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seen Oct 19 at 4:34

Jun
25
answered Complex analysis is more “real” than real analysis
Jun
24
comment What should I learn first, Mathematica or MatLab?
Want to define a function? Stick an underscore after the argument names. Want to apply a function to some arguments? Use square brackets. Want to get the $i$th element of an array, use double square brackets. While one might quibble about what is or isn't natural, saying my claim that Mathematica notation "can look very unnatural" is outrageous is, well, interesting.
Jun
24
comment How to establish this inequality without using induction?
Seeing as you defined the $a_n$ by induction it's going to be completely impossible to prove any non-trivial property of them without using induction in some form or other. Maybe you're trying to rule out one particular induction proof.
Jun
22
comment logic\math question
If you look at the following link you'll see a number of examples that fit the pattern, some of which suggest answers other than 90: oeis.org/…
Jun
20
comment Writing a chain of implications in English
@porton I see your point and I like your emphasis on clarity. But an expert mathematician will realize that an empty set of propositions contains none that fail to follow. And a non-expert mathematician won't even notice that there is an issue. So I'm not sure who, in reality, would have a problem.
Jun
20
comment Writing a chain of implications in English
It seems clear to me. Maybe it's worth saying "Each of these four assertions..." just in case readers are tempted to include the following paragraphs.
Jun
20
comment Writing a chain of implications in English
Each of these assertions implies the following ones.
Jun
20
answered Neighbourhood of a matrix
Jun
20
comment Using dimensional analysis to evaluate $\frac{d}{dx}x^n$
You don't need to prove everything from scratch. When students first compute derivatives of monomials they often do the product rule first so they know monomials will have derivatives before they compute them. As for not appealing to heuristics, my entire PhD was based on using heuristics from physics to prove mathematical results, once they were appropriately formalised. Dimensional analysis is completely formalisable in mathematics.
Jun
20
comment Category-theoretic description of the real numbers
Also maths.mq.edu.au/~street/EffR.pdf
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 :-)