Arcadian Functor

occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

Wednesday, January 23, 2008

Brave New World

I joined Facebook just to look at a few photos, but then momentarily found myself drowning in the temptations of procrastination, so I joined the group The Petition for Alexander Grothendieck to Return From Exile. A local paper ran a feature last week on the Evil of Facebook, but the evils discussed sounded more like a list of shallow 20th century social constructs, ever present on the web due to the conditioning of its participants.

But a Brave New World could lead to an even braver New World, if we so desired. We can create the benevolent Big Sister, one who looks back on the 20th century respectfully, but in dismay. For instance, much funding for such sites clearly comes from advertising revenue. How can we remove advertising from the forums of the future? Personally, I don't want to discuss life, the universe and everything whilst being bombarded by pictures of women wearing underwear that doesn't fit properly. I went into one of those popular lingerie stores once and was disgusted to find that none of the expensive items was well made, or fitted. That implies that their revenue is generated entirely by trend value, gladly paid by women, many of whom are technically living below the poverty line and should be spending the money on more fresh fruit and vegetables.

I don't see why Alexander Grothendieck would want to return from exile.


Anonymous Tony Smith said...

Since I am not big on registering with commercial big-money sites requiring javascript etc,
I am ignorant about "The Petition for Alexander Grothendieck to Return from Exile" (why don't the people there just set up an open blog about it?),
even though I am ignorant,
here are some thoughts as to
why Grothendieck might want to come out of exile now::

Allyn Jackson, in Notices of the AMS November 2004, said:
"... David Mumford of Brown University ... said ... “He [Grothendieck] doesn’t think concretely.”
Consider by contrast the Indian mathematician
Ramanujan, who was intimately familiar with properties of many numbers, some of them huge. That way of thinking represents a world antipodal to that of Grothendieck.
“He really never worked on examples,” Mumford observed. “I only understand things through examples and then gradually make them more abstract. I don’t think it helped Grothendieck
in the least to look at an example. He really got control of the situation by thinking of it in absolutely the most abstract possible way. It’s just very strange. That’s the way his mind worked.”
Norbert A’Campo of the University of Basel once asked Grothendieck about something related to the Platonic solids. Grothendieck advised caution. The Platonic solids are so beautiful and so exceptional, he said, that one cannot assume such exceptional beauty will hold in more general situations.
“L’Enterrement (II) ou La Clef du Yin et du Yang”, which Grothendieck notes is the most personal and deepest part of Récoltes et Semailles, ... uses the “yin-yang” dialectic to analyze different styles of doing mathematics, concluding that his own style is fundamentally “yin”, or feminine. This style is captured in one especially evocative section called
“La mer qui monte…” (“The rising sea…”). He likens his approach to mathematics to a sea:
“The sea advances imperceptibly and without
sound, nothing seems to happen and nothing is disturbed, the water is so far off one hardly hears it. But it ends up surrounding the stubborn substance, which little by little becomes a peninsula, then an island, then an islet, which itself is submerged, as if dissolved by the ocean stretching away as far as the eye can see” ...".

In his 24 May 2003 article "The Rising Sea: Grothendieck on simplicity and generality I" Colin McLarty said:
"... Grothendieck describes two styles in mathematics. ...
Grothendieck ... finds Serre “Super Yang” against his own “Yin” ...
Serre concisely cuts to an answer.
Grothendieck creates truly massive books ...
offering set-theoretically vast yet conceptually simple mathematical systems adapted to express the heart of each matter and dissolve the problems ...[like a]... “rising sea” ...[in which]... the theorem is “submerged and dissolved by some more or less
vast theory, going well beyond the results originally to be established” ...".

Maybe Grothedieck would be happy to read this blog, which describes so many connections between
the Yin of category and topos theory etc (conventionally thought of as abstract)
the Yang of E8 root vector polytopes, Clifford and division algebras, etc (conventionally thought of as concrete examples)
it seems that the rising sea of abstraction and the rocky islets of examples are really a unified whole,
that Grothendieck might like to participate in their harmonious synthesis.

If Grothendieck is on the web where he
"... is now said to be living in a shepherd's hut in the Pyrénées, where he lives on a vegetarian diet and is sustained by meditation. ..."
(quote from David Berlinski, in his review of David Ruelle's book "The Mathematician's Brain")
then if he does read this blog maybe he might think that it exemplifies
“Discovery ...[as]... the privilege of the child ... the child who has no fear of being once again wrong, of looking like an idiot, of not being serious, of not doing things like everyone else.”

That last quote is from Récoltes et Semailles (page 1), as quoted in Allyn Jackson's November 2004 Notices of the AMS article, which goes on to say:

"... Grothendieck puts the highest value on the innocent, childlike curiosity that gives birth to the creative impulse, and he mourns the way it is trampled on by competitiveness and the desire for power and prestige. ...
For the work of discovery and creation, Grothendieck saw intellectual endowment and technical power as secondary to the child’s simple thirst to know and understand.
This child is inside each of us, though it may be marginalized, neglected, or drowned out. ...
One aspect of this childlike curiosity is a scrupulous
fidelity to truth. Grothendieck taught his students
an important discipline when writing about
mathematics: never say anything false. ... It was acceptable to be vague, but when one gives precise details, one must say only things that are true. Indeed, Grothendieck’s life has been a constant search for truth. ...".

Tony Smith

January 23, 2008 7:18 PM  
Blogger Kea said...

If you are not on Facebook, check out Lieven's post.

January 24, 2008 9:36 AM  
Anonymous Tony Smith said...

When I went to the suggested link, I found:

"... We recently spotted Grothendieck in the “Gentleman’s Choice” bar in Montreal, Quebec. ... After a couple of rounds (on us) we were able to convince him to return from exile, under one stipulation - we created a facebook petition with 1729 mathematician members ... 1729 being of course the taxicab-curve number. ...
Last week Grothendieck, or “the ‘Dieck” as we affectionately refer to him, returned to Montreal ... , he explained how he has generalised the theory of schemes even further, to the extent that the Riemann Hypothesis and a Unified Field Theory are both trivial consequences of his work. ...".

In light the fact that 1729 is associated with Ramanujan, and in light of Allyn Jackson's November 2004 Notices of the AMS article saying:
"... Consider ... the Indian mathematician
Ramanujan, who was intimately familiar with properties of many numbers, some of them huge. That way of thinking represents a world antipodal to that of Grothendieck. ..."
also saying
"... Grothendieck’s mode of thinking ... seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”
But Grothendieck must have known that 57 is not
prime, right? Absolutely not, said David Mumford
of Brown University. “He doesn’t think concretely.”...".

it is my opinion that the 1729-member Facebook effort is not serious.

If it were, it would be easy for one human plus 1728 sock-puppets to satisfy the criteria.

Further, I wonder why they did not use a genuinely Grothendieck-related number, such as
the Grothendieck Prime 57 = 3x19.

Still further, although a "... generalised the theory of schemes ..." may well be related to "... the Riemann Hypothesis and a Unified Field Theory ...",
I do not believe that Grothendieck (or anyone else who is informed and serious) would consider them to be "trivial consequences" of anything.

Tony Smith

January 24, 2008 10:39 AM  
Blogger Kea said...

Dear me, Tony, can't we have a bit of fun? AFAIK, Grothendieck is still in the Montpellier district, and not in Canada, but I was hoping it wouldn't be necessary to say so.

And if it is sneakily genuine...

January 24, 2008 10:47 AM  

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