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!
http://worldofdrjustice.blogspot.com/2014/12/a-new-proof-of-existence-of-coffee-cups.html
http://worldofdrjustice.blogspot.com/2014/12/a-new-proof-of-existence-of-coffee-cups.html
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.
http://worldofdrjustice.blogspot.com/2011/12/minimum-axiomatization-for-reality-part.html
"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.
http://worldofdrjustice.blogspot.com/2014/03/on-brilliance-and-boredom.html
http://worldofdrjustice.blogspot.com/2014/03/on-brilliance-and-boredom.html
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
R x R 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 |
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 |
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.
http://worldofdrjustice.blogspot.com/2014/01/language-and-math.html
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| 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.
http://worldofdrjustice.blogspot.com/2014/01/what-is-mathematics-expanded.html| 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.
http://worldofdrjustice.blogspot.com/2014/01/a-dive-to-depths-expanded.html
Further choice morsels
here:
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