My post was not about the existence of infinitesimals, but just how I found Bell’s particular argument for them not convincing.

I don’t offer any justification of infinitesimals along the lines of the Bell quote, just a pragmatic justification.

@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”}

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.

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.

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.

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.)

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.