Sticky hosts research into Bergsonian axioms for physics. [Note we have posted a new (as of June 2024) intro/overview of the project here.]

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 that this insight alone shows a way out of the block-universe philosophy’s paradoxes. 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 boolean algebras, nested so that an earlier point’s algebra is always a 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 regions of spacetime with nested algebras, taken to represent the observables within those regions. These are von Neumann algebras rather than boolean algebras, but algebras of the latter type can be derived from them (more precisely, from their projection lattices) in a natural way.

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 continue to post work on this project as it becomes available.

Negative result brings project’s current phase to an end


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; what is needed is a $B$ with “densely-and-rigidly-nested” subalgebras. 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 rigidly nested. 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 do not retain their von Neumann algebras’ rigidity of inclusion. 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.

Main Conjecture


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.

Mathematical Papers


“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 and Self-Constructing Continua


“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.

Bergson’s paradox, and Cantor’s


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.

Generic Quantum States


Here is the latest version of “Noncommutative analogs of random-real forcing”, the paper that introduces generic quantum states as noncommutative analogs of random real numbers.

Abstract: The measure algebra used in random-real forcing is isomorphic (as is well known) to the projection lattice of a commutative von Neumann algebra $\mathcal{R}$, and it is not difficult to show that a generic filter $G$ on this lattice induces a normal state $\omega_G$ on $\mathcal{R}$, with which it is interdefinable. We show that this construction also works when carried out with noncommutative von Neumann algebras, inducing generic states on them that can be seen as noncommutative analogs of random reals. We consider whether these forcing notions can or must collapse $2^{\aleph_0}$.