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Jun
19
revised Smooth transition between two quaternions?
Incorrect spelling
Jun
19
suggested suggested edit on Smooth transition between two quaternions?
Jun
19
comment Category-theoretic description of the real numbers
I don't know much about toposes. But curiously the people who've written about this construction seem to work with toposes a lot. I feel like in some sense this construction gets to the very heart of what real numbers are about. It seems like a generalization of reasonable (in some sense) strategies for sharing $M$ objects between $N$ people. I haven't worked out the details of that yet though...
Jun
19
answered Category-theoretic description of the real numbers
Jun
19
comment Category-theoretic description of the real numbers
It's not clear that the reals are the optimal way to formulate notions of the continuum and there are alternatives like the non-standard reals. There are also very different points of view. The standard view is that the real line is a collection of individual points and notions of continuity are an additional structure imposed on this collection. But there are other points of view in which the real line is no longer a collection of distinct points and continuity is built in right from the start. Eg. logicandanalysis.org/index.php/jla/article/viewFile/63/25
Jun
19
answered Using dimensional analysis to evaluate $\frac{d}{dx}x^n$
Jun
19
comment Using dimensional analysis to evaluate $\frac{d}{dx}x^n$
"how can one use physical units to argue anything about the behaviour of the mathematical functions?" Physical units are no less mathematical than dimensionless numbers. What else could they be? They are quantities with formal rules for manipulation just like the rest of mathematics. With formal rules come theorems. See here: terrytao.wordpress.com/2012/12/29/… Tao is using dimensions to reason about pure mathematics.
Jun
19
comment Using dimensional analysis to evaluate $\frac{d}{dx}x^n$
No dimensional analyst would claim that your expression is constant because it has no dimension. There is a dimensioned constant in your expression, $c$. If you account for your dimensioned constants correctly dimensional analysis isn't just a heuristic.
Jun
17
comment What application is there for a non-Hausdorff topological space?
@MJD We can't easily put our intuitions on display and see where they differ so I think we're going to have to agree to differ on this one.
Jun
17
comment What application is there for a non-Hausdorff topological space?
Great answer. But..."The point of abstract topological spaces is not that they are intuitive" I beg to differ. There is very good intuition for these things. Topologies can be seen as characterising sets of states of a system compatible with a measurement. I explain a bit about it here: mathoverflow.net/a/19156/1233 IMO, in the very place where you say "There is no intuition for that" you've just made a great start on explaining the intuition :-)
Jun
17
comment How big is infinity?
Great answer. Arguably Euclid's proof of the infinitude of the primes doesn't need anything to do with cardinalities or collections of integers. Saying that there are infinitely many primes is shorthand for saying that no matter what bound you pick there's always a bigger prime.
Jun
17
comment What languages to learn for maths?
See also: mathoverflow.net/questions/8056/…
Jun
17
comment Tuple and vector. What do I have here?
There's absolutely no problem referring to your tuple as a vector as long as you're clear and unambiguous about what it is. The word 'vector' is already used in many ways in mathematics and there is no one single canonical definition. Your proposed definition is close enough to some existing uses that it's a natural choice. But your use of the word is unlikely to confuse anyone.
Jun
17
revised Why can't we define more elementary functions?
added 10 characters in body
Jun
16
comment What is the value of $c$ in this problem of sequence and series?
Any problem that requires solving equations could be written as "what is the value of c?" so your title isn't going to attract people who have specific interest in the area you're asking about. I thought it was going to be a question about the speed of light. So how about mentioning arithmetic and geometric progressions in your title?
Jun
16
answered Why can't we define more elementary functions?
Jun
13
answered Why are the limits of the integral $0$ and $s$, and not $-\infty$ and $+\infty$??
Jun
12
revised Number of digits of the number of digits of the number of digits of $2014^{2014}$
Number of "numbers" is non-standard
Jun
12
suggested suggested edit on Number of digits of the number of digits of the number of digits of $2014^{2014}$
Jun
12
answered Intuitively, why does $\int_{-\infty}^{\infty}\sin(x)dx$ diverge?