Jair Taylor
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 1d comment What is the current state of formalized mathematics? Personally, I wouldn't say that the axioms are illogical unless they are inconsistent. The axioms are just the rules of the game we've agreed to play - you might like them or dislike them, but this is a matter of aesthetics, not logic. 1d comment What is the current state of formalized mathematics? Hmm, I am not sure where you get this impression. Proof verification systems like Mizar and coq have had impressive results far exceeding the Principia - proving the Jordan Curve Theorem, for example. Apr 28 comment Why do we study real numbers? Historical reasons in some sense, I suppose. I could imagine some kind of alien civilization that does not use integers at all and only does computations in the symmetric group, say, but it seems very unlikely. It's hard to get very far without making some reference to a number - the order of the group, or the number of subgroups, etc. Personally, I would guess numbers are inevitable as a starting point for math because they are simplest useful things that it's possible to study logically. Apr 21 comment Find a recurrence relation and associated generating function for the number of different binary trees with n leaves Hint: Every tree is either a single leaf or a root with two subtrees. If $f(x)$ is the generating function this means $f(x) = x + f(x)^2$... Apr 19 comment Set equipped with some operation which is such that it is not known is it a group and is of some importance in mathematics Just because you haven't encountered any does not mean they do not exist. I would not be surprised if there was a nice example. At the very least you could construct artificial examples based on well-known conjectures. e.g., take $G = \{ -2 + \liminf_{n\rightarrow \infty} p_{n+1} - p_n\}$ under addition where $p_n$ is the $n$-th prime. The Twin Prime conjecture says this is the group $\{0\}$. Apr 17 comment Summing $n$ numbers so that they equal $0 \mod{n}$ I would find the formula for the number of tuples with $\sum a_i = k(n+1)$ and then sum over $k$. Apr 17 comment Summing $n$ numbers so that they equal $0 \mod{n}$ Ah, I see. I think you mean the ordered tuple $(a_1, \ldots, a_n)$ not the set $\{a_1, \ldots, a_n\}$ since the order matters here. Apr 17 comment Summing $n$ numbers so that they equal $0 \mod{n}$ This is not very clear. If you're taking a sum of all $a_1$ through $a_n$ you will get the same thing no matter what the order - it's either $0$ mod $n$ or not $0$ mod $n$. So I don't know what you are asking. Apr 17 answered What is an intuition behind conjugate permutations? Apr 15 accepted References for chromatic symmetric functions of hypergraphs Apr 14 comment Conceptual proof relating linear fractional transformations to matrices Hmm, I don't think I saw this the first time around. Nice answer - not too late for a +1 and accept! Apr 14 accepted Conceptual proof relating linear fractional transformations to matrices Apr 13 comment Is it acceptable to use “But” in a proof that doesn't use contradiction? That proof is not really the best example, though, since it's not written in full sentences like you will generally see in papers. Apr 13 comment Is it acceptable to use “But” in a proof that doesn't use contradiction? I would use 'but' to indicate that something is somewhat surprising, in a loose sense. In the proof you link to, they are essentially writing that "$b_{k+1}$ is nonnegative but less than or equal to $a_{m+1}$" which I think is pretty standard English. It helps to give your reader a slightly better feel for what the important parts of the proof are, as well as breaking up the monotony of "and... and... and... " Apr 9 answered Function that maps the “pureness” of a rational number? Apr 8 answered How to know if one problem is more difficult than another one? Apr 7 comment Interesting representation of $e^x$ Very cool if true! Apr 7 awarded Yearling Apr 6 comment Combinatorics Books Peter Cameron's Combinatorics: Topics, Techniques, Algorithms is also a good choice. Apr 5 comment Is there any interesting interpretation of Taylor coefficients of $e^{-\log(1-x)}$? No problem. Let me know if anything is unclear (or incorrect).