Slides from my talk at Rutgers are now available here. The idea was just to advertise to physicists there the point of view that is all too familiar to regular readers here. The final speculative comments about relations to mathematics shouldn’t be taken too seriously, these are things I hope to work on and write about much more in a few months once my current book project is completed.

**Update**: Interestingly, my Princeton advisor Curt Callan yesterday gave a talk at the KITP with a bit of a similar theme, starting off by arguing that the success of the standard model made future progress in HEP very difficult. His answer to the problem is quite different than mine (his involves trying to make contributions to biology). The first question at the end (from David Gross) is about the relation to new mathematics.

If you use \documentclass[handout]{beamer}, it should shrink your pdf document from 125 to 32 pages (eliminating the strip-tease feature).

Thanks Ian,

That’s a useful fact I didn’t realize. Will change that in the morning.

I like pacing through the talk bullet by bullet. It makes the material easier to digest.

I’m not so sure that the current obstacles in experimental HEP mean an end of being able to conduct experiments at all. Rather, I have the impression that only the paradigm of the particle accelerator is coming to an end. Instead, we have to think about how to conduct new types of experiments that allow us to probe effect occurring far beyond the electroweak scale. For instance, some people are working on probing Planck-scale physics with optomechanical oscillators. This is actually not that crazy, at least far less crazy than expecting the Chinese to build yet another accelerator and hoping for a miracle to show up there.

I’ve changed the pdf file to one with single-pages.

G.S.,

For you and anyone else who likes the pauses, that version is at

http://www.math.columbia.edu/~woit/rutgers-withpauses.pdf

Bernd,

I wasn’t intending to claim that there are no possibilities for non-accelerator based experimental progress (or, lower energy accelerator-based things like neutrino experiments). Such things should certainly be pursued, my point was just to provoke thought about what can be done in the absence of new data.

Did you get any interesting questions?

lun,

Some good questions, mostly people who sensibly wanted to challenge my provocatively negative take on prospects for progress (for instance: what about cosmology? what about neutrino masses? what about previous fallow periods in the history of the subject?). No string theorists there who wanted to argue about that.

There are also possibilities for accelerator-based progress.

Magnets technology is progressing fast. Superconducting cavities: same thing.

A lot of progress in wakefield plasma accelerator with a lot of R&D at SLAC and CERN.

Do not forget the possibility of ILC and a muon collider.

Sure: very expensive projects but all this could push the end of collider physics decades in the future.

Accelerators are needed anyway for all the precision physics measurements. Just to name a few: mu2e, g-2 , pi to 3e, B-factories…

Anyway: Peter I liked your perspective in the talk.

As for the neutrinos, there are powerful experimental efforts. The Fermilab’s DUNE project will clear up a lot of things: hierarchy (non and inverted) and CP phase in the lepton sector. The Japanese are progressing towards Hyper-K with similar goals. This stuff will be also useful for cosmic neutrinos and proton decay.

I do not see a grey future for HEP right now. Maybe in >50 years.

Reading thru now, got to page 15 “There is one science that does not rely on empirical testing to make progress: mathematics.” and aren’t those fighting words at an institution that employs Doron Zeilberger?

@Peter:

I also scrolled through the slides and found a very basic statement (not central to your talk) that is not so clear to me: On slide 24 you write “Observables correspond to self-adjoint linear operators on H”. This depends a lot on your definition of observables. I think it does not cover the Berry phase which also gives you a measurable quantity of quantum mechanical systems. Is this not considered to be an observable?

Jake: (At the risk of being OT.) After refreshing myself on DZ’s opinions, yes they are rather amusing in juxtaposition to PW’s talk. Thank you.

Gregor,

The Berry phase is a matrix element of an observable. The operator, not the matrix element is the observable. This is not in conflict with the idea that self-adjoint operators are observables.

Doron ZeilbergerNever heard of him before. Damn! I suppose he is using Coq now

There is one science that does not rely on empirical testing to make progress: mathematics.The judgment of whether something is “empirical” or “à priori” (I am at a loss to find the best contextually relevant antonym of “empirical”) is really on a sliding scale.

“Do you judge this statement to be a theorem of this hopefully self-consistent system”is a call performed by resource-constrained, imperfect judges that moreover have to rely on external memory like paper and state-tracking systems like blackboards. Such is life…Many thanks for the slides of your talk. I’m not a scientist, but I found this a very helpful distillation of your blog and point of view that gives me more of a feeling that I am getting a glimpse of the issues. I also think I agree that going over the details of the mathematical physics with mathematical insight and rigor is likely to be very helpful, but what is the role of “physical intuition” (like Einstein’s thought experiments e.g. about the equivalence principle) in this?

Hasn’t mathematics itself gone through several episodes in just the past century of redefining what exactly is a proof (Russell’s Paradox/incompleteness, computer proofs, categorification), and what kind of mathematical tools and objects can or should be used in proofs?

Even more, is there any danger that “mathematical proof” can become so difficult that progress slows or halts, by analogy with the increasing difficulty of high-energy physics experiments? Is there a “human intelligence limit” here somewhere? How much of the increasing difficulty, assuming it is real, is due to mere complexity, and how much is due to increasing abstraction/sophistication? Can computer-assisted proof help with this?

I’m a mathematical Platonist like Godel, but I’m not sure I’m a radical Platonist in your sense. How does a radical Platonist learn which mathematical entities give the most insight into physical reality without experimental decision?

Gregor,

Peter Orland is right that this is still an operator on states. This kind of operator is very interesting though, not of the same sort as local fields, which have an interpretation in terms of a representation of a Heisenberg Lie algebra (in physics terms, canonical commutation relations). Things like Wilson loop and ‘t Hooft loop operators indicate that fields may naturally be quantized in a quite different way than matter fields.

Michael Gogins,

I don’t think the basic idea of what a proof is has changed in quite a long time. It’s fundamentally just the idea that you need to make your assumptions and claimed logical implications completely explicit.

The remarkable thing about mathematics is that it continues to progress, despite the use of ever more difficult to absorb formalisms. I find this really striking and well worth thinking about it. Naively one would expect progress to have ground to a halt, as the whole thing became too difficult for humans to master, but this hasn’t happened.

I don’t think you can understand the deep connections between physics and math by just looking at one of the two subjects. As for whether you can find new relations between math and physics of the sort I expect, the point is that you already know a lot about the physics side (the SM), and the argument is that you should focus on that. All experiment has been telling us for many years is “pay better attention to the SM”. Would be great if new experiments told us something more than that, but this may not happen.

Very nice. I am going to have to try to go through your book at some point, though I worry that I’m not sufficiently math-literate, in the sense that I don’t really know how enough of the vocabulary to think intelligently about phrases like “the tangent space at the identity of the group”. I know separately what a tangent space, an identity, and a group are, but mathematicians have a way of speaking that I haven’t learned. If you have any advice on a “how to read about modern math for non-mathematicians”, I’d greatly appreciate it.

The book I’m writing does try to convey exactly what you’re asking about, at least the first half of it, which tries to explain basic examples of Lie algebras, groups, representations in a concrete way, while also taking somewhat of a mathematician’s point of view and language. The second half of the book things aren’t so simple, I’m trying to work out basic examples of QFT using some different mathematics than usual. This I suspect both mathematicians and physicists will have to struggle a bit to follow.

I’m working through your book notes (haven’t gotten to QFT parts, yet) and am enjoying it, but I must say it’s a struggle. Biologist here, with some physics background. If a reader hasn’t prepared with a couple books on lie algebras, group theory, and the basic math of QM (linear algebra, hilbert space, operators) before trying your book, I’d think he/she would get lost quickly.

Is calling mathematics a “non-empirical science” also intentionally provocative? There are quite a few learned people who would consider maths a separate field and feel mathematicians have a different métier from that of scientists. Furthermore, they would say maths and the natural sciences are all the better for this differentiation, while acknowledging the former’s “unreasonable effectiveness”. I’m on the fence about this distinction myself, and just wonder why you favor looking at it as a “science” vs….well, something else.

LMMI,

Yes, that’s intentionally a bit provocative, I’m well aware that many if not most people have a different view of mathematics. But from the “radical Platonist” point of view, mathematics is not different than other sciences, it is the science of the study of certain kinds of objects and their relations, objects with a deep connection to the physical world. The only thing different is the role of empirical experiment (arguably there are also “experiments” in math, e.g. numerical calculations checking examples of a number theory conjecture, but these are “non-empirical” in the sense of not measuring something about the physical world).

Vaguely relevant is this comic, putting the sciences on a scale together…

https://xkcd.com/435/

Peter,

You *are* aware of the popup text (when you hover the mouse over the comic) in that particular xkcd, right? 🙂

Best, 🙂

Marko

vmarko,

Yes. Gell-Mann liked to say that. Until a mathematician explained to him about SU(3)…

Your take on mathematics continuing to progress w/o empirical data, of course raises the question of what is progress. Is it finding additional answers to ad hoc puzzles, or finding new understanding through the solution of puzzles?

Eli Rabett,

Mathematics progresses by discovering new interesting mathematical objects, and better understanding the ones we know about. Solving specific problems (like the Poincare conjecture or Fermat’s last theorem) are in some sense just evidence that one has better understood the relevant objects.

I know you’re not a fan of Tegmark’s multiverse enthusiasm, but your “radical Platonism” sounds something like Tegmark’s Level 4 multiverse (the mathematical multiverse). Am I misunderstanding one of you?

As a mathematician I would make the argument that math is NOT an empirical science. Even if one is a Platonist, since the Platonic objects are not accessible via anything but math. Now that I’m thinking about it perhaps mathematics is really the multiverse, if it is Platonic. In the Platonic universe of mathematical objects there would have to be contradictory ones, which were internally self consistent. Say, the universe where the continuum hypothesis is true and the one where it isn’t. Two different universes, with different rules. Not a problem for us mathematicians, you just have to decide which one you want to live in. Kind of a big problem for physicists, who are supposed to describe the universe we do actually live in 🙂

Will,

In one sense I’m on-board with Tegmark, identifying the physical world with a mathematical structure. In another sense, our visions are opposite. He sees a multiverse of different physical laws corresponding to the infinity of all possible mathematical constructs (which to me is an empty idea). I see ultimately one specific set of physical laws unifying physics (something not too far from the SM + GR) corresponding to the fact that mathematics is a highly structured subject, with a unifying fundamental set of ideas (which we don’t yet completely understand).

Excellent set of slides. I think your lessons from math on slide 17 could well be applied to other disciplines where models are gradually claiming to replace hard experiments and simulation is becoming an impressive if misleading source of hype and funding.

[Peter Woit wrote:] “mathematics … continues to progress, despite the use of ever more difficult to absorb formalisms. … Naively one would expect progress to have ground to a halt, as the whole thing became too difficult for humans to master, but this hasn’t happened.”

I don’t think the naive expectations are necessarily so different from what is happening. The training time to get to the frontier and make meaningful contributions is, as expected, getting longer and longer. More and more giant papers appear that very few people understand, and take years for the community to digest.

Total progress has accelerated as the number of PhD’s has exploded, but progress per mathematician may be slowing down as higher complexity and diminishing returns set in. This is masked somewhat by the availability of ever more powerful technology and the new growth areas that creates within mathematics. The increases in productivity from LaTeX and email and Skype and github (or similar tools) are one-time jumps so that one expects a slowdown to set in at some point, unless funding is pumped in to increase the number of trained specialists. If there is a salvation through AI coming online that would also be a kind of validation of the Horgan thesis, and would probably only be temporary.

Voevodsky’s program of proof formalization is based exactly on the “incomprehensible complexity” thesis and he gives some examples from his own research experience.

p.s. Peter, I had meant to write a long reply to Brian Conrad at the Mochizuki thread, but did not get there before you closed the comments? Is there any possibility of reopening it? The thread on ABC at Cathy O’Neil’s blog is also closed, so no ability to continue there. I guess I can start a blog if necessary but was hoping to answer under the blogpost where Conrad’s comments appeared.

random reader,

It’s true that the trend is for mathematics research to get more specialized, with new developments understood by smaller fractions of the community (the number of people who understand the proof of FLT is pretty small). And I’m somewhat mystified how students manage to get to the frontiers of the subject and make a contributions in a finite amount of time (then again, as I get older, the ability of some of the young to quickly absorb difficult ideas never ceases to amaze). But it’s undeniable that significant progress keeps happening, of a sort that hep-th isn’t seeing. We’ll see if this continues, I’m not optimistic that more Ph.D. students or proof formalization are that helpful.

I’ve changed the parameter that sets the date for closing old comments, so you can still comment on that old posting. One problem with this is that typically no one looks at old comment threads, so an ongoing discussion there is not very likely. I would encourage you in the idea of setting up your own blog and thus having a place you can easily write more extensively. I promise to link to it!

Mathematical progress includes the development of ever-improving language for efficiently organizing the mathematics, so that conceptual complexity is reduced in some ways at the same time as overall difficulty and amount of knowledge increases. Scheme theory, derived geometry, homological/homotopical algebra, and perfectoid spaces all are complicated (and interrelated) formalisms with heavy upfront investment for learners, but they systematize and simplify what was done before, allowing the same human cognitive capacity to achieve more in finite time.

Other than that, however, the different impression of progress rates between math and physics (or math and any other field with well understood benchmarks of progress, such as nuclear fusion, AI, space travel, lifespan extension etc) is largely because of a cultural difference in standards of progress. In math one can always formulate and explore new problems that look solvable and define solutions of those as progress. Every once in a while an old and difficult benchmark problem falls to new tools but that’s a rarity. Theoretical physics has this to a degree, but a much lesser one because there is a much smaller set of problems considered fundamental. “Have you unified GR and QM yet” (or predicted new particles, or understood dark matter, etc) is a very hard standard to meet.

In math there is also a tendency to accept weak notions of “solve”, such as qualitative or nonconstructive existence proofs, or proofs of concept that don’t necessarily blossom into a working tool after finding some in-principle solution to a problem. In

physics, particularly experimental physics, it is hard to accept something that claims to be predictive of some particle mass but does not actually lead to a precise number that can be checked against experiment. In math the equivalent is often considered a full solution.

You will be aware I’m sure that Lubos has gone through your slides point by point. Less the hysterics, some of what he says is valid. Have a look at any random day of hep-th and ignore the popular press. The field has moved on quite a bit from the pair of pants diagram and the Randall-Sundrum model.

You have also no doubt heard rumours of gravitational waves detected by LIGO. On the verge of this massive experimental discovery you’re announcing that empirical confirmation of new physics models is impossible and we should instead turn to noodling around with group theory and see what happens with the massive web of conjectures comprising the geometric Langlands program. Certainly these things are interesting mathematics but does this honestly seem like a promising direction for physicists? What should physicists do about it? Most are far less mathematically inclined than you and care about predicting the results of real experiments – check any day on hep-ph. What kind of career do you propose for the workaday phenomenologist?

Radioactive,

I think it should be obvious there’s little point to me responding to Lubos’s ranting, his agenda is just to misinterpret and misrepresent what I have to say. Obviously the few words that fit on a slide can’t convey the actual arguments I’m trying to make (much fuller versions are available of course in many places, for instance my take on the hype problem is http://www.math.columbia.edu/~woit/wordpress/?cat=8, not just the two particular examples in the talk). As I pointed out at the beginning, the intent of the talk was to point to where the actual arguments were, and to be provocative, not make a bunch of carefully worded, fully hedged and defended claims.

There’s no argument in my talk that experiment is dead. Obviously whatever experimental avenues are available should be explored, and I’ve no expertise to advise experimentalists about what they should be doing. The question I was raising was that of, if you’re a theorist interested in certain questions, with no experimental help in sight, what can you do? The Munich conference I was referring to was based on the assumption that a significant part of theoretical activity is now “post-empirical”. A perfectly defensible attitude towards this is that it’s a mistake, that theorists should should stick to thinking about things with some reasonable connection to experiment. A lot of people read my book as making that argument, but that’s not really what I think, and the talk was an attempt to explain why. I do think it is possible to make progress by trying to better understand the internal logic and mathematical structure of a theory, even without help from experiment telling one if one is on the right track. And if you try and do this, there’s a lot you could learn from mathematicians about how to make progress.

“see what happens with the massive web of conjectures comprising the geometric Langlands program”

That is not quite accurate. Geometric Langlands is not like some neighboring subjects where there are vast pyramids of conjectures that go far beyond any proven theorems and with no hope for an imminent proof. There is a massive web of both conjectures and theorems, and the weight of the latter is considerable. Lots of the conjectural picture, although still developing, has been proved, and the time span between conjecture and proof has been far shorter for geometric Langlands than classical arithmetic Langlands (because the geometric theory is easier and more structured). The subject is in flux but in a very healthy way where more and more of the expected picture is falling into place, leading to new refinements of that picture, and further cycles of conjecture-proof-application.

Physics unlike mathematics has both logical and experimental tests to pass. Working on mathematical structure is what string theorists do and have done for years. You are advocating a more rigorous mode of working – axiom, conjecture, theorem, proof, consequences – instead of trying to leap over steps as physicists are wont to do. But it seems if people actually followed your approach then we would be more or less in the same place.

If everything was formulated nice and consistently what then? Streater and Wightman’s axioms had very little effect on the important developments of the 20th century. Do you think that the mental labour involved in proving rigorous theorems about string theory would stop people from hyping it in the popular press?

Radioactive,

I’m not advocating that physicists adopt mathematician’s standards of rigorous proof, just that, especially when they can’t make contact with experiment, they devote more attention to making clear what they understand and what they don’t. Sure, leap over steps and see what happens (little known fact: mathematicians do this too), but be aware that you’re doing it and make that clear to others. Would this not make any difference? Maybe. But my experience with string theory has been that the extreme difficulty in figuring out exactly where the subject is, what is understood and what isn’t, has not only made the public hype problem worse, but makes it difficult to make progress (with the progress needed sometimes just achieving clarity on what doesn’t work).

As for QFT, again, I’m not arguing for applying full rigor to what is well understood. An example of what I have in mind is the question of whether the electroweak theory (chiral gauge theory) has a well-defined formulation outside of perturbation theory. Can you even in principle consistently do non-perturbative calculations? This appears to be irrelevant to contact with experiment, and the general assumption is that this is an uninteresting technical issue, but maybe there’s something to be learned there. For people to be motivated to work on it though, there has to be some perception that there is a problem there, and some idea of what would count as progress.

Peter,

Interesting you should mention chiral gauge theories – there is a recent article by D M Grabowska and D B Kaplan

http://arxiv.org/abs/1511.03649

which combines two nice ideas in lattice field theory in a proposal to non-perturbatively regulate chiral gauge theories.

wrote:random readerIn physics we call this working on M-Theory. 😉

“Streater and Wightman’s axioms had very little effect on the important developments of the 20th century.”

Is this attitude fair? Bisognano and Wichmann took those axioms as hypothesis and published their theorem a year before Unruh published his result. Also, the irony of this comment appearing on this blog in particular is, Streater and Wightman published their book more or less motivated by their disappointment with a talk by Geoff Chew.

As to what concerns workaday physicists, sure, e.g. Lawrence’s collider experiment paradigm is about studying effective field theories at different scales of contractable minkowski space. Ultra cold physics cares about field theories on stratified manifolds and non-abelian braid statistics. Many important insights there have genealogies traceable to VFR Jones and to Mandelstam’s (Kadanoff-Ceva inspired?) string localized excitations studied via DHR theory of superselection sectors in the Haag-Kastler axioms (Frohlich credits D.Friedan’s talks as inspiration for his paper on non-abelian statistics- I should mention, BTW, it takes a lot of chutzpah to walk into Greg Moore’s physics department and tell them they need to do more math). Shoot, over and above Jones’ huge influence via Witten, there’s a paper Jones presented at a conference where he asked for something like the Wegner’s self-dual lattice models but with Hopf Algebras from finite non-abelian groups. So along came this guy named Kitaev, and…

It sometimes feels like we say the phrase “contact with experiment” as if there’s some static, God given collection of experiments, faithfully streaming new phenomena at physicists who would then be surely remiss if they didn’t spend all their time, like Beatrice in the Paradisio, staring at this God given stream of Data. I think we say it this way because it’s an honest reflection of a workaday phenomenologist’s experience- new kinds of experiments can sometimes take an entire career to bare fruit. You must publish or perish, and attempts at rigorizing interacting field theories is, notoriously, a graveyard of ideas. Sure, you can listen to the siren song of mathematical physics but only after your men have tied you to the mast. True, an individual who dives off his boat in pursuit of their song will surely perish. But, also, someone eventually realized those were wales and made an industry out of fueling lamps with the stuff. If there were high priests of lay-internet discussion of sirens back then, maybe no one ever would have.

Hi, maybe not unrelated to the content of your talk, Smolin in a recent paper (http://arxiv.org/pdf/1512.07551) commented on the role of Mathematics in HEP, and what lesson should we draw from Einstein’s General Relativity.