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.