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Mathematics professor at Cambridge

May
12
comment What does it mean for a mathematical statement to be true?
If someone asks you why it is true, then you will deduce it from something more basic. And if they ask you why that is true, you will deduce it from something more basic still. What you won't say (unless you want to leave your questioner very dissatisfied) is that it's just a Platonic fact about the mathematical universe. Obviously these "why" questions have to come to an end eventually, and ... bingo ... there are your postulates.
May
8
comment Measuring how “heavily linked” a node is in a graph
Alternatively, take a lead from Google and work out the eigenvector with largest eigenvalue. That gives you a pretty good measure of which vertices are most linked, taking into account the linking of the neighbours.
May
7
comment Does this approach for the Poincare conjecture work?
I should add that it is clear from a glance at the paper that he is not a crackpot: it looks like real mathematics that's wrong, rather than weird stuff that doesn't even count as mathematics.
May
7
comment Does this approach for the Poincare conjecture work?
I agree that the God sentence is not evidence enough on its own. But there is another important piece of evidence: the paper is short and has been in the public domain since January. If it were correct, it would surely have been hailed as a spectacular twist in the story of the Poincaré conjecture. Given that it hasn't, I am predisposed not to believe the paper, and against that background the God sentence increases my scepticism.
May
6
comment Does this approach for the Poincare conjecture work?
I'm not hugely encouraged by the final sentence of the introduction to the paper: "This essay is devoted to the meditation of God’s Word among the inhabitants of the earth."
May
6
comment Examples of common false beliefs in mathematics
I'd think of that not as a belief that people are likely to have so much as a statement that looks moderately plausible and doesn't have an obvious counterexample. In other words, it's not something that people unthinkingly assume, because they don't tend to think about it at all. But perhaps I'm wrong about that and it is routinely used as a lemma by inexperienced algebraists.
May
5
comment What practical applications does set theory have?
I think of this proof as "morally" a pure existence proof, but it happens that it can be made constructive. (I'd say the same about any proof that begins, "Let q_1,q_2,... be an enumeration of the rationals," when it doesn't matter in the slightest what the enumeration is.) I don't have a formalization of this view though.
May
5
comment Examples of common false beliefs in mathematics
By an amusing coincidence, I came across this for the first time a couple of days ago. There was a Cambridge exam question in 2008 where you had to show that products and subspaces of separable metric spaces were separable, and then you were given a topological space and asked to show that its square was separable, and that a certain subspace of it was not separable. I had to stare at it for about a minute before I understood why I had not just proved a contradiction.
May
4
comment Help with a double sum, please
I enthusiastically second that.
Apr
29
comment Choice vs. countable choice
I'm in the same minority. I don't find it "obviously true" in any useful sense that the empty set exists, for instance. I just don't know what "exists" means over and above our use of the axiom in mathematics.
Apr
29
comment A question of Erdős on equidistribution
No idea what the answer is, but thanks for drawing my attention to this nice problem. (I'm hoping someone will say that it's still open.) How easy is the counterexample to Khintchine's conjecture?
Apr
28
comment Wanted: A constructive version of a theorem of Furstenberg and Weiss
OK that gives me a better idea of what you are asking -- enough to see that I don't know the answer I'm afraid. (My gut reaction is that what you're asking for looks hard, but I don't know much about this area so this opinion should not be taken seriously.)
Apr
28
comment Wanted: A constructive version of a theorem of Furstenberg and Weiss
It's certainly possible to produce a constructive upper bound. Are you really looking for the best possible upper bound?
Apr
27
comment How to know if somebody else is also working on your problem?
"If the other guy has already completed a certain amount of (say, not yet published) work on that specific topic, knowing this would help you to avoid waisting time to try to re-do something that has already been done (at least with the same methods)." I wouldn't underestimate the value of that kind of waste of time ...
Apr
22
comment A Simple Generalization of the Littlewood Conjecture
I can't quite follow what your argument is, but I would point out that it is a known fact that if a sequence $d_1,d_2,...$ grows geometrically (in the strong sense that there is a positive constant c such that all ratios are at least 1+c) then there must exist a real number s and a positive epsilon such that for every n the distance from $sd_n$ to the nearest integer is at least epsilon.
Apr
22
comment Can curves induced by analytic maps wiggle infinitely across a line?
Very nice. My mistake was to try to use the no-isolated-zeros theorem rather than imitating its proof.
Apr
22
comment Can curves induced by analytic maps wiggle infinitely across a line?
My comment does not imply that the question isn't straightforward. I am struggling to understand your argument, however. Are your "real analytic" functions complex-valued? I should add that the fact that I am struggling to understand your argument is compatible with its being clear and correct.
Apr
22
comment A Simple Generalization of the Littlewood Conjecture
Hmm, I have now thought a bit more and I think this paper is irrelevant -- though it is interesting in its own right. I like your question too.
Apr
22
comment Can curves induced by analytic maps wiggle infinitely across a line?
Oh dear, this question looks as though it has the potential to obsess. I thought I had an answer but it collapsed as I tried to write it down.
Apr
19
comment Subset of the plane that intersects every line exactly twice
If my memory serves, it's open even for G_delta sets (but known that they cannot be F_sigma sets).