To combine quantum mechanics and general relativity into one pleasing theory has been the main challenge to fundamental physics for going on a century. It has proved difficult (to say no more than that) and it has left us today with a spectrum of competing approaches, from the mainstream ones pursued by thousands to the pet projects of individual visionaries. Bergsonian axioms lie hard against the latter endpoint. But it is worth stressing the uniqueness of the axioms’ motivations, lest they be assimilated to projects quite unlike them, and so overlooked by their natural constituency — namely, opponents of the “multiverse” philosophy who suspect that its wrongness is deeper and more widespread than commonly acknowledged.
Many of today’s approaches follow a common template. A genius with solid math and physics training identifies an abstract structure, either new or hitherto neglected, that seems to fit nicely into fundamental physics, and seems exquisitely beautiful to its identifier; the identifier carries out an impressive analysis confirming that the structure does fit, more or less; and then various factors determine the larger or smaller audience that will find the approach worth pursuing. Amplituhedrons and Octonions come to mind as recent examples of such structures; maybe even the String itself is an example.
The Bergsonian approach does not follow this template. For one thing, its structures are not particularly beautiful, new, or neglected (in the fields they are drawn from). But the greatest divergence from the template is that neither Bergson himself nor the present axiomatizer ever set out to find more elegant or effective physics theories. Our motivation has instead been a deep misgiving about something behind physics theories, namely the “possible worlds” idea that seems, from one angle, to form the background to all of them.
Of course, possible worlds are the foreground of some recently popular theories — the string-theoretic “landscape,” the “multiverse.” What is insufficiently stressed is that even theories that deny or are mute about the reality of other worlds can be seen as implicitly granting their existence as unrealized possibilities. Scientists themselves may have little reason to remark on this, but philosophers of science have started drawing attention to its importance. To take one example, L. Ruetsche, in Interpreting Quantum Theories:
“The content of a [scientific] theory is given by the set of [possible] worlds of which that theory is true.” This idea is sufficiently deep-seated and widespread that I’ve called it the standard account. (p. 6)
The paradox that ensues — call it Bergson’s paradox — is that this standard account of science lays out the timelessly-existing ensemble of all possible “block universes,” and implies that we ourselves are bits encoded into one or more of these universes, which is insane, as it contradicts our immediate sense of time and freedom and consciousness. (Of course, many people find this viewpoint sane and non-contradictory; God bless them.)
Bergson realized that this paradox cannot be escaped by altering, augmenting, or otherwise jazzing up a scientific theory within the standard account. Instead, the whole framework of a timelessly-existing ensemble of all possible worlds must be scrapped, and replaced by a framework in which the world is the continual creation of genuinely new possibilities. But it was not possible to formalize this idea until set theorists developed rigorous machinery for enlarging the universe of possible structures. Now that this machinery exists, it is not particularly hard to write down axioms for a model of set theory that “constructs itself” over time, or (more narrowly) axioms for a self-constructing family of continua. Whether such a family can exist, and be structurally similar to the world we actually observe, is now the question.
Before we turn to this question we should remark that a new and more satisfying solution to another infamous paradox, Russell’s paradox of the “set of all sets,” emerges once we countenance the idea of new sets being constructed over time. The consensus solution to Russell’s paradox is to declare “Thou shalt not reason about the class of all sets as though it were a set itself,” and this works in the sense that it keeps contradictions out of mathematics. But many regard this as an ad hoc prohibition rather than a satisfying solution, and wonder, “why, if the totality of all sets has a well-defined extension, is it not a set in a more extensive totality?” (W. W. Tait, “Constructing cardinals from below,” p. 10.) The Bergsonian axioms allow the totality of all sets now to be a set in a more extensive totality that becomes possible later.
The potential charm of Bergsonian axioms, then, is that while they may have the air of a too-clever-by-half philosophical contrivance, their mathematical upshot may include constraints on physics that coincide remarkably with our current picture of the world, and yield hints on how to refine that picture. If our mathematical conjectures are borne out we may be in a position to say: “Notwithstanding the cheekiness of this solution to the Bergson and Russell paradoxes, it would resolve them in a far more satisfying way than the other solutions on offer — but only if the world were exotically structured with type III von Neumann algebras in a curved spacetime that is continuous but sprinkled with discrete ‘collapse’ points.” The punch line, of course, would be that the world manifestly is structured in this exotic way (though we do not know all the details). So our approach would — assuming we were able to back up the claim just stated — explain why it must have this exotic structure, rather than a simpler Newtonian or Maxwellian one.