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  Witten's philosophy of exceptional nature

+ 4 like - 0 dislike

In various of his talks and writings, Edward Witten has been revealing -- always in passing -- a philosophical perspective saying that:

Since nature (reality) is exceptional in that it has existence, it is plausible that it is the exceptional among all mathematical structures -- such as the exceptional examples in the classification of simple Lie groups, the exceptional Lie groups -- that play a role in the mathematical description of nature.

One place where I heard him say this, years back, was, in the context of grand unification, in

In the archived writeup of this talk the relevant passage appears on page 5:

Describing nature by a group taken from an infinite family does raise an obvious question – why this group and not another? In addition to the three infinite families, there are five exceptional Lie groups, namely G2, F4,E6,E7, and E8. Since nature is so exceptional, why not describe it using an exceptional Lie group? 

While I am sure I have seen similar passages in other articles by Witten, right now I don't remember in which articles that was. But for an exposition that I am wriiting, I would like to cite these, or the one most pronounced version that is available.

John Baez in TWF 66 expresses the same sentiment:

one may argue that the theory of our universe must be incredibly special, since out of all the theories we can write down, just this one describes the universe that actually exists. All sorts of simpler universes apparently don't exist. So maybe the theory of the universe needs to use special, "exceptional" mathematics for some reason, even though it's complicated

Does anyone happen to have citation details for further instances of the thought expressed in these quote above?

I am collecting relevant material in the nLab entry universal exceptionalism.

asked Aug 2, 2015 in Resources and References by Urs Schreiber (6,095 points) [ revision history ]
edited Aug 2, 2015 by Urs Schreiber

In the same ·"exceptional philosophy":  http://xxx.lanl.gov/pdf/1505.01635v1.pdf

+juancho , these authors discuss exceptional mathematical structure appearing in physics, but do they express the philosophical sentiment that I am asking about? 

I am aware that I am asking a question that is more in the philosophy of physics than in physics itself. By the rules it might better belong to a philosophy forum (and I cross posted to Phil.SE) but since I feel that non-physicists will hardly have the insight required to discuss this, I felt like I have to ask this here.

@UrsSchreiber I don't understand why there should be only one exceptional group, and not a multitude of groups describing the universe.

@UrsSchreiber from my personal point of view, this question is exactly fine and nice to have here on PO.

Hi Urs. I'm curious to know why you're collecting such things. 

"nature is exceptional in that it has reality". Pretty unconvinced by that assertion. Any mathematical model that allows intelligent beings to evolve, will have solutions where those intelligent beings consider self-evident the existence of the mathematical model that enables their existence. But it's better that Witten doesn't concern himself with such thoughts. If the search of exceptional physics is what drives him to create such excellent physics and mathematics, let him believe whatever is that he wants to believe.

Are they descriptive or constructive theories ? At this point of thinking, it is unknown if the exceptionnal tools emerge or if they were ( are ) "at work" at the design step, if any. A pattern of the former case might be the limited size of each theory field of applicability, when unification attempts introduce difficult assumptions. Otherwise, there is also a hidden "beauty" attribution question, is it in the mathematician eye or inherent to the object and just somehow discovered. In the former case, considering the uncertainty ( and the physicists simplifications of calculations ), the unicity of the solution is not implicit, even if one efficient has been already found. Answers may come from a rigorous construction of QFT, still explaining wholly the experiments and from the resulting possible reduction of the postulates set. It is prudent not to be universalist yet.

5 Answers

+ 4 like - 0 dislike

A 1986 paper by Witten http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0267.0306.ocr.pdf says after (41): 

 Having found a hint that $E_6$ can play a role, it is natural to wonder whether one can go further and base a grand unified theory on the biggest exceptional group $E_8$. 

 The 2003 slides http://www-hep.phys.cmu.edu/beauty2003/beauty2003/talks_web/Edward_Witten.pdf say on p.17:

Or we could use the group E8. This group is worthy of describing nature as it is the biggest and most splendid of the exceptional simple Lie groups.

answered Aug 2, 2015 by Arnold Neumaier (15,787 points) [ revision history ]
+ 4 like - 0 dislike

Found one more:

Edward Witten, Deconstruction, \(G_2\) Holonomy, and Doublet-Triplet Splitting, arXiv:hep-ph/0201018

has this on p. 13:

[...] arise in compactifying from eleven to four dimensions on a compact seven-manifold \(X \) of \(G_2 \)-holonomy. This seems like an interesting starting point for making a model of the real world, which is certainly exceptional, [...] \(G_2 \) which is the smallest of the compact exceptional Lie groups,

(my boldface).

answered Aug 2, 2015 by Urs Schreiber (6,095 points) [ revision history ]

It is not my business, but "the real world is certainly exceptional" is exceptionally wrong idea. Those who say such things do not understand what the physics is about. We never deal with the whole and unique "real world" but with tiny and distorted pieces of it.  No way to reduce our particular and subjective descriptions to some unique exceptional group since the number of subjects is infinite, in my opinion.

Edit: Our perception is like a movie, it consists of inclusive "frames" with a certain exposition time and ordered in time (we perceive it as "memory"). Then we build our mathematical models of this. And in our mathematical wold the Sun shines all the time and as we like, whereas "reality" is highly dependent on the "brightness", exposition time, light wavelengths, etc. etc. What is transparent in one region of wavelengths may be non-transparent in another region. Observable "sizes" are also dependent on the wavelengths and inequalities here are in order. worse, we observe "object" due to their internal changes, but we think of them as of unchangeable "points". That is why we encounter the quantum realm with no "usual" microscopic certainty where "determinism" only exists for somehow averaged (inclusive) variables. That is why "more is different" whereas in mathematics we can count things as countable, as if they were always identical. Thus, there are many "levels" of perception and all of them are equally "realistic" since they "exist". This makes the phrase "the real world is certainly exceptional" meaningless as overly simplified and primitive way of doing physics.

"the real world" = the ultimate theory of all, the one we are looking for, if any.
"is certainly" = prudence is our friend
"exceptional" : the main point of the claim.
One may understand it in the frame of the string theories with their parameters sets or smell the wonder of the theorist in front of the efficiency of his tools. That's all. This doesn't help a lot to drive a lab day by day but with a little hindsight, one must remember that the labs ancestors were full of spirits and other daemons. If today, the lab physicist became a very great magician, he may (also) thank the theorists. To get all friends, the measure object became central. Sciences need both.

+ 0 like - 0 dislike

Exceptional groups appear in Physics as groups for grand unified theories. Other important role of the exceptional groups in Physics is as holonomy groups for compactifications in SUGRA, Superstrings, M-Theory and F-Theory.  This physical application of the exceptional holonomy discovered by Dominic Joyce can be observed for example in http://arxiv.org/abs/hep-th/9407025   .

answered Aug 2, 2015 by juancho (1,130 points) [ no revision ]

Sure, that's what the references listed above are about. But what I was after is explicit expressions of the philosophical sentiment expressed at times by Witten and saying that exceptional groups play a role in the physics of the real world "because reality is exceptional", in that it is real. Possibly this is not the right thing to discuss here on this group, in which case I apologize.

Then there are three powerful arguments on favor of the "exceptional natural philosophy":

Exceptional Gran Unified Theories,

Compactifications with Exceptional Holonomy,

Exceptional Khovanov Homology..

+ 0 like - 0 dislike

"Since nature (reality) is exceptional in that it has existence"

In what sense does existence makes nature exceptional?

Any mathematical model that allows intelligent beings to evolve, will have solutions where those intelligent beings consider self-evident the existence of the mathematical model that enables their existence. A mathematical model that allows stable structures. and intelligent beings to evolve can be considered anthropically favourable, but that is far from being exceptional in any sense of the term

answered Sep 24, 2015 by CharlesJQuarra (555 points) [ revision history ]
edited Sep 24, 2015 by CharlesJQuarra
+ 0 like - 2 dislike

I actually believe any theory describing the ultimate nature of reality is not  a physical theory. Physical theories only describe the content of our senses and their farthest reaching conclusions. There is still to wonder what we are missing. 

Instead of entering here in detail about my personal considerations about the World, I suggest any foundational principle must be as symmetrical as logically possible. This is the most general World. The special ones are our senses. I know my addition to this questions is not physicalist in the sense of a real (peer accepted/prediction maker/falsifiable/indeed real) physical theory but is my insight about why this considerations are utterly fruitless (unless you are playing with a helluva complicated mathematical apparatus, Im looking at you dear Witten).  

answered Dec 13, 2017 by hector parra [ no revision ]

Most of what is done in physics is far away from what our senses can perceive. 

You can estimate the lack of utter fruitlessness of modern physics from the fact that you are able to contribute an utterly fruitless answer to this forum, which is possible only because physics extracts from reality a large part of its ultimate nature.

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