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 the quote above?

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

  • In the same line of thoughts, one can easily argue that a good symmetry group of space-time has to be a simple group (in the mathematical sense) that intertwine rotation, boosts, and translations too. Notice that Galilean group is a direct product of rotations, boosts, and translations, while Poincaré group intertwine only rotations and boosts, but not translations. Looking for such unification, some people (see possible kinematics by Bacry and Levy-Leblond) proposed that it should be possible to have a new theory of (special) relativity where the symmetry group is the de Sitter one.
    – sure
    Commented Aug 25, 2015 at 14:12
  • Also, let me point that it is not because such symmetry "group" could potentially explain physical interactions (and I insist on physical) that you would be allowed to find it: you're part of nature too, and there's no reason that nature allows you to find the good mathematical object describing it. (Now, if you believe that mathematics or abstractions are not part of nature, then notice that if it was possible to explain everything, then it would also be possible to explain why it is so.) That is, I deny that you could even write down or construct such symmetry group, even if it "exists".
    – sure
    Commented Aug 25, 2015 at 14:19
  • Also, mathematical objects are not the interpretations we put on them to make sense of them. The concept of group (or Lie algebra) is a convenient idea to make sense of symmetries, but it is by no mean the only one. You don't know if some "twisted" idea that overlap with the one of group but is neither contained nor containing it wouldn't make "more sense" to encode (physical) symmetries. In particular, the classification we use suffers the same problem: one group that is exceptional for one classification might be part of a canonical infinite family in another. There's no unicity of meaning.
    – sure
    Commented Aug 25, 2015 at 14:32
  • The argument is a little odd. Since we don't know how generic "theories we can write down" are the "incredibly special" sounds like a case of the base rate fallacy. And we don't know that "simpler universes apparently don't exist" either. The conclusion that exceptional groups are involved may well be true, but the reasoning is like the Pythagorean argument that heavenly bodies must move uniformly along circles. Because they are perfect, and so are the circles.
    – Conifold
    Commented Sep 15, 2015 at 1:01

2 Answers 2


Since Nature is exceptional in that it has existence; it is plausible that it is exceptional amongst all mathematical structures

The word exceptional can only make sense if there were other possibilities; and for it to be properly exceptional, a multitude of possibilities.

This suggests that Lewises plural worlds might have a bearing on this; and this too, because what little I've seen of it, is thinking through the concepts of modality; but there, something neccessarily exists, if it exists in all possible worlds.

There is another possibility - Aristotle, states that the only eternal motion in place is circular, and that this is a continua; this is interesting, since Lie groups are, in a sense, generalised spherical motions which are continua ie smooth.


This is the 'Anthropic Principle'. It shouldn't seem odd that the universe is able to support us because, if it didn't, we wouldn't be thinking about it. Some argue that the odds hint for the existence of other universes with different laws.

  • No, the "principle of exceptional nature" is not the anthropic principle. Rather the opposite. Think about it. Commented Dec 15, 2015 at 5:46
  • My mistake. Witten's premise, that nature is exceptional, is wrong because of the anthropic principle, so he is wrong to assume the need for an exceptional math.
    – amI
    Commented Dec 21, 2015 at 20:32
  • No, that does not make sense either. Commented Dec 22, 2015 at 14:41
  • Sorry. Is Witten saying that you can't build a calculator without E8 ruling the parts? Can't other universes co-exist? Can't math in one universe encompass Lie groups that are beyond it?
    – amI
    Commented Dec 23, 2015 at 23:20
  • I think it's all stated clear enough. Please don't further add to the noise. Commented Jan 5, 2016 at 0:28

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