Some links related to the foundations of math and physics:

- Kevin Hartnett at Quanta has a long article on Jacob Lurie and his work on infinity categories. Unfortunately Lurie didn’t participate in the article himself, so comments are only from others. The article does a good job of giving at least a vague sense of what these very abstract foundational ideas are about, as well as examining the math community’s struggle to absorb them. Lurie’s work on this is spread out over more than 900 pages here and more than 1500 pages here. Recently he has been putting together an online textbook/reference version of this material as Kerodon, which is modeled after and uses much of the same software as Johan de Jong’s Stacks project.
- At Mathematics without Apologies, Michael Harris has some comments on a recent discussion of the Mechanization of Math, held here in New York at the Helix Center. A video of the discussion is available here.
- In the new (November) issue of the AMS Notices John Baez has a review of a recent collection of articles about the foundations of mathematics and physics. The book, Foundations of Mathematics and Physics One Century After Hilbert, contains contributions about both math and physics, although in his review Baez concentrates on issues related to physics. He notes “The elephant in the room is string theory.”

The same issue of the Notices contains an informative long article about Michael Atiyah and his career, written by Alain Connes and Joseph Kouneiher (Kouneiher is the editor of the book reviewed by Baez).

After reading the review by Baez it seems to me that “The elephant in the room is string theory.” is somewhat contradictory. Apparently the whole book adresses the lack of understanding regarding string theory. But, and please forgive my ignorance, the metaphor “the elephant in the room”…doesn’t it say actually, that the elephant is ignored?

Heiko242,

It’s not true that “the whole book addresses the lack of understanding regarding string theory”. The only article directly addressing string theory is Witten’s. Baez I think explains his “elephant in the room” claim when he writes:

“Given its remarkable impact on mathematics, it is natural to ask what string theory has achieved toward its original goal: becoming a true theory of physics, one that makes experimental predictions we can test. The volume under review does not address this.”

That’s the elephant in the room: the last 35 years have been dominated by a specific proposal for fundamental physical theory (unification via string theory), but, while this subject has interesting connections to mathematics, it has none to experiment and the real world. Wikipedia defines “elephant in the room” as

“an important or enormous topic, problem, or risk that is obvious or that everyone knows about but no one mentions or wants to discuss because it makes at least some of them uncomfortable or is personally, socially, or politically embarrassing, controversial, inflammatory, or dangerous.”

which is an accurate characterization of the way the problems with string theory get ignored in contexts where they are a central issue.

I just looked at my old blog posting about the Witten article (which was also published in Physics Today):

https://www.math.columbia.edu/~woit/wordpress/?p=8068

There I used the “elephants in the room” line, getting it from an Ashtekar interview where, discussing the problems with string theory he says: “These are elephants in the room which are not being addressed.”

In other math news, a report of the ICM Structure Committee makes for some interesting reading: https://www.mathunion.org/fileadmin/IMU/Report/SC/2019/structure_committee_final.pdf

I found in the context of the above the following 1988 article by David Gross:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC282459/pdf/pnas00301-0008.pdf

From p.8374:

”Our critical colleagues denounce these efforts, indeed all of string theory, and call it by the dirtiest name they can come up with – recreational mathematics. Although I resent being called a recreational mathematician, I admit that there is a valid (albeit small) point to these criticisms. They remind us of the danger, in following the Diracian dictum, of turning into mathematicians. This for some theorists is an ever-present temptation.”

I feel that by not having Lurie contributing to the Quanta article about his work we missed reading there about his motivation in developing this theory, like, what were the problems and limitations he saw with the current view or tools of mathematics. But Lurie has given many talks about his work, so maybe Kevin Hartnett could have tried (maybe he did try) to find the pieces that formed his motivation and make it clearer for the readers where all this is heading to? With Grothendieck´work the motivation were the Weil conjectures for example. I’m not sure if the same case happened with MacLane – Eilenberg work that introduced categories for the first time. Some of the comments in the article refer to the non-clarity of what is this really about.

Zoyiver,

I don’t think we have to ask Lurie why mathematicians are interested in higher categories and n-stacks and such. It’d be interesting to know why he personally chooses to work on them, of course, but a lot of people were already interested before he started work on them. Topologists have been thinking about these ideas since at least the 1970s; they’ve long known that the standard homotopy category is deficient and were looking for a good replacement. Likewise, people have known that derived categories are defective since they were first invented. In algebraic geometry, the idea of using higher categories goes back at least to Grothendieck’s letter to Quillen in the early 80s. Mathematical physicists, I think, got turned on to them mainly through the Baez-Dolan Hypothesis and Dan Freed’s papers in the early 90s. Point being, the general ideas and motivations (some of them, anyways) aren’t new and didn’t come from Lurie. But until he got involved, the ratio of theorems to conjectures and motivation wasn’t very high.