[FRIAM] The Weil Conjectures

Wed Nov 6 15:27:31 EST 2019

Hi, Frank, 

Last Thursday, in order to stir up a conversation amongst the mathematicians
and the humanists, I forwarded the attached to the FRIAM list.  It is a book
about the intellectual/emotional relationship between the two siblings, the
mathematician Andre Weil, and his sister Simone.  When I got home, I
realized that absolutely nobody had mentioned it.  Now, the experience of
being ignored is not ENTIRELY unfamiliar to me, but I did begin to wonder if
the message had never gone out on the list.   So, if you don't get this from
the list, but only directly, could you forward it on to the list?  You might
want to edit a bit, first, to take out some of the nesting.  

Oh, and . don't forget to read it yourself.  You were one of the people I
had in mind when I wrote it. 

Nick 

Nicholas S. Thompson

Emeritus Professor of Psychology and Biology

Clark University

 <http://home.earthlink.net/~nickthompson/naturaldesigns/>
http://home.earthlink.net/~nickthompson/naturaldesigns/

From: Nick Thompson [mailto:nickthompson at earthlink.net] 
Sent: Thursday, October 31, 2019 3:21 PM
To: Friam (Friam at redfish.com) <Friam at redfish.com>
Cc: 'Jon Zingale' <jonzingale at gmail.com>
Subject: Document8

Dear Phellow Phriammers, 

So I have been reading, Olsson, K. The Weil Conjectures
<https://www.amazon.com/Weil-Conjectures-Math-Pursuit/dp/0374287619> : On
the pursuit of math and the unknown. New York, NY Farrar Straus Giroux.
2019.  Not only does the book evoke the chasm that mathematical genius
excavates around itself, but it also has many interesting meditations on
conjecture, metaphor, and scientific creativity generally.  I keyed in some
passages, attached them, and will copy them in below 

Love to hear some comments tomorrow.  Also, I am hoping for a continuation
of our discussion of Darwin's Origin as a metaphor.  

Nick 

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Selections From, Olsson, K. The Weil Conjectures: On the pursuit of math and
the unknown. New York, NY Farrar Straus Giroux.  2019. 

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PP 41-2

            The word conjecture derives from a root notion of throwing or
casting things together, and over the centuries it has referred to
prophecies as well as to reasoned judgments, tentative conclusions,
whole-cloth inventions, and wild guesses.  "Since I have mingled celestial
physics with astronomy in this work, no one should be surprised at a certain
amount of conjecture," wrote Johannes Kepler in his Astronomia Nova of
1609."This is the nature of physics, of medicine, and all the sciences which
make use of other axioms beside the most certain evidence of the eyes."
Here conjecture allows him to press past the visible, to sacrifice the
certainty of witnessing for the depth and predictive power of theory.
There's another definition of conjecture that means something inferred from
signs or omens (for example, from a Renaissance work on occult philosophy:
"Whence did Melampus, the Augur, conjecture at the slaughter of the Greeks
by the flight of little birds.").

            Elsewhere it's hokum, claptrap, bull: "Conjecture, which is only
a feeble supposition, counterfeits faith; as a flatterer counterfeits a
friend, and the wolf the dog" wrote one early Christian theologian.  So it's
a word with contradictory meanings since at times conjecture carries the
weight of reasoning behind it, and at other times it's a wild statement, an
unfounded claim.  Good thinking or bad, clever speculation or a reckless
mental leap. 

            In contemporary mathematics, conjectures present blue-prints for
theorems, ideas that have taken on weight but haven't been proved.  Couched
in the conditional, they establish a provisional communication between what
can be firmly established and might turn out to be the case.  More than a
guess, conjecture in this sense is a reasoned wager about what's true.  

            A rough draft.  A trial balloon.  It seems to me laced with
optimism, a bullishness about what could, in the future, come more fully to
sight. 

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P 54

            Strategies for tackling problems, from Polya's How to Solve it:
Do you know a related problem.  Look at the unknown! Here is a problem
related to yours and solved before.  Could you use it? 

            But what about the problem of too many related problems? My
weakness for juxtaposition: I'll sense that one thing might be illuminated
by another thing and go chasing the other thing. .For better or worse, a
light paranoia goads me along.  Maybe it's all connected! This and this and
this and - look, over there - that. There's the bringing together of
disparate elements that informs a conjecture, and then there's the mental
nausea brought on by the fact that there's too much out there to know.  Not
grasping but googling.  I can't always tell one from the other.  

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Pp 95-6

            At last Andre goes on the offensive,, that is to say, he answers
Simone's repeated requests with a long, technical description of some of
this mathematical work, a treatise in the form of a letter.  .He knows full
well that she won't understand these 'thoughts", as he calls the: " I
decided to write them down, even if for the most part they are
incomprehensible to you.  He plunges into a density of terms shouldn't know,
with only minimal efforts to say what he means by quadratic residues, nth
roots of unity, extension fields, elliptical functions. 

In the first have of his letter he sketches a historical context for his
work, starting with the 109th century watershed in algebra, that leap by
which mathematicians inverted the problem of solving equations within given
domains by construing domains in which given equations had solutions.  He
alludes to a time when questions about numbers began to rub up against
questions about equations or functions in new ways. "Around 1820,
mathematicians (Gauss, Abel, Galois, Jacobi) permitted themselves, with
anguish and delight, to be guided by the analogy between the division of the
circle . and the division of elliptical functions," he writes. 

            Anguish and delight! As he's laying out his none too explanatory
explanation of his research, Andre emphasizes the role of analogy in
mathematics-which his sister might appreciate even if the rest of it flies
right over her head.  Here, analogy is not merely cerebral.  The hunch of a
connection between two different theories is something felt, a shiver of
intuition.  For as long as the connection is suspected but not entirely
clear, the two theories engage in a kind of passionate courtship,
characterized by "their conflicts and their delicious reciprocal
reflections, their furtive caresses, their inexplicable quarrels," he
writes. "Nothing is more fecund than these slightly adulterous
relationships." 

            Analogy becomes a version of eros, a glimpse that sparks desire.
"Intuition makes much of it; I mean by this the faculty of seeing a
connection between things that in appearance are completely different; it
does not fail to lead us astray quite often."  This, of course, describes
more than mathematics; it expresses an aspect of thinking itself -- how
creative thought rests on the making of unlikely connections.  The flash of
insight, how often it leads us off course, and still we chase after it.  

`

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