Selling Infinitesimals
May 6th, 2008 by WaltJ. L. Bell’s A Primer of Infinitesimal Analysis (an intro to synthetic differential geometry) begins with a series of quotes to motivate why we should think of the reals as containing infinitesimals. The quotes all involve the idea that philosophically speaking the continuum is an indivisible whole. This quote from Hermann Weyl is a typical example:
A true continuum is simply something connected in itself and cannot be split into separate pieces; that contradicts its true nature.
I find this line of reasoning completely unconvincing as motivation for allowing the reals to have nilpotent infinitesimals. I can grant, for the sake of argument, that maybe its unnatural that our model of the line can be split cleanly into two or more parts, but to me this is an argument for constructivism, not infinitesimals.
May 6th, 2008 at 10:40 pm
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.
May 7th, 2008 at 1:17 am
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..
May 7th, 2008 at 2:32 am
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 :-|.
May 7th, 2008 at 4:23 pm
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.
May 8th, 2008 at 8:15 am
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.)
May 8th, 2008 at 12:24 pm
Infinitesimals make it easier to write computer programs to illustrate what you are doing.
May 9th, 2008 at 9:12 pm
Dominic: Synthetic differential geometry allows nilpotent infinitesimals, to formalize arguments like “take a quantity so small that its square is zero”.
Todd: I like that Moerdijk and Reyes book a great deal, and my view of SDG is strongly colored by it. Sadly, my local library doesn’t have it.
May 11th, 2008 at 5:23 am
About Constructivism…
Most people don’t see much difference between “It’s either true or false” and “It can’t be both,” but you can deny the Law of the Excluded Middle and still get tenure among the Constructivists, while your prospects as a non-non-contradictionist are bleak:
“Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned.” (Avicenna)
In spite of such dire threats, a few brave logicians still explore a “paraconsistent” logic whose origin is usually attributed to Francisco Miró Quesada, although Miró Quesada himself traces its developement all the way back through Plotinus to Meinong and the very obscure A.C.N Da Costa, who apparently coined the Portugese equivalent of “paraconsistent.” (Alternative Logics, Paul Weingartner ed. p.24ff)
This branch of logic attracts its share of odd ducks along with a few reputable academics and, fittingly, a few hybrid odd-duck-reputable-academics like Richard Sylvan (né Routley), who changed his name to reflect a love of forests and published 200 academic papers during his long tenure as Professor of Logic at the National University of Australia in Canberra.
Professor Sylvan’s “noneism” extended Meinong’s already wonderfully extensive universe beyond its original fund of round squares and other such “subsistents,” with the eminently practical objective of providing a solid philosophical basis for the dreams of ecological dreamers like himself.
Apart from environmental utopias, paraconsistent logic offers an unusually straightforward solution to the paradoxes that constantly arise in set theory and the foundations of mathematics: Russell’s self-including sets, for example, subsist happily with their own impossibility, and more sophisticated versions of this sort of application motivated the development of “dialetheism” by Graham Priest and his students at the University of Melbourne.
Before we dismiss paraconsistency as a peculiar Australian heresy, it’s worth remembering that classical monstrosities like Russell’s self-including sets often reappear in new disguises, however carefully the foundations of mathematics are reformulated to avoid them, and the mathematical community deals with these uninvited guests more or less on the same paradigm that tells the Federal Reserve when to save a bank: your local savings-and-loan may very well go under, but Chase Manhattan is too big to fail.
Likewise your favorite journal will let you perish before they publish your self-contradictory article, but when a hybrid of truth and untruth appears in the foundations of mathematics, nobody dreams of abandoning “the paradise that Cantor has created for us,” and instead the latest true-and-untrue intruder is usually shooed out of the garden with a mixture of hand-waving and ad hoc formalism.
Advocates of infinitesimals distinguish themselves as unusually inventive apologists by accomodating to the relevant paradoxes with a useful little monstrosity of their own, and the embarrassment of ex nihilo creation of a positive measure out of points of probability disappears, but at the price of a dangerous rapprochement with Meinong’s jungle, where the infinitesimals are equally useful for building round squares.
May 11th, 2008 at 9:25 am
Walt –
Are you also saying that you reject the notion of infinitesimals? In that case, I would counter with: In what meaningful sense do infinite decimals exist? In a finite world such as ours, where we all die eventually, we have no way of demonstrating the existence of any infinite decimal. We ASSUME they exist, and all of us with mathematical training have done so for so long that we now take their existence for granted. In other words, the real line is already assumed to include entities none of us have ever seen, nor could we ever see, not even in principle.
BTW — There was a serious academic argument in England in the first half of the 19th century about whether or not negative numbers existed, with people publishing books pro and con. Augustus de Morgan produced a convoluted argument to show it was OK to use them even if you could not prove they existed.
May 11th, 2008 at 9:34 am
Also, another nice book is the one by Anders Kock:
@BOOK{kock:book81,
AUTHOR = “Anders Kock”,
TITLE = “Synthetic Differential Geometry”,
PUBLISHER = “Cambridge University Press”,
YEAR = “1981,
series = “London Mathematical Society Lecture Note Series 51″,
address = “Cambridge, UK”}
May 15th, 2008 at 3:42 pm
I recently typed up some notes on smooth infinitesimal analysis based on Bell’s book and Moerdijk and Reyes’s (although I don’t go into any construction of the models). People here might find it interesting. They’re at:
http://www.math.cornell.edu/~oconnor/sia.pdf
I don’t offer any justification of infinitesimals along the lines of the Bell quote, just a pragmatic justification.
May 15th, 2008 at 9:45 pm
I did a post on the Kock book here.
My post was not about the existence of infinitesimals, but just how I found Bell’s particular argument for them not convincing.