It seems to appear that I have somehow managed to pass the 100,000 score milestone for reputation on MathOverflow. A hundred kiloreps! Does this qualify me for micro-celebrity status? I have clearly been spending an inordinate amount of time on MO… Truly, it has been a great time.
MathOverflow, an eternal fountain of mathematics, overflows with fascinating questions and answers on every imaginable mathematical topic, drawing unforeseen connections, seeking generalizations, clarification, or illustrative examples, questioning assumptions, or simply asking for an explanation of a subtle mathematical point. The mathematics is sophisticated and compelling. How could a mathematician not immediately plunge in?
I first joined MathOverflow in November 2009, when my colleague-down-the-hall Kevin O’Bryant dropped into my office and showed me the site. He said that it was for “people like us,” research mathematicians who wanted to discuss mathematical issues with other professionals, and he was completely right. Looking at the site, I found Greg Kuperberg’s answer to a question on the automorphism tower problem in group theory, which was one of the first extremely popular questions at that time, the top-rated question. I was hooked immediately, and I told Kevin on that very first day that it was clear that MathOverflow was going to take a lot of time.
I was pleased to find right from the beginning that, although there were not yet many logicians participating on MO, there were nevertheless many logic questions, revealing an unexpectedly broad interest in math logic issues amongst the general mathematical community. I found questions about definability, computability, undecidability, logical independence, about the continuum hypothesis and the axiom of choice and about large cardinals, asked by mathematicians in diverse research areas, who seemed earnestly to want to know the answer. How pleased I was to find such a level of interest in the same issues that fascinated me; and how pleased I was also to find that I was often able to answer.
In the early days, I may have felt a little that I should be a kind of ambassador for logic, introducing the subject or aspects of it to those who might not know all about it yet; for example, in a few answers I explained and introduced the topic of cardinal characteristics of the continuum and the subject of Borel equivalence relation theory, since I had felt that mathematicians outside logic might not necessarily know much about it, even when it offered connections to things they did know about. I probably wouldn’t necessarily answer the same way today, now that MO has many experts in those subjects and a robust logic community. What a pleasure it has become.
I feel that mathoverflow has enlarged me as a mathematician. I have learned a huge amount here in the past few years, particularly concerning how my subject relates to other parts of mathematics. I’ve read some really great answers that opened up new perspectives for me. But just as importantly, I’ve learned a lot when coming up with my own answers. It often happens that someone asks a question in another part of mathematics that I can see at bottom has to do with how something I know about relates to their area, and so in order to answer, I must learn enough about this other subject in order to see the connection through. How fulfilling it is when a question that is originally opaque to me, because I hadn’t known enough about this other topic, becomes clear enough for me to have an answer. Meanwhile, mathoverflow has also helped me to solidify my knowledge of my own research area, often through the exercise of writing up a clear summary account of a familiar mathematical issue or by thinking about issues arising in a question concerning confusing or difficult aspects of a familiar tool or method.
Mathoverflow has also taught me a lot about good mathematical exposition, both by the example of other’s high quality writing and by the immediate feedback we all get on our posts. This feedback reveals what kind of mathematical explanation is valued by the general mathematical community, in a direct way that one does not usually get so well when writing a paper or giving a conference talk. This kind of knowledge has helped me to improve my mathematical writing in general.
Thanks very much again, MathOverflow! I am grateful.
A few posts come to mind:
- My highest-voted answer is to the question What are some reasonable-sounding statements that are independent of ZFC?, a very early question on MO asked by Anton Geraschenko, the founder of MathOverflow. In my answer, I explained that the the implication $\kappa<\lambda\implies 2^\kappa<2^\lambda$ is independent of ZFC. I have become fascinated by the reception of this answer, which surprised me, and I now often mention it in talks on the philosophy of set theory, for it provides an example of a principle that is (1) extremely appealing to non-set-theorists; (2) affirms many pre-theoretic expectations that one might have for a concept of “size” in mathematics; (3) known to be consistent with the axioms of set theory; and yet (4) almost universally rejected by set-theorists when it is proposed as a fundamental principle. So it provides a counterpoint to what might seem otherwise to be a reasonable philosophical position, that the first three properties would be grounds for general acceptance.
- My very first question, Does every set admit a rigid binary relation? led to a paper by Justin Palumbo and myself, where we introduce the rigid relation principle, which asserts that indeed every set has a rigid binary relation, and which, we prove, is neither provable in ZF without the axiom of choice nor is it equivalent to the axiom of choice. Thus, it is a new weak choice principle.
- There was a fascinating flock of questions on the topic of infinite chess, including especially Johan Wästlund’s question Checkmate in $\omega$ moves? and Richard Stanley’s question on the decidability of infinite chess, which I pondered for some time. Eventually, Philipp Schlicht and I proved that The mate-in-$n$ problem of infinite chess is decidable, and Cory Evans and I discovered very high Transfinite game values in infinite chess, finding new finite positions with value $\omega$, infinite positions with values up to $\omega^3$, and proving that the omega one of 3D infinite chess is true $\omega_1$.
- I became truly obsessed with Ewan Delanoy’s question, who asked for an easy way to see that the countable models of set theory were not linearly ordered by embeddability (on our companion site math.stackexchange). After having worked furiously on the problem, I finally discovered to my astonishment, contrary to all our initial expectations, that this collection of models actually was linearly ordered after all! This work led directly to my theorem showing that every countable model of set theory is isomorphic to a submodel of its own constructible universe, and to a question, Can there can be an embedding $j:V\to L$ from the set-theoretic universe $V$ into the constructible universe $L$, when $V\neq L$?, which remains open.
- Cole Leahy’s extremely interesting question on Pointwise algebraic models of set theory led to our joint paper on Algebraicity and implicit definability in set theory, and to further open questions on Imp, the inner model of implicitly constructible sets.
There have been so many more great questions and posts. If you are inclined, feel free to post comments below linking to your favorite MO posts!
Concerning the MO reputation system, I suppose some might suspect me of harboring unnatural thoughts on reputation — after all, I once proposed (I can’t find the link now) that the sole basis of tenure and promotion decisions for mathematics faculty, as well as choice of premium office space, should be: MO reputation, ha! — but in truth, I look upon it all as a good silly game. One may take reputation as seriously as one takes any game seriously, and many mathematicians can indeed take a game seriously. My honest opinion is that the reputation and badge system is an ingenious piece of social engineering. The designers must have had a good grasp on human psychology, an understanding of the kinds of reasons that might motivate a person to participate in such a site; one thinks, for example, of the intermittent reward theory. I find it really amazing what the stackexchange designers have created, and who doesn’t love a good game?
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