A new (2024) overview of the Bergsonian axioms project.

We have disproved a certain conjecture that had taken a central place in the Bergsonian axioms project. This brings the current phase of the project to a close. It also provides a good moment to briefly take stock of the project and to suggest what a future phase might look like.

Bergson’s key philosophical insight is that the world is the continual creation of new possibilities, rather than the successive realization of pre-existing possibilities. We developed the Bergsonian axioms to formalize this idea — or, more precisely, the idea of a universe of mathematical possibilities constructing itself over time. There is a natural recipe for candidate structures to satisfy the axioms; it takes as inputs a boolean algebra $B$ and a function $F$ from the set of $B$’s subalgebras into $B$. The most commonly-used boolean algebras do not work in the recipe because they have too many symmetries. What is needed is a $B$ with “densely-nested” subalgebras, but without the “flexible homogeneity” property that prevents satisfaction of the axioms. We know of no such $B$, but several years ago we came across an example of rigidly-nested von Neumann algebras (“simple subfactors”) and we were later advised by their inventor R. Longo that they can be densely nested as well. The task then became to derive boolean algebras from these von Neumann algebras, in the hope that they too will prove, if not rigidly nested, then at least free of the flexible homogeneity property. The cleanest method is to take the boolean completions of the algebras’ projection lattices. We proved that if “new possibilities” were obtained this way, they would be random quantum states, which raised the question whether the physical world could be a model of the Bergsonian axioms. We investigated the advantages of an affirmative answer (see the paper “Bergson’s not-even-wrong theory, now with extra math!”). However, in the last week or two we have proved that these boolean completions of projection lattices are flexibly homogeneous. This closes our main avenue of research. We will certainly continue to maintain this site; millions have found Bergson’s underlying intuition to be compelling, and will continue to do so; in a world of eight billion people, at least a handful will surely be drawn to the idea of formalizing this intuition; and we would like to spare them at least some wheel-reinvention. We would also suggest to them the investigation of substructures of the projection lattices we have focused on; it is common in the theory of forcing to pass to a particular sub-poset of one’s forcing poset, thereby obtaining a forcing notion with very different properties. It must be said, though, that direct use of the projection lattices would have made for an elegant solution (if one may speak counterfactually about facts that hold a priori!), and that poking around for sub-posets here would have a somewhat ad-hoc character.

This is the proof that, if the continuum hypothesis holds, the structures from AQFT (algebraic quantum field theory) that we had hoped might yield a structure satisfying the Bergsonian axioms cannot, in fact, do so.

A new (as of March 2021, anyway) overview of the Bergsonian axioms project, explaining its advantages as a theory (or an eventual theory, anyway) of physics.

The following conjecture is the critical one for the proposed connection between self-constructing continua and AQFT models. It is used in an essential way by “AQFT as a possible source of self-constructing continua.” If the conjecture turned out to be false, the case for pursuing the Bergsonian Axioms project would be gravely weakened.

Conjecture: If $R \subseteq S$ is a simple subfactor inclusion of injective von Neumann factors, then every partition of $R$’s projection lattice is also a partition of $S$’s projection lattice.

Partition here means maximal set of pairwise-disjoint projections (projections $P, Q$ are “disjoint” when $P \wedge Q = 0$). For the definition of simple injective subfactors see R. Longo’s paper of that title (1987). A stronger version of the conjecture would replace “simple” with “irreducible” (meaning that $R$ has trivial relative commutant in $S$).

We have very little in the way of leads on proving (or disproving) this conjecture. The requirement that every partition of $R$’s projection lattice be a partition of $S$’s projection lattice is satisfied by an inclusion $\mathcal{R} \subseteq \mathcal{S}$ of separably-acting commutative type II von Neumann algebras (this is easy to show if we consider that in this case the lattices are measure algebras), but these are not factors and it is not obvious how this situation might apply to the factor case. On the other hand, if there existed a type I factor $\mathcal{N}$ “between” $\mathcal{R}$ and $\mathcal{S}$, i.e. $\mathcal{R} \subseteq \mathcal{N} \subseteq \mathcal{S}$, then the requirement would be violated. But if there exists such an $\mathcal{N}$ then $\mathcal{R} \subseteq \mathcal{S}$ is not irreducible, hence not simple.

“Self-constructing continua” is the mathematical core of the Bergsonian axioms project; it should be accessible to anyone familiar with the boolean-algebraic approach to forcing (used in standard set-theory texts like T. Jech’s or J. L. Bell’s). It presents the Bergsonian axioms, argues that they capture the intuitive notion of a self-constructing mathematical universe, develops a forcing construction to produce candidate models of the axioms, and shows that the boolean algebras most commonly used for forcing do not work when used as inputs to this construction.

“AQFT as a possible source of self-constructing continua” shows that some models of algebraic quantum field theory would, assuming a Main Conjecture about the projection lattices of von Neumann subfactors, yield boolean algebras suitable as inputs to the forcing construction in “Self-constructing continua.”

This Main Conjecture is, alas, false, at least assuming the continuum hypothesis (CH): this is shown in “Under CH, the boolean completion of a type III factor’s projection lattice is the standard continuum-collapsing algebra.”

“Noncommutative analogs of random-real forcing” shows that random-real forcing is equivalent to using the projection lattice of a commutative von Neumann algebra to force a generic state on that algebra, and it shows that generic states will also be forced when the von Neumann algebra used is noncommutative.

“AQFT as a possible source of self-constructing continua” (pdf)

Abstract: A forcing construction was developed to produce self-constructing continua in the paper of that title; these continua formalized the idea that the totality of real numbers might grow “organically,” but two problems identified there prevented various forcing posets from working when used in the construction. We show here how a forcing poset that avoids both problems can, under a conjecture about von Neumann subfactors, be obtained from certain models of AQFT (algebraic quantum field theory). The simple subfactors originally identified by R. Longo provide our main tool.

The Possible and the Real (Bergson, 1930)

Bergsonian Axioms for Physics (general introduction)

Bergson’s Paradox, and Cantor’s

Bergson’s not-even-wrong theory, now with extra math! (A 2021 overview/update of the project that is more technical than the other papers listed here and less facetious than its title would indicate)

This paper argues that the famous set-theory paradoxes and the paradoxes of the “block-universe” framework have a common root — unwarranted belief in a timelessly fixed totality of mathematical possibilities — and introduces the Bergsonian axioms project that seeks a unified solution to all of them.