[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; 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:
*
[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 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]
[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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