A. Neumaier said:
I have a strong focus on what has been discovered, rather than on speculations what one might (or more likely might not) discover.
Exactly. It is like someone outside in the dark who has lost their keys and is searching for them, but then deliberately chooses to only search in the small area illuminated by a street light purely because it is the only place they are able to see anything at all. New mathematics isn't discovered by those cautious researchers who dare not tread into the dark - or worse, are so afraid that they actively prevent other researchers from doing so as well.
Instead, a level of fearlessness and a certain amount of boldness is required to venture out directly into the dark and commit illegal mathematical procedures from the point of view of contemporary mathematics. This is of course historically exactly what mathematical outlaws like Euler and Gauss did (e.g. with complex numbers and non-Euclidean geometry) and so ended up inventing the core theory underlying most of modern mathematics. It is impossible to say a priori if such an endeavor will be successful.
Almost any form of naively employed likelihood analysis in the case of the success of such situations is premature due to the unstated presumptions made about the choices that such researchers make in their discovery strategies, purely in order to carry out a successful likelihood analysis; in fact, the process of unforeseen mathematical discovery is a Pareto-distributed phenomenon and thereby already a priori defies almost all commonly employed conventional statistical tools to treat likeliness.
A. Neumaier said:
I don't see any evidence why quaternions should ever be more relevant in physics than they are now - namely for doing certain computations in SO(3) and SU(2), and for classifying certain families of simple Lie groups.
The use comes from them being natural mathematical objects - e.g. such as complex numbers, polynomials and polyhedra are natural mathematical objects. Looking at the history and philosophy of mathematics and physics, it has been shown time and time again that natural objects are prime candidates used to marry different separate fields in mathematics; the most well-known example of course is Descartes naturally unifying geometry and algebra and so inventing analytic geometry, eventually leading to analysis.
Not recognizing this vitally important aspect of natural objects in the relationship between mathematics and physics is actually a failure of academia and education. Suffice to say, unification based on natural objects was for much of history the goal of the main school of pure mathematics in the pre-Bourbakian era, which sadly died in the divorce between mathematics and physics; I am merely someone who in the modern era still follows that school of thought.
A. Neumaier said:
It is pure speculation that such a theory would be useful.
It is equally pure speculation that it wouldn't be useful. Moreover, 'usefulness' is a very contigent concept which means many different things to different people: with theories this applies as well, just because some mathematical theory isn't directly useful in some practitioners opinion doesn't mean it isn't useful at all.
By that logic the theory of differential forms isn't useful either because vector calculus does largely the same thing and has the added benefit of being far more widely known. Funny enough I have actually met physicists - usually working in applied physics or experimental physics - who have made exactly such claims.
In any case, finding the correct use for some form of mathematics is a matter of creativity and trial and error, not a matter of conforming to strict rationality or routine logic as the more mundane applications of mathematics as are.
A. Neumaier said:
World class mathematical physics would be the rigorous construction of QED or QCD. This needs tools quite different from what you speculate about.
It is funny that you would say so as what I am speculating about is actually part of a practically abandoned research direction in constructive QFT; the reason for abandoning this direction wasn't lack of results but instead lack of funding due to explosively dominant competing programs (read: string theory). The researchers moved onto applied mathematics, made their mark there and the original connection of their work with physics was forgotten. This just shows how much of a fad-ridden endeavor contemporary theoretical physics has become.
