An effort by Chris Isham and Andreas Doring, of the Imperial College of Science, Technology & Medicine, “to develop a fundamentally new way of constructing theories of physics,” is another topic being discussed by John Baez & company in The n-Category Cafe. In the previous post below, I reproduced a comment I made over there regarding the ad hoc inventions of real and imaginary numbers, but, of course, my comment did not refer to our new system of physical theory. Otherwise, I’m sure it would have been deleted post haste.
However, in writing about their new system of physical theory, Isham and Doring, surprisingly, refer to the role of the invented real and complex (imaginary) numbers, as “critical mathematical ingredients that are taken for granted,” in quantum theory. But it isn’t the numbers per se, that is the focus of their raised eyebrows, but the use of these numbers as a concept of the continuum that gives them pause. They warn:
The a priori imposition of such continuum concepts could be radically incompatible with a quantum gravity formalism in which, say, space-time is fundamentally discrete…
They also make clear that, while they are interested in new interpretations of quantum theory, new insight into the foundations of existing quantum theory isn’t their main goal. Rather, what they are trying to do is to step back even further in search of an entirely new system of physical theory. They write:
However, although we are certainly interested in such conceptual issues, the main motivation for our research programme is not to find a new interpretation of quantum theory. Rather, the goal is to find a novel structural framework within which new types of theory can be constructed, and in which continuum quantities play no fundamental role.
Whoa! “New types of theory…in which continuum quantities play no fundamental role.” Did I read that right? It almost makes one think of rushing to the phone to give them a call. “Hey, fellas,” I would probably blurt out, “we already have just that sort of thing. Come on over and take a look!” But, after a sobering pause, it becomes clear that such a dialog can never take place, not because we couldn’t give them a call, or shoot off an email, but because there is a great gulf betwixt them and us. A gulf of sophisticated mathematics, demarcated by the steep walls of professional sophistication and the associated language of esoteric communications.
Their “basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system.” What is a topos? Well, that’s the gist of the discussion going on at The n-Category Cafe. The authors tell us that “classical physics arises when the topos is Sets, the category of sets,” and that “other types of theory employ a different topos.” However, their particular strategy is more general than this, because it is “based on the intrinsic logical structure that is associated with any topos.” Heavy stuff and, accordingly, they caution the uninitiated physicists: “topos theory is a sophisticated subject and, for theoretical physicists, not always that easy to understand.”
Nevertheless, one need not be a mathematician, or a theoretical physicist, to understand how difficult such a huge undertaking is. Certainly, overhauling the mathematical formalism of current physics is something no one could take lightly, but they are funded by the Foundational Questions Institute to do exactly this sort of thing. They write:
What is needed is a formalism that is (i) free of prima facie prejudices about the nature of the values of physical quantities - in particular, there should be no fundamental use of the real or complex numbers; and (ii) `realist’, in at least the minimal sense that propositions are meaningful, and are assigned `truth values’, not just instrumentalist probabilities. However, finding such a formalism is not easy: it is notoriously difficult to modify the mathematical framework of quantum theory without destroying the entire edifice.
Of course, in our approach, we are even more ambitious, because we are claiming that the foundation of the mathematical structure, underlying quantum theory, is not only the root of the problems plaguing theoretical physics, but, once understood in the proper manner, the new insight that results actually comprises a new framework, or system, for constructing new types of physical theories, in a simple and a straightforward manner, with no need to resort to a mathematical formalism of any kind; that is, we take the position that discovering the properties of numbers and mathematics will naturally lead us to nature’s physical structure. Hence, we too take the realist view in physics, but we take the realist view in mathematics, as well.