Not Even Wrong

A couple weeks ago rumors were circulating that Yitang Zhang was claiming a proof of a longstanding open conjecture in number theory, the “no Landau-Siegel zeros” conjecture. Such a proof would be a very major new result. Zhang was a little-known mathematician back in 2013 when he announced a proof of another very major result, on the twin prime conjecture. Before that, he had a 2007 arXiv preprint claiming a proof of the Landau-Siegel zeros conjecture, but this was never published and known to experts to have problems such that at best the argument was incomplete.

Zhang now has a new paper on the arXiv, claiming a complete proof. The strategy of the proof is the same as in the earlier paper, but he now believes that he has a complete argument. At 110 pages the argument in the paper is quite long and intricate. It may take experts a while to go through it carefully and check it. Note that this is a very different story than the Mochizuki/abc conjecture story: Zhang’s argument use conventional methods and is written out carefully in a manner that should allow experts to readily follow it and check it.

For an explanation of what the conjecture says and what its significance is, I’m not competent to do much more than refer you to the relevant Wikipedia article. For a MathOverflow discussion of the problems with the earlier proof, see here, for consequences of the new proof, see here.

Update: I’m hearing that the above is not quite right, that what Zhang proves is weaker than the conjecture, although strong enough for many of its interesting implications. Perhaps someone better informed can explain the difference…

Update: Davide Castelvecchi at Nature has a news story here.

Update: Via David Roberts on Twitter, Zhang answers some questions about the paper here.

The arXiv today has a very comprehensive survey of a conventional point of view on where “Physical Mathematics” is in 2022 and where it is going, written by a group of six authors. “Physical Mathematics” is a term popularized by one of them, Greg Moore (see here and here, with some commentary here), and it’s an expansion of a Snowmass white paper. A separate paper by Nikita Nekrasov covering the material listed in Section 10 is advertised as forthcoming with the title “The Ghosts of Past and Future Ideas and Inspirations on Interface of Physics and Mathematics”.

The term “Physical Mathematics” is a play on the more conventional name of “Mathematical Physics” to describe work being done at the intersection of math and physics. In its usage by Moore et al. it refers to a point of view on the relation of math and physics which heavily emphasizes certain specific topics that have been worked on intensively during the last four decades. These topics mostly have roots in seminal ideas of Witten and his collaborators, and involve calculational methods developed in quantum field theory and string theory research. The huge volume of this research is reflected in the fact that the survey reference section contains 62 pages giving 1276 separate references. A major problem for anyone taking up an interest in this field has been the sheer scale and complexity of all this work, and this survey should be helpful in providing an overview.

While some of these 1276 papers could equally well be simply characterized as “Mathematics”, it’s hard to describe exactly what makes a lot of the rest “Physical Mathematics” rather than “Physics”. Part of the answer is that these are not physics papers because they don’t answer a question about physics. A striking aspect of the survey is that while a lot of it is about QFT, the only mention at all of the QFT that governs fundamental physics (the standard model) is in a mention of one paper relevant to some supersymmetric extensions of the SM. The only other possible connection to fundamental physics I noticed was about the landscape/swampland, something only a vanishingly small number of people take seriously.

Also striking is the description of the relation of this field to string theory: while much of it was motivated by attempts to understand what string/M-theory really is, section 3.1 asks “What Is The Definition Of String Theory And M-Theory?” and answers with a doubly-boxed

We don’t know.

with commentary:

This is a fundamental question on which relatively little work is currently being done, presumably because nobody has any good new ideas.

In the background of this entire subject is the 1995 conjecture that there is a unique M-theory which explains various dualities as well as providing a unified fundamental theory. After nearly 30 years of fruitless looking for this, the evidence is now that there is no such thing, and maybe the way forward is to abandon the M-theory conjecture and focus on other ways of understanding the patterns that have been found.

I share a faith in the existence of deep connections between math and physics with those doing this kind of research. But the sorts of directions I find promising are very different than the ones being advertised in this survey. More specifically, I’m referring to:

• the very special chiral twistor geometry of four-dimensions (no twistors in the survey)
• the subtle relation of Euclidean and Minkowski signature (only a mention of the recent Kontsevich-Segal paper in the survey)
• the central nature of representation theory in quantum physics and number theory (very little representation theory in the survey)

Looking back at Greg Moore’s similar 2014 survey, I find that significantly more congenial, with a more promising take on future directions (in particular he emphasizes the role of geometric representation theory).

There’s a long interview with David E. Kaplan (not the same person as David B. Kaplan…) by David Zierler at the AIP Oral Histories site. The whole thing is quite interesting and I recommend reading it, but I do want to point out that it shows that I’m a voice of moderation on the string theory issue. Some extracts follow:

About Ann Nelson and string theory in the 1990s:

She was extremely dismissive of string theory, and thought it was—you know, there was—my impression from her and from other people of that generation that weren’t doing string theory was that the string theorists were colluding in a sense, or were dismissing anything but string theory, and deciding that if you did string theory then you’re much smarter than the people who are not doing string theory. There was some unhappiness in the theoretical field. And the cancellation of the SSC probably added to that tension between the two.

But I don’t think she came of it from taking a side. I think she looked at the situation and said, “String theory is total bullshit.” In the mid-’80s, there were some realizations—there were some consistency checks that kind of worked in string theory, and people got super excited. Oh, my god, string—yeah, it could be the, you know, underlying thing to particle physics. But that was it.

The successes after that were few and far between. But there was an obsessive—like we’re studying the theory of quantum gravity. And it was deridingly called the theory of everything. And then they took that on, you know. We’re studying the theory of everything. And then the young people who want to do the greatest stuff would go to string theory. And there was a concern and some upset by the people not doing string theory that they’re absorbing a lot of people to do this crap, which is not very physics like. “It’s I believe the theory, and so I’m going to study all aspects of it, and maybe one day we’ll connect it with the physical world.” As opposed to I believe in the phenomenon, and I’m trying to explain that and more, and so I’m going to try out different theories and see what they’re consequences are.

And now I look back, and it’s obvious that string theory was bullshit in the sense of there were so many people working on it, and they were not manifesting any real progress externally. It was all internal consistency checks and things like that. And so at the time, you know, whenever it came up—and it didn’t come up much because there were no string theorists in Seattle—she was just very dismissive, like, you know, “What are those people doing? I don’t know what they’re doing.” [laugh]

About being a postdoc at SLAC:

There were a lot of string theorists at Stanford. I didn’t understand any of those talks. Or sometimes when the talks were not in strings, Lenny Susskind would yell at the speaker that this is bullshit or whatever, da, da, da, da—you know, abusive at some level. So Stanford was weird in that way.

About realizing what was going on in string theory, his evaluation of past (Strominger-Vafa) and current claims about string theory and black holes:

But—so I don’t—and it’s part of probably why I didn’t understand—I didn’t think of myself as a physicist because there’s a lot of physicists working very hard on what? I don’t know what they’re working on. It’s not—you know, I used to just think I’m too stupid to understand what they’re working on. And finally reading some of those papers, they’re not what—it’s stupid. There’s a lot of stupid stuff in there. String theory really is just stupid. It’s unbelievably stupid. There’s so many people who are working on it that don’t actually know physics that they can’t even describe a physical characteristic of the thing they’re calculating. They’re missing the whole thing.

…

So that’s when I realized string theory is like a video game. There are people just addicted to it. That’s all that’s happening. And it’s couched in the theory of everything and da, da, da, da.

So that’s all. I just kind of—I learned quite a bit about these things. And then I saw the people like Lenny Susskind, who was terrorizing people who work on regular physics, as just a plain asshole. That there are actual people who are deciding string theory’s important, wanting to do string theory, and they’re even protecting the field. And some of those people are talking about how entropy now of a black hole can be described as a geometric thing, an entanglement, and that Hawking’s paradox about evaporating black holes is really wormholes, virtual wormholes coming from the inside to the outside, and all kinds of language. And you could test information theory of black holes using atomic physics experiments. And it’s literally bullshit.

There are people—prominent people—in physics who say, “I’m applying for this money from the DOE, but I know it’s bullshit.” And then there are experimental atomic physicists who don’t know and are shocked to learn that “What? String theorists don’t have a Hamiltonian? They don’t actually have a [laugh] description? What am I testing?”

So I have converted a little bit to the opinions of my predecessors, only because I’ve actually done the work. I’ve actually tried to understand black holes of late, and I’ve gone back to those papers which are the breakthrough, celebrated, amazing papers about black holes, and there’s nothing in them. It’s really—it’s just a very simplistic picture where, look, if you take this hyper-simplistic picture, these numbers match these numbers, which means thinking about a black hole having entropy is correct, da, da, da, da, da.

No matter that the black hole they’re talking about is extremal. It doesn’t actually Hawking radiate. It’s a totally hyper-supersymmetric, multiple charges, free parameters. So now that I’ve finally dug into it, I realize that—not that all humanities fields are bad. But it’s much more like a humanities field where there are the prominent people in the field, and they decide what’s interesting. And that if you impress those people, you can get ahead. But that dictates then what research is done. And they’re not going to discover anything in that context. They’re not going to get anywhere. There’s not a lot of people doing—you know—thinking outside the box or just thinking diff…you know, doing different things, you know.

About the argument that string theory must be worthwhile because lots of people are doing it:

Zierler:

What is your response to a string theorist who would say, and I know this because one has said this to me, “Look, four people were doing this in 1968, 20 people were doing it in 1984, 1,000 people were doing it in 2000, and now there’s 6,000 people who are doing string theory all over the world. And that’s proof that there’s something here that’s worthwhile”? What is your response to that line of reasoning?

Kaplan:

[laugh] Take those numbers, continue the exponential, and apply it to Christianity—

Zierler:

[laugh]

Kaplan:

—and Islam and Judaism and Buddhism. Give me a fucking break. They’re describing a religion that can attract and addict people. That is exactly the kind of statement that shows it’s bullshit and non-scientific. They’ve proven it for me that they are not about discovering something. They’re about dominating the field for the purpose of what? That’s proof? Give me a break. Give me a fucking break. Slavery was very popular, and became widely used. Nazism. Come on. You can take extreme examples and show that that is so non-scientific and sick that the progress they have made is to get more people to work on something that isn’t producing anything. Oh, man, I wish you didn’t tell me that. [laugh]

About the current state of the field:

There are so many things to think about. I don’t know what narrowed our field. I don’t see it as we’re dying because we’re coming to the limits of what we can do, the limits of what we can calculate in string theory, and the limits of how big of a ring we can build. I think most people are just doing useless stuff.

…

And so that’s why I—the whole depression or whatever, that’s a product of the non-willingness to feel stupid by the majority of our field. Expertise is more important to them than discovery. And that’s what I think is happening. And so what we’re seeing is not the death of the field, but the death of a direction that is being committed to by 98% of them.

A question that has always fascinated me about mathematics is that of how the field manages to stay healthy and not degenerate in the way I’ve seen theoretical physics do as it lost new input from experiment. On Twitter, Ash Joglekar gave a wonderful quote from von Neumann that addresses this question. The quote was from a 1947 essay “The Mathematician” (available here and here). von Neumann argues that:

…mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to an empirical science.

but warns

As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from “reality” it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l’art pour l’art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities.

which describes all too well what has happened to string theory. What saves a field from this? “Men with an exceptionally well-developed taste”? He poses the general question this way:

What is the mathematician’s normal relationship to his subject? What are his criteria of success, of desirability? What influences, what considerations, control and direct his effort?

Normally mathematicians are loath to debate this kind of “soft” topic, but the rise of computer software capable of producing proofs has recently led several first-rate mathematicians to take an interest. Each year the Fields Institute in Toronto organizes a Fields Medal Symposium, structured around the interests of a recent Fields Medalist. This year it’s Akshay Venkatesh, and the symposium will be devoted to questions about the changing nature of mathematical research, specifically the implications of this kind of computer software. Last year Venkatesh wrote an essay exploring the possible significance of the development of what he called “Alephzero” (denoted $\aleph(0)$):

Our starting point is to imagine that $\aleph(0)$ teaches itself high school and college mathematics and works its way through all of the exercises in the Springer-Verlag Graduate Texts in Mathematics series. The next morning, it is let loose upon the world – mathematicians download its children and run them with our own computing resources. What happens next – in the subsequent decade, say?

Among the organizers of the conference is Michael Harris, who has written extensively about mathematical research and issues of value in mathematics. Recently he has been writing about the computer program question at his substack Silicon Reckoner, with the most recent entry focusing on Venkatesh’s essay and the upcoming symposium.

One of the speakers at the symposium will be Fields medalist Tim Gowers, who will be addressing the “taste” issue with Is mathematical interest just a matter of taste?. Gowers is now at the Collège de France, where he is running a seminar on La philosophie de la pratique des mathématiques.

I’ve tried asking some of my colleagues what they think of all this activity, most common response so far is “why aren’t they proving theorems instead of spending their time talking about this?”

Update: For yet more about this happening at the same time, there’s a talk this afternoon by Michael Douglas on “How will we do mathematics in 2030?”.

Update: The talks from the Fields Institute program are now available online.

Terry Tao is one of the organizers of a planned February workshop at UCLA involving many of the same people, much the same topic.

Heading to Oxford today, this evening I’ll give a talk there on Unified Theories of Physics. On Saturday I’ll try to find some way to get to the HTLGI Festival in London despite a national rail strike, where I’ll give a talk on Saturday and be on two panel discussions Sunday.

I’ll post slides after the talk tonight, one theme of which will be the failure of a series of attempts to extend the Standard Model, all of which were started in the mid-1970s (GUTs, SUSY, string theory). An opinion piece by Sabine Hossenfelder appeared yesterday in the Guardian, which takes a similar point of view on the current fate of extensions of the SM, but I strongly disagree with a lot of what she has to say.

The bad theory activity she points to has been going on for decades, but in recent years it seems to me to be a lot less popular. Most influential theorists have (quietly) agreed with her that particle physics is dead. In attacking bad model building in particle physics, I think she’s going after a small group of stragglers, not the center of theoretical activity (which has problems much more worth discussing).

What I most disagree with her about though is her treatment of HEP experiment and experimentalists. Yes, one can find people who have used bad theory to make bad arguments for building a new machine, but I don’t think those have been of much significance. For more on the current debate about this, see here. At the present time though, no one is spending money on building a new energy frontier machine any time soon. Money is being spent on running the LHC at high luminosity (CERN) and studying neutrinos (US), as well as studying the possibilities for going to higher energy. All of these activities are valuable and well-justified.

The LHC has been a huge success so far, with the old claims that it was going to see extra dimensions an embarrassment which doesn’t change the science that has happened. The discovery of the Higgs was a huge advance for the field, and the on-going effort to study its properties in detail is important. Another huge advance for the field has been the careful investigation of the new energy range opened up by the LHC, shooting down a lot of bad theory. Pre-LHC, the most influential theorists in the world heavily promoted dubious SUSY extensions of the SM, making these arguably the dominant paradigm in the field. LHC experimentalists have blown huge holes in that bandwagon, in some sense by doing exactly what Hossenfelder complains about (looking for evidence of badly motivated theories of new particles). In this story they’re not the problem, they’re the solution.

I’ll be busy this week with the talks mentioned and with attending math talks in Oxford, so little time to discuss more here or do a good job moderating a discussion. So, behave.

Update: The slides from the Oxford talk are here.

Update: Sabine has a blog entry more carefully explaining her point of view here.

Update: Some coverage of this at Physics World.

Update: More discussion of this from Ethan Siegel here, response from Sabine here.