[a continuation of this]
Writes J. L. Synge, in Relativity: The Special Theory (2nd edn. 1965), p. 163:
According to this hypothesis [viz. that of a unique mathematical structure for nature], the mathematical formulae of physics are discovered not invented, the Lorentz transformation, for example, being as much a part of physical reality as a table or a chair.
Hear, hear! The Realist raises his glass with a sigh, basking in the warm glow of these purling words. -- But suddenly, the author surprises us with a basin of ice-water in the face:
But this hypothesis of a unique mathematical structure for nature is actually very naïve. It is the product of the eighteenth century, a period when mathematics was understood much less than it is today, and it is unacceptable to any physicist who has thought about mathematics, or any mathematician who has thought about physics. [Oh, snap!] When understood properly (i.e. as mathematicians understand them) these concepts exist in the human mind and not in nature; it is a meaningless waste of time to debate whether the ratio of two measured lengths is rational or irrational, or whether matter is continuous or discontinuous, because the concepts of irrationality and continuity belong to a world of the intellect, a world of mathematics, and not to the real world in which phenomena occur and are measured by pieces of apparatus.
(… slow burn…)
And again, p. 207:
Is matter really discrete or continuous? … That question must be regarded as quite meaningless. For ‘continuous’ is a mathematical word, not a physical word, and has only a very vague bearing on nature; we must not try to attach physical meanings to mathematical concepts which involve infinite processes.
[…] The above remarks merely underline the philosophical attitude [quoted above]. It is a theme that bears repetitition.
( Somehow in these last bits, I detect the tones of Dolores Umbridge…)
And p. 308:
We use now this mathematical representation, now that, seeking those representations which are convenient to work with and which yield at least some correct physical predictions.
Now, we quite agree that it doesn’t make sense to say whether a numerical physical measurement is rational or irrational. And certainly, in physics, infinities are tricky. The continuum may indeed not be what the doctor ordered for the texture of spacetime. And we agree that it may be convenient to employ this or that formalism for this or that particular problem. (We shall develop this point further in our discussion of the wave picture vs. the matrix methods in quantum mechanics: but shall draw a very different moral from that of Professor Synge.) And yet we hold to our Realist position: which has, moreover, practical consequences, and is not simply a matter of private preference in ontology.
The Realist claim is that mathematical objects are as real as physical objects. That is not to claim that any given mathematical object -- be it the continuum, E8, or Klein bottles -- is actually instantiated in this particular physical cosmos, let alone that it applies everywhere and across the board. Thus, take the continuum. Our own spacetime may well be granular, not continuous; the cosmos might be finite, both in diameter and duration. It might even be downright cellular, as in Stephen Wolfram’s view.
Furthermore, the continuum itself is among the most mysterious of mathematical objects; it is, after all, the eponym of the spectacularly counterintuitive status of the Continuum Hypothesis. It sticks in the craw even of some mathematicians (though not the ones we prefer to share a beer with), let alone physicists or shopkeepers.
However. Considered purely as an object of group theory, E8 has calculable values along certain dimensions of assessment (algebraic properties): Let us call the dimensions of assessment "alpha, beta, gamma" … , and the values we calculate "alpha-0, beta-0", etc. Thus, for E8 (reading the values out of Wiki), alpha might be “What is its dimension?”, and alpha-0 turns out to be 248; beta might be “What is its rank?”, and the answer “8”; “What is its center?” -- “Trivial”; “What is the order of its Weyl group?” -- “696729600” (but you already knew that) …
Now: Suppose that spacetime does turn out to have six extra compactified dimensions, and that heterotic string theory is indeed the gospel truth. Now, those in a position to know, inform us that in that case there are only two possible choices for the gauge group of our world. Recall that earlier gauge groups we’ve met were things like the rotation of a wheel, things whose reality is familiar even prior to their employment as gauge groups: but now we must choose between SO(32) and E8 X E8. The choice must be made on the basis of certain physical predictions. Assessment-dimension lambda, for instance, might turn out to be physically measurable (specifying, say, the spin of a graviton, or the mass of a magnetic monopole, or what have you); lo and behold, lambda-0 in E8 X E8 turns out to correspond what we have measured, whereas the different value that falls out of SO(32) does not; and so forth for some other assessments.
All right then: The Realist thesis in this case says simply that adoption of E8 -- which is, in this view, pre-existent, and independent of physics just as it is independent of politics -- is a package deal. Not every fact and feature about E8 will be physically interpretable, let alone measurable; but for any that is -- say, omega -- the value as measured experimentally must turn out to be omega-0. If it doesn’t, we have a problem. E8 being pre-existent, and not invented by ourselves for our own convenience, it cannot be toyed and tinkered with, selecting some features and rejecting others in a Procrustean attempt to match experiments. And Synge’s picture of using Newton or Einstein, merely as the mood moves us, as though one were an Allen wrench and the other a pair of pliers, won’t do at all. If the mathematical structures we seem dimly to glimpse behind the veil of the world were analogous, not to mountains, but to man-made tools, then we could always kludge one up for the occasion. Then physics would be the theoretical equivalent of toenail clippers and pinking shears.