[FOM] Godel's First Incompleteness Theorem as it possibly relates to Physics

[Vaughan Pratt]

Some of the answers to Brian Hart's question are assigning "Theory of 
Everything" one or another intuitively plausible meaning, such as the 
19th century notion "that all future observations are predictable within 
the theory" which 20th century quantum mechanics showed to be a lost 
cause.  As used today in physics ToE refers more narrowly to a theory 
purporting to unify the nuclear, electromagnetic, and gravitational 
forces.  A theory that will predict the weather is not what physicists 
have in mind for this notion.  Nor will naively amalgamating the 
separate theories of these forces pass muster here.

An intriguing example of a candidate ToE is presented in the Physics 
arXiv paper "An exceptionally simple Theory of Everything" by maverick 
physicist Antony Lisi.  Lisi identifies "the 222 quantum numbers of all 
gravitational and standard model fields" with 222 of the 240 roots of 
the simple Lie group E_8, the largest of the five exceptional simple Lie 
groups found by Wilhelm Killing in 1887.  (Killing claimed six but 
Cartan in his 1894 thesis constructing them showed that two were 
isomorphic.)

E_8 has a number of remarkable properties listed at 
http://en.wikipedia.org/wiki/E8_(mathematics), some of which suit it as 
an environment within which to find various ToE's.  Lisi's instantly 
famous but still unpublished physics arXiv submission is a word play on 
the property that E_8 is the largest of the five exceptional cases of 
simple Lie algebras, having complex dimension 248 (hence real dimension 
496).  E_8 is far from simple in the intuitive sense, having 248 
symmetries neatly represented by an 8-vertex Dynkin diagram.

While the high dimensionality of E_8 might seem to undermine its claim 
to being a reasonable framework for a ToE, Lisi claims to embed not only 
the structure and dynamics of the Standard Model gauge group but a Higgs 
boson as well as 22 new bosons (making "Lisi" the answer to "Who ordered 
that?" for any future I.I. Rabi experimentally stumbling across one of 
those bosons).  The usual parameters still have to be put in by hand 
however, as well as the (presently unknown) masses of Lisi's 22 new 
bosons, greatly weakening its claim to being a ToE.  Furthermore the 
fermions accommodated via Lisi's approach are non-chiral.

E_8 is not so much a ToE as an environment embedding enough structure to 
reflect, via Lisi's approach, a wide range of phenomena in the standard 
model plus classical (but not quantum) gravity plus some promising 
bosons.  Formulating the relevant features of E_8 as a ToE that can be 
judged as to whether it meets the conditions of Gödel's first 
incompleteness theorem might benefit from some collaboration between 
physicists and logicians.

Whether a high-quality ToE can be formulated entirely within E_8 still 
seems wide open, perhaps because E_8 is sufficiently complicated as to 
make it hard to demonstrate the absence of certain desirable features of 
a good ToE from E_8.  For example, while it seems on the face of it 
rather implausible that E_8 would somehow embed any of the usual 
hand-inserted parameters, how would one go about proving this?   Some of 
them might conceivably turn out to be encoded somehow within it; if so 
Lisi's techniques would attract a lot more attention.

Paul Budnik wrote:
> I created a half hour video "Mathematical Infinity and Human Destiny"
> (http://video.google.com/videoplay?docid=-8677521434864225474) that 
> offers philosophical speculation about this and related ideas along with 
> scenes of nature as metaphors for the philosophy.

Those with less than half an hour to spare for this may find the 
transcript at http://www.mtnmath.com/movies/math.html more convenient.

Vaughan Pratt