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:
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.
[Update Oct 2013] Continued here.
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