User katz - MathOverflow

Intuitionistic logic as quantization of classical logic?

2013-04-23T13:12:43Z

A classically trained mathematician is more likely to be familiar (at least anecdotally) with an area of mathematical physics such as deformation quantization than with Intuitionistic logic. It is helpful in any sense (philosophical or mathematical) to think of Intuitionistic logic as a quantization of classical logic? Has anyone explored such an approach? Note that the idea itself of comparing the passage from classical to intuitionistic logic to denying commutativity is not new; see Richman's "Interview with a constructive mathematician" at http://www.ams.org/mathscinet-getitem?mr=1400617

Answer by katz for Leibnizian calculus textbook

2013-05-10T07:31:37Z

If what you are looking for is "multivariable calculus with infinitesimals", you might be interested in Stroyan's book at http://library.wolfram.com/infocenter/Books/6877/ The table of contents is here: http://homepage.math.uiowa.edu/~stroyan/MultiCalc/iMultiCalcTOC/iMultiCalcIntro.html

Answer by katz for Leibnizian calculus textbook

2013-04-18T09:24:05Z

Does Vakil's book "real analysis through modern infinitesimals" http://www.google.co.il/books?id=hyFjtJ3Wq24C&source=gbs_navlinks_s contain the additional material you are looking for?

Answer by katz for In what ways did Leibniz's philosophy foresee modern mathematics?

2013-04-18T08:03:58Z

Daniel Geisler speculates that "because of Leibniz's philosophical reflections, he foresaw aspects or parts of modern mathematics" and asks: "Can anyone elaborate on these connections and recommend any references?"

Several responders mentioned the connection to Robinson's theory. On the other hand, François Brunault rightly cautioned: "The statement that someone (even Leibniz) foresaw parts of modern mathematics is potentially controversial because of its subjectivity. I think most historians of mathematics now insist on the fact that the works by earlier mathematicians should also be studied from the point of view of that time, before extrapolating possible connections."

François Brunault is correct in suggesting that there is resistance among historians of mathematics to the idea of seeing continuity between Leibniz and Robinson. Indeed, the prevalent interpretation of Leibnizian infinitesimals is a so-called syncategorematic interpretation, pursued notably by R. Arthur and many other Leibniz scholars. On this view, Leibnizian infinitesimals are merely shorthand for ordinary ("real") values, assorted with a (hidden) quantifier, viewed as a kind of a pre-Weierstrassian anticipation. These scholars rely on evidence drawn from various quotes from Leibniz where he refers to infinitesimals as "useful fictions", and explains that arguments involving infinitesimals can be paraphrased a l'ancienne using exhaustion. In this spirit, they interpret the Leibnizian "useful fictions" as LOGICAL fictions, denoting what would be described in modern terminology is a quantified formula in first-order logic.

For example, Levey writes:

"The syncategorematic analysis of the infinitely small is ... fashioned around the order of quantifiers so that only finite quantities figure as values for the variables. Thus,

(3) the difference $|a-b|$ is infinitesimal

does not assert that there is an infinitely small positive value which measures the difference between~$a$ and~$b$. Instead it reports,

($3^*$) For every finite positive value $\varepsilon$, the difference $|a-b|$ is less than $\varepsilon$.

Elaborating this sort of analysis carefully allows one to express the now-usual epsilon-delta style definitions, etc."

This comment appears in the article

Levey, S. (2008): Archimedes, Infinitesimals and the Law of Continuity: On Leibniz's Fictionalism. In Goldenbaum et al., pp.~107--134. The book is

Goldenbaum U.; Jesseph D. (Eds.): Infinitesimal Differences: Controversies between Leibniz and his Contemporaries. Berlin-New York: Walter de Gruyter, 2008, see http://www.google.co.il/books?id=tWWuQ9PHCusC&q=

I personally find it hard to believe Levey is talking about Leibniz, but there you have it. Whether or not Levey's analysis stems from a "Desire To Preserve The Orthodoxy of Epsilontics Against The Heresy of Infinitesimals", as Yemon likes to put it, is anybody's guess.

What the "syncategorematic" view tends to overlook is the presence of DUAL methodologies in Leibniz: both an Archimedean one, and one involving genuine "fictional" infinitesimals. On this view, Leibnizian infinitesimals are PURE fictions (rather than logical ones). Such a reading is akin to Robinson's formalist view, and sees continuity not merely between Leibniz's and Robinson's mathematics, but also their philosophy. This view is elaborated in a text entitled "Infinitesimals, imaginaries, ideals, and fictions" by David Sherry and myself, to appear in Studia Leibnitiana, and accessible at http://arxiv.org/abs/1304.2137

Answer by katz for Curvature of contour lines of a scalar field

2013-04-21T13:55:24Z

The formula for computing the curvature of a curve defined by an implicit equation can be found in my notes at http://u.math.biu.ac.il/~katzmik/egreglong.pdf on page 32. It is closely related to the Reiss relation in algebraic geometry. See references there.

Answer by katz for Why can an infinite-order polynomial be written as the ratio of two finite-order polynomials?

2013-04-18T16:20:38Z

I am not sure how much background you have in analysis but it may be helpful to keep in mind that "the assumption about the roots outside the unit circle" is required not "for the infinite order polynomial to have this ratio representation", which seems to be a separate assumption, but rather for the series to converge for all points in the unit disk.

Is there a lower bound for variance in terms of curvature?

2013-04-15T07:33:39Z

If the Gaussian curvature of the metric $g= f^2(x,y)(dx^2+dy^2)$ is nonzero then $f$ cannot be constant. This can be expressed by stating that the (probabilistic) variance $Var(f)$ of $f$ is nonzero (in a suitable domain). If the metric $g$ has nonzero Gaussian curvature, then $f$ is nonconstant and therefore $Var(f)$ is nonzero. Can this conclusion be quantified? Namely, assume $g$ has curvature bounded away from zero on a suitable disk. Can one get a lower bound for $Var(f)$? This has immediate applications to a stronger version of Loewner's torus inequality with isosystolic defect term a la Bonnesen, see http://arxiv.org/abs/arXiv:0803.0690 and http://arxiv.org/abs/1105.0553

Note 1. The Gaussian curvature of the metric $g= f^2(x,y)(dx^2+dy^2)$ is given by $K=\frac{-1}{2f^2}\left(\frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2}\right)\log f$, so that the problem involves partial differential inequalities for the Laplacian.

Note 2. To restate the curvature hypothesis more carefully: we have a metric $r$-disk (for the metric $g$) where the curvature is bounded below by an $\epsilon>0$. The problem is to obtain a lower bound for $Var(f)$ in terms of $r$ and $\epsilon$ (certainly the estimate will become weaker as $r$ gets smaller).

Note 3 (reformulation along the lines of Robert's suggestion). On a fixed domain $\Omega$ of unit area in the $xy$-plane, we consider metrics $g=f^2(dx^2+dy^2)$ with the property that $\Omega$ contains a subdomain (say, a disk) $D$ on which Gaussian curvature $K\geq C>0$ and such that the $g$-area of $D$, i.e., $\int_D f^2 dxdy$, is at least $A>0$. We want to know whether there is a lower bound for $Var_\Omega(f)$ in terms of the constants $C$ and $A$. Is there an optimal lower bound, and if so does a rotationally symmetric metric on a disk $D$ attain it? A similar question for $K$ negative and bounded away from $0$.

Answer by katz for $f^3,f^2$ are the cube and quadratic of f respectively and both infinite differentiable on $R$,how to show so is $f$

2013-04-16T14:59:54Z

If $f^2$ has a zero $p$ of finite order, then $f^3$ also has finite order at $p$, and vice versa. If $f^2$ has a zero of infinite order at $p$, then its square root $f$ does, as well, and hence $f$ is smooth at $p$. Otherwise, both $f^2$ and $f^3$ have finite orders, respectively $a$ and $b$, at $p$, and therefore $f=f^3/f^2$ is smooth of finite order $b-a$ at $p$. Therefore $f$ is always smooth.

Answer by katz for geometric interpretation of Lie bracket

2013-04-17T08:19:06Z

It is not entirely correct to assert that "the Lie bracket measures the change in Y along integral curves of X", at least not in the context of Riemannian geometry which is of course Spivak's context. Note that the same phrase could be applied to $\nabla_X Y$ as well, so at best the phrase is ambiguous. Thanks to Peter for the interesting reference which I hope to study further. Note that interpretations of the Lie bracket in terms of actual infinitesimals have been worked out in various contexts, so that the 4-step procedure becomes literally correct without taking limits.

Note also that you can push forward the standard coordinate fields in the plane by an arbitrary diffeomorphism and obtain random-looking vector fields that Lie-commute by construction. From the Riemannian viewpoint, it is odd to insist that one of them "does not change" along the other.

Lapses of "the early proponents of the doctrine of limits"

2013-04-03T16:43:14Z

I have a question that I have been wondering about for a long time without finding any answer. Concerning the period around 1900, Robinson commented in his 1966 book that "there is in the writings of this period an noticable contrast between the severity with which the ideas of Leibniz and his successors are treated and the leniency accorded to the lapses of the early proponents of the doctrine of limits. We do not propose here to subject any of these works to a detailed criticism."

I have always found the last sentence a bit disappointing. Robinson could have indeed subjected the "lapses" of these early proponents to a detailed criticism. The further we are removed from them in time, the harder it becomes to carry out such a criticism. Is anyone aware of such "lapses" and/or discussion thereof in the literature?

Note 1. As per request by Joël, here is a larger quote, borrowed from Stroyan's site http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/Lecture1/HTMLLinks/Lect1_7.html "The history of a subject is usually written in the light of later developments. For over half a century now, accounts of the history of the Differential and Integral Calculus have been based on the belief that even though the idea of a number system containing infinitely small and infinitely large elements might be consistent, it is useless for the development of Mathematical Analysis. In consequence, there is in the writings of this period an noticable contrast between the severity with which the ideas of Leibniz and his successors are treated and the leniency accorded to the lapses of the early proponents of the doctrine of limits. We do not propose here to subject any of these works to a detailed criticism. However, it will serve as a starting point for our discussion to try to give a fair summary of the contemporary impression of the history of the Calculus..." I don't have Robinson's book in front of me right now, so can't provide a lengthier quote.

Answer by katz for Does some type of curvature require the space be an embedded manifold in a higher-dimensional space?

2013-04-16T13:19:43Z

Here is an attempted answer-cum-interpretation, modulo adjusting some dimensions: A closed surface of negative Gaussian curvature cannot be isometrically embedded in $\mathbb{R}^3$. So in this case, at least $n+2$ dimensions are needed (where $n=2$).

Does the derivative of log have a Dirac delta term?

2013-04-15T08:31:48Z

Dirac writes down the following formula on page 61 of his "Principles of quantum mechanics": $\frac{d}{dx}\log x = \frac{1}{x} -i\pi\delta(x)$, see http://adsabs.harvard.edu/abs/1947pqm..book.....D for the exact reference (but no text). What is the best way of formalizing this to a mathematician's satisfaction?

Answer by katz for Geometry of Hopf fibrations and the fibration of Steifel Manfiolds over Grassmannians

2013-04-15T15:18:22Z

Apriori you don't get a Riemannian submersion because the corresponding subsets of the unit spheres in the $k$-planes will not be "constant distance apart" as they are in the symmetric rank 1 case.

Answer by katz for injectivity radius of hyperbolic surface

2013-04-15T13:44:01Z

I didn't find the word "congruence subgroup" on this page, so I thought I would comment that using congruence subgroups may be more elementary than appealing to residual finiteness of 3-manifolds.

Answer by katz for Minimize diameter of a tree

2013-04-15T09:17:40Z

You may not be able to decrease the diameter at all with a single cut, for example for the Y-shaped tree (Mercedes sign-shaped tree). At any rate in order to decrease the diameter you will have to deal with the midpoint of the tree (midpoint of the longest imbedded path). It seems the number of edges you will have to remove depends on the valence at the midpoint (or adjacent vertices).

Answer by katz for Lapses of "the early proponents of the doctrine of limits"

2013-04-15T07:19:07Z

This is mainly a comment on the discussion following the original post. Leibniz did not use the $\Sigma^\infty$ notation. If you replace $\infty$ by an infinite hyperinteger, and interpret the sum as a hyperfinite sum, the calculation becomes correct. Not only would one "dare" send such a proof to a modern referee, but in fact the literature is full of such routine calculations. Similarly, the other sums, notably in Euler, have interpretations in terms of averaging theory that make them correct, as detailed by Laugwitz, Kanovei, and others. To summarize, Leibniz's calculation only seems "wrong" from a post-Weierstrassian viewpoint where "infinite numbers" are a no-no, and illustrates well some of the misconceptions that were dealt with in our articles "Ten misconceptions from the history of analysis" (http://arxiv.org/abs/1202.4153), "Is mathematical history written by the victors?" (http://u.cs.biu.ac.il/~katzmik/bairetal.html), and a number of other recent publications, see http://u.cs.biu.ac.il/~katzmik/infinitesimals.html

Answer by katz for Extensions of Carathéodory's theorem

2013-04-14T14:00:05Z

The statement of the theorem only uses the linear structure of the ambient space, so the Banach space structure does not affect its validity.

One way of proving the theorem is by applying Helly's theorem. The latter seems to be more readily amenable to generalisations because of obvious connections to topology.

Answer by katz for Surfaces filled densely by a geodesic

2013-04-14T13:49:58Z

Clairaut's relation shows that any simply connected surface of revolution has this property, whether convex or not (for the reason stated by Robert).

Answer by katz for On closed simple curve with curvature at most 1

2013-04-14T07:50:06Z

I would check work from the 1980s on the cut locus being a tree on surfaces. Consider the inward normal exponential map of the simple curve, take its cut locus, the latter is a tree, take a leaf of the tree. This point will be at least distance 1 away from the boundary if the (signed) curvature of the curve is less than 1. This works for all curves (convex or not).

For the cut locus, see for example Itoh, Jin-ichi, Essential cut locus on a surface. Proceedings of the Fifth Pacific Rim Geometry Conference (Sendai, 2000), 53–59, Tohoku Math. Publ., 20, Tohoku Univ., Sendai, 2001.

See also Zamfirescu, Tudor, On the critical points of a Riemannian surface. Adv. Geom. 6 (2006), no. 4, 493–500.

Answer by katz for Was Desargues more an Euclid or an Eudoxos?

2013-04-12T08:47:10Z

Desargues certainly pioneered original mathematics. The notion of a point at infinity in projective geometry is usually attributed to him. Kepler apparently did not work in projective geometry but rather in astronomy and pioneered a number of mathematical techniques such as infinitesimals. I am not aware of any interactions between Desargues and Kepler, but Desargues did play an interesting role of attempting to resolve a dispute between his junior colleagues Fermat and Descartes.

I now see that wiki attributes the notion of the point at infinity to Kepler, citing Coxeter. This seems like a novelty to me. Kepler did talk about points at infinity, but not in the context of projective geometry as we understand it, but rather as a way of developing a unified technique for treating conic sections through a kind of a continuity principle. This is closer to calculus than projective geometry.

Answer by katz for Nonstandard analysis in probability theory

2013-04-12T08:12:08Z

The answers given earlier are excellent. I would merely like to supplement them by the observation that the success of NSA and IST in probability and related fields is attested to by the fact that new books continue to be in demand and are being published in this area, in some of the most prestigious series, such as the 2013 book by F. Herzberg entitled "Stochastic Calculus with Infinitesimals", see http://link.springer.com/book/10.1007/978-3-642-33149-7/page/1

Answer by katz for About Sectional Curvature

2013-04-12T08:01:25Z

To supplement the answer given at SE, I would mention that the definition via the Taylor expansion of the distance function is really a slick shortcut rather than a definition. It relies on the fact that the Gaussian curvature at $x$ of the surface obtained as the image of the plane spanned by $v$ and $w$ is the same as the sectional curvature in this 2-dimensional direction. So to convince yourself of the formula, you just need to understand it for surfaces. Trying to relate it to the general formula in terms of the curvature tensor is a red herring.

Answer by katz for Non-ZFC set theory and nonuniqueness of the hyperreals: problem solved?

2013-04-11T12:02:18Z

The usefulness of the hyperreals stems from such tools as saturation and the transfer principle. These tools are available in other superfields R' of R only to the extent that one can construct morphisms between the hyperreals and R'. Even if some limit ultrapower model of $\mathbb R^*$ is maximal and unique in some weak sense, this does not really affect its applications, certainly not in physics where the most likely model to be used is the simplest one, namely $\mathbb R^{\mathbb N}$ modulo an ultrafilter on $\mathbb N$. The precise relationship to the surreals is being discussed at http://mathoverflow.net/questions/127041/surreals-and-nsa-some-foundational-issues but ultimately this discussion has little bearing on whether "Robinson's program has been completed successfully" as you put it.

Answer by katz for Nonstandard definition for the generator of a standard Ito diffusion

2013-04-11T11:14:57Z

try F. Herzberg at http://link.springer.com/chapter/10.1007/978-3-642-33149-7_7

Answer by katz for Is there an observer dependent mathematics?

2013-04-11T10:29:35Z

Perhaps this idea can be used to motivate the phenomenon of "local triviality, global nontriviality". Here the role of "observer" is played by a (local) coordinate chart, where, say, the bundle over a manifold looks trivial, whereas global nontriviality transcends each individual observer. That's why you can't comb the sphere.

Answer by katz for The Origin of the Musical Isomorphisms

2013-04-11T09:39:57Z

The musical isomorphisms already appear in 1971 on page 21 of BERGER M. et al, Le spectre d'une variete riemannienne, Lecture Notes in Math. 194, 1971, Springer, see http://ci.nii.ac.jp/naid/10003477917/

However, I am not convinced that this is the first place where they appear, or that Berger was the inventor. One would have to trace German textbooks in Riemannian geometry from the 1960s or perhaps earlier. Interesting question!

Answer by katz for How helpful is non-standard analysis?

2013-04-11T08:19:12Z

I just came across a 2013 book by F. Herzberg entitled "Stochastic Calculus with Infinitesimals", see http://link.springer.com/book/10.1007/978-3-642-33149-7/page/1 where probability and stochastic analysis are done without having to develop the complexities of measure and integration theory first. Ever since E.Nelson, such an approach is called "radically elementary" and it really is. What this proves is the new result that stochastic calculus can be done without measure theory.

To give a historical parallel, recall that Leibniz's mentor in mathematics was Huygens. When Huygens first learned of Leibniz's invention of infinitesimal calculus, Huygens was sceptical, and wrote to Leibniz that he is merely doing what Fermat and others have done before him in a different language. What Huygens failed to recognize immediately (but did recognize later) was the generality of the methods and the lucidity of the presentation of Leibniz's new approach. The Nelson-Herzberg approach to stochastic calculus is in a way more significant than merely a new "result", since it provides a new methodology.

Euler's mathematics in terms of modern theories?

2013-04-09T14:43:14Z

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in operation in their writings a conception of mathematics which is quite extraneous to that of Euler." Ferraro concludes that "the attempt to specify Euler's notions by applying modern concepts is only possible if elements are used which are essentially alien to them, and thus Eulerian mathematics is transformed into something wholly different"; see http://dx.doi.org/10.1016/S0315-0860(03)00030-2.

Meanwhile, P. Reeder writes: "I aim to reformulate a pair of proofs from [Euler's] "Introductio" using concepts and techniques from Abraham Robinson's celebrated non-standard analysis (NSA). I will specifically examine Euler's proof of the Euler formula and his proof of the divergence of the harmonic series. Both of these results have been proved in subsequent centuries using epsilontic (standard epsilon-delta) arguments. The epsilontic arguments differ significantly from Euler's original proofs." Reeder concludes that "NSA possesses the tools to provide appropriate proxies of the inferential moves found in the Introductio"; see http://philosophy.nd.edu/assets/81379/mwpmw_13.summaries.pdf (page 6).

Historians and philosophers thus appear to disagree sharply as to the relevance of modern theories to Euler's mathematics. Can one meaningfully reformulate Euler's infinitesimal mathematics in terms of modern theories?

Note 1. I just noticed a related thread at http://mathoverflow.net/questions/24396/would-eulers-proofs-get-published-in-a-modern-math-journal-especially-consideri

Answer by katz for Would Euler's proofs get published in a modern math Journal, especially considering his treatment of the Infinite?

2013-04-10T14:01:55Z

In answering this question, it is helpful to make a distinction between, on the one hand, what Reeder calls the "inferential moves" that Euler makes (see related thread http://mathoverflow.net/questions/126986/eulers-mathematics-in-terms-of-modern-theories), and on the other, the mathematical objects he manipulates (infinitesimals, infinite integers, etc). This allows Reeder to observe that modern infinitesimal theories are far more successful in formalizing Euler's procedures ("inferential moves") than are $\epsilon,\delta$ techniques.

Traditional scholars like Ferraro (see thread linked above) were trained on the basis of conceptual frameworks that are inadequate to the task of making such an evaluation, and tend to receive the work of scholars like Laugwitz with hostility.

Laugwitz argued for an essential coherence of infinitesimal reasoning in both Cauchy and Euler, modulo certain "hidden lemmas" that need to be made explicit to meet a modern standard. I would adopt an optimistic position that many of Euler's greatest contributions are immediately publishable in contemporary journals, provided minimal changes are made so as to clarify the nature of the objects as well as the "hidden lemmas".

The verdict is still out on whether the MATHEMATICAL community (as opposed to that of the HISTORIANS of mathematics) will in the end side with Reeder's analysis or Ferraro's analysis.

Answer by katz for Definable collections without definable members (in ZF)

2013-04-09T16:17:30Z

The question admits an answer that may be somewhat more "concrete" in the sense of having "applications", and seems somewhat relevant (correct me if I am wrong). Namely, Kanovei and Shelah have constructed definable models of the hyperreals where each of the nonstandard elements is actually undefinable. Thus the set of nonstandard elements is definable, but none of its members are.

Comment by katz

2013-05-13T13:21:45Z

Frankel and I constructed metrics on a disk such that curve length grows very rapidly and the "obvious" curves in fact get very long before much of the area is swept out. Liokumovich et al. recently published some generalisations of this. It is possible that even the weaker version of your question admits of counterexamples.

Comment by katz

2013-05-13T07:53:28Z

Just to make sure I understand what you are looking for: it seems plausible that by applying coarea to a suitable distance function one might be able to get a non-optimal bound similar to the one you asked for. Are you interested in the optimal value $2\pi$ for boundary length, or does the coarea argument get stuck on diameter issues?

Comment by katz

2013-05-12T13:46:24Z

Can elaborate why you formulated the question in terms of "area between pi and 2pi" rather than in terms of the Cheeger constant, and how this might affect the answer?

Comment by katz

2013-04-28T11:45:50Z

Thanks for a very interesting answer. Is there a home for this "Antonio Montalbán and me showed"?

Comment by katz

2013-04-24T12:19:49Z

I am not too familiar with this material and am curious to find out more. Can you summarize why one might think of a Galois connection more in terms of intuitionistic logic as a quantisation of classical logic than vice versa, for example? The wiki page is written in such general terms that it is hard to tell how something like this could be applied.

Comment by katz

2013-04-23T17:56:04Z

Very interesting. I looked at the wiki article on "Galois connection" (incidentally, you could fix the link). Is this notion related to "equivalence of categories"? What would be a concrete example to illustrate the power of this notion as a means of clarifying the theory?

Comment by katz

2013-04-23T15:25:13Z

I note that in your "Practical Foundations" you use a similar analogy: "In classical logic, as in classical physics, particles enact a logical script, but neither they nor the stage on which they perform are permanently altered by the experience. In the modern view, matter and its activity are created together, and are interchangeable (the observer also affects the experiment by the strength of the meta-logic)."

Comment by katz

2013-04-23T15:04:45Z

Here is a relevant quote from MacTutor: "When the Dutch Mathematical Association announced a prize question in 1927 they gave Heyting an ideal topic on which to compete. They asked for a formalisation of Brouwer's intuitionist theories and Heyting's outstanding essay was awarded the prize in 1928. This essay was then polished and expanded by Heyting and published in 1930." See www-gap.dcs.st-and.ac.uk/~history/Biographies/…

Comment by katz

2013-04-22T12:22:05Z

@Robert: thanks for your comments. Which reference would you cite for this?

Comment by katz

2013-04-18T09:05:26Z

I don't feel bad! I feel enlightened :-)

Comment by katz

2013-04-18T08:56:19Z

Good point. Did you ever try seeing what Bos has to say about this? See ams.org/mathscinet-getitem?mr=469624

Comment by katz

2013-04-18T08:43:28Z

You might want to tag this in differential geometry and/or Riemannian geometry, as well.

Comment by katz

2013-04-18T07:17:25Z

Oops, mea culpa, etc. This is what happens when a commoner thinks he knows something about analysis :-)

Comment by katz

2013-04-17T18:00:03Z

"Vanishing of infinite order" means that it tends to zero faster than any polynomial $x^n$. The sine does not affect the order of vanishing at the origin.

Comment by katz

2013-04-16T16:11:48Z

@Toby Bartels: I've thought about this for a few days but I am still not sure what point you are trying to make. Surely there IS a correct formula for the second derivative of a composite function, and there is a coherent theory of higher order differentials.