A quiver springs his voice and breast
The March 2007 issue of Notices of the American Mathematical Society is out. In it we find:
- Fascinating conversations with three of the four Fields medalists (guess which one declined to be interviewed?)
- An obituary of George Dantzig (linear programming pioneer), which I found incredibly frustrating for two reasons. First, the article repeatedly sidesteps the most interesting questions about Dantzig’s career: what were those open problems that he solved mistaking them for homework exercises? What impact did his WWII work actually have? Second, just as nothing in biology makes sense except in the light of evolution, so nothing in optimization makes sense except in the light of computational complexity — a topic this 19-page article somehow assiduously avoids.
- A poem by Bill Parry (1934-2006), which stirred my soul as Walt Whitman never did back in 11th-grade English, and which I here reproduce in its entirety.
Argument
As he cleaned the board,
chalk-dust rose like parched mist.
A dry profession, he mused as morosely
they shuffled settling tier upon tier.Now, almost half-way through the course,
(coughs, yawns, and automatic writing)
the theorem is ready.Moving to the crucial point,
the sly unconventional twist,
a quiver springs his voice and breast;soon the gambit will appear
opposed to what’s expected.
The ploy will snip one strand
the entire skein sloughing to the ground.His head turns sympathetically
from board to class.
They copy copiously.
But two, perhaps three pause and frown,wonder will this go through,
questioning this entanglement
— yet they nod encouragement.
Then the final crux; the ropes relax and fall.His reward: two smile, maybe three,
and one is visibly moved.
Q.E.D., the theorem is proved.This was his sole intent.
Leaving the symbols on the board
he departs with a swagger of achievement.
Follow
Comment #1 February 9th, 2007 at 12:17 pm
Hi Scott, you may find this article amusing:
http://www.eetimes.com/news/design/showArticle.jhtml?articleID=197004661
the first commercial quantum computer easily solves NP-hard problems!
Comment #2 February 9th, 2007 at 10:50 pm
Aren’t the Bill Perry poem, the George Dantzig story, and the Fields Medal interviewees all speaking about the same topic, namely, the touching faith of the young that P does equal NP? That faith is at the heart of the Dantzig story. The professor assures the youth that a certificate exists, and buoyed by this faith, the student finds the certificate in polynomial time. To embrace this story, therefore, is to embrace the community-binding ideal that students should have faith that P=NP, and moreover, that students are themselves the embodiment of that P=NP algorithm! I would never condemn that faith — perhaps I still embrace it myself! As one becomes older, however, it slowly becomes easier to embrace Delmore Schwartz’ Calmly We Walk Through This April’s Day
—————The above is intended as a special treat for all you Star Trek fans! 😉
Comment #3 February 10th, 2007 at 6:21 am
The Snopes page cites the two published problems that Dantzig solved:
Dantzig, George B., “On the Non-Existence of Tests of `Student’s’ Hypothesis Having Power Functions Independent of Sigma”, Ann. Math. Stat. 11, 186 (1940).
Dantzig, George B. and Abraham Wald, “On the Fundamental Lemma of Neyman and Pearson”, Ann. Math. Stat. 22, 87 (1951).
Comment #4 February 11th, 2007 at 8:59 am
Scott, thank you for pointing everyone to the Fields Medalist interviews. Multiple comments in that interview made it into my database of quotes (which doesn’t happen very often). Here are some of them:
————
Okounkov: “We [mathematicians] can be better neighbors. We shouldn’t build high fences out of sophisticated words and a ‘you wouldn’t understand’ attitude. We should explain what we know in the simplest possible terms and minimal generality. Then it will be possible to see what grows in the next field and use the fruits of your neighbor’s labor.”
Okounkov: “It is easier to generalize an example than to specialize a theory.”
Okounkov: “Some people can manage without dishwashers, but I think proofs come out a lot cleaner when routine work is automated. This brings up many issues. I am not an expert, but I think we need a symbolic standard to make computer manipulations easier to document and verify. And with all due respect to the free market, perhaps we should not be dependent on commercial software here. An open-source project could, perhaps, find better answers.”
Tao: “Problem-solving and theory-building go hand in hand, though I tend to work on the problems first and then figure out the theory later.”
Tao: “Mathematics has expanded at such a rate that it is no longer possible to be a universalist such as Poincaré and Hilbert. On the other hand, there has also been a significant advance in simplification and insight, so that mathematics that was mysterious in, say, the early twentieth century now appears routine, or more commonly, several difficult pieces of mathematics have been unified into a single difficult piece of mathematics, reducing the total complexity of mathematics significantly. Also, some universal heuristics and themes have emerged that can describe large parts of mathematics quite succinctly; for instance, the theme of passing from local control to global control, or the idea of viewing a space in terms of its functions and sections rather than by its points, lend a clarity to the subject that was not available in the days of Poincaré or Hilbert. So I remain confident that mathematics can remain a unified subject in the future, though our way of understanding it may change dramatically.”
Tao: “I think the next few decades of mathematics will be characterized by interdisciplinary synthesis of disparate fields of mathematics; the emphasis will be less on developing each field as deeply as possible (though this is of course still very important), but rather on uniting their tools and insights to attack problems previously considered beyond reach.”
————
These quotes do a pretty admirable job of summarizing the cognitive and mathematical objectives of quantum system engineering (which is why I too the trouble to enter so many of them!).
And Tao’s comment “There has also been a significant advance in simplification and insight [in that] several difficult pieces of mathematics [are being] unified into a single difficult piece of mathematics” made me LOL. 🙂
Comment #5 February 13th, 2007 at 5:32 am
Terence Tao said (see above): “Some universal heuristics and themes have emerged that can describe large parts of mathematics quite succinctly; for instance, the theme of passing from local control to global control, or the idea of viewing a space in terms of its functions and sections rather than by its points, lend a clarity to the subject that was not available in the days of Poincaré or Hilbert”
If we apply this to computational complexity, then it is pretty clear (IMHO) that “local control” is logic, and “points” are gates. So I would be interested to learn, in the context of computation, what emerging themes map onto Tao’s notions of “global control”, “functional spaces”, and “sections”?
And are these emerging mathematical tools sufficiently powerful to offer a reasonable prospect of understanding computational complexity “quite succinctly”?
The above questions were, to me, the most interesting questions provoked by Tao’s commentary.