Toby Bartels
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1d |
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Is there a convenient differential calculus for cojets? Thanks, that looks promising! |
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Jun 10 |
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Explanations for mathematicians, about the falsifiability (or not) of string theory I view string theory as a research programme rather than as a theory of physics as such. To the extent that it is falsifiable, it has been falsified (see Carlo's comment above), and this is damning to a theory of physics. But as a research programme into areas that are very difficult to test experimentally, I cannot fault it. It's a promising approach, and the lack of observed supersymmetry and extra dimensions so far doesn't prove much, since we can't go to high enough energies. The only problem, in my mind, is premature confidence in it. It's an interesting idea, not the last word. |
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Jun 4 |
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Is every representable map a submersion? I have seen the term "carrable" for what is here called "representable", in Paul Taylor's work. I have no idea where that comes from. At least it is unique! |
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May 18 |
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Is there any proof that you feel you do not “understand”? I learnt the epsilon-delta definition of limits in 1990 in the regular calculus class at a regular high school in Nebraska. (I can't recall if we learnt the definition of the Riemann integral or did all of the proofs, but these were in the book.) I didn't learn that this was unusual until years later. |
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May 18 |
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Is there any proof that you feel you do not “understand”? Another way to see the uses of choice in the theorem is to use nonstandard analysis. This works just like ultrafilters, except that you talk about a hyperpoint in the infinitesimal neighbourhood of a standard point p instead of an ultrafilter that converges to p. Like the ultrafilter approach to topology, nonstandard analysis simply doesn't work without the ultrafilter lemma, so that's enough for the Hausdorff case; but the non-Hausdorff case requires an arbitrary choice of p in each space, just as with ultrafilters. |
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May 18 |
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Is there any proof that you feel you do not “understand”? I had learnt the ‘first’ and ‘second’ fundamental theorems under the opposite names, and I'd almost forgotten that I once read a calculus textbook that had them switched, until I read your post now that also has them switched (switched compared to how I learnt them, I mean). I wonder how that came about? |
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May 18 |
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Is there any proof that you feel you do not “understand”? Thanks! I never understood the Sylow theorems either (except for purposes of exams, of course), but now that you've explained how to look at them, I think that I'll try again. |
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May 18 |
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Is there any proof that you feel you do not “understand”? To prove the FTC, you first need a definition of the integral, and some textbooks don't even have that! |
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May 18 |
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Is there any proof that you feel you do not “understand”? You don't need the ultrafilter lemma to set up the basic theory of topology via nets or filters, but you need it to set up the basic theory of topology via universal nets or ultrafilters. And the proof is much cleaner using universal nets or ultrafilters, since you merely need to show that each of either converges. (If instead you try to show that each net has a convergent subnet or that each proper filter has a convergent proper refinement, then you find yourself needing choice again.) And yes, convergence of all nets and proper filters (if convergent at all) is unique in a Hausdorff space |
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May 18 |
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Is there any proof that you feel you do not “understand”? The dot-product proof relies on the equivalence between the geometric meaning of the dot product (which is obviously rotation invariant) and the algebraic formula for it (which gives Dustin's result). But proving this equivalence, which relies on proving that the geometric dot product is distributive, is as complicated as proving the Pythagorean Theorem directly: upload.wikimedia.org/wikipedia/commons/thumb/a/aa/… |
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May 9 |
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Category of categories as a foundation of mathematics Andrew's link again doesn't work; use this one: ncatlab.org/nlab/show/… In general, if you wish to link to a section within an nLab page, you should first (if it hasn't already been done) edit the page to give the section a permanent title. Otherwise, your link will stop working when the page is edited to add, delete, or rearrange sections. |
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May 1 |
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Counterintuitive consequences of the Axiom of Determinacy? This really highlights Douglas Zare's point that what's intuitive depends on whether AC is intuitive. I find (1) and (2) quite intuitive, and I gather that many more people did around a hundred years ago. (3) also seems reasonable, although I could imagine its going either way. ((4) and (5) are a little outside where I feel good intuitions.) |
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Apr 23 |
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When is the class of functions between sets a set? Just for the record, in reply to one of David's comments above, there are set theories in which the class of functions between two given sets is not a set; ZF, ETCS, and SEAR all have strongly predicative versions, in which the power set axiom is removed and (if replaced at all) replaced by something too weak to give function sets. (Of course, if we have classical logic, then function sets are equivalent to power sets, so even removing power set and replacing it with function sets is an option only when using intutitionistic logic.) |
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Apr 17 |
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Was the early calculus inconsistent? Yes, the correct chain rule for second derivatives is $d^2y/dx^2 = (d^2y/du^2)(du/dx)^2 + (dy/du)(d^2u/dx^2)$. But unlike the chain rule for first derivatives, you can't derive this by treating $d^2y$ and the rest as if they were elements of an infinitesimal-enriched continuum obeying the ordinary rules of algebra. At the very least, this is annoying (and I have found it so since since high school); but more than that, it suggests that the $dx^2$ that appears in $d^2y/dx^2$ is not really the square of $dx$ in such a continuum. |
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Apr 15 |
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Was the early calculus inconsistent? @Vladimir: This does not contradict your point (or indeed any of the text of your latest comment), but one must be careful with $d^2y$, $dx^2$, and the like. Second derivatives in Leibniz's notation don't work as well as first derivatives, because the chain rule $d^2y/dx^2 = (d^2y/du^2) (du/dx)^2$ is false. I would prefer to write $\partial^2y/\partial{x}^2$ myself, on the grounds this is the coefficient on $dx^2$ in an expansion of $d^2y$ (in $d^2x$ and $dx^2$), analogous to the coefficients that are partial derivatives, rather than the ratio of $d^2y$ to $dx^2$. |
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Apr 7 |
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Was the early calculus inconsistent? OK, if I write anything up carefully, I'll put it at ncatlab.org/nlab/show/Cauchy+sum+theorem (or at least somewhere easily available from there) and put a note here in case anybody else is still following along later, but otherwise I will keep it to private email. |
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Apr 4 |
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Was the early calculus inconsistent? And I still think that (6.2) is missing the hypothesis that $n$ is infinite; it must be implicit. The way that it's phrased, it really makes it look like $r_n(x)$ is only required to be infinitesimal when $x$ is infinitesimal, which of course is not the case; but it is the case that $r_n(x)$ is only required to be infinitesimal when $n$ is infinite. So at best, it is confusingly written, although now that you've explained why you wrote it that way, I can understand it. (But earlier I really thought that you had meant to write ‘$n$ infinite’ and had written ‘$x$ infinitesimal’ by mistake.) |
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Apr 4 |
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Was the early calculus inconsistent? You offered that formula (apparently on page 57 of the Springer version) as an explanation of the ‘straightforward formalisation’ in Robinson's nonstandard analysis of Cauchy's $x = 1/n$ counterexample to the claim that $\sum_{k=1}^n \frac{sin(k x)}{k}$ converges everywhere, which I had said was difficult to do. And although I put forth a possible formalisation in a later comment, I don't think that it's straightforward, and (6.2) doesn't help. |
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Apr 3 |
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Was the early calculus inconsistent? Sure, I agree with that. But I still don't understand why you write (6.2) in that way. (And it still needs the clause ‘for $n$ infinite’, otherwise there is no reason why $r_n(x)$ should ever be infinitesimal.) And it doesn't help me understand how $1/n$ makes sense as a possible value of $x$ in a context where $n$ has yet to appear. |
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Apr 3 |
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Was the early calculus inconsistent? It makes no difference whether one interprets $\forall x$ as meaning for every standard real number $x$ or for every hyperreal number $x$, or whatever. Cauchy is not saying anything of the form $\forall x, \Phi(x)$ in the first place. He is saying something more like $\forall n, \forall x, \Psi(x,n)$, where $\Psi(x,n)$ states my correction to (6.2): ‘if $n$ infinite then $r_n(x)$ infinitesimal’. I'm not sure that even this correctly interprets what he meant, but at least it allows one to meaningfully substitute $1/n$ for $x$. |
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Apr 3 |
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Was the early calculus inconsistent? To my mind, the real problem with understanding what Cauchy means is that he talks about $f_n(x)$ converging to $f(x)$ ‘toujours’, and we want to interpret this as $\forall x, \Phi(x)$, where $n$ appears only as a bound variable in $\Phi(x)$ (which states that $f_n(x)$ converges in $n$ to $f(x)$). To show explicitly that this is false, one proves $\exists x, \neg\Phi(x)$ by finding a specific $\xi$ and proving $\neg\Phi(\xi)$. In this context, it makes no sense to use $1/n$ as $\xi$, since $n$ has no meaning outside of the formula $\Phi(x)$, yet the quantifier is outside. |
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Apr 3 |
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Was the early calculus inconsistent? Yes, I don't really understand (6.2). If this is supposed to state that $f_n$ converges (in $n$) to $f$ ‘everywhere’, then this should read ‘if $n$ infinite then $r_n(x)$ infinitesimal’ (where $r_n(x)$ was just defined as $f(x) - f_n(x)$), rather than ‘if $x$ infinitesimal then $r_n(x)$ infinitesimal’ as you wrote. Choosing $x$ to be the infinitesimal $1/n$, of course, is the example that shows that $\sum_{k=1}^n \frac{\sin(k x)}{k}$ fails to converge ‘everywhere’, but whether $x$ is infinitesimal doesn't go into the definition of convergence itself. |
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Apr 2 |
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Was the early calculus inconsistent? Technicality: Say that $r$ is continuous in $h$ when $h = 0$. (To require $r$ to be jointly continuous when $h = 0$ makes $f$ continuously differentiable, not just differentiable.) |
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Apr 2 |
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Was the early calculus inconsistent? Todd: As far as I know, no. But one should ask him. |
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Apr 2 |
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Was the early calculus inconsistent? Last comment for Dr. Katz: Can you tell me where page 57 in the Springer book corresponds to in the arXiv version? |
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Apr 2 |
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Was the early calculus inconsistent? Although I can vaguely see now how an interpretation in nonstandard analysis might go. (Pick a single infinite integer H, then use both n = H and x = 1/H. Then the sum of sin(kx)/k, as k goes from 1 to n, is not always infinitely close to 0.) I'm not sure that this is a fair interpretation either, but at least it is an interpretation. |
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Apr 2 |
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Was the early calculus inconsistent? @katz: I've read and liked your Ten Misconceptions […] (which I cited at ncatlab.org/nlab/show/epsilontic+analysis too), but it didn't show me how to me interpret that example in nonstandard analysis. (I actually expected you to agree that it doesn't work in nonstandard or epsilontic analysis, that either interpretation is ahistorical.) |
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Apr 2 |
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Was the early calculus inconsistent? Robinson (whom Lakatos also read) interpreted Cauchy's infinitesimals as his infinitesimals (quite reasonably), how does one interpret x = 1/_n_ precisely? (keeping in mind that this is the same n as appears in the series). |
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Mar 23 |
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Why do I need densities in order to integrate on a non-orientable manifold? A great source for intuition and physical meaning of pseudoforms is Theodore Frankel's _Geometry of Physics_: books.google.com/books/about/… (Google Books has somehow got the wrong title). |
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Mar 23 |
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The “ds” which appears in an integral with respect to arclength is not a 1-form. What is it? This is also not a good answer, since we want to know what $ds$ is on the ambient space, not just what it is on the curve. |
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Feb 13 |
accepted | What is an ideal-supporting algebra? |
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Feb 12 |
revised |
What is an ideal-supporting algebra? typo |
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Feb 12 |
answered | What is an ideal-supporting algebra? |
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Jan 22 |
awarded | ● Necromancer |
