Death to triangles! (and life to ultrafilters…)

The (in)famous rallying cry of Jean Dieudonné is the title of a new blog by “Nick Bornak” (hmmm, I wonder where the name came from…). The author, a student of mathematics and philosophy, started the blog in order to have a place to write about mathematics (for that, WordPress is pretty much the only way to go at the moment).

One of his major interests is nonstandard analysis, the very concept of which I used to despise until I read this brilliant post by Terence Tao. If a Fields Medalist says it’s okay, then it must be okay! :-) Actually, what Tao’s post helped me to realize was that so-called “nonstandard analysis” can be regarded as just another item in the “soft” vocabulary of (“standard”) analysis, like asymptotic notation, or the definition of continuity in terms of inverse images of open sets. Since I absolutely adore “soft” mathematics (to the point where one of my missions in life is to “mollify” as much “hard” mathematics as I possibly can), you can see why I would find this point of view so appealing!

Before this, I used to see nonstandard analysis as, at best, an obfuscatory way of formulating ordinary analysis; or at worst, a sort of contrary position in the ontology/epistemology of mathematics, like constructivism (except perhaps on the “other extreme”). The way nonstandard analysis was described — as a way of making Leibniz’s infinitesimal reasoning rigorous — irritated me to no end. Didn’t these people realize that the work of the likes of Cauchy, Weierstrass, and Cantor had already made that reasoning rigorous? Not to understand this — simply because of Leibniz’s “different” vocabulary — is to exhibit the sort of over-concrete literal-mindedness that is all too typical of certain historians — particularly historians of mathematics — and, for that matter, music theorists. It thus seemed as if (not for the first time, alas) a whole field had been founded on a simple lack of intellectual agility!

Luckily, thanks largely to Terry Tao, I’ve since moved beyond this. Perhaps Nick Bornak will over time be able to still further illuminate the virtues of the “nonstandard” way of thinking. (That word “nonstandard” really is a turn-off, I have to say — to me it carries a suggestion of crackpottery.)

(Incidentally, though I’m a great admirer of Dieudonné, I don’t have anything against triangles. In fact, I think I could discuss them in a way that he would approve of. But that’s for another occasion…)

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One Response to Death to triangles! (and life to ultrafilters…)

  1. Because of the way the non-standard number system
    once appeared (quite unexpectedly) in my research,
    I’ve arrived at a more algebraic point of view about
    non-standard numbers which differs in outlook from
    managing epsilons or making infintessimal arguments
    rigorous or the standpoint of logical model theory.
    Just to be clear, I am in no way claiming that
    this point of view is better (or worse) than the
    others, but rather offer it with the thought that
    seeing the same subject from different perspectives
    can lead to deeper understanding and that seeing
    the same mathematical structure arise in several
    contexts can suggest that it is something
    interesting which is worthy of further examination.

    A well-known characterization of the real number
    system is as the unique complete ordered field (up
    to isomorphism). Hence, if one wants to extend
    this field, one will have to give up either
    ordering or completeness (or perhaps both).

    If one gives up ordering, one can construct an
    algebraic field extension of the real numbers,
    which has the pleasant property of being
    algebraically complete as well as complete in the
    sense of convergence. One can then proceed
    to extend the operations of analysis to functions
    over this larger field and finds that, even when
    one is only interested in some question involving
    real numbers (say deriving trigonometric
    identities or computing a definite integral),
    one can often save labor by first working with
    complex analysis and then taking the real part
    at the end of the day. (As Watson once quipped,
    the shortest path between two real facts often
    passes through the complex plane.)

    Suppose instead that one wants to extend the
    real numbers to a larger ordered field. One
    then needs to give up completeness and arrives
    at non-standard analysis. The epsilons arise
    naturally — one statement of completeness is
    Archimides principle, the negation of which is
    tantamount to the assertion that the field
    contains infinitessimal elements.

    The basic methods of non-standard analysis are
    reminiscent to me of complex analysis. Just as
    in the latter, one analytically extends real
    analytic functions to complex functions in a
    unique manner so too, in the latter, one extends
    real functions defined in first order logic
    to non-standard functions in a unique fashion.
    Just as the principle of persistence of relations
    allows one to carry over functional equations and
    differential equations to the complex plane, so
    too does the transfer principle work.
    Just as one might take the real part at the end
    of some complex operations, so too one often
    takes the standard part after some non-standard
    operations to obtain a result in good old
    real analysis.

    As for the ultrafilters, they arise from looking
    at limits of sequences algebraically. (I learned
    this from Ebbinghaus’ wonderful book on numbers.)
    When talking about taking limits of sequences,
    what we are considering is assigning numbers to
    sequences is some reasonable way. As for what the
    term “reasonable” might denote in the last
    sentence, we presumably don’t want to assign
    numbers to sequences haphazardly, but in a way
    which is consistent with the structure; at the
    least, we might want the limit of a sum of sequences
    to be the sum of limits and likewise the limit of
    products to be the product of limits, for instance.

    Algebraically, this desideratum about sums and
    products says that the limit operation should be
    a morphism from the ring of sequences to that of
    real numbers. Now, in real analysis, we only
    obtain such a morphism for the proper subalgebra
    of convergent sequences; let’s see if we can do
    better by picking a different field as target of
    the morphism. A basic result of commutative algebra
    states that a morphism from a ring to a field is
    isomorphic to quotienting by a maximal ideal.
    Hence, we should llok for the maximal ideals of
    the ring of real sequences. A little work shows
    that these are in one-to-one correspondence with
    the maximal ideals of the Boolean algebra of
    subsets of the integers, which are trivially
    related to ultrafilters. This too, is a parallel
    to complex analysis — just as one can obtain
    complex numbers by extending the real number
    system so as to make every polynomial have a
    root, so too one can obtain the non-standard
    nunmber system as an extension in which every
    sequence has a limit.

    By the way, I am no more put off by the term
    “non-standard” than by the term “imaginary”
    (which could suggest that complex analysts are
    dayreamers) or the term “degenerate” or the
    term “irrational” — mathematicians have a
    habit of using negative, even pejorative terms
    to describe new concepts which challenge
    naive intuition.

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