Math Porn

As we remarked earlier (Mathsex) the intersection of mathematics and literal pornography is a set of measure zero.   (A perhaps/perhaps-not  related fact, is that mathematicians themselves are entirely sexless.   They certainly do not reproduce biologically;  in fact, they do not even reproduce academically, most great mathematicians having been lousy lecturers -- and I mean really, really bad.  Instead, each individual mathematician-to-be  receives an Annunciation -- from which archangel, I alas do not know, never having received one, despite fervent prayers.)

Even “porn” in the journalistic sense, not of actual pornography, but of tawdry crowd-pleasing shallow presentations of deep subjects, seldom if ever is to be found in the same bar-booth with Mathematics.  (For a list of subjects that do so lend themselves to marketplace exploitation, consult our definitive document:  Funporn.)

Yet it is now our sad duty to report, that we have, for the very first time in our young life (young with respect to the afterlife, that is;  w.r.t. any of you-all whippersnappers, ancient), encountered an actual specimen of “math porn” (although in a very limited sense, as we shall see):  Popular Lectures on Mathematical Logic (sic, sic, sic), published in English in 1981, by the philosopher-logician Hao Wang.

There is a long tradition of lectures to the general educated public on scientific topics, by men preëminent in their field.  These were often later collected and published as volumes, sometimes with “Popular Lectures” in the title, by such true luminaries as Mach, Kelvin, Helmholtz (these in physics, though, note;  not mathematics).  The heyday for this activity was the late nineteenth century.   Their success presupposed a pool of educated laymen keenly interested in learning more about the intellectual forefronts of the day.

In America, the tradition survives sparingly, in university towns.   While our family lived in Princeton, I used to attend evening lectures at the IAS, held in their largest lecture hall.  Occasionally the audience was overflowing -- standing, or sitting on window-ledges.

Wang’s book likewise originated in public lectures -- though not perhaps to the general public, since they were given at the Chinese Academy of Science  (in 1977; translated  and published in English  four years later).   But by no stretch of semantics do they qualify as “Popular Lectures”, nor even as “Introductory Lectures for Logic Majors”.   (Of course, Wang himself may not be responsible for the English title of his book;  it may have been cooked up by some hunchbaked, drooling drone in the marketing department.)   Already on the eighth page of the introductory lecture, we read this (typical) passage:

Around 1960, Hanf numbers appeared, and Scott proved that measurable cardinals yield nonconstructable sets.  Results often mentioned as being impressive  are Morley’s theorem on categoricity in power, and applications to algebraic problems by Ax and Kochen.

Solovay soon proved the consistency without dependent choice  of the proposition that every set of reals is Lebesgue measurable.  (He has to assume that there are inaccessible cardinals;  it remains an open problem whether this stronger assumption can be avoided.

(There is no definition of any of these terms, like “inaccessible cardinals”, in early pages.  You’re supposed to already know this stuff.)   Now, for a non-specialist, it will be hard to judge just where that passage stands in the spectrum of difficulty;  but as a thumbnail comparison:  at Harvard, I took introductory logic in the philosophy department, and then straight logic in the math department, and we never got anywhere near these topics.

Lest  for some reason  Wang might have shoved all the hard stuff into the very first lecture, so as to clear the auditorium of lightweights, and become more pedagogical later on, I opened the book at random, happening upon this:

Exercise 2.  Find a direct proof of Corollary 1.3 without appeal to Lemma 1.1

You do not pose such homework in a “popular lecture”.  The book’s title amounts to false advertising  -- publishing-fraud.

~

The infraction perpetrated by the Wang book is minor -- a scattershot gallimaufry of sketches-for-an-idea-for-an-article on unrelated subjects, mislabeled as a popular introduction to a single subject -- and may well be laid at the door of the publisher rather than the author.    But now we must notice something more serious:  actual pandering to the public’s ignoble propensities, in the matter of math.  The book is titled The Math Instinct:  Why You’re a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs), by Keith Devlin.

It has been well remarked, “As a rule, perjury in subtitles should be forgiven”.  This formulation was by Tony Rothman of Princeton, writing in American Scientist (March 2007), reviewing a biography by Siobhan Roberts, of the solid geometer Coxeter, King of Infinite Space, bone-headedly subtitled The Man Who Saved Geometry.  (It didn’t need saving, and he didn’t do it.)   But in the “Why You’re a Mathematical Genius” case, it is not just something tacked on by the publisher, it goes to the heart of the book:  “Devlin tries to make the subject less intimidating by demonstrating that math is all around us.”   That formulation is already cognitively-impaired;  “math” is all around us only in the sense that relativistic quantum mechanics is all around us.  Stuff is all around us, and a very few people (not you, not me) are able to make sense of some of it, somewhat (though less than is popularly assumed) by deploying a mathematical armamentarium:  neither Nature’s intricate hidden design, nor the occasional successes of experts, make you a mathematical “genius”.   And actually, few would be fooled by such transparent flattery (though it is of a left-handed sort, since the reader’s putative “genius” is, with the next breath, set at the level of lobsters);  one wonders why the publishers thought the public would be suckered-in by such fluff, rather than nauseated.   The answer no doubt is that such an approach is in tune with the general trend in America to “celebrate” one another’s narcissism.   That frame of mind does not make for keen exposition; as the reviewer in American Scientist remarks (Nov 2005, p. 572),

Devlin seems uncertain what his readers will need to have explained.  He reminds us that bats are mammals, but doesn’t define Ohm’s Law.

The case is the sadder since Devlin himself knows better.  He once wrote a very good semi-popular survey (only “semi-“, since equations do appear), Mathematics:  The New Golden Age (1999).   In this case, the rather grand subtitle is actually warranted: for while the great age of math as a handmaiden to physics is behind us, in the past half-century or so  it has effloresced to an astonishing extent as a multiply-connected enterprise in its own right -- with just enough new practical uses in cyptography and (if this one prove more than a dream) string theory, to keep it anchored.

~

It is now our happy privilege, likewise to report, that the other book occupying us this weekend, is precisely the opposite to the sugar-coating of the Wang packaging:  The title austere, uninviting;  the content as propaedeutic and intuitive as it is possible to be.  We refer, with reverence, to a book likewise originating in (semi-) public lectures (at UCLA, in the early 1980s):  Richard Feynman’s QED, published by Princeton in 1985.

The title might inspire a spurious sensation of familiarity:  quod erat demonstrandum, in which case it might fit a set of lectures on high school geometry.  But no, the acronym denotes something much more fearsome, something …. (but send the children to bed, before you scroll down)

…

…  nothing less than …

(horresco referens)

…

 =>  QUANTUM ELECTRO-DYNAMICS.

For years (decades, actually) I avoided this slender volume, though it stood on my shelves, owing to the misapprehension that, based on its titled subject-matter (which presupposed basic quantum mechanics) it must be significantly more difficult than the Big Red books, aimed at physics-major freshmen, on which I was weaned.  Yet not so.  Feynman, a master of exposition, offers the most intuitive account possible, of a highly non-intuitive subject.   Intuitive, yet not dumbed-down:  it takes a genius to tackle that.

But nota bene:   He never endeavors to coax you into thinking something is easier than it is, nor to flatter you that you have understood something when you have not -- both being common expository strategies of popularizers of science in this Age of Self-Esteem.

Rhetorically -- engagingly -- Feynman adopts the opposite strategy, hinting at what you are up against, though not bullying.  In the opening of the introductory lecture, he makes the quite accurate though socially/intellectually scandalous observation that

Everybody who comes to a scientific lecture  knows they are not going to understand it, but maybe the lecturer has a nice, colored tie to look at.

The second lecture, raising this observation to the status of a meme, opens thus:

This is the second in a series of lectures about quantum electrodynamics, and since it’s clear that none of you were here last time (because I told everyone that they weren’t going to understand anything), I’ll briefly summarize the first lecture.

The next begins:

This is the third of four lectures on a rather difficult subject -- the theory of quantum electrodynamics -- and since there are obviously more people here tonight than there were before, some of you haven’t heard the other two lectures  and will find this lecture almost incomprehensible.  Those of you who have heard the other two lectures  will also find this lecture incomprehensible,  but you know that that’s all right:  as I explained in the first lecture, the way we have to describe Nature  is generally incomprehensible to us.

This stance, though fey in a way, is yet preferable to that of the Theory-of-Everything charlatans who babble on about Beauty, implying that they can share their vision by mere dermatological osmosis.   There was one such on NPR’s science show last week  -- a string theorist yet, which is to say:  a specialist in a fashionable but drastically unintuitive math-packed would-be-physical theory, which, additionally, may well be actually wrong as a description of the real world -- or, worse, “not even wrong”.  But the interviewer kowtowed and pandered, and the interviewee luxuriated in his 15 famous minutes, as in a warm bubble-bath, complete with rubber duck.

A far more prominent string-theorist, Frank Wilchek, of the IAS, has a similar piece in a recent issue of Nature / Physics (Nature, by the way, seems to be in some ways following Scientific American down the path of corruption), calling on us all to “celebrate Beauty”.  -- By all means, say I, let us do so, and preferably in the altogether;  but don’t go calling it physics.

~

[Update & after-reflection]  The title (and thesis) of Devlin’s book are misconceived in a rather subtler and more substantial way than the undraped pandering of the subtitle:  The Math Instinct.     Because, in fact, there isn’t any.

I satirized the thesis that there is one -- honed, like the other instincts, by Natural Selection -- here:

The Urysohn Metrization Theorem:

an Adaptationist Account

However, the Ultradarwinians were not our real quarry in that sotie, but rather the mathematical Nominalists.

Devlin’s title echoes that of an earlier and much better work of scientific popularization, Steven Pinker’s The Language Instinct.   Here, the title is no mere gimmick from the marketing department:  the author is well aware that, properly understood (something that itself is no easy task), the positing of a language instinct constitutes a highly contentful, counterintuitive, scientifically testable, and in some quarters bitterly controversial hypothesis.   It is one that Chomsky and those in his intellectual wake  have been investigating with great ingenuity, for over half a century now.   They have made a telling case (though one that has become more difficult to understand as the theory itself advanced, rather the way Galilean mechanics are more intuitively accessible than Quantum Field Theory), one which need not here be passed in review.

With mathematics, matters lie quite otherwise. …

[To be continued, if reader interest warrants.  In the meantime, consult our counterthrenody to the thesis of universal mathematical genius:   Oligophrenia Mathematica.  ]

[Update Oct 2013]  Continued here.