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. He argued (see “The possible and the real“) that only this insight can free us from the paradoxes of the block-universe philosophy. 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 revive 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 are constructed 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 any structure can satisfy them is not. The technique that lends itself to producing candidates for the 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 (like the “Cohen real” and “random real” algebras) 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.

Along these lines we have developed a “random-state” forcing, a noncommutative analog of random-real forcing that induces a new state on an algebra of quantum observables in AQFT. We have shown that this forcing is not equivalent to Cohen or random-real forcing, and may therefore be free of the properties that keep those forcing notions from satisfying the Bergsonian axioms.

The central question of our research project is thus: Can we find a variant of random-state forcing that yields continua satisfying the Bergsonian axioms? We will post work on this project as it becomes available.