Lex Fridman’s latest podcast features a nearly four hour long conversation with Edward Frenkel, under the title Reality is a Paradox – Mathematics, Physics, Truth & Love. Normally I’m fairly allergic to hearing mathematicians or physicists publicly sharing their wisdom about the larger human experience (since they tend to have less of it than the average person), and I’m pretty sure I’ve never before listened to a podcast/interview longer than an hour or so. But in this case I listened to and enjoyed the entire thing. Besides sharing Frenkel’s deep interests in the relation of representation theory and quantum mechanics, and views on the unity of mathematics (and physics…), I envy his positive and thoughtful outlook on life and his openness to a range of human experience. The interview left me with a lot to think about and I recommend it highly.

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I mostly enjoy the podcasts by Lex Fridman, also the one with you. So i will listen to the one with Frenkel. I’m not a physicist but i am very interested in the subject. I just recently read the book by Sean Carroll The biggest idea in the universe in which he explains also the mathematics (calculus and linear algebra) behind physics for the layman. The second volume will go deeper into quantum mechanics. How do you feel about this translation work to a greater audience? I think he is doing very important work to make this subject (physics including the mathematics) available to the public.

erik jan bosch,

I don’t want to change the topic here to Sean Carroll and his recent book (which I haven’t read). Almost universally I’ve found physicist’s attempts to write not about physics but about things like the meaning of life to be not worth paying attention to (with Carroll’s “The Big Picture” a good example). That he’s now sticking to the worthwhile (and too difficult for me) task of trying to explain real ideas about physics to a wide audience is a big improvement.

Frenkel is a very unusual example of a scientist who has something to say about the human condition worth listening to, and it’s noteworthy that he doesn’t ground it in “I’m a scientist, know things you don’t about math/physics, so you should pay attention to my views on life in general”.

There was also an interesting interview with Frenkel on the Numberphile podcast, this one only an hour and twenty minutes for those with less time to spare đź™‚

https://www.youtube.com/watch?v=lMwLCSgvxtw&t=2s&ab_channel=Numberphile2

For those with even less time to spare there is this 56 minutes interview of Edward Frenkel by Brian Keating: https://youtu.be/b67o24J3bAs

Frenkel certainly seems like a jolly fellow. The interview sounds almost calculated to attract your specific interest, with his first mathematics epiphany being exposure to representation theory.

It seems Oulipo is having a moment. Not only does Frenkel mention the group, but it also pops up in Sarah Hart’s “Once Upon a Prime”, as reviewed in the Economist. Long ago I read Perec’s “Life, A User’s Manual” (in translation) based solely on the title.

On the subject of complex numbers.

The version of the story of complex numbers that Edward Frenkel gives is historically incorrect, and significantly so. (It’s not necessarily the case that Frenkel is at fault there; it may be that that is how the story had come to Frenkel.)

Generally, what tends to happen in physics/mathematics is this: there is a concept that has many contributors, and as succesive authors retell the story, relying on old memory only, the attribution gets shifted more and more to one persion.

My information source for the story of complex numbers:

‘A Short History of Complex Numbers’ Author: Orlando Merino

https://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf

According to the above account:

Cardano’s contribution to the overall history consisted of developing a systematic method for finding the roots of cubic equations. According to Merion Cardano avoided the class of cases that would involve considering the square root of a negative number. While Cardano’s contribution is vital, it is historically wrong to attribute complex numbers to Cardano.

Rafael Bombelli went a step further:

There are cubic equations that obviously do have a root, but the Cardano formula involves the root of a negative number. Bombelli noticed that if you allow that root of negative number then there in the cases with obvious root that factor subsequently drops out of the calculation, and then it all works out. I infer that at that stage the root of a negative number was admitted, but only in cases where it is seen to drop out of the calculuation.

My impression:

Over several generations of mathematicians there was a gradual lessening of suspicion. Over time mathematicians came to see the root of a negative number something on par with other mathematical concepts.

My point is:

Frenkel comes to the story of complex numbers from musings of how important it is to retain a child-like willingness to consider things that appear to make no sense. Then he enthousiastically gives the Cardano-centered version of the story of complex numbers to illustrate his belief.

Now, I share the concept of child-like openness that Frenkel professes. It’s just: the true history of complex numbers is totally *not* the story to illustrate that.

There is a quip by Mark Twain that goes as follows:

“In the real world, nothing happens at the right place at the right time. It is the job of journalists and historians to correct that.”

It’s funny because it’s true.

Human memory is fallible; human memory is prone to restructuring memories to suit the belief system.

When physics authors write a “historical introduction” they always end up doing the very thing that Mark Twain described. When I read a “historical introduction” in a physics textbook there are almost always significant deviations from the actual history. (Often the deviation will have accumulated from author to author.)

Frenkel’s home page is worth studying.

https://www.edwardfrenkel.com

It seems that Frenkel has a newsletter that people can subscribe to via email.

Cleon Teunissen: Full agreement that complex numbers came about as a result of the need to explain how the Cardano formula, expressed in terms of square roots of a negative number, could possibly produce a positive real number. Tignol (

Galois’ Theory of Algebraic Equations, World Scientific) makes the case very powerfully. However, I think you are not being entirely fair to Cardano. He does not introduce a symbol for the square root of -1. But, in the article you cite, Merino cites a famous example of Cardano’s fromArs Magnaas follows, involving aquadratic, not a cubic, namely $x^2 – 10 x + 40$: “to divide 10 into two parts, the product of which is 40″, and completing the square. (Cardano does not actually write down the quadratic.) This leads to $x = 5 \pm \sqrt{-15}$, and, remarking that we must ”put aside the mental tortures involved”, finding that the product of the roots equals 40. After some discussion, he concludes, “So progresses arithmetic subtlety, the end of which, as is said, is as refined as it is useless.” (See T. R. Witmer’s translation ofThe Great Art, or the Rules of Algebra, G. Cardano, pp. 219-220.) The best account of the long trial to nail down the Cardano formulas, at least in English, is perhaps in M. Livio,The Equation That Couldn’t Be Solved.Off topic: the MJR site was put to rest until the end of summer, seems to be an strange and very unpopular decision. The site was gaining tremendous action and while it hosted some asinine posts, some of the mathematics related posts were very thoughtful and valuable

Z Y,

I’ll be surprised if it gets revived. Whenever I would go take a look, my impression was that the level of dumb was pretty overwhelming. Who would want to spend the time it would take to try to find the few interesting comments? What sensible person who wanted to have an intelligent discussion would comment there?

The fundamental problem with an anonymous site like that is it needs a good moderation system to stop the idiots and trolls from overwhelming the place. I never understood the moderation system in place there, but whatever it was, it was nowhere near good enough. Like everything else in our brave new world, maybe there’s an AI solution in the future…

And yes, off-topic so I’ll moderate out any but the most amazingly insightful discussion of what happened at that site.

Getting back to your summary and link to the Fridman interview… I’m exchanging dueling Youtube links with a friend in the social sciences/humanities. His was Martin Rees on multiverses. Mine was the shortest one here, the interview of Edward Frenkel by Brian Keating.

How would you rate this for a heuristic? “Rees says different universes may be a millimeter apart. In Langlands, our representations may be a prime number apart.”

Obviously I’m using representation and prime quite loosely, as Rees, I assume, was with millimeter. I think still 120 odd years since quanta, it’s difficult to convey the challenge of discreteness when things appear continuous. (Einstein’s middle period frequent quoting of Ernst Mach is one way in.)

My other comments relevant to your summary would be about Frenkel’s use of the term Platonic, and getting beyond it in math foundations. Scanning his book, I don’t think he’s going a direction I would from that (ethics in Greek drama for one), but he is wonderfully open and engaging.

Frenkel’s discussion of math & physics ideas not easy to popularize is close to crystalline in the shortest video. As I haven’t read the book fully, I’ll suppose his philosophical direction is good too. Thanks for posting about it.

kitchin,

The problem with the multiverse of Martin Rees, e.g. at

https://theconversation.com/the-multiverse-our-universe-is-suspiciously-unlikely-to-exist-unless-it-is-one-of-many-200585

is that he has no evidence and no viable theory to point to. He and others going on about this do not seem to be referring to anything well-defined (closest might be the string theory landscape, which is highly problematic).

About deep structures unifying math and physics, when Frenkel points to the Langlands program, he is pointing to something very specific, which we don’t fully understand, but do know a great deal about. If you want to propose thinking of universe as at the real place, other universes at different primes p, at least there is something specific, well-defined, with a lot of interesting structure that you are pointing to. Maybe this it is useless to think about this as a “multiverse”, but at least there is something well-defined to talk about.

Your own experience that you haven’t found mathematicans’ insights on non mathematical valuable is of course inarguable, but the statement

“their wisdom about the larger human experience (since they tend to have less of it than the average person)” makes me sad. I read that as separating the experience of doing (in whatever form) mathematics from “human experience “, and I disagree strongly. Math has an existence beyond us (IMO), but our experience of it is by definition human. I find it rich and rewarding (which is why I am a mathematician) and in my experience it’s deeply connected to the rest of my experience – i see mathematics in the world, and the practice of mathematics informs my worldview. I value a wide range of experience, but mathematicans are among those whose experiences I’m most interested in personally. I’d like to see more of musings on the wider world of the character of, to name a few exemplary cases, the writings of Poincare, Penrose, Ulam, Gromov, and Thurston. I’d also enjoy seeing more mathematicians engaged with the wider world; Frenkel’s doing his part, as are you (PW). I think we have a lot to contribute.