In last week’s podcast,, “The Chaitin Interview II: Defining Randomness,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on his method of describing true randomness:. If no theory is simpler than the data you are trying to explain, then the data is random. They also discussed the work of true randomness but also on how Ray Solomonoff (1926–2009), another algorithmic information theory founder, who pursued the “shortest effective string of information that describes an object.” But now, for a lighter touch, we learn that a musical comedy was made of Fermat’s Last Theorem.
This portion begins at 19:24 min. A partial transcript, Show Notes, and Additional Resources follow.
Robert J. Marks: If you give a test where all the problems are simple, you get kind of a histogram with a little peak. If you make them all hard, you get another peak on the other end. So an ideal test should have a gradient. I tell the students that there’s going to be some simple ones, some medium ones, and some hard ones. Sometimes I ask questions which I don’t know the answer for, so I tell them, if you get the answer to some of the harder questions, we have a publication. I think that that’s kind of what you did at the Columbia entrance test, right?
Gregory Chaitin: Yeah. Well, that reminds me of a joke of my late friend Jacob Schwartz, a mathematician at Courant Institute. He floated the idea of putting Fermat’s last theorem in an important exam in mathematics, in the hope that some undergraduate would come up with a a wonderful short proof — Fermat claimed he had a short proof — not knowing that this was an immensely hard problem that many famous mathematicians had worked on unsuccessfully for a long time. But that’s not how it was solved. It was solved with a very fine, sophisticated mathematician working on it in secret for years, and it’s a very long proof, [Andrew] Wiles’s proof.
Note: “Fermat’s last theorem is a theorem first proposed by Pierre de Fermat in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica by Diophantus. The scribbled note was discovered posthumously, and the original is now lost. However, a copy was preserved in a book published by Fermat’s son. In the note, Fermat claimed to have discovered a proof that the Diophantine equation x^n+y^n=z^n has no integer solutions for n>2 and x,y,z!=0.” – “Fermat’s Last Theorem,” Wolfram MathWorld
Robert J. Marks: Amazing. I guess Fermat was wrong when he said he could fit the proof in the margin.
Gregory Chaitin: You know, that’s an interesting historical question. When Fermat said he had a proof, he always had a proof, I think. The only case that was left hanging was that one. He was a superb mathematician, Fermat, so I personally think he had a proof, but we haven’t figured it out. It’s based on different ideas than Wiles’s proof, because those concepts didn’t exist at that time. But I could be wrong.
Note: Pierre de Fermat (1607–1665, pictured) “effectively invented modern number theory.” He “devised a wide range of conjectures and theorems. He is also given credit for early developments that led to modern calculus, and for early progress in probability theory.” – The Story of Mathematics
Gregory Chaitin: By the way, there’s a lovely musical comedy about all of this called Fermat’s Last Tango and it’s available on the web. It’s a conflict between the ghost of Fermat, who doesn’t want Wiles to find the proof, and Wiles — and Wiles’s wife, who would like Wiles to come back to earth because he’s working all the time in secret, and she doesn’t get to see him very much, nor do his children.
It’s great fun. It’s a musical comedy. It’s written by someone who knows mathematics, so the jokes are all good math jokes.
Robert J. Marks: I’ve got to ask, was it successful? It seems that the audience would be somewhat limited.
Gregory Chaitin: People were falling off their seats. It was wonderful. They’re not terribly sophisticated math jokes but they’re all correct, and the songs are correct, and the history that they give. There’s a song where Fermat is taunting Wiles: Your proof has got a hole.
It’s very clever, and they also have Heaven where there are the ghosts of Euclid, Gauss, Pythagoras looking over all of this. And the different styles of music, it’s great fun. They’re dancing also. It’s wonderful.
The Clay Mathematics Institute, the one that has the Clay prizes for a million dollars for those very hard problems, paid the money to make a DVD, and then somebody put it on YouTube.
Gregory Chaitin: I recommend it highly. Oh, and then there’s a song where Fermat is taunting Wiles again, to try to keep him from finding his proof, saying mathematics is a young man’s game, and how old are you?
Note: The Clay Mathematics Institute offers a number of programs to assist the development of young math leaders. That includes seven Millennium Prize Problems. Rules. More information. One of the problem s was solved by Grigoriy Perelman of St. Petersburg, Russia (the Poincaré Conjecture, 2010). But he declined the $1 million in prize money so it was used to assist young math scholars.
Previous: Chaitin’s discovery of a way of describing true randomness. He found that concepts f rom computer programming worked well because, if the data is not random, the program should be smaller than the data. So, Chaitin on randomness: The simplest theory is best; if no theory is simpler than the data you are trying to explain, then the data is random.
How did Ray Solomonoff kickstart algorithmic information theory? He started off the long pursuit of the shortest effective string of information that describes an object. Gregory Chaitin reminisces on his interactions with Ray Solomonoff and Marvin Minsky, fellow founders of Algorithmic Information Theory.
Here are the stories, with links, to an earlier recent podcast discussion with Gregory Chaitin:
Gregory Chaitin’s “almost” meeting with Kurt Gödel. This hard-to-find anecdote gives some sense of the encouraging but eccentric math genius. Chaitin recalls, based on this and other episodes, “There was a surreal quality to Gödel and to communicating with Gödel.”
Gregory Chaitin on the great mathematicians, East and West: Himself a “game-changer” in mathematics, Chaitin muses on what made the great thinkers stand out. Chaitin discusses the almost supernatural awareness some mathematicians have had of the foundations of our shared reality in the mathematics of the universe.
How Kurt Gödel destroyed a popular form of atheism. We don’t hear much about logical positivism now but it was very fashionable in the early twentieth century. Gödel’s incompleteness theorems showed that we cannot devise a complete set of axioms that accounts for all of reality — bad news for positivist atheism.
You may also wish to read: Things exist that are unknowable: A tutorial on Chaitin’s number (Robert J. Marks)
Five surprising facts about famous scientists we bet you never knew: How about juggling, riding a unicycle, and playing bongo? Or catching criminals or cracking safes? Or believing devoutly in God… (Robert J. Marks)
- 00:27 | Introducing Gregory Chaitin
- 01:12 | Chaitin’s landmark paper published in his teen years
- 02:04 | Chaitin’s definition of randomness
- 08:43 | Metaphysics
- 10:30 | Chaitin’s philosophical interest
- 19:24 | Fermat’s Last Theorem
- Gregory Chaitin’s Website
- “On the length of programs for computing finite binary sequences,” by Gregory Chaitin, published when he as a teenager. (Journal of the ACM (JACM) 13, no. 4 (1966): 547-569).
- Unravelling Complexity: The Life and Work of Gregory Chaitin, edited by Shyam Wuppuluri and Francisco Antonio Doria
- Karl Popper
- The Logic of Scientific Discovery by Karl Popper
- Hermann Weyl
- Gottfried Wilhelm Leibniz
- Leibniz’s Discours de Métaphysique (in English)
- “A Theory of Program Size Formally Identical to Information Theory” by Gregory Chaitin (Journal of the ACM (JACM) 22, no. 3 (1975): 309-440).
- Ray Solomonoff
- Marvin Minsky
- Fermat’s Last Tango on YouTube
(Portions of this transcript have been altered to clarify the content).