nLab exceptional naturalism





physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

Exceptional structures



In the philosophy of science and particularly the philosophy of physics, the philosophical sentiment which expresses the following perspective on the description of physics by mathematics might deserve to be called exceptional naturalism or similar:

Since nature (reality) is exceptional in that it has existence, it is plausible that it is the exceptional structures 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, hence in physics and specifically in phenomenology.

In that this perspective provides a priori grounds for the way the universe is, it may be contrasted with empiricism.


Sources where a sentiment of exceptional naturalism has been expressed include the following:

David Olive (in Olive & Landsberg 1989) writes:

The results…add the stunning new question as to why nature has an apparent predilection for exceptional structures such as the E groups.


I sense that the trail of exceptional structures is growing even warmer.

Baez 95, in a context of the role of the monster group in physics, writes:

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

Ramond 01 writes, in a context of heterotic string theory and M-theory:

Nature relishes unique mathematical structures.


The Exceptional Algebras are most unique and beautiful among Lie Algebras, and no one should be surprised if Nature uses them.

Witten 02a, in the context of M-theory on G2-manifolds says, with regard to the exceptional Lie group G2:

…arise in compactifying from eleven to four dimensions on a compact seven-manifold XX of G2-holonomy. This seems like an interesting starting point for making a model of the real world, which is certainly exceptional

Witten 02b, in the context of grand unified theory and heterotic string theory says:

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?

Ramond 03 writes in a context of grand unification:

In the Lie garden, one also finds five rare flowers, the exceptional algebras: G2, F4, E6, E7 and E8, their rank indicated by the subscripts. In view of Nature’s fascination with unique structures, they merit further study.

Boya 03 writes in view of the various occurrences of the octonions in M-theory:

If the current M-theory is a unique theory, one should expect it to make use of singular, non-generic mathematical structures. Now it is known that many of the special objects in mathematics are related to octonions, and therefore it is not surprising that this putative theory-of-everything should display geometric and algebraic structures derived from this unique non-associative division algebra.

Toppan 03 writes:

There is a growing interest in the logical possibility that exceptional mathematical structures (exceptional Lie and superLie algebras, the exceptional Jordan algebra, etc.) could be linked to an ultimate “exceptional” formulation for a Theory Of Everything (TOE). The maximal division algebra of the octonions can be held as the mathematical responsible for the existence of the exceptional structures mentioned above.

Moore 14 writes, in a survey of the state of mathematical physics applied to fundamental high energy physics (“Physical Mathematics”):

it must be said that much of Physical Mathematics has a predilection for special, sporadic, and exceptional structures. [[]] I cannot forecast what stormy weather our field is destined to endure, but I can confidently forecast abundant moonshine in the years ahead. (section 11 “Exceptional structures”)

A related comment in the context of F-theory GUT phenomenology requiring a point with E-type symmetry is in (Vafa 15, slide 11):

The landscape concept typically goes against things being exceptional. Here we seem to have evidence to the contrary.

Penrose 2015, thinking of twistor space, voices the idea that nature realizes very special mathematical spaces (in the first few minutes of the video recording here).

Harvey & Moore 2020 write:

Physicists who wonder about the ultimate structure of our universe can be divided into two classes. The first class believes that our universe is chosen at random from a huge ensemble of possible universes. The second class believes that our universe and the fundamental laws that govern it are based on some beautiful and exceptional mathematical structure. String theory provides evidence for both points of view… The authors of the present paper are philosophically inclined towards the point of view of the second class of physicists and are thus interested in studying special points in the landscape of string compactifications that involve beautiful and exceptional mathematical structures. Certainly among these special points the ones associated to moonshine are amongst the most beautiful.


Last revised on February 3, 2023 at 20:03:50. See the history of this page for a list of all contributions to it.