer please leave a comment.
g) I also thought about The Ketchup problem and other problems, that I won't go into since I already blogged about them here
4) I was on the math team in high school, but wasn't very good at it. I WAS good at making up math team problems. I am now on the committee that makes up the Univ of MD HS math competition. I am still not good at solving the problems but good at making them up.
5) From 9th grade on before I would study for an exam by making up what I thought would be a good exam and doing that. Often my exam was a better test of knowledge than the exam given. In college I helped people in Math and Physics by making up exams for them to work on as practice.
6) I was good at reading, understanding, and explaining papers.
7) I was never shy about asking for help. My curiosity exeeded by ego... by a lot!
8) Note that items 5,6, and 7 above do not mention SOLVING problems. The papers I have written are of three (overlapping) types:
a) I come up with the problem, make some inroads on it based on knowledge, and then have people cleverer (this is often) or with more knowledge (this is rarer) help me solve the problems.
b) I come up with the problem, and combine two things I know from other papers to solve it.
c) Someone else asks for my help on something and I have the knowledge needed. I can only recall one time where this lead to a paper.
NOTE- I do not think I have ever had a clever or new technique. CAVEAT: the diff between combining known knowledge in new ways and having a clever or new technique is murky.
8) Over time these strengths and weaknesses have gotten more extreme. It has become a self-fulfilling prophecy where I spend A LOT of time making up problems without asking for help, but when I am trying to solve a problem I early on ask for help. Earlier than I should? Hard to know.
9) One aspect is `how good am I at math' But a diff angle is that I like to work on things that I KNOW are going to work out, so reading an article is better than trying to create new work. This could be a psychological thing. But is that nature or nurture?
10) Could I be a better problem solver? Probably. Could I be a MUCH better problem solver? NO. Could I have been a better problem solver I did more work on that angle when I was younger? Who knows?
11) Back to the Quintic: I had the following thought in ninth grade, though I could not possibly have expressed it: The question of, given a problem, how hard is it, upper and lower bounds, is a fundamental one that is worth a lifetime of study. As such my interest in complexity theory and recursion theory goes back to ninth grade or even further. My interest in Ramsey Theory for its own sake (and not in the service of complexity theory) is much more recent and does not quite fit into my narrative. But HEY- real life does not have as well defined narratives as fiction does.
12) Timing and Luck: IF I had been in grad student at a slight diff time I can imagine doing work on algorithmic Galois theory. Here is a talk on Algorithmic Galois theory. Note that one of the earliest results is by Landau and Miller from 1985---I had a course from Miller on Alg. Group Theory in 1982. This is NOT a wistful `What might have been' thought. Maybe I would have sucked at it, so its just as well I ended up doing recursion theory, then Ramsey theory, then recursion-theoretic Ramsey Theory, then muffins.