Comments for Ars Mathematica

Comment on Selling Infinitesimals by sigfpe

sigfpe — Thu, 08 May 2008 19:24:41 +0000

Infinitesimals make it easier to write computer programs to illustrate what you are doing.

Comment on Selling Infinitesimals by Todd Trimble

Todd Trimble — Thu, 08 May 2008 15:15:38 +0000

Walt — I don’t have this book, but on the face of it I agree with you. It looks more like Bell is trying to prepare the reader for the inevitable appearance of constructive or intuitionistic logic if one takes nilpotent infinitesimals seriously. In other words, he’s adducing some philosophical arguments which are merely consistent with (or support some necessary consequences of) this approach.

To me the main arguments for Smooth Infinitesimal Analysis are pragmatic. Some examples of the pragmatics are given in the Introduction to Models for Smooth Infinitesimal Analysis by Moerdijk and Reyes; they give several examples of “synthetic arguments” used by geometers like Elie Cartan and Sophus Lie, and also the Ambrose-Palais-Singer theorem on connections and sprays, where a rigorous language for working directly with infinitesimals was clearly desired but unavailable at the time. (Significantly, they don’t mean invertible infinitesimals as in Robinson’s approach.)

Comment on Selling Infinitesimals by Kenny Easwaran

Kenny Easwaran — Wed, 07 May 2008 23:23:54 +0000

Domenic - there has been discussion by various philosophers of probability who suggest using Robinson’s hyperreals for probability theory in order to avoid giving probability 0 to non-empty events. However, if we consider a point selected randomly from the hyperreal interval [0,1], rather than the real interval, then clearly this probability should be less than any hyperreal interval, but that rules out any hyperreal value other than 0. I’ve been working on other arguments against this suggestion in one section of my dissertation.

As for the topic of the main post, I agree with Walt - I don’t see how this motivates infinitesimals, but I do see how it motivates various notions of constructive topology and the like.

Comment on Selling Infinitesimals by Domenic Denicola

Domenic Denicola — Wed, 07 May 2008 09:32:11 +0000

Constant probability measure, that’s exactly what I meant. And as I said, I haven’t played around with it, so maybe it’s not reasonable… actually the more that I think about it the less reasonable it becomes, so maybe you should completely disregard that comment of mine :-|.

Comment on Selling Infinitesimals by Jan

Jan — Wed, 07 May 2008 08:17:45 +0000

I find the argument for non-standard analysis of Terry Tao (see http://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/ )
infinitely more convincing than arguments from the non-divisibility of the continuum. I recently wished it was taught this way, it would have shortened a model solution I made for students in a Differential Geometry class enormously, exposing the ideas behind it more clearly.

The non-splitting of the line makes me think more in terms of non-commutative geometry, but that is maybe due to my background in physics.

As for the probability measure on an unbounded interval: isn’t the bell/Gauss curve precisely such a measure? I thought a constant probability measure on the unbounded interval is impossible I believe, but I think (again, from physics) that such a measure has no “right” to exist. I may be confusing concepts here, I’m not too good in statistics..

Comment on Selling Infinitesimals by Domenic Denicola

Domenic Denicola — Wed, 07 May 2008 05:40:25 +0000

I have always been extremely fond of infinitesimals. The hyperreals are of course a nice way of doing it, but I’m thinking that this is something more? (”Synthetic differential geometry” is not something I’ve heard of in connection to them.)

For me the simplest motivation is that the probability of picking any given point on a continuum should not be zero—it should instead be infinitesimal. I haven’t played with this enough yet, but ideally one should be able to use this idea to construct a probability measure on the unbounded interval (-\infty, +\infty), which unless I’m phrasing my intent incorrectly is impossible with the usual techniques.

Comment on Proof Style by George Butler

George Butler — Tue, 06 May 2008 15:41:55 +0000

Proofs and art have a lot in common. The old masters guide contemporary art. For polish and beauty I recommend Grundlagen der Analysis by Edmund Lanudau. Any mathematician can improve their proofs by trying to reach the standards he set in this timeless work.

Comment on Drexelmath by manu

manu — Tue, 06 May 2008 02:40:29 +0000

hi walt -

my name is manu, i’m a pure math grad student in argentina. i’ve been a silent reader of arsmath for a long time now. great site.

Viele danke for the open access link. There i found yet another application of geometry (which i was completely unaware of): “Information Geometry”, i.e. methods of differential Geometry applied to Statistics, Probability and Information theory… seems to have been founded nearly 30 years ago!

oh, by the way: I gathered some courage and opened my very own wordpress blog on math :P. it’s small, it’s humble (i’m just at math). questions, problems, exercises - mainly stuff that comes to me during showers or while having lunch. or while coming back from school and always linger on the top of my head.

be my guest.

cheers
manu

Comment on 2008 Abel Prize by Guest

Guest — Sun, 20 Apr 2008 01:20:42 +0000

I really like Tits very much and I am not afraid to admit it.
But not that other Cox-Zucker Thompson.

Comment on Groups of Order Sixteen by Jacob Freeze

Jacob Freeze — Wed, 16 Apr 2008 22:15:12 +0000

A few links into the references on this post turned up an unfamiliar way of appreciating the rarity of non-solvable groups:

If you call a number “solvable” if every group of that order is solvable, then…

A positive integer n is a non-solvable number if and only if it is a multiple of any of the following numbers: a) 2^p(2^2p-1), p any prime. b) 3^p(3^2p-1)/2, p odd prime. c) p(p^2-1)/2, p prime greater than 3 such that p^2+1 = 0 (mod 5). d) 2^4*3^3*13. e) 2^2p(2^2p+1)(2^p-1), p odd prime.