*Dijkgraaf and Witten on math vs physics, 22 minutes*

On May 29th, the IAS used the opportunity of the publication of a new book by Graham Farmelo, "

*The Universe Speaks in Numbers*", to organize a mini-conference about the relationships between mathematics and physics. Talks by the first female winner of the Abel Prize for mathematics Karen Uhlenbeck, by Freeman Dyson, Nima Arkani-Hamed, Farmelo, Dijkgraaf, Kyle Cranmer, Thomas Lam, Greg Moore, Natalie Wolchover and a few others may be found on the IAS YouTube channel.

The director himself also decided to record a conversation that would become the most watched YouTube video ever (see above). He chose an employee as his talking buddy carefully – it was Edward Witten – and indeed, the video is already approaching 1,000 views, close to 6.2 billion earned by a Despacito video. ;-)

At the beginning, Dijkgraaf shyly tries to persuade the viewers that Witten is qualified to speak about mathematics and physics. Robbert checked that no one is throwing tomatoes at them because of this potentially blasphemous statement.

Fine. Witten says that the gap between math and physics isn't due to personalities but really due to the different logic and motivation of both fields. The gap grew for 25 years after the Second World War, largely because physics was being advanced by looking at rather dirty experiments that mathematicians don't like – and also because quantum field theory became the main theoretical foundation of physics.

Mathematicians have had trouble to develop a rigorous foundation of e.g. \(D=4\) quantum field theories – QFTs are simply too mathematically hard for mathematicians, a major example of a framework where the physicists' relaxed attitude to rigor and the theoretical physicists' somewhat higher intelligence relatively to the mathematicians leads to consequences. In fact, almost no one seems to be working on the axiomatic foundations of QFTs today. Dijkgraaf tried to persuade Witten to repeat many of Dijkgraaf's opinions about the future of thinking or about the conceptual scheme of all the knowledge but you may see that in almost all cases, Witten was very careful and at least ambiguous about any potential agreement with Dijkgraaf.

For example, Dijkgraaf thinks that quantum field theory is ill-understood — the cooperation between the patchy partial understanding looks too chaotic to him – and we will discover some completely new or "more global" understanding of quantum field theories in the future. Well, it's possible but like Witten who politely stayed silent, I am skeptical when it comes to such far-reaching statements. I would personally bet that quantum field theory won't undergo new huge revolutions in the way how we define it – that it is "more than 50%" settled subject when it comes to its foundations. And if the network of relationships between various QFT ideas looks chaotic to Dijkgraaf, it's his psychological problem but this network is probably here to stay.

This view of Dijkgraaf is one of the "politically correct" views that are promoted in the string-theory-friendly but otherwise pop science mainstream media. It's probably not a coincidence that a chap who is aligned with these views has become a director of the research-only institute where Einstein has worked for a long time. I may have shared Robbert's intuition at some point but I think that I have largely moved on – and it seems that so did Witten and others. In particular, I can't imagine how the complex and diverse "perspectives" on QFTs and their properties could ever be "undone" or replaced with something totally different – except if the change may be described as a disappearing knowledge or a symptom of a dumbed down mankind. In other words, I feel that by his seemingly innocent comments directed against the "untamed complexity of insights in QFT", Dijkgraaf sounds like a "Smolin Lite" and this attitude is still "Wrong Lite" – and a manifestation of some math-phobia if not knowledge-phobia. Certain complicated insights are here with us to stay. Sorry, Robbert. They could disappear if physicists stopped doing physics and started to say "OM" (or some equation-free would-be philosophical clichés, like the clichés popular among the eukaryotes) which is simpler and "less messy" – but it's also less physics, unless you add at least some equations of (the noncommutative) OM-theory.

OK, so I am obviously with Witten on these questions. My reason is that QFT really looks transparent enough. Mathematicians don't know how to rigorously treat path integrals and renormalization but physicists, at their level of rigor, do know how to deal with these things and QFT still "is" some theory described by explicitly constructed degrees of freedom on an explicitly real spacetime. This theory leads to Green's functions and other observables that exhibit patterns (like the patterns of some QFTs explained by the associahedrons etc.) and new patterns may be found in the future. But those are really aspects of the "solutions", not a part of the definition. General QFTs are probably going to be defined in ways that just cosmetically differ from the present ones. A limit of a lattice QFT or whatever. There's no room for genuine new mystery – which may only hide in quantum gravity which we currently

*can't*fully define in terms of explicit degrees of freedom living on an explicit spacetime now.

Since the 1970s or so, Witten implicitly said, mathematics and physics got closer again – because the power of experiments to make quick changes had been largely depleted and because string theory and perhaps a few analogous developments were expanding and they're clearly close to be a common topic exciting both physicists and mathematicians. But this drift shouldn't be overstated, there's still a gulf in between math and physics.

Robbert asked Ed about Ed's predictions that were made or could have been made 30 years ago. Witten had expected a more unifying definition of the laws of string theory to be found. It really wasn't. Instead, most of the progress took place in the second part of the research which isn't the "search for the very laws of physics", namely in "the research into the solutions of the theory". In particular, the strongly coupled regime of string/M-theory was largely understood much more than before the mid 1990s – but surprisingly enough, this understanding was achieved without a huge progress in the "writing of the basic laws".

After 11:11, you see another theme where the two men aren't quite on the same frequency and I'm closer to Witten again. Dijkgraaf is convinced that the geometry must be shown to be emergent or an "afterthought". Well, I used to say similar things but I switched to thinking that

*some*geometry must be "totally and fully present" in any formulation of a theory that is in any way equivalent to string theory (or QFT) as we know it today. You may be very free to choose the background geometry that is the starting point to organize the Hilbert space and observables (T-duality, mirror symmetry, string-string duality, AdS/CFT, ER=EPR, and other generalized dualities guarantee lots of freedom) but you shouldn't choose "no geometry" because "no geometry means no physics".

So I think some geometry will always exist in the initial form of the laws of physics – and the new progress may be "what kind of degrees of freedom" may be associated with the geometry and why various choices are really physically equivalent to each other. Strings, bilocal wormholes etc. have been added to generalize the point-like particles in QFTs. Ironically, the BFSS Matrix theory seems "most distant" from what I say. But it still has the spatial coordinates of the D0-branes in it etc. The spacetime isn't eliminated. Just the second quantization from QFT is replaced with the wave functions for block-diagonal matrices.

Dijkgraaf forced Witten to pick some high points – links between math and physics that have excited Witten. Witten picked some technical answers.

Also, Dijkgraaf later said that string theory has helped to bring different subdisciplines of mathematics closer to each other and that much of the separation of mathematics into subbranches is an artificial, sociological effect, not a reflection of true long-term borders in the realm of mathematics. I think that I completely agree with Dijkgraaf on this one. And so does Witten, it seems. When asked "why it's so", Witten said that "it's because the Universe was created by a mathematician". Laughter.

Steve Shenker has discovered the Shenker-Murphy law that says that "you always avoid one type of mathematics and it's the mathematics that will be relevant for your next paper". So Dijkgraaf obviously wanted to know which part of mathematics Witten is avoiding LOL.

To address this funny question, Witten mentioned that he always found the number theory – and the arithmetic Langlands program – to be too abstract which is why he chose the geometric Langlands program which could be advanced in his lifetime. (Witten kept on pushing his campaign to persuade everyone that he's primarily a full-blown physicist whose heart is full of blood and milk, not a damn mathematician.) He believes that number theory may become important in the physics-like thinking in the distant future – and he is therefore jealous about the future generations. These guesses look so sensible to me...

At the end, since 18:00, Witten was asked a big question and answered that he found it extremely likely that physics was on the right track and he considered it implausible that string theory (that addresses so many physics questions and that works at this demanding enough level, and that also sheds so much light on mathematics) could be just a series of coincidences that ultimately have nothing to do with the Universe. I completely share this intuition. Witten admits that this kind of thinking may be considered as "not being scientific evidence".

Dijkgraaf said that we would never get rid of string theory because it's already a part of QFT that has been proved essential to understand the Universe etc. Witten agreed but saw Dijkgraaf's comment as some kind of "a partial surrender" or an attack on the primary purpose of string theory, i.e. to be a "theory of everything". So Witten authoritatively and importantly said that string theorists shouldn't forget that the real primary – and still totally viable – goal was to find the correct vacuum that describes all the observed phenomena.

String theory is primarily our successful framework unifying all the known forces and all types of physical objects and phenomena – and the process of completely proving and settling this statement is still ongoing. I feel that Dijkgraaf might be among those who have agreed with the eukaryotes to classify this

*absolutely essential*point as being politically incorrect. And if I am right, it's just too bad when a director of a crucial institute is being manipulated by the lowly life forms in this way.

"If string theory is not right, who is living in the Universe inhabited by string theory? ;-)" It is a quote that Dijkgraaf attributed to Witten. Witten said that it couldn't all be a coincidence as a response to this quote.

**Wolfram and Gell-Mann**

If you have the lust and nerves for very insightful, informative, Wolfram-centric, and tense memories of Murray Gell-Mann as remembered by Stephen Wolfram, click at the link.

They've known each other very well – Wolfram has spent years at Caltech as an ingeniuous kid-particle physicist. Wolfram claims credit for some things associated with Gell-Mann, e.g. for marketing the Santa Fe Institute as a "complexity" institute. In 2012, Gell-Mann no longer recognized Wolfram. I guess that the evolving appearance of the latter was the main reason. Wolfram also says things I couldn't agree with – like some mysterious words about "muon's not being understood" (muon is as understood as the electron – they really

*are*completely analogous and only differ by the mass which also implies muon's instability), his implicit denial of the importance of quantum mechanics (and, somewhat smaller, importance of Gell-Mann–Hartle consistent histories), his idea that the cellular automatons are very important, and more.

The text makes it clear that Gell-Mann has wanted to influence the world at many levels – well, just like Wolfram himself, I would add – but it hasn't quite worked out. It's bad it didn't. I do think that a guy like Gell-Mann who spoke 13 languages should influence the environments where languages are considered important, for example. Also, there's a fun idea involving birds that Wolfram suggested – or at least the idea emerged in my skull after I read Wolfram's memories. Feynman used to pick "names of birds" as something that he was taught by his father to be useless. This story could have been "improved" by Feynman and Feynman could have deliberately chosen birds because Gell-Mann was a keen ornithologist or birdwatcher! The bird name story of Feynman's could have been an attack on Gell-Mann, too! ;-)

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