Tuesday, January 6, 2015

Miscellanea mathematica (quater renovata)

An epigrammatic index to various essays.  For the context, click on the link:

“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.”

A near-synonym of the math-word deep, but shorn of all irrelevant aesthetic echo, is:  “highly nontrivial”.   The term is decidedly commendatory, though to a layman it might sound like faint praise.

But what then, of the java, that flows from the mug whose reality we have just proved???  Why, it flows to fuel the brains of mathematicians!

A classic joke runs: 

The Axiom of Choice is obviously true;  the Well-Ordering Principle, obviously false;  and Zorn’s Lemma -- who can understand it?
It would be worth your while to obtain a Ph.D. in mathematics, simply to be able to get that joke (which contains deep truths).   Nothing else in the universe is nearly so funny.

"Some it-from-bit proponents  stretch this logic still further.  They look on the universe as a giant computer simulation."

“I do not know whether it can deal with jelly-like or cloud-like entities, with mushy viscous messes  that do not break up into manifest units.  I suspect that nothing is beyond the technical ingenuity of men…” http://worldofdrjustice.blogspot.com/2012/04/ontology-of-logic-updated.html

I personally attended Professor Langlands IAS lecture series (autumn 1999);  in the first of these he stated that he'd wanted to be a physicist, but physics was "too difficult", so he had to settle for being a humble mathematics professor at the Institute for Advanced Studies.   Nor was this a pose;  his whole manner is that of straightforward humility, very … Canadian.

To vary Nestroy’s celebrated epigram -- “Bis die Topologie gehts noch, aber von da bis sheaf theory  zieht sich der Weg.”

We may define pure mathematics as the subject in which Bertrand Russell does not know what he is talking about, though what he says is none the less true.

A more adequate symbol would be simply a rectangle with opposite edges identified -- or better yet, the toroidal covering-space which carpets RR with infinite replications of this patch-sample.  (Already how distantly we have left behind the donut!)

Mathematicians, like philosophers, and unlike anyone else (including even lexicographers), are given to a certain semantic Akribie --  an extraordinary self-critical attention to their own use of language.

iz here sumware
“Koestler concluded that his hours spent by the prison window  scratching equations  had brought mystical insights into another realm of being.”

It is one of the few major novels whose protagonist is presented as being a mathematician.    Now, mathematicians are (if you please) god-like beings;  yet with very few exceptions (Galois,  Erdös ..) they do not lead colorful lives.  The man who settled Fermat’s Last Theorem, for instance, Andrew Wiles, is … um …. ahh… actually, I cannot think of a predicate -- he just is.

The quirky, philosophically-minded Intuitionist mathematician Brouwer, harbored similar “mysterian” views on ultimate indefinabilty.

In the stylistic spirit of minimalism (and of that pointilliste Wittgenstein), we shall begin with a Delphic  epigram:
Logicism:  a kind of reductionist minimalism.

Differentiable Penguins
Note that dismissive back-of-the-hand to mere “fun”,  which nonetheless figures first in the list of motivators, in the quotation from Ian Stewart  with which we began this essay.   For full-time professionals, mathematics is just too darn hard to be nothing but fun.  You don’t devote your entire life to crossword-puzzles or sudoku.

One frequently meets statements along these lines  (in the present instance, reporting the work of Freedman and Donaldson on h-cobordism):
     It’s true topologically, but not smoothly, for dimension four.
Are the post-modernists here positing some abstruse new varieties of truth -- topological and smooth?

"The fundamental theorem of enumeration, independently discovered by several anonymous cave dwellers, states that the number of elements in a set  is the sum over all elements of that set  of the constant function 1."

In focusing on definition, I am inadvertently revealing the déformation professionelle of one who used to earn his bread (or rather his hardtack; the profession is ill-paid) as a lexicographer.   For, rather than trying to say what a thing “is” (and here the Korzybskian strictures against the copula  have their full force), we may say, pragmatically rather than ontologically, what a thing is for.

Seen in that perspective, the ugling-duckling of a phrasing, “Why is mathematics?”, spreads the wings of a swan.

One day, for a wager, Cantor,
after much ribbing and banter …

Mathematicians are given  
to chiseled concision.

Mathcat hath thpoken!

The very truth-predicate itself has been questioned within mathematics (albeit, by a rabble of Nominalists).   Thus, for a comparatively straightforward proposition “Catalan’s constant is transcendental”, a constructivist will not accept that this is either true or false.

... algebra, not in the sense of Galois or of André Weil, but of those unpleasant little sentences with x’s in them, relating to draining bathtubs which (against all reason) are simultaneously being replenished from the tap. http://worldofdrjustice.blogspot.com/2012/12/mathsex-updated-for-holiday-season.html

Our purpose is twofold;  indeed, the twin goals “can be thought of” as dual to each other.  The ostensible aim is (lightly) mathematical:  to provide pithy thumbnail sketches of complex fields of research.   The more substantial project takes place rather in the lexicographic ‘conjugate space’ which maps the items so defined.

Hard at work  in the math library

Compare a formulation we likewise favor, “Infinity is big.”   That epigram is double-edged.  First, it mimics the naïve astonishment that the novice feels, not only upon being introduced to the idea of infinity, but even large-but-finite things like a googolplex.  (As a child, I marveled over that one, much as I marveled over the brontosaurus, and for the same reasons.)

“Attempts to extend the geometry of second-order surfaces  and the algebra of quadratic forms  to objects of higher degrees  quickly leads to  the detritus of algebraic geometry, with its discouraging hierarchy of complicated degeneracies, and answers that can be computed only theoretically.”

Here, the answer to “What is Topology?”  is not simply knitted into a sampler and tacked to the drawing-room wall, but used as an actual heuristic for finding your way in a different field.

“Not a ‘triplex of mutually orthogonal rabbit-slices’, dammit!  I mean three  separate  rabbits !!”

~  Posthumous Endorsement ~
"If I were alive today, and in the mood for a mystery,
this is what I'd be reading"
(Ich bin Georg Cantor, and I approved this message.)

Such ontological excrescences are even more thewless than “perfect” numbers, since at least the latter are independent of their inscriptional base.   (You can think of the writing of one of God’s own integers in any base  as representing a tragic demotion from the Platonic sphere, sort of like a soul’s being incarnated in the body of a frog.)

the topic of definite integrals -- painful but necessary, rather like a rectal exam

Which brings us back to the vexing question of the Riemann Hypothesis.   Its acceptance or non-acceptance cannot be left simply to each individual whim of the moment.    Rather, we must settle it the way all things are settled:  by majority vote (subject to Republican filibuster).

Such beauty and such elegance are perceptible only to the mind prepared -- otherwise it is like playing Bach to a baby.

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, 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.)

In practice, we know as little of this  as a starfish knows of the stars. 

Anthem of the Oligophreniacs:  “If I Only Had a Brain”

Intuitionism -- initially a sort of mathematical vegetarianism -- is by no means dead.

No new theorem of any importance came out of the immese effort at systematization of Nicolas Bourbaki

As so often when some movement of math has been seen streaking off westwards  out into the void, presumably never to been seen again by mortal man, it reappears shining in the east, reborn in some applicable form -- thus suggesting, you will notice, that the global topology of the noösphere is toroidal. 

“Too large a generalisation  leads to mere barrenness.  It is the large generalisation, limited by a happy particularity, which is the fruitful conception.”

The so-called “abstract” groups (MacLane himself uses the sneer-quotes here) mean to lift aloft from Groups of Transformations, in that they retain the laws (associativity, inverses, and all that) while becoming agnostic as to the nature of the elements …

Our bow to Gleason’s semantic precisionism  is not by way of fetishizing fine distinctions.

Trans-cosmic Pi Day

“Lobschevsky’s colleagues  failed to understand his work.  Since they did not want to write negative reviews, they simply ‘lost’ the text.”

In Vergleichende Anatomie der Engel (1825),  Fechner argued that the angels, as the most perfect beings, must be spherical, since the sphere is the most perfect form.

The final anguish  of the Asian bride  suggests the depth  of the Riemann Hypothesis.

Further choice morsels  here:

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