*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
And:

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; 3^{rd}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 L^{2}to L*spaces motivates the Riesz interpolation theorem.”*^{p}
~

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; 3^{rd}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:

*

[Cf.

__Language and Math__]
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
[Cf. Oligophrenia mathematica]

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

[Cf. Minimalism in Mathematics]

*meta*-wannabe, a*wanna*wannabe: 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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