In our most recent podcast, “The Chaitin Interview III: The Changing Landscape for Mathematics,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on many things mathematical, including whether math is invented or discovered. This time out, Chaitin talks about why he thinks great books on math, advancing new theorems, aren’t written much any more:
This portion begins at 02:49 min. A partial transcript, Show Notes, and Additional Resources follow.
Robert J. Marks: You don’t hear the word “scholarship” very much anymore in academia.
Gregory Chaitin: And people don’t write books. In the past, some wonderful mathematicians like G. H. Hardy (1877–1947, pictured in 1927) would write wonderful books like A Mathematician’s Apology (1940) or his book on number theory (1938).
Note: Hardy was the mathematician who received gifted young mathematician Srinivasa Ramanujan’s letter from India and “ in 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration.” Ramanujan (1887–1920) was prompted to write to Hardy after reading one of his books. (MacTutor)
Ramanujan was plagued by ill health and Hardy recounted at one point: “I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. ‘No,’ he replied, ‘it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”
On broader issues, Hardy also said, “I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations,’ are simply the notes of our observations.” – A Mathematician’s Apology. So he seems akin to those who think that mathematics is discovered, not invented.
Here’s a short clip from a film made about Ramanujan’s letter and its consequences:
Robert J. Marks: And they’re just beautiful papers. Claude Shannon’s paper is just a wonderful paper in the founding of Shannon Information Theory, and Lotfi Zadeh in 1965 wrote a wonderful paper on the founding of fuzzy or soft logic, and I don’t see those sorts of papers anymore. Everybody’s interested in number of papers more than quality.
Gregory Chaitin: Yeah. Well nobody can work on a difficult project, which may be years until you come up with something or maybe you’ll never come up with something. So if you want to try that kind of stuff you have to do it in parallel with more normal stuff.
I think that Andrew Wiles (pictured in 2005), when he was working in secret for years in his attic on Fermat’s Last Theorem, couldn’t have stopped publishing altogether so he probably had more routine mathematical questions he published on. He probably cursed them because it was taking time away from his great project. But the great project could fail, it could have failed.
Robert J. Marks: He would need something to fall back on. There’s an old academic joke that the Dean can’t read, but the Dean can count. They don’t look at the contents of the paper, but they just count the paper numbers. I think that’s unfortunate and there’s a pressure to reduce the quality of papers.
Gregory Chaitin: There also used to be professors who were wonderful teachers. The students adored them. They learned a lot from them. But they weren’t research mathematicians, for example, and you can’t do that anymore. You don’t get credit for being a wonderful teacher as far as I know or for writing wonderful books. You have to have refereed papers if I’m not mistaken.
I was able to not play according to the current rules because I was being paid to do industrial development work for IBM at IBM research. So I worked on hardware, I worked on hardware design. I worked on the software for new hardware. That was a lot of fun, and I was doing my mathematical research as a hobby basically. So that was how I created an ecological niche for myself, because I had a rather unconventional story as a mathematician. I don’t have a degree. I only have honorary degrees.
Karl Popper was a professor, at first in New Zealand. He had a terribly heavy teaching load. Then fortunately he managed to go to the London School of Economics. There he didn’t have that outrageously heavy teaching load.
Robert J. Marks: We even see this from, I guess, William Sealy Gosset (pictured in 1908) an employee of the Guinness Brewing Company (1899–1937). Gosset came up with this incredible math which is still taught today to undergraduates. He couldn’t associate his name with the article, so he published under the name Student T — and we still refer to it as the Student T or the T statistic. He was also under pressure not to take time away from his work.
Note: Gosset (1876–1937) has been said to be “One of the greatest minds in 20th Century statistics was not a scholar. He brewed beer.
“Guinness brewer William S. Gosset’s work is responsible for inspiring the concept of statistical significance, industrial quality control, efficient design of experiments and, not least of all, consistently great tasting beer.
“But Gosset is certainly no household name. Books and articles about him are sparse, and he is rarely discussed among history’s most important statisticians. Because he used a pseudonym, his name isn’t even familiar to most people who frequently use his most famous discovery. Gosset is the “student” of the Student’s T-Test, a method for interpreting what can be extrapolated from a small sample of data. – Priceonomics
Robert J. Marks: Today it’s exactly the opposite, by the way. If you’re a mediocre teacher and you publish a lot, you get all sorts of accolades as long as your teaching is acceptable. That’s unfortunate, but that’s the way things have evolved.
Gregory Chaitin: Yeah. Well what can you do?
Robert J. Marks: You got to play the game, I guess, right?
Gregory Chaitin: Well, you don’t have to. Look at Elon Musk. He’s my great hero. He’s a wonderful engineer and he’s a wonderful entrepreneur and he doesn’t follow the rules.
Next: Why Elon Musk, and others like him, can’t afford to follow the rules
Note: The photo of Andrew Wiles in 2005 is copyright C. J. Mozzochi, Princeton N.J.
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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.
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- 00:23 | Introducing Gregory Chaitin
- 00:39 | Is math discovered or invented?
- 02:49 | The pressure to publish papers
- 08:31 | A human-computer symbiosis?
- 13:22 | Computer software proofing mathematics
- 19:45 | Bureaucratic obstacles to genuine research
- The Chaitin Interview Part I
- The Chaitin Interview Part II
- Gregory Chaitin’s Website
- Unravelling Complexity: The Life and Work of Gregory Chaitin, edited by Shyam Wuppuluri and Francisco Antonio Doria
- Proving Darwin: Making Biology Mathematical by Gregory Chaitin
- Henri Poincaré, 19th century French mathematician
- Georg Cantor, German mathematician
- A Mathematician’s Apology by G.H. Hardy
- Claude Shannon, mathematician, “the father of information theory”
- Lofti Zadeh, world-renowned computer scientist
- Andrew Wiles, English mathematician
- Karl Popper, Austrian-British philosopher
- William Sealy Gosset, statistician and chemist
- Elon Musk, engineer and entrepreneur
- Kurt Gödel, Austrian-born mathematician
- Alan Turing, mathematician and philosopher
- Stephen Wolfram, computer scientist and physicist
- Leonard Euler, Swiss mathematician and physicist
- Carl Friedrich Gauss, German mathematician and physicist
- Srinivasa Ramanujan, Indian mathematician