- Last week a review of the Mochizuki IUT papers appeared at Math Reviews, written by Mohamed Saïdi. His discussion of the critical part of the proof is limited to:

Theorem 3.11 in Part III is somehow reinterpreted in Corollary 3.12 of the same paper in a way that relates to the kind of diophantine inequalities one wishes to prove. One constructs certain arithmetic line bundles of interest within each theatre, a theta version and a q-version (which at the places of bad reduction arises essentially from the q-parameter of the corresponding Tate curve), which give rise to certain theta and q-objects in certain (products of) Frobenioids: the theta and q-pilots. By construction the theta pilot maps to the q-pilot via the horizontal link in the log-theta lattice. One can then proceed and compare the log-volumes of the images of these two objects in the relevant objects constructed via the multiradial algorithm in Theorem 3.11.

Saïdi gives no indication that any one has ever raised any issues about the proof of Corollary 3.12, with no mention at all of the detailed Scholze/Stix criticism that this argument is incorrect. In particular, in his Zentralblatt review Scholze writes:

Unfortunately, the argument given for Corollary 3.12 is not a proof, and the theory built in these papers is clearly insufficient to prove the ABC conjecture….

In any case, at some point in the proof of Corollary 3.12, things are so obfuscated that it is completely unclear whether some object refers to the q-values or the $\theta$-values, as it is somehow claimed to be definitionally equal to both of them, up to some blurring of course, and hence you get the desired result.After the Saïdi review appeared, I gather that an intervention with the Math Reviews editors was staged, leading to the addition at the end of the review of

Editor’s note: For an alternative review of the IUT papers, in particular a critique of the key Corollary 3.12 in Part III, we refer the reader to the review by Scholze in zbMATH: https://zbmath.org/1465.14002.

Since the early days of people trying to understand the claimed proof, Mochizuki has pointed to Saïdi as an example of someone who has understood and vouched for the proof (see here). Saïdi is undoubtedly well aware of the Scholze argument and his decision not to mention it in the review makes clear that he has no counter-argument. The current state of affairs with the Mochizuki proof is that no one who claims to understand the proof of Corollary 3.12 can provide a counter-argument to Scholze. Saïdi tries to deal with this by pretending the Scholze argument doesn’t exist, while Mochizuki’s (and Fesenko’s) approach has been to argue that Scholze should be ignored since he’s an incompetent. The editors at PRIMS claim that referees have considered the argument, but say they can’t make anything public. This situation makes very clear that there currently is no proof of abc.

- At one point the American Institute of Mathematics (founded in 1994 with financing from John Fry) was supposed to move from its location behind a Fry’s Electronics store to a castle in Morgan Hill modeled on the Alhambra (see here). This never worked out, and last year Fry’s Electronics declared bankruptcy. The latest news is that next year AIM will move to Caltech, for more see here.
- I’ll never understand why places like MIT continue to teach undergraduate courses on a failed speculative idea about physics.
- There has been a lot of coverage in the press of claims by a group analyzing old CDF data to have come up with a dramatically better value for the W mass (one seven sigma away from the SM value). While this would be really wonderful if it were true, unfortunately that doesn’t seem very likely. There isn’t a well-motivated theoretical reason for this discrepancy, this is a very challenging measurement, and the new value seriously disagrees with several previous measurements at CERN. For an informed discussion of this from someone who was on CDF and has worked on these sorts of analyses, see Tommaso Dorigo’s blog post.
- It will be interesting to see how well the LHC experiments can ultimately do this measurement. The LHC is about to start up again after a long shutdown, with beam commissioning starting on Friday.

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As a regular reviewer for Mathematical Reviews, this is disappointing. Presenting a biased pro-IUTT review while linking to an “alternative review” that describes the consensus of mainstream mathematicians, gives the false impression that the debate is ongoing. As a resource used widely by mathematicians all over the world, MR has a responsibility to play referee here, rather than providing access to arguments for both sides and letting everyone decide for themselves what the truth is.

It seems we have learned nothing from the Trump years. It’s not enough to give both sides of a disagreement an equal platform. To maintain credibility, one must be willing to take a side when the preponderance of evidence is clear.

(In fact, it’s worse than that, because even with the editor’s note, MR comes across as more on Mochizuki’s side than not.)

I review papers for MR on a voluntary basis because I appreciate MathSciNet as a useful resource. But credibility is a fundamental part of what makes it a useful resource.

S,

I’ve found it very hard to understand why MR published the Saïdi review despite being aware that there were well-know problems with the proof that the review did not address. The current editor’s note improves the situation but I agree that as it stands the review gives the wrong impression, not making clear that the consensus of the expert community is that there is no proof of Corollary 3.12.

I agree that there is quite some evidence that the CDF W-mass measurement is problematic (eg comparison with earlier measurements).

There is, however, also a well known effect that new measurements have a tendency to be consistent with earlier measurements even when the latter measurements are wrong.

All credit to CDF for following a robust analysis procedure that precludes the possibility of artificially agreeing with earlier measurements. They did the best measurement they thought they could do and published it.

I wouldn’t bet much on it being right but that’s another story.

Link to the MIT course is not working

Mike,

Works for me, but did change link to https in case that was causing the problem.

It works now, thanks.

It would actually be great if MIT taught a course on failed speculative theories in physics, perhaps using Helge Kragh’s book as a textbook. If students only learn about successful theories they won’t get how hard it is to be right, and how many great physicists had theories that didn’t pan out.

Two other takes on the W mass:

https://resonaances.blogspot.com/2022/04/how-large-is-w-boson-anomaly.html

https://non-trivial-solution.blogspot.com/2022/04/do-we-have-finally-found-new-physics.html

Prof. Baez has made here a wonderful suggestion. I wonder if such a course might be further enriched by an account of theories that “failed” in their original form and context, but which contained conceptual elements that re-emerged in subsequent decades or centuries, i.e., as components of a superseding theory. Examples might include: (a) in the fifth century BC Greek thinker Leucippus and his pupil Democritus envisioned small indivisible particles, “atoms”; (b) Descartes proposed a “corpuscular” theory of light (1637) at odds with the wave nature of classical light; (c) Einstein posited a cosmological constant to provide an outward pressure opposed to overall gravitational collapse, at odds with Hubble’s observations; and (d) Veneziano strings as a theory of the strong force, re-conceptualized as a renormalizable model inclusive of gravitons.

I would respectfully disagree with the negative assessment of the handling of reviewing IUT by the colleagues of MathSciNet. Compared to the peer review process, the purpose of a review in MathSciNet (as well as zbMATH Open) is different; to quote the Guidelines for Reviewers (https://mathscinet.ams.org/mresubs/guide-reviewers.html):

It is frequently very difficult to find reviewers willing to write a high-standard review, and obviously especially so in this peculiar case (zbMATH Open was very lucky that Peter Scholze agreed for this specific volume). Since our services are fundamentally based on the willingness of the reviewers to provide appropriate reviews, editorial policy will in most cases refrain from interfering or even rejecting detailed reviews contributed by an expert in the field (who Mohamed Saïdi certainly is). With this long-term practice in mind, adding an Editor’s note as in the case of the MathSciNet review is already a very rare measure, and a quite indicative statement.

Ideally, of course, the validity of a theorem should not need to be settled in the framework of our reviewing services; but given the situation as it is, I think that both reviews contributed to the clarification of the current status.

John Baez says:

That’s a great idea, and it sounds like it would be a great course. One problem with this idea is that the course would most likely be listed under Science, Technology, and Society (STS) and not under Physics (Course 8), so most physics majors woudn’t end up taking it.

John Baez,

Perhaps teaching a course on the details of a failed speculative theory would be a good idea, if you explained to the students that it was a failure and why it failed. Teaching such a course and presenting it to undergraduates as a success is what I don’t understand at all.

Olaf Teschke,

Thanks very much for contributing your perspective. My understanding (which may not be correct) is that the problem MR had in this situation is that they were unable to find any expert willing to write a review including a discussion of the problem with the proof. Saïdi was willing to write a review, but not to discuss the problem with the proof. It seems they decided to publish his review since the alternative would be no review.

But a review that doesn’t discuss a well-known flaw in the proof clearly violates their own standards as you’ve quoted them above: that the proof is flawed is highly relevant to whether a reader would want to read the paper, and is crucial for the background of the paper and for its evaluation. The editor’s note improves the situation, but this still is a flawed review that does not meet their stated standards.

Ironically, the first page of the linked-to notes for the MIT string-theory course gives the argument against string theory in capsule form.

I first read about the W anomaly in Quanta. Why they don’t just go straight to Tomasso Dorigo or Adam Falkowski every time one of these things heats up is a bit of a mystery to me. Maybe because the wet blanket or reality doesn’t get eyeballs.

Kevin S Van Horn,

I also found that pretty weird. You could interpret “we don’t know what string theory implies” as an argument for further research in the subject, but to me it seems an extremely strong argument against teaching the subject to undergraduates.

Lot of other strange things there, such as “Fact 2” which tells us that “the global picture is still lacking today because the field is very young”. Over 50 years old “very young”? I can understand why people try and make excuses for the failures of the subject, but I don’t understand the phenomenon of making absurd excuses.

I see the MIT course notes have the standard sum of all integers is -1/12 to get d=26, without any real justification. This put off multiple students when I took this course long ago.

Just to add a bit in this various section: the restart from LHC comes along with some just-around-the-corner hype: https://www.bbc.com/news/science-environment-61149387

Sam,

At least for that there’s a perfectly legitimate mathematical explanation, whether or not the instructor explains it. What’s really odd is that this is a physics department teaching fundamentals of physics to its students, but scheduling one course with no known connection to physics. “String theory” has become institutionalized as an ideology disconnected from conventional science, with, at least at MIT, indoctrination part of the undergraduate curriculum.

Peter : OT. A talk by David Gross on string tehory and unification at KITP in 2022

https://online.kitp.ucsb.edu/online/snowmass-c22/gross/rm/jwvideo.html

would be interested in your views.

Shantanu,

I did watch that. Gross repeated the now 45 year old argument about SUSY GUTs unification of couplings (from what I remember, when I watched this, in the discussion afterwards Mike Peskin pointed out that that unification only works well in the simplest one-loop calculation, that it gets worse if you do a more precise calculation).

He did admit that things haven’t worked out, pointing to no SUSY at LHC and the landscape. He did also say “I see little hope that we can achieve the original goal” absent some new theoretical breakthrough. He argued that people should try to solve QCD via string theory and address the “What is string theory?” question, but he has been saying this for decades, people have been trying for decades, nothing has changed.

The one new thing I heard from him was a mention of the bootstrap program and a comment that “other UV completions than string theory may exist and we should be open to this possibility”.

About IUT, what I’ve been trying to understand is if that work has other merits on its own.

Ie. even if the proof of Corollary 3.12 is wrong, so the overall proof is wrong, does Mochizuki’s vast construct contribute to math in some other collateral way?

If I’m right this has happened before with proofs that turned out to be wrong, and I’ll be genuinely puzzled if such complex work has nothing else to offer and all its value is hanging on a single point.

Mochizuki’s latest endeavor seems to be regularly updating this document (https://www.kurims.kyoto-u.ac.jp/~motizuki/Essential%20Logical%20Structure%20of%20Inter-universal%20Teichmuller%20Theory.pdf), which is now 140 pages. I doubt this will convince anyone; if anything, his writing has become even more turgid and depressing.

@tulpoeid: No, there is *nothing* useful in IUT. If there was, it would’ve been successfully extracted by ambitious young people in the ten (!) years since the IUT papers were first posted.

Hermann Vile,

That document just gets stranger and stranger. Last time I looked at it, see here

https://www.math.columbia.edu/~woit/wordpress/?p=12220

it was at only 65 pages. It has kept the same structure:

1. No reference to Scholze or Stix by name, no reference to the original document they wrote about the problems with the proof, no reference to later efforts by Scholze to further explain why the proof is invalid (e.g his Zentralblatt review and the discussion on this blog).

2. Out of three long sections, the only one with technical content is the third. Mochizuki seems to think it’s a good idea to precede that by two long sections about elementary mathematics and logic supposedly not understood by his critics, as well as a long rant about mathematical ethics (given the PRIMS publication story, this is especially rich). These two sections do an excellent job of completely destroying his credibility.

tulpoeid,

If you look at Scholze’s Zentralblatt review, he summarizes the mathematical content of the rest of the IUT papers. Kirti Joshi has written a series of papers, summarized in

https://arxiv.org/abs/2111.06771

which attempt to put Mochizuki’s ideas in a more conventional context, where it should be more clear if anything useful can come out of them.