Sunday, December 19, 2010

Topoi and Timepiece

            Only once have I been obliged to drop a course because I just wasn't getting it, and never would.  That was algebraic topology.  And the aspect I found most off-putting was Category Theory.  Category Theory aims to be maximally general, not really "about" any one specific thing; but it had its cradle in algebraic topology, and the latter field has some genuine proving to do, about real topological objects, studying these algebraically in much the same way that Euclid's geometry became algebrized as analytical geometry.  But as the course went on,  category theory seemed to dissolve into unsalted abstraction, rather than illuminating anything concrete in the actual subject-matter.
I expressed my frustration to a classmate, Stephen Kudla (now a professor of mathematics at U.Toronto).  To my surprise, his eyes lit up, and he spoke glowingly of Category Theory, how it made everything clear.  At that point I realized:  He's in the right field; I'm not.

            A certain reserve towards Category Theory was then widespread among working mathematicians.  They referred to it as "diagram-chasing", "abstract nonsense" (that term not quite as derogatory as it sounds to an outsider), and "the higher macramé".  For many years I just forgot about it.

            Yet of late, one runs across assertions to the effect that the theory, still flourishing rankly like most mathematical growths, is now finding applications in ... physics, of all things.  That is as surprising as a report that general relativity has been applied to the problem of building a better brownie.  So it needed another look.  There's an introduction with the practical he-man title Categories for the Working Mathematician, by Saunders MacLane -- suggestive of all the carpentry and plumbing texts that our son now has scattered about the house.  But now one learns that Category Theory itself has spun off yet a further area of endeavor, yet more abstract, called Topos Theory.  So I checked out a volume with the (to the rhetorician) charming title of Topoi (2nd edn. 1984), considerably more euphonious than the alternate plural, topuses (anathema to the classicist).   Its self-described aim is to be a sort of Topuses for Doofuses, requiring no specific prerequisites (although of course  it helps if you have multiple Ph.D.s) -- "a fully introductory exposition, aimed at a wide audience.  Everything -- set theory, logic, and category theory -- begins at square one."  Lend it to your charmaid when you're done!  It sounded promising.

            Alas, 'twas not to be.  The diagrams swirled like a dog chasing its tail, but nothing mathematical or philosophical  was illuminated for me, let alone anything in physics.  In most math or physics books, there are many things of which I get a tantalizing glimmer, and would love to study and come to understand, granted merely that I be ten times smarter and have all the time in the world.  Here, nothing even presented itself as a possible object of interest.   Worse, about midway through, it develops that Topos Theory takes place in the context of (Brouwer/Kripke) intuitionistic logic -- a distinctly minority taste among mathematicians, and to me, the least attractive mathematical idea anyone has ever had.  Worse still, in Kripke semantics, it turns out,

Truth is temporally conditioned.  A sentence is not true or false per se, as in classical logic, but is only so at certain times ... We speak of sentences as being "true at a certain stage" or "true at a certain state of knowledge.   [p. 188]

This sounds like the Paris Hilton school of epistemology -- "If I haven't heard ot if, it doesn't exist."  Further, it smells of atheism.

            After I'd finished the book, life seemed just a little bit less worth living.


            By mere happenstance, the next thing I read after putting aside that book with a sigh, was an article with the startling title, "Is Mathematical Truth Time-Dependent?" -- startling because it appeared in the American Mathematical Monthly (as opposed to, say, the National Enquirer), and was dignified by inclusion in the widely-cited Tymoczko omnibus volume on the philosophy of mathematics.  I tensed every nerve.

            And then read a brief and shallow essay  about the evolution, not of truth, but of standards of proof -- a thrice-told tale.  It’s basically a bait-and-switch: the question in the title is never even addressed.  Only in the concluding paragraph does the author even recall the question supposedly at the heart of the article, and weakly concludes, "Perhaps mathematical truth is eternal, but our knowledge of it is not" -- a statement no-one has ever disputed, since it applies to our knowledge of elephants as well.  It is as though you wrote an article titled "Did O.J. Kill Kennedy?", and then prattled on about brownie recipes or whatnot, only to conclude that, as regards O.J. and all that, it all needs further research.

            So: "Is Mathematical Truth Time-Dependent?"
            Short answer:   No.
            Long answer:   No --   effing   --      way ………..


  1. The statement "Barack Obama is the POTUS" is temporally conditioned: it is true now, but it wasn't true twenty years ago and it won't be true twenty years from now.  Why does this "smell of atheism"?

    The statement "God intervenes frequently and obviously in human affairs" is true of the earlier books of the Bible much more than the later ones.  I once met a yeshiva student who believed that God (rather than our view of Him) actually changed during the Biblical period, because the Bible says so.  Thus, for me, temporally-conditioned truth smells of overly-dogmatic theists and their Biblical literalism.

    Mathemtical truths can be eternal only to the extent that they are divorced from the physical universe.  1+1=2 is "true" given a whole bunch of usually-unstated background assumptions (countable objects, etc.)  One icecube plus one icecube equals two icecubes, but one drop of water plus another drop equals one bigger drop, not two drops that maintain their separate identities, so even arithmetic truths about sets of H₂O molecules can be viewed as temporally (or at least thermally) conditional.

  2. > Mathemtical truths can be eternal only to the extent that they are divorced from the physical universe.

    That's like saying that the equations of physics are true only if divorced from football.
    That two water-droplets meld is a contingent fact of terrestrial chemistry; nothing to do with mathematics.