Standard hosts research into Bergsonian axioms for physics.

The goal is to axiomatize Henri Bergson’s insight using mathematics that were not available to him during his lifetime, then to find whether structures used in physics supply models of the axioms, and (if so) whether they supply all such models, which could help explain some of the universe’s more baroque structural properties.

Bergson’s central insight was that the world is the continual creation of new possibilities, rather than the successive realization of pre-existing possibilities. For present purposes we will simply take it for granted that this merits enthusiasm; the best arguments for it are in Bergson’s own paper “The possible and the real.” The tools needed to make this insight rigorous did not exist during Bergson’s lifetime, but starting in the 1960s, set theorists developed a robust account of how the universe of mathematical possibilities could expand. For example, there is a core set of real numbers that must be possible because they are definable (more precisely, constructible in the sense established by Gödel); yet it is consistent to suppose that further real numbers with various properties could be added to the continuum.

This suggests a way to resurrect Bergson’s insight as a precise theory: identify each spacetime point with a distinct continuum, in such a way that more real numbers are possible at later points than at earlier ones, and write axioms to capture the idea that the new real numbers arise out of older ones in a smooth, philosophically innocuous way. This is carried out in the paper “Self-Constructing Continua.”

The resulting Bergsonian axioms are clear enough; the matter of whether they can be instantiated is not. The technique that lends itself to producing candidates for our axioms is set-theoretic forcing. Applied to our project, it would associate spacetime points with (infinite) boolean algebras, nested so that an earlier point’s algebra is always a complete subalgebra of a later point’s; a generic filter G on the outermost algebra would then be postulated, and we would identify a spacetime point with the set of real numbers constructible from G‘s restriction to that point’s algebra.

There are two immediate difficulties with this plan.

The mathematical difficulty is that none of the well-known types of boolean algebras works. While some of them can be used to associate each point with a different continuum in the manner just described, these continua will fail to satisfy the Bergsonian axioms. (This failure is shown in the “Self-constructing continua” paper.)

The scientific difficulty is that our whole setup seems alien to any actual theory of physics. Or, at least, to any physics that an undergraduate is likely to encounter. Algebraic quantum field theory (AQFT) does have similarities to our setup. It too associates parts of spacetime with nested algebras, and the randomness of the states that result from measurements might be construed as genericity in the sense of set-theoretic forcing. But there is a disanalogy: AQFT’s local algebras are C* algebras rather than boolean algebras.

It turns out that each local algebra R of AQFT is generated by a fundamental substructure, its projection lattice, which also generates, in a different way, a boolean algebra of subsets of the Hilbert space H on which R acts. Specifically, the range of each projection operator in is a distinct closed subspace of H. Taken together they form a lattice of subspaces, whose inherent boolean structure we can bring out by considering it as the base for a topology on H (excluding its null vector) and then taking its regular open algebra.

This does not quite work: although the regular open algebra A of an earlier point does embed into the algebra B of a later point, it does not do so as a complete boolean subalgebra, which is needed for a generic filter on B to induce a generic filter on A by restriction. To get around this problem we have defined a subset of projections that are “back-compatible” (with respect to an AQFT setup) and we have shown that they do not succumb to this particular problem.

The central question of our research project is thus: When the back-compatible projections of past-cone regions in an AQFT setup are defined appropriately and used for forcing, can the continua of the resulting forcing extensions satisfy the Bergsonian axioms? We hope to have a working paper on back-compatible projections available here soon.