The San Francisco Chronicle had a story Saturday about a young man in Danville, Evan O’Dorney, who submitted a math solution to become a finalist for the $100,000 Intel Talent search for students. That has sent me back to review the tape Eric Cohen made of his father Paul J. Cohen at the 2006 Godel Centennial in Vienna, Austria.
Regarding his childhood, it was said that when Paul was a kid the librarian in Brooklyn tried to refuse Paul from checking out certain math books in their collection because they did not believe such a young person could possibly understand them.
Beyond being some sort of advisor or noodge to Steve and Eric’s effort to make a film about their father and the solving of “the continuum hypothesis” I also have an idea that it would be interesting to produce a one-man show about mathematicians, perhaps having an actor try to memorize this address and embue it with as much emotional variety as Paul shows here.
Steve and Eric started their film project shortly before Paul’s initial illness; sadly, the film became something of a documentation of his medical decline, rather than about the math per se; I remember accompanying them from one floor of Stanford Hospital to another, Paul being rolled in a bed, and that the camera was rolling as well. He died in March of 2007. Understandably, the sons have let the project sit since then (besides uploading this lecture — seen 15,000 times already; besides helping sort his books and papers, and helping their mother with the estate).
I like the anecodote earlier in this lecture about “whether it is true or not you are going to keep telling it.”
Young Evan O’Dorney suggests a method to determine whether a fraction will accurately predict irrational numbers like square roots:
Jill Tucker of the Chronicle actually says : “The problem involved the ability to determine whether a fraction will accurately predict irrational numbers like square roots.” Whereas the Intel Contest website bio for Evan says
‘(He) compared continued fraction convergents with iterated linear fraction transformations…(and) in studying two methods for approximating the square root of a non-square integer. One method (continued fractions) is more accurate, while the other (iterated linear transformation) is faster. He discovered exact conditions under which the iteration method produces the same values as the continued fraction method infinitely often”.
Combined together it has me at least skimming if not getting the wikipedia entry on square roots.
Over the course of my five years or so trying to follow and abet Steve and Eric (and I will let alone recollections of trying to stand up to a certain standard while in the presence of Paul) I go in and out of feeling I have any knack for this topic. I read Amir Aczel’s “Mystery of the Aleph” and for a minute there thought I got it. Likewise, “Logicomix: An Epic Search for Truth” by Christos Papadimitriou and Apostolos Doxiadis .
I’m not sure if I want to imply that square roots are synonymous with irrationals. The essence of set theory is the difference between people with rudimentary educations (i.e. took the SATs and got in to college) recalling very few examples of irrationals — square root of two, pi, et cetera — and think of them as exceptional when in fact, as is the actual math, for people who study it, and this is almost the entire point in synecdoche, THE REVERSE IS TRUE, there are many more, infinitely more, irrationals than integers, yes?
Over the years of describing this project to lay people I generally say that Paul’s work has something to do with the nature of infinity, or that it has something to do with the nature of the number line, or that it has something to do with the fact there are types of infinities, or that there are infinitely many types of infinities, or that there are infinitely more irrationals than integers, or that there is an infinite number of points between zero and one, or something. But to pick it up cold again, as Jill Tucker’s story yesterday on Evan O’Dorney sparked me to do, and be accurate in describing any of the math here, can be treacherous. Steve and Eric (and Charles, their younger brother, who has an advanced degree in science but works in finance) are actually very good at explaining math to people. The point would be to make a film that has something for the average viewer but that the math people would appreciate as well.
If time were infinite, I suggest gentle reader do as I do and watch the entirety of the six-part Cohen lecture and then read the Amazon comments on Aczel and “Logicomix.”
Back in the hypothetically infinite of free time world, or perhaps already at it in a parallel world (where we are also simultaneously listening to complete works of The Eels), we should check out David Foster Wallace’s non-fiction book on infinity.
Steve or Eric had a brief correspondence with Wallace in which he declined their invitation to contribute to a lay film project on Paul J. Cohen and the continuum hypothesis; he claimed to be unqualified. “Everything and More: a Compact History of Infinity.”
this is up my alley a play on alan turing, but it came up on a link suggested by my email server based on my mention of “passing strange” indeed:
Someone named East Bay Guy added some math discussion on point to the more than 150 comments on the Chron’s posting of the Evan O’Dorney story. He also provided a link to more info on Evan’s actual work:
For any math people out there, below is a link to the theorems that O’Dorney set out to prove regarding continued fractions. (The link does not actually contain the proof.)
Continued fractions are an interesting branch of mathematics. For instance, consider the square root of 5: 2.2360679775….
This is an irrational number, and the digits to the right of the decimal point will never repeat. This number can be written, however, as the following continued fraction:
2 + (1/(4+1/(4+1/ … )))
You can calculate the part after the plus sign as follows:
– Start with 5.
– Take reciprocal = 0.2.
– Add 4 (giving 4.2).
– Take reciprocal = 0.238…
– Add 4 (giving 4.238…).
– Take reciprocal (giving 0.235955…)
– Continue this as long as you want.
This series approaches 0.2360679…; and when you add 2, you get the square root of 5. The interesting thing here is that this irrational number — a number that appears to have a haphazard collection of digits — is equal to a continued fraction that contains nothing but the integers 2, 1 and 5.
I was still mulling over the distinction between irrationals and square roots.
I think what he may have done, Evan, is notice a pattern between the two existing ways to calculate square roots and articulate something interesting and hopeful about his observation.
This looks like an outline of a talk that young Mr. O’Dorney gave to the Berkeley Math Circle a year ago, the link suggested by “East Bay Guy” above.
Click to access EvanContinuedFractionspdf.pdf
Someone else’s comment on Chronicle story, ccortezz (pun on nautical depths) has me fixing to riff on his math: does he mean half percent i.e. one in two hundred or five hundredths percentile meaning one per twenty thousand? If he is saying that Evan has achieved more in the math world than all other young U.S. math students (and Evan I found placed first in US in “math olympian”, top twenty in world) then it is probably more like one in a million or if he has done the most ever, over a two hundred year period, more like one in ten or a hundred million, like “0.5 %”%% or more. I took the PSATs once and got in the top five percent I think it was which was called “National Merit Honorable Mention” i.e. an order below the famous semi-finalist, which was top half percent. I remember that my Gunn class had 19 semifinalists and Paly had 25 and stole all the headlines but when you calculated it as a percentage of the respective’s senior class it came out identical to two decimals i.e. 19 out of 411 versus 25 out of 535 or whatever. We didn’t have Intel Scholars back then but we did have Andy Grove’s daughter (plus two kids of Paul Cohen, of course).
“It appears Evan is among .05% of the world population. Don’t look down on us kid and enjoy the travels of life.” he said. I also liked the lady who said she was “square rooting” for him to win the award.
For a while there I could easily recite the 19 National Merit Semifinalists in my class — five percent of our high school class was in the top half percent nationally, ten times the average concentration or distribution??
The actual article:
I posted thus, on the Chronicle’s site:
I appreciate this addendum and the article itself. Evan reminds me both of Taylor Eigsti who was a music prodigy that I booked into a concert series when he was 15 and Paul J. Cohen, my classmates’ father, a Stanford math professor who, when he was a kid, the story goes, the public librarian tried to refuse him from checking out certain math books they did not believe young Paul could possibly understand. Jill Tucker’s article also has me reading wiki entries on irrationals versus square roots.
Evan O’Dorney did indeed win the $100,000 top prize in the Intel Talent Search, for his work with the methods of calculating square roots of integers.
Charles Cohen points out that his father won the Westinghouse Talent Search or was a finalist and suggests that the Intel Award referenced above is the same contest, expanded and re-named. Here is a bio on Paul from Stanford:
We are coming up on the fourth anniversary of his passing.
I’d like to produce a one-man show in the theatrical realm in which an actor memorizes this speech and performs it.
Maybe Rinde Eckert would do that.
I should write him.
i found this tribute to paul cohen. wasn’t aware of the 2001 book about Hilbert’s solvers, “the honors class.”
the page could be updated to mention the birth of his first grandchild. mazel tov.
Now there is a new thread here about an eight-year-old with the confusingly similar name Evan Doherty who can land a 720; sounds Euclidian to me.
evan o dorney looking better: