Not Even Wrong

Recently Jean-Pierre Bourguignon recently gave the Inaugural Atiyah Lecture, with the title What is a Spinor? The title was a reference to a 2013 talk by Atiyah at the IHES with the same title. Bourguignon’s lecture is not yet online, but I realized there are lectures explaining what a spinor is that I can highly recommend: my own, in this semester’s course on the mathemematics of quantum mechanics. I’m closely following the textbook I wrote.

Teaching this course this past academic year has made me all too aware of things that are less than ideal about the book, and I unfortunately haven’t had time to get to work on making any significant improvements. Going through the material on spinors though, I’m pretty happy with how that part of the book turned out, think it provides a clear explanation of a beautiful and important story, one that is not readily available elsewhere.

One aspect of this that I emphasize is the remarkable parallel between

• The usual story of canonical quantization, which is based on an antisymmetric bilinear form on phase space, giving an algebra of operators generated by $Q_j,P_j$ acting on the usual quantum state space.
• Replacing antisymmetric by symmetric, you get the Clifford algebra, generated by $\gamma$-matrices, acting on the spinors.

For a table summarizing precisely this parallelism, see chapter 32 of the book.

For more video from my office, I recently had a long conversation with Reza Katebi, who has a Youtube channel of interviews called The Edge of Science.

Sean Carroll has a new interview up with Frank Wilczek in which they discuss, among other things, the problematic current state of fundamental physics. On the topic of string theory, here’s the discussion:

0:58:34.8 SC: Well, some of this worry has come out of string theory, many of our colleagues for the last several decades have pointed to string theory as the most promising way forward. As far as I know, you have not done a lot of work directly on conventional string theory. What is your feeling about that approach to moving beyond quantum field theory?

0:58:54.8 FW: Well, I think it has produced a lot of attractive work that’s intellectually rich and has spun off into fertile mathematics, but I don’t see that it’s been converging towards informative assertions about the physical world…

0:59:18.7 SC: That’s very elegantly stated, actually, yes.

0:59:21.0 FW: That you can check. And for me personally, I’ve kind of voted with my feet, I think there are more promising things to think about, that’s partially a sociological statement, but I think it’s, string theory is getting plenty of attention, it doesn’t need me. I’m happier doing things that other people aren’t doing, but that’s a personal statement, and so far, I haven’t regretted my choice, but I watch what people… I watch the subject and I watch what people are doing and I wish them good luck, and if and when things that I think are promising insights into the physical world emerge, I will pay a lot of attention.

1:00:12.3 SC: Sure Do you think that the rest of the field has voted with their feet in a slightly too uniform way, do you think that too much of our intellectual effort is going in that particular direction?

1:00:21.8 FW: I do, but I might be wrong, so I don’t want to discourage. Plenty of people are doing other things, so it’s not as if the rest of the world is feeling the lack of input from people who are working on string theory, it’s fine, people can work on string theory and it doesn’t hurt anything. I feel… Well, it’s going to sound, I don’t want to be patronizing, the people who do it are mostly adults and they know what they’re doing, but students and people who are thinking about what they’re going to do should go into it with open eyes. They should realize that the prospect of making an impact in our understanding of empirical science or technology are not… The prospect that you’ll make impact like that is probably not optimized by going into string theory.

1:01:20.8 SC: Yeah, no, actually, I think that we’re in exact alignment here. I feel a need to defend the string theory against unfair criticisms, but I do worry a little bit about the fact that it seems hard these days to connect it directly to empirical reality.

1:01:36.2 FW: Yeah, well, some nice ideas are coming off, as coming out as spin-offs, very, very clever people do string theory and they do clever things. So as I said, there’s been a lot of fruitful mathematics, there have been new techniques that have proved somewhat useful in condensed matter, although certainly not proportional to the amount of effort that’s going into it, and the future may look different, they may be real breakthroughs that come out of string theory that wouldn’t have come otherwise. But so far, the amount, I would say, other people may disagree, and I might be very unpopular among some of my colleagues for saying this, but I think the output compared to the input has been pretty disappointing on the empirical side.

I find it kind of remarkable that Carroll, known for defending string theory and string theorists, here reacts to Wilczek’s pretty negative characterization of string theory with “I think that we’re in exact alignment here.”

About current hot topic work on the black hole information paradox:

1:02:33.5 SC: Right. You have been involved in productive ways on the black hole information problem, which a lot of string theorists care about… What is your current feeling on the state of that problem? Do you think we’re making real progress?

1:02:50.8 FW: I think progress is being made in the sense that more intellectually coherent pictures are being drawn and some surprising connections to error correction and really interesting new chapters of quantum theory are emerging. On the other hand, it is a very esoteric problem, nobody’s going to produce… I don’t see a way, but who knows, but nobody has produced an experimental system to which these ideas apply in any reasonably direct way. So what does it mean to solve a problem like that? I’m not even sure what it means, where you can’t check. Many hypotheses go into it, the distance between the models and actual black holes that were phenomena you can observe are vast and many things could go wrong along the way in making these models.

1:04:04.7 FW: So I guess, yeah, it’s wonderful that people are making progress at the field, they’re making progress and have a literature that they enjoy, and it really is interesting from any point of view, it’s good, and maybe I should leave it at that, but how should I say? I don’t think it’s… I don’t think it’s the pinnacle of physics, let me put it… Let me put it that way.

On SUSY, Wilczek acknowledges

I’m a supersymmetry diehard.

which he certainly is.

If you look back at his many talks about prospects for the future, you’ll see that pre-LHC he was arguing

By ascending a tower of speculation, involving now both extended gauge symmetry and extended space-time symmetry, we seem to break though the clouds, into clarity and breathtaking vision. Is it an illusion, or reality? This question creates a most exciting situation for the Large Hadron Collider(LHC), due to begin operating at CERN in 2007, for this great accelerator will achieve the energies necessary to access the new world of of heavy particles,if it exists.

In the current interview and elsewhere, Wilczek makes clear the reason he believes in SUSY is his 1981 calculation with Dimopoulos/Raby showing that in SUSY versions of GUTs you could get the coupling constant evolution to overlap at the same energy. He’s still quite devoted to this argument, for him it’s of greater significance than the usual hierarchy problem arguments.

He has by now lost multiple bets that the LHC would see SUSY particles, including ones with Garrett Lisi in 2009 and Tord Ekelöf in 2012. At this point, even diehards like Wilczek acknowledge that chances that the LHC will see SUSY are slim. Another problem is that increasingly sensitive proton decay experiments have also ruled out a large part of the proton lifetimes predicted by the SUSY models Wilczek favors. He puts his faith in further proton decay experiments and a new, expensive collider. This is pretty much exactly the sort of thing that causes Sabine Hossenfelder to go ballistic over arguments for a new collider.

For a flavor of the SUSY discussion, here’s one piece of it, with Carroll starting off with a quite peculiar argument for SUSY:

0:29:28.8 SC: Yeah, maybe you can opine on this, but the way that I like to say it is, we could, in the space of all possible worlds that we live in, only one of them, we could have found supersymmetry already at the LHC very easily, but the fact that we haven’t doesn’t mean it’s not there. Maybe it’s less likely that it’s there, but it’s easy also to imagine theories where supersymmetry is real, and we just haven’t seen it yet.

0:29:53.1 FW: Right. So supersymmetry, as I said, there have to be… For supersymmetry to be valid, there have to be these superpartner particles that are the particles that the particles we know about turn into when they move into the quantum dimensions, but we don’t know what their masses are. We know some of their properties, but not their masses, and they could be very heavy. If they’re very, very heavy, we lose the benefit of improving… The benefit that supersymmetry would otherwise give in improving how the couplings unify, but okay, maybe that was a cruel joke on the part of nature. I want to think not, but the alternative is that they’re just a little bit too heavy to have been produced easily and identified easily at the LHC, and we just have to work a little bit harder and spend a little more money on…

An article by Steven Weinberg entitled On the Development of Effective Field Theory appeared on the arXiv last night. It’s based on a talk he gave in September, surveys the history of effective field theories and argues for what I’d call the “SM is just a low energy approximation” point of view on fundamental physics. I’ve always found this point of view quite problematic, and think that it’s at the root of the sad state of particle theory these days. That Weinberg gives a clear and detailed version of the argument makes this a good opportunity to look at it carefully.

A lot of Weinberg’s article is devoted to history, especially the history of the late 60s-early 70s current algebra and phenomenological Lagrangian theory of pions. We now understand this subject as a low energy effective theory for the true theory (QCD), in which the basic fields are quarks and gluons, not the pion fields of the effective theory. The effective theory is largely determined by the approximate SU(2) x SU(2) chiral flavor symmetry of QCD. It’s a non-linear sigma model, so non-renormalizable. The non-renormalizability does not make the theory useless, it just means that as you go to higher and higher energies, more possible terms in the effective Lagrangian need to be taken into account, introducing more and more undetermined parameters into the theory. Weinberg interprets this as indicating that the right way to understand the non-renormalizability problem of quantum gravity is that the GR Lagrangian is just an effective theory.

So far I’m with him, but where I part ways is his extrapolation to the idea that all QFTs, in particular the SM, are just effective field theories:

The Standard Model, we now see – we being, let me say, me and a lot of other people – as a low-energy approximation to a fundamental theory about which we know very little. And low energy means energies much less than some extremely high energy scale 10¹⁵−10¹⁸ GeV.

Weinberg goes on to give an interesting discussion of his general view of QFT, which evolved during the pre-SM period of the 1960s, when the conventional wisdom was that QFTs could not be fundamental theories (since they did not seem capable of describing strong interactions).

I was a student in one of Weinberg’s graduate classes at Harvard on gauge theory (roughly, volume II of his three-volume textbook). For me though, the most formative experience of my student years was working on lattice gauge theory calculations. On the lattice one fixes the theory at the lattice cut-off scale, and what is difficult is extrapolating to large distance behavior. The large distance behavior is completely insensitive to putting in more terms in the cut-off scale Lagrangian. This is the exact opposite of the non-renormalizable theory problem: as you go to short distances you don’t get more terms and more parameters, instead all but one term gets killed off. Because of this, pure QCD actually has no free parameters: there’s only one, and its choice depends on your choice of distance units (Sidney Coleman liked to call this dimensional transvestitism).

The deep lesson I came out of graduate school with is that the asymptotically free part of the SM (yes, the Higgs sector and the U(1) are a different issue) is exactly what you want a fundamental theory to look like at short distances. I’ve thus never been able to understand the argument that Weinberg makes that at short distances a fundamental theory should be something very different. An additional big problem with Weinberg’s argument is its practical implications: with no experiments at these short distances, if you throw away the class of theories that you know work at those distances you have nothing to go on. Now fundamental physics is all just a big unresolvable mystery. The “SM is just a low-energy approximation” point of view fit very well with string theory unification, but we’re now living with how that turned out: a pseudo-scientific ideology that short distance physics is unknowable, random and anthropically determined.

In Weinberg’s article he does give arguments for why the “SM just a low-energy approximation” point of view makes predictions and can be checked. They are:

• There should be baryon number violating terms of order $(E/M)^2$. The problem with this of course is that no one has ever observed baryon number violation.
• There should be lepton number violating terms of order $E/M$, “and they apparently have been discovered, in the form of neutrino masses.” The problem with this is that it’s not really true. One can easily get neutrino masses by extending the SM to include right-handed neutrinos and Dirac masses, no lepton number violation. You only get non-renormalizable terms and lepton number violation when you try to get masses using just left-handed neutrinos.

He does acknowledge that there’s a problem with the “SM just a low-energy approximation to a theory with energy scale M=10¹⁵−10¹⁸ GeV” point of view: it implies the well-known “naturalness” or “fine-tuning” problems. The cosmological constant and Higgs mass scale should be up at the energy scale M, not the values we observe. This is why people are upset at the failure of “naturalness”: it indicates the failure not just of specific models, but of the point of view that Weinberg is advocating, which has now dominated the subject for decades.

As a parenthetical remark, I’ve today seen news stories here and here about the failure to find supersymmetry at the LHC. At least one influential theorist still thinks SUSY is our best hope:

Arkani-Hamed views split supersymmetry as the most promising theory given current data.

Most theorists though think split supersymmetry is unpromising since it doesn’t solve the problem created by the point of view Weinberg advocates. For instance:

“My number-one priority is to solve the Higgs problem, and I don’t see that split supersymmetry solves that problem,” Peskin says.

On the issue of quantum gravity, my formative years left me with a different interpretation of the story Weinberg tells about the non-renormalizable effective low-energy theory of pions. This got solved not by giving up on QFT, but by finding a QFT valid at arbitrarily short distances, based on different fundamental variables and different short distance dynamics. By analogy, one needs a standard QFT to quantize gravity, just with different fundamental variables and different short distance dynamics. Yes, I know that no one has yet figured out a convincing way to do this, but that doesn’t imply it can’t be done.

Update: I just noticed that Cliff Burgess’s new book Introduction to Effective Field Theory is available online at Cambridge University Press. Chapter 9 gives a more detailed version of the same kind of arguments that Weinberg is making, as well as explaining how the the Higgs and CC are in conflict with the effective field theory view. His overall evaluation of the case
“Much about the model carries the whiff of a low energy limit” isn’t very compelling when you start comparing this smell to that of the proposals (SUSY/string theory) for what the SM is supposed to be a low energy limit of.

Our semester at Columbia started earlier than usual this year, with first classes this week, my first class yesterday. This semester I’m teaching the second half of a year-long course on the mathematics of quantum mechanics. There’s a Youtube channel with the lectures for the first half of the course, and now also for the second half. The course is largely following the textbook I wrote based on teaching this is earlier years. The first lecture yesterday was a summary of a point of view on canonical quantization explained in the first semester and in the book. This point of view is essentially that Hamiltonian mechanics is based on a Lie algebra (functions on phase space with Poisson bracket the Lie bracket), and canonical quantization is all about the essentially unique unitary representation of (a subalgebra of) that Lie algebra. On Thursday I’ll start on the fermionic version of canonical quantization, which has a very much parallel structure, giving a super-Lie algebra and spinors.

A few other items:

• John Baez’s This Week’s Finds in Mathematical Physics was an unprecedented project conducted over 17 years, providing a wealth of fantastic expository material on topics in math and physics. It started in 1993, and on its twentieth anniversary I wrote an appreciation (in an appropriate font) here. John has now announced that this material has been typeset (2610 pages!) and he is editing it, to be released in batches. The first part is now available, on the arXiv as This Week’s Finds in Mathematical Physics (1-50). As I find time, I’m looking forward to reading through these, encourage everyone interested in math and physics to do the same.
• Frank Wilczek has a new book out, and there’s an interview with him at Quanta. You can see a conversation between him and Brian Greene here on Friday.
• Another physicist with a new book is Jesper Grimstrup, whose Shell Beach: The search for the final theory I’ve just finished reading and enjoyed greatly. The book is quite personal and non-technical, with topic Grimstrup’s life as a theorist pursuing a unified theory. His career story is quite interesting, giving insight into the ways academic theoretical physics is challenging for young theorists trying to pursue non-mainstream research programs. Several books have appeared in recent years aimed at putting this kind of physics research in a human and philosophical context, telling you what it has to do with the meaning of life. There’s some of that in this book too, of a much more compelling sort than what you see elsewhere. Grimstrup has a website here, and in recent years has ended up leaving academia and trying to fund his research with donations. I can think of a lot worse things you could do with your money than send him some.

  I’m quite sympathetic to the underlying theme that he describes pursuing (together with Johannes Aastrup) in the book, that of bringing together the insights of loop quantum gravity and non-commutative geometry. More recently they’ve been working on some new ideas for formulating QFT non-perturbatively that seem worth investigating. There’s a survey blog post here.

Update: Another bit of private math/physics funding news. The IAS has announced establishment of the Carl P. Feinberg Cross-Disciplinary Program in Innovation

Scientific research at the Institute is traditionally driven by the collaboration and independent projects of a full-time Faculty and a revolving class of more than 200 researchers at various stages in their careers. The Carl P. Feinberg Cross-Disciplinary Program in Innovation will build on this successful model with the recruitment of mid-career scholars who have pioneered foundational developments in new areas. Bringing together scholars with such unique insights—which may not be obviously connected to the existing themes of the past 20 or 30 or 40 years—ensures that IAS will remain agile and responsive to new intellectual developments that do not yet fit the mold of what graduate students and postdocs generally know. In order to close this knowledge gap, the program will feature intense, focused workshops and “master classes.”

“Since its founding, the Institute has served as a world center for investigations into the fundamental laws of nature. We are currently in the middle of a grand symbiosis of ideas, from the equations of general relativity to the quantum information of black holes,” stated Robbert Dijkgraaf, IAS Director and Leon Levy Professor. “This revolutionary program will provide a dedicated space and the necessary flexibility to accelerate these exciting developments, and will surely forge new connections across fields.”

I first wrote here in 2015 about DAU, the unusual film project based to some extent on the life of Landau. Parts of the film first were shown in Paris early in 2019, and this past year started appearing on the DAU website. I’d been looking forward to seeing Gross, Yau, Rovelli and others in the film, so paid to watch one of the first parts, DAU. Degeneration, when it became available last year. It’s over six hours long, for a review, see here. I ended up doing a certain amount of fast-forwarding, was disappointed to only see Nikita Nekrasov and Dmitri Kaledin, none of the other math/physics world figures I had heard had participated.

DAU largely was funded by Russian oligarch Sergei Adoniev. For an excellent article discussing the project and its context in current Russian culture, see Sophie Pinkham’s article Nihilism for Oligarchs.

There wasn’t much physics in DAU. Degneration, but evidently it plays a significant role in other parts of the film. According to the DAU website,

Real-life scientists, who were able to continue with their research in the Institute, included: physicist Andrei Losev; mathematicians Dmitri Kaledin and Shing-Tung Yau; string theorist Nikita Nekrasov; Nobel-Prize winning physicist David Gross; neuroscientist James Fallon; and biochemist Luc Bigé. “One group was researching string theory and another researching quantum gravity. These groups hated each other. One stated there were 12 dimensions, the other claimed there were 24. The string theory group believed there couldn’t be 24 dimensions. The quantum gravity group believed that the other scientists were narrow-minded,” explained Khrzhanovskiy.

Now available is a part which seems to more centrally involve physics, DAU. String Theory, which is described as follows:

Nikita Nekrasov is a scientist, a theoretical physicist who studies our world and other possible worlds. He refuses to make a choice between mathematics and physics, between one woman and another, as he ponders the existence of the multi-universe. At scientific conferences, attended by eminent foreign scientists and a rising younger generation of physicists alike, Nekrasov gets carried away debating the beauty of string theory. He attempts to explain to all of his women – Katya, the librarian, Zoya, the scientific secretary, Svetalana, the head of department – about the theory of his own polygamy, and the possibility of having enough feelings to satisfy everyone.

Multiple universes have always been advertised with “in some other universe you’re dating Scarlett Johansson”, relating the idea to multiple partners in this universe is an innovation.

I haven’t yet watched DAU. String Theory, will likely find time for that soon. I’m worried that I’ll still not get to see Gross, Yau, Rovelli and others though, and lack the time and energy to look through all the other parts of the film. I’d like to crowd-source a solution to this problem: if anyone watching these things can let the rest of us know in which parts (at what times) well-known math/physics personalities appear, that would be greatly appreciated.

Almost exactly twenty years ago I started writing a short article about the problems with string theory. I had been thinking about doing this for quite a while, and the timing of entering the twenty-first century seemed appropriate for evaluating something that had long been advertised as “a piece of 21st-century physics that had fallen by accident into the 20th”. The piece was done in a week or two, after which I sent it around to a group of physicists to ask for comments. The reaction was mostly positive, although at least one well-known theorist told me that publicly challenging string theorists in this way would be counter-productive.

One person who wrote back was Phil Anderson, I’ve quoted some of what he wrote to me in this posting. He suggested I send it to Gloria Lubkin at Physics Today, and evidently talked to her about it. I did do this, and after not hearing anything back for a week or two, decided to go ahead and post the article to the arXiv, where it appeared as String Theory: An Evaluation.

Rereading that article today, there’s little I would change. Its argument is even more valid now than then. The problems of the theory and how it was pursued evolved over the next twenty years in ways far worse than what I could have imagined back then. In particular, the “multiverse” argument explaining away why string theory predicts nothing is something I could not have conceived of in 2001. The tribalistic sociology that has led to a large group of people calling themselves “string theorists” when what they do has nothing to do with string theory is also something I would have thought impossible.

In many ways, twenty years of further failure have had less than no effect. Lubos Motl is still arguing that string theory is the language in which God wrote the universe, and Michio Kaku has a new book about to appear, in which it looks like string field theory is described by the God Equation. Ignoring these extreme examples, string theory remains remarkably well-entrenched in mainstream physics: for example, my university regularly offers a course training undergraduates in string theory, and prestigious \$3 million prizes are routinely given for work on the subject. The usual mechanisms according to which a failed scientific idea is supposed to fall by the wayside for some reason have not had an effect.

While string theory’s failures have gotten a lot of popular press, the situation is rather different within the physics community. One reason I was interested in publishing the article in Physics Today was that discussion of this issue belongs there, in a place it could get serious attention from within the field. To this day, that has not happened. The story of my article was that I finally did hear back from Lubkin on 2/21/2001. She told me that she would talk to the Physics Today editor Stephen Benka about it. I heard from Benka on 5/6/2001, who told me they wouldn’t publish an article like that, but that I should rework it for publication as a shorter letter to the editor. I did this and sent a short letter version back to them, never heard anything back (a few months later I wrote to ask what had happened to my letter, was told they had decided not to publish it, but didn’t bother to let me know). In 2002 an editor from American Scientist contacted me about the article, and it ended up getting published there.

Looking back at how Physics Today has covered string theory and related speculation over the past 25 years, I did a search and here’s what I found:

• Reflections on the Fate of Spacetime (Witten) (April 1996)
• String Theory Is Testable, Even Supertestable (Kane) (Feb. 1997)
• Duality, Spacetime and Quantum Mechanics (Witten) (May 1997)
• Large Extra Dimensions: A New Arena for Particle Physics (Arkani-Hamed, Dimopoulos, Dvali) (Feb. 2002)
• Is string theory phenomenologically viable?(Gates) (June 2006)
• The case for extra dimensions (Randall) (July 2007)
• String theory in the era of the Large Hadron Collider (Dine) (Dec. 2007)
• String theory and the real world (Kane) (Nov. 2010)
• What every physicist should know about string theory (Witten) (Nov. 2015)

The only thing I could find anywhere during those 25 years indicating to Physics Today readers that none of this speculation had worked out was a short opinion column by Burt Richter

• Theory in particle physics: Theological speculation versus practical knowledge (Richter) (October 2006)

It seems to me that those now in charge of Physics Today should be thinking about this history, their role in it, and what they might be able to do to make up for this heavily one-sided coverage of a controversial issue.

A few recent items of interest:

• Martin Greiter has put together a written version of Sidney Coleman’s mid-1990s lecture Quantum Mechanics in Your Face, based on a recording of one version of the lecture and copies of Coleman’s slides.

  It’s often claimed that leading physicists of Coleman’s generation were educated in and stuck throughout their careers to a “shut up and calculate” approach to the interpretation of quantum mechanics. Coleman’s lecture I believe gives a much better explanation of the way he and many others thought about the topic. In particular, Coleman makes the crucial point:

  The problem is not the interpretation of quantum mechanics. That’s getting things just backwards. The problem is the interpretation of classical mechanics

  He ends with the following approach to the measurement problem:

  Now people say the reduction of the wave packet occurs because it looks like the reduction of the wave packet occurs, and that is indeed true. What I’m asking you in the second main part of this lecture is to consider seriously what it would look like if it were the other way around—if all that ever happened was causal evolution according to quantum mechanics. What I have tried to convince you is that what it looks like is ordinary everyday life.

  While some might take this and claim Coleman as an Everettian, note that there’s zero mention anywhere of many-worlds. Likely he found that an empty idea that explains nothing, so not worth mentioning.

• For the past year the CMSA has been hosting a Math-Science Literature Lecture Series. The talks have typically been excellent surveys of areas of mathematics, given by leading figures in each field. To coordinate with Tsinghua, the talks have often been at 8am here in NYC, and several times the last couple months I’ve made sure to get up early enough to have breakfast while watching one of the talks. All of the ones I’ve seen were very much worth the time, they were the ones given by Edward Witten, Andrei Okounkov, Alain Connes, Arthur Jaffe and Nigel Hitchin. Jaffe’s covered some of the ideas on Euclidean field theory that I’ve been spending a lot of time thinking about. For a more detailed version of his talk I highly recommend this article.
• Peter Scholze has posted at the Xena blog a challenge to those interested in formalizing mathematics and automated theorem proving (or checking): formalize and check the proof of a foundational result in his work with Dustin Clausen on “condensed mathematics”. As part of the challenge, he provides an extensive discussion of the motivation and basic ideas of this subject, which attempts to provide a replacement (with better properties) for the conventional definition of a topological space.

  Spending a little time reading this and some of the other expositions of the subject convinced me this is a really thorny business. Scholze explains in detail his motivation for making the challenge in part 6 of the posting. My suspicion has always been that most of the value of computer theorem checking lies in forcing a human to lay out clearly and unambiguously the details of an argument, with that effort likely to make clear if there’s a problem with the theorem. It will be fascinating to see what comes of this project.

  I see there’s also a blog posting about this at the n-category cafe.

• On the topic of theorems that definitely don’t have a clear and unambiguous proof, supposedly PRIMS will be publishing Mochizuki’s IUT papers in 2021. Mochizuki and collaborators have a new paper claiming stronger versions of Mochizuki’s results.

In a remarkable article entitled Contemplating the End of Physics posted today at Quanta magazine, Robbert Dijkgraaf (the director of the IAS) more or less announces the arrival of the scenario that John Horgan predicted for physics back in 1996. Horgan argued that physics was reaching the end of its ability to progress by finding new fundamental laws. Research trying to find new fundamental constituents of the universe and new laws governing them was destined to reach an endpoint where no more progress was possible. This is pretty much how Dijkgraaf now sees the field going forward:

Confronted with the endless number of physical systems we could fabricate out of the currently known fundamental pieces of the universe, I begin to imagine an upside-down view of physics. Instead of studying a natural phenomenon, and subsequently discovering a law of nature, one could first design a new law and then reverse engineer a system that actually displays the phenomena described by the law. For example, physics has moved far beyond the simple phases of matter of high school courses — solid, liquid, gas. Many potential “exotic” phases, made possible by the bizarre consequences of quantum mechanics, have been cataloged in theoretical explorations, and we can now start realizing these possibilities in the lab with specially designed materials.

All of this is part of a much larger shift in the very scope of science, from studying what is to what could be. In the 20th century, scientists sought out the building blocks of reality: the molecules, atoms and elementary particles out of which all matter is made; the cells, proteins and genes that make life possible; the bits, algorithms and networks that form the foundation of information and intelligence, both human and artificial. This century, instead, we will begin to explore all there is to be made with these building blocks.

In brief, as far as physics goes, elementary particle physics is over, from now on it’s pretty much just going to be condensed matter physics, where there at least is an infinity of potential effective field theory models to play with.

Dijkgraaf ends with an argument indicating that human intelligence is outmoded, artificial intelligence is our future:

Science concerns all phenomena, including the ones created in our laboratories and in our heads. Once we are fully aware of this grander scope, a different image of the research enterprise emerges. Now, finally, the ship of science is leaving the safe inland waterways carved by nature, and is heading for the open ocean, exploring a brave new world with “artificial” materials, organisms, brains and perhaps even a better version of ourselves.

Along the same lines, today also brings an article in the New York Times by Dennis Overbye, Can a Computer Devise a Theory of Everything? The article discusses the new MIT Institute for Artificial Intelligence and Fundamental Interactions and Max Tegmark’s hopes that AI will “discover all kinds of new laws of physics”. My guess is that this will work just fine if you give up on the 20th century understanding of what a “law of physics” is and follow Dijkgraaf’s lead. The problem then may be not so much “will we understand the new laws of physics found by AI?”, but rather that of them not being interesting enough to be worth understanding…

Update: To clarify the point I was trying to make about the Dijkgraaf piece arguing against the “end of physics”, compare it to the similar 1996 piece Gross and Witten published in the Wall Street Journal (a summary is here, an extract here). Gross and Witten were strongly disagreeing with Horgan, whereas it seems to me that Dijkgraaf implicitly agrees with Horgan that fundamental physics has hit a wall and theorists are moving on to do something else.