Wednesday, July 24, 2013

A mathematical scratchpad (even further scratched)

[There simply isn’t time, at least before retirement, to integrate each thought-balloon as it bubbles up -- a proto-insight or pre-idea -- into the appropriate essayistic context in finished form.   Yet to leave these on the desktop equivalent of a desk drawer is to tempt the Reaper.   Therefore I shall place some of them here -- philosophical post-it notes;  mathematical Zettel.]

[Cf. De stultitia]

Our focus  in the essay of that name, is on the plight of those sorry souls (99.9999999 % of us) who fail to grasp what Grothendieck, or Witten, or whom-have-you, saw easily enough.
Distinct from, though related to, this, are questions of which (at the forefront of science) we are permitted a glimpse,  but which nobody understands.  As:

On cosmogenesis:

The whole vast imposing structure  organizes iteself  from absolutely  nothing.
This is not simply  difficult to grasp.   It  is    incomprehensible.
-- David Berlinski,  “Was There a Big Bang?” (1998), collected in :  The Deniable Darwin (2009), p. 229


All this leaves us  where we so often find ourselves.  We are confronted with certain open questions.  We do not know the answers, but what is worse, we have no clear idea -- no idea whatsoever -- of how they might be answered. 
But perhaps that is where we should be left:  in the dark, tortured by confusing hints, … and a sense that, dear God, we really do not yet understand.
-- David Berlinski,  “God, Man, and Physics” collected in :  The Deniable Darwin (2009), p. 270

The hardest part of a subject is the beginning.  Once a certain stage is passed, we gain confidence  and feel that, if need be, we could carry on by ourselves.
-- John Synge & Byron Griffith,  Principles of Mechanics (1942, 1959), p. 506

Alas, that has not been my experience at all.
Any technical subject is like a whirligig, which rotates faster and faster until the centrifugal force throws you off.   It’s like the Peter Principle, everyone eventually reaching his own personal level of incompetence;  only, in math and in physics, these levels stack indefinitely towards heaven, so that a few of us can ascend quite a ways, before we are finally out of our element.

[Cf.  Any Ideas? ] 

Recent years have seen striking developments in the conceptual organization of mathematics.  There developments use certain new concepts  such as “module”, “category”, and “morphism”  which are algebraic in character.
-- Saunders MacLane & Garrett Birkhoff, Algebra (1967; 3rd edn. 1999), p. vii

The reason they speak here of new “concepts” rather than additional structures  is that the notions referred to do not exist merely within algebra, but serve to organize other mathematical fields as well.


In an exterior view of the finished product, we see structure mathematics as largely logical or deductive:  P entails Q.
But from the interior standpoint of the practicing mathematician (and here, though we refer to the ‘actio’ sense of mathematicizing as opposed to the actum or product, the interest is not psychological but ideational), a key verb is rather motivate:  P motivates Q.   An illustration of this special vocabulary:  “The desire to extend Fourier L2 to Lp spaces  motivates the Riesz interpolation theorem.”


More vocabulary from the conceptual domain:  thrust, as in the following passage

The Heisenberg uncertainty principle:  The mathematical thrust of the principle can be formulated in terms of a relation between a function and its Fourier transform.  The basic underlying law, formulated in its vaguest and most general form [i.e., its most intuitive formulation], states that a function and its Fourier transform cannot both be essentially localized.
-- Elias Stein & Rami Shakarchi, Fourier Analysis (2003), p. 158

~   ~   ~

[Cf.  On Depth]

When I made my original discovery of radiation from black holes, it seemed a miracle that a rather messy calculation should lead to emission that was exactly thermal.  However, joint work with Jim Hartle and Gary Gibbons  uncovered the deep reason.
-- Stephen Hawking, in: Stephen Hawking & Roger Penrose, The Nature of Space and Time (1996), p. 44

For the mathematician, contrasting with messy  are simple and elegant -- yet in the following, even these don’t get you to the yonder side, where Depth dwells:

Having derived the equation for the vibrating string, we now explain two methods to solve it:
(1) using traveling waves;
(2) using the superposition of standing waves.
While the first approach is very simple and elegant, it does not give full insight into the problem.
-- Elias Stein & Rami Shakarchi, Fourier Analysis (2003), p.  8

String theory is sometimes described as a theory that was invented backwards … People had pieces of it quite well worked out  without understanding the deep meaning of their results. … Math is funny that way.  Formulas can sometimes be manipulated, checked, and extended  witnout being deeply understood.
--Steven Gubser, The Little Book of String Theory (2010), p. 2

[Cf. Consilence in Mathematics]

Horizontal consilience:

… the structure theorem for finitely generated groups -- a fine illustration of conceptual unification.
-- Saunders MacLane & Garrett Birkhoff, Algebra (1967; 3rd edn. 1999), p. vi

Mathematics is a coherent, interlocking whole, and advances in one area  often lead to advances elsewhere.
-- Ian Stewart,  How to Cut a Cake (2006), p. 89

Commercial Break
A private detective  confronts the uncanny;
an ecclesiastical mystery:


Expressing himself in the language of fluxions and fluents, Newton managed to conceal his insights in a notation that was miraculously maladroit.  Not so Leibniz.  The language of mathematics and mathematics itself  are mutually sustaining.
-- David Berlinski, Newton’s Gift (2000), p. 57

Note:  The first clause of that observation does not actually relate to the point about notation (as opposed to vocabulary), and is silly in itself.  The concepts were new, so obviously any term for these would either be an out-and-out neologism, or a semantic hijacking of an extant word.   There is nothing lexically more rebarbative about fluent and fluxion than about derivative, differential, infinitessimal. 
Betrand Russell, in The Principles of Mathematics (1903):

Mathematics is the class of all propositions of the form ‘p implies q’ …

The appended dribble of dots replace additional uninteresting clauses, which rob the sally of its epigrammatic pithiness, while yet failing to throw any light upon the subject.   It is a definition for people with no interest in the dark loamy richness of actual math as such;  and worthy of the author whose massive Principia Mathematica could as well have been titled Why Math Isn’t Interesting After All.
The characterization becomes even less interesting when you reflect that the expression “p implies q”, in logician’s lingo, is mere ‘material implication’ -- what would be better dubbed immaterial implication, since it involves no notion of causation or even logical entailment (and is thus immaterial to any actual problem), but is neither more nor less than another way of saying “either not-p, or q”.

Two citations illustrating the insight that axiomatizations, though perhaps logically prior, are pragmatically post-hoc:

The order of nature, and the order of logical dependence, are not the same as the order of our discoveries.
-- Morris Cohen & Ernest Nagel,  An Introduction to Logic and Scientific Method (1934)

Not all axiom systems are formal systems, and formalization need not lead to axiomatization.
The axiomatic method is an orderly way of summarizing experience.
-- Hao Wang, Popular Lectures in Mathematical Logic  (1981), p. 11

I am not a mathematician, but a math groupie;  not even a math wannabe (as I once was, back in Math 55), but a math wannedabe.
And in fact, considered coldly, I did not then  even rise to the level of a wannabe, but only a meta-wannabe, a wannawannabe:  someone who wished that his dearest wish was for mathematics, but who, truth to tell, was more interested in history and literature.

[Cf. Minimalism in Mathematics]
On Ramanujan’s notebooks:

There were thousands of theorems, corollaries, and examples.  For page after page, they stretched on, rarely watered down by proof or explanation, almost aphoristic in their compression, all their mathematical truths  boiled down to a line or two.
-- Robert Kanigel, The Man who Knew Infinity, p. 204

The reasons for this were twofold.  Ramanujan himself was not particularly aphoristic.   But he had never absorbed the modern notion of proof, which would take up so much more space;  and as a poor man in India, he suffered from a shortage of paper.

Für psychologisch tiefgreifende Krimis,
in pikanter amerikanischer Mundart,
und christlich gesinnt,
klicken Sie bitte hier:

[Sui generis]

On Ramanujan, who grew up in India, and in mathematics  was largely self-taught:

He was like a species that had branched off from the main evolutionary line  and, like an Australian echidna or a Galapagos tortoise, had come to occupy a biological niche all his own.
-- Robert Kanigel, The Man who Knew Infinity (1991), p. 61

The unusual career of Ramanujan  is one of the most celebrated biographies in the history of mathematics.   His achievements in the face of relative intellectual adversity as a child of modest means in the rural subcontinent,  are indeed inspiring, and warm the hearts of those in quest of Diversity -- whence, for those who can decipher the trobar clus of modern peri-academic patois, the book’s subtitle,  “A Life of the Genius Ramanujan”.   (Genius he indisputably was;  but the word these days is mainly used to celebrate anyone other than straight white males -- a “genius at basketball” or whatever.)
Yet the larger lesson is not how divergent Ramanujan was, but how much in the mainstream of things:  He did not found a new field of mathematics, he worked within number theory.   And this fact in turn reminds us of two characterizations of math as a whole, on which we have often dwelt:
            (a)  It is not something we invent out of whole cloth, it is something we discover.   This must channel our discoveries, just as the facts of the actual universe  discipline physics.
            (b)  Mathematics has already, for at least two hundred years, been uniquely rich conceptually  among human endeavors.   In a landscape embracing Cantorian set theory, algebraic geometry, and topos theory, it is next to impossible to come up with something unprecedentedly deep and strange;  in any case, Ramanujan did not.   To return to the metaphor:  We certainly treasure our quirky friend the echidna;  but only to someone whose zoological experience extended no further than a European barnyard, would he seem all that aberrant.   In a world of social insects, benthic hypothermophiles, and communal slime-molds, the echidna seems like just one more furry friend.

~  Posthumous Endorsement ~
"If I were alive today, and in the mood for a mystery,
this is what I'd be reading: "
(My name is Ramanujan, and I approved this message.)
~         ~

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