Gauss saw things in terms of sheets embedded in three-space,
the natural abstraction from his experience as a land-surveyor. Riemann shifted the view to that
interior to the space.
-- John Derbyshire, Prime
Obsession (2004)
I recently attended a screening of the 2012 German film “Die
Vermessung der Welt” (literally geo-metry,
in the etymological Greek sense) presented to a select circle of the deutschgesinnt. What moved me to tear myself away
momentarily from my usual daytime occupation of hand-to-hand rooftop combat with our
adversaries, was the fact that the film was billed as a sort of dual biography
of two scientific figures very much worth biographizing: Alexander von Humboldt, the naturalist
brother of the philologist Wilhelm, and Carl Gauss, probably the greatest
mathematician who ever lived.
We watched it off a DVD;
had I known that the original theatrical version was 3-D, I would have
lowered my expectations accordingly.
(Actually, 3-D could be put to very good use in the exploration of the
Gaussian geometry of manifolds, but that was not its use here.)
The movie begins, as all Gauss sagas must, with the tale of
how the young schoolboy, given a pensum
along with his fellows of
reckoning up the sum of the integers from one to a hundred, shot back an answer instanter, by finding a clever
shortcut, rather than, as John von Neumann would have done, simply adding the
series instantly in his head.
(That’s a joke.) Our eighth-grade algebra class was
regaled with this (and wisely so;
it’s one of the few things I remember), and our son, in the Princeton
Friends School at a tender age, was instructed in the same as well. But in the film, the anecdote was
given what I take to be a possibly Germanic twist, for the scene opens with the
explicitly filmed rhythmic
thwack,
thwack,
thwack
of a supple and vicious-looking cane upon the bared buttocks
of a lad of around nine; graphic
enough as it was, but probably even more disquieting in a theatre, with
Surroundsound and 3-D.
Somehow, this sequence alone marked the movie out as not of American
provenience -- here, you might be sent to prison for even watching it. (Later, after Gauss has solved the
arithmetic problem, the scene is repeated with Gauss as victim, for any viewers
who didn’t manage to come to climax during the first sequence.)
So: a rather
pornographic presentation of what was in reality an utterly asexual and indeed incorporeal milestone in the annals of
mathematical awakening. But
as long as we’re here, let us dwell -- as the film alas did not -- for just a
moment on the math part.
That sequence 1 + 2 + 3 + … + 100 equals, as it happens, 5050. That fact is of no mathematical interest whatsoever, but
belongs rather to the Museum of Particular Results. (We presented a jolly fable of this notion here. Be sure to click on that essay,
it’s full of woodchucks.) Of
marginally more interest is the shortcut found by young Gauss: pair the outermost integers in turn and
you get 50 × 101. That is clever enough; but at the lowest level, it might be simply one of an
unrelated jumble of dodges used by a Calculating Idiot-Savant, and thus belong
to the Museum of Particular Tricks, just one step up from brute-force addition. A significant step up from this
recognizes that the trick is (with some tiny extra cleverness) generalizable to
any sequence 1 + 2 + … + n. Now it has risen to the level of
a general trick, and thus belongs to
the Museum of Particular Algorithms. But then this
finding generalizes to the idea of summation-formulas überhaupt: for
instance the sum of the squares of
the first n integers, or the cubes,
or any power. The resulting
infinite collection of formulas belongs to the Museum of Particular Strokes of
Genius. Striving to
generalize these, you eventually wind up with Analytic Number Theory, and its
sought-after crown jewel, the Riemann Hypothesis; which is where things stand today.
None of this is even hinted at in the movie; but really, such a development is the only reason to treasure that Gaussian
anecdote: otherwise the whole thing
can seem a mere transient bit of precious cleverosity -- as it did (in the
film’s telling) to Gauss’s schoolfellows, who give him a beating for his trick,
and no doubt to the bulk of the audience. And this is the “math porn” aspect of the presentation: Even in the absence of any mathematical
understanding whatsoever, we spectators are nonetheless supposed to be
tremendously impressed with young Gauss, who is presented as a romantic and
tragic figure, his attraction being thus, not Gaussian, but Byronic.
 |
| A superb mathematician -- and you can take that to the bank! |
*
Für psychologisch
tiefgreifende Krimis,
in pikanter
amerikanischer Mundart,
und christlich gesinnt,
klicken Sie bitte
hier:
*
~
Thus far, the perspective is that of male narcissism. That stance applies as well to the
portrayal of von Humboldt, as he goes stalking about the Amazon in his
seven-league boots, freeing slaves as he goes, and making immortal discoveries.
Both portraits involve a
certain taste of algolagnia (in the naturalist’s case, it involves his naked
back and an electric eel -- all for science, you understand). There is nothing explicitly
homoerotic, though perhaps a touch of a repressed version of that, by
implication, when von Humboldt goes apeshit upon discovering his handsome
French traveling companion
dallying with a local squaw.
Subsequently, the movie tosses a bouquet in the direction of
unearned autogynophilia as well, in the incident of young-man Gauss,
still all sturm-und-drangy, brought wisdom from the Tree of Knowledge by a
chance remark of a comely though uneducated Fräulein posing Evelike with an apple.
Mathematically, the scene will almost certainly have soared
over most of the audience’s heads.
Gauss chats about measuring the Earth (Vermessung der Welt) by adding up
triangles; the lass objects that
the Earth is not flat …. (not a
Euclidean surface, as we say in the trade) … portentous pause … Gauss,
reflecting, says, Well, you’d need lots of leeetle weeentsy triangles
(infinitessimal, mathematicae linguâ). She sensuously/attentively pares the
apple; and the penny drops, the
scales fall from his eyes, and he rushes off to scribble calculations.
What just happened -- and the viewer may well be excused for
having missed it -- is that Gauss has (apple-prompted, like Newton and gravity)
just discovered Gaussian curvature, differential geometry, and much of modern
mathematics. This episode will be
utterly opaque to anyone coming fresh to the movie; apparently, we are expected to have read the book, as with
the Harry Potter movies (the latter of which were incoherent, and would have
baffled anyone who hadn’t already read the series). Which, indeed, the director had cause to suppose, since the
movie is based upon a novel that was a humongous German bestseller.
* * *
~ Commercial break ~
For a mini-movie of
our own, try this:
We now return you to
your regularly scheduled essay.
* * *
The depiction of the wisdom-from-the-mouth-of-babes Mädchen
is likewise Byronic -- namely it recalls
Lord Byron’s daughter, Ada the countess of Lovelace, an associate of
Babbage. She has been exalted by
those who go hunting in history for neglected heroines; a summary can be found here:
In fact, though, Ada is a reasonably admirable and realistic
role-model for girls, since she did hang around a really smart guy and did work
hard and did achieve some understanding if not any actual original results,
which is all that most of us can ever hope to do. Gauss, by contrast, is no role-model at all, for anyone,
since none of us have been born with his genius, which is almost unexampled in
history. Indeed, for any actual
stellar mathematician, his example is yet worse, since he was notorious for
hoarding results. Hopeful young
mathematicians would make the pilgrimmage to Göttingen to present their results
(much as young Gauss himself is shown as doing, in a singularly infructuous
interview with Immanuel Kant), only to be told that he himself had discovered
those results long ago, and had them in his drawer, but had never bothered to
publish them. (His dismissal of Bolyai in this regard is notorious.)
Lagniappe:
Mathematically inclined lasses seeking ipsigeneric role-models would do
better to follow Noether, Kovalevskaya, Julia Robinson, or Ingrid
Daubechies. Though, once you reach
that level, you have come to realize
that pure mathematics is entirely genderless, and even (so we have argued here
and there in this series of essays) extraspecific.
Note:
Eventually, after an hour or so, weary of its pieties, and disinclined
to take in yet another sex scene
(Memo to directors: That is not why moviegoers flock to a film about
mathematicians and scientists), I walked out. So maybe I missed some dazzling final mathematical
exposition. But I doubt it.
[Footnote:
For another psychologically attuned analysis of movies, click here.]
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