# Prospectus

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bergsonian.org hosts research into Bergsonian axioms for physics.

Note that a main conjecture underlying most recent work on this project has been disproved. Consequently the project will be in limbo for the foreseeable future.

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 chief “coincidence” that has motivated research on Bergsonian axioms is that, under a certain conjecture about von Neumann algebras, the nested boolean algebras derived from AQFT models can have a property (rigid inclusion) that lets them evade the mathematical difficulty noted above. The way in which they would evade the difficulty is shown in “AQFT as a possible source of self-constructing continua.” Moreover, forcing with these algebras has been shown to be a noncommutative analog of random-real forcing that induces a new “random state” on an algebra of observables in AQFT; this suggests that we may eventually be able to argue that Bergson’s insight can be scientifically vindicated if and only if the world grows through the emergence of random quantum states — which it manifestly does!

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

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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

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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

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“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.”

“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

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“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.

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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

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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 $2^{\aleph_0}$.

# Self-Constructing Continua

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Here is the latest (pdf) version of the formalization (within set theory) of the idea of a self-constructing mathematical universe. The axioms are presented along with a schema for generating candidates to satisfy the axioms; the main necessary condition for a forcing set to succeed in this schema is derived.

# The possible and the real

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#### by Henri Bergson

##### Translated by DVM, from “Le possible et le réel”, pp. 99-116 in La pensée et le mouvant, P.U.F./Quadrige, 6th edition, according to which it first appeared in the Swedish journal Nordisk Tidskrift in 1930. The French text is online here.

I think that the big metaphysical problems are generally ill-posed, and that they resolve themselves when one corrects their givens, or maybe that they are problems posed in phantom terms, which vanish as soon as one inspects the formula’s terms up close. They are the offspring of our tendency to regard creation as mere assembly. Reality is global and indivisible growth, gradual innovation, persistence — like a rubber balloon inflating little by little and taking on unexpected forms at each instant. But our intelligence represents its origin and evolution as an arrangement and rearrangement of pieces trading places; it could then, in theory, foresee any state of the system. For in positing some definite number of stable elements, we give ourselves all their possible combinations in advance. And that’s not all. Reality as we perceive it directly has a fullness that never ceases to swell and that knows no emptiness. It has extension, just as it has persistence, but this concrete extension is not the infinite and infinitely divisible space that intelligence takes for its own workspace. Concrete space has been extracted out of things. They are not in it; it is in them. Only, as soon as our thoughts reason about reality, they make space into a container. Just as their habit is to bring pieces together in a relative void, they imagine that reality fills up some kind of absolute void. Now, if the misunderstanding of radical newness is the origin of poorly posed metaphysical problems, the habit of going from emptiness to fullness is the source of vacuous problems. And it is easy to see that the second error is already latent in the first. But I would first like to define it more precisely.

I am saying that there are pseudo-problems, the very problems that have been the most agonizing in metaphysics. For me they come down to just two. One has spawned the theories of being, the other the theories of knowledge.

The first consists in asking oneself why there is being, why something or someone exists. The nature of what exists doesn’t matter: just say that there is matter, or mind, or both, or say that matter and mind aren’t enough and manifest a transcendent Cause — in any case, once you have considered existences, causes, and causes’ causes, you feel yourself being dragged down the road to infinity. You only stop to ward off dizziness. People are always confirming, believing themselves to have confirmed that the difficulty remains, that the problem is still open and will never be solved. It’s true that it won’t, but it should never have been posed. It is only posed on the assumption of a nothingness that precedes being. One thinks “there could have been nothing” and then one is shocked to discover that there is something, or Someone. But analyze this sentence: “there could have been nothing.” You see that you’re dealing with words, not with ideas, and that this “nothing” has no meaning. “Nothing” is a term of everyday language that can only have meaning on humanity’s home turf, in the domain of activity and manufacture. “Nothing” designates the absence of whatever we were looking for, or wanted, or expected. Even supposing that experience did in fact present us an absolute void, it would be limited, it would have contours, it would in short be something. But in reality there is no void. We perceive and conceive only fullness. One thing disappears only because another has replaced it. Disappearance thus means substitution. Only we say “disappearance” when we see only one of substitution’s two halves, or rather, one of its two faces, the one that interests us. By doing so we note our preference for directing our attention to the object that has departed, and away from that which has replaced it. Then we say that nothing remains, meaning by this that what is does not interest us, that we are interested in what is no longer there or in what could have been there. The idea of absence, or of nothingness, or of nothing, is thus inseparably linked to that of disappearance, real or possible, and that of disappearance is itself only one aspect of the idea of substitution. These are all ways of thinking that are useful in practical life; it is particularly important to our productivity that our thinking be able to fall behind reality and remain fixed, when necessary, on what was or on what could be, instead of being monopolized by what is. But when we move from the domain of manufacture to that of creation, when we ask ourselves why there is being, why something or someone, why the world or God exists and why not nothingness, when in short we pose ourselves the most agonizing of metaphysical problems, we virtually accept an absurdity. For if all disappearance is substitution, if the idea of disappearance is only the truncated idea of substitution, then talk of the disappearance of everything is talk of a substitution that isn’t a substitution, a self-contradiction. Either the idea of the disappearance of everything has exactly as much existence as a circular square — the existence of sound, flatus vocis — or, if it does represent something, it gets across the motion of an intelligence going from one object to another, preferring the one it just left to the one it finds before it, and means by “absence of the first” the presence of the second. The whole was put forth, and then one by one each of its parts was made to disappear, without any attention paid to what replaced it. It is thus the totality of present things, only configured into a new pattern, that one has before oneself when one wants to totalize the absences. In other words this so-called mental picture of an absolute void is, in reality, a picture of absolute fullness in a mind forever jumping from part to part, having vowed never to consider anything but the emptiness of its own dissatisfaction, rather than the fullness of things. Which amounts to saying that the idea of Nothing, when it isn’t just a word, implies as much matter as that of All, with, in addition, an operation of thought.

I’d say as much for the idea of disorder. Why is the universe orderly? How has law imposed itself upon the lawless, form upon matter? How is it that our thought finds itself at home amid material things? This problem, which has become for the moderns the problem of knowledge, after having been for the ancients the problem of being, is born of the same kind of illusion. It vanishes if we consider that the idea of disorder has a definite sense only in the domain of human industry, or, as we have said, of manufacture, but not in that of creation. Disorder is simply the order we weren’t looking for. You cannot make one order disappear, even in thought, without making another spring up. If there is no free will or teleology, it’s because there is mechanism; when mechanism lets down its guard, then will, caprice, and teleology rush in. But when you expect one of these orders and find the other, you call it disorder, formulating what is in terms of what could or should be, and reifying your disappointment. Disorder thus always comprises two things: first, external to ourselves, an order; second, internal to ourselves, the picture of a different order that alone interests us. Making something disappear again means substitution. And the idea of making all order disappear, that is, of achieving absolute disorder, contains a veritable contradiction, since it consists in letting only one face of an operation remain which, by hypothesis, has two. Either the idea of absolute disorder is only a combination of noises, flatus vocis, or, if anything does answer to it, it is the motion of a mind jumping from mechanism to teleology, from teleology to mechanism, and marking its place at each step by indicating the place where it is not. So, in order to make order disappear, you give yourself two or more of them. Which amounts to saying that the concept of an order coming to add itself onto an “absence of order” implies an absurdity, and that the problem vanishes.

The two illusions that I have just pointed out are in reality only one. They consist in believing that there is less in the idea of void than in the idea of fullness, less in the idea of disorder than in that of order. In reality, there is more intellectual content in the ideas of disorder and nothingness (when they do represent something) than in those of order and existence, because they imply many orders, many existences, and, beyond that, a mind juggling unconsciously with them.

Well, I find the same illusion in the case before us. There are plenty of misunderstandings, plenty of errors, at the bottom of the doctrines that misunderstand the radical newness of each moment of evolution. But above all there is the idea that the possible is less than the real, and that, for this reason, the possibility of things precedes their existence. They would thus be representable in advance; they could be thought before being realized. But the opposite is true. If we leave to the side closed systems, subject to purely mathematical laws and locked off due to the fact that abiding never gets a foothold in them — if we consider the whole of concrete reality or simply the world of life, and even more so that of consciousness, we find that there is more, and not less, in the possibility of each of the successive states than in their reality. The possible is simply the real with, in addition, a mental act that casts its image into the past once it has been produced. It’s just that our intellectual habits prevent us from seeing this.

That’s what my interlocutor forgot when he questioned me on the theater of tomorrow. He may also have been playing unconsciously on the meaning of the word “possible”. Hamlet was surely possible before it was realized, if you mean by this that there was no insurmountable obstacle to its realization. In this particular sense, we call possible whatever is not impossible; and it is self-evident that a thing’s non-impossibility is the condition for its realization. But possibility understood this way has nothing to do with virtuality, with ideal preexistence. Close the gate and you know no one will cross the road; it does not follow from this that you can predict who will cross when you open it. Yet from the completely negative sense of “possible” you pass surreptitiously, unconsciously, to the positive. Just now possibility meant “absence of obstacle”, and now you make it into a “preexistence in the form of an idea”, which is something completely different. In the first sense of the word, it was a tautology to say that something’s possibility precedes its reality: you simply mean by this that the obstacles, having been surmounted, were surmountable. But in the second sense it’s an absurdity, because clearly any mind in which Shakespeare’s Hamlet were composed in the form of possibility would thereby have created it in reality: it would thus have been, by definition, Shakespeare himself. In vain you imagine that this mind could have sprung up before Shakespeare. What you forget to think about are all the play’s details. To the extent that you fill them in, this predecessor of Shakespeare finds himself thinking everything that Shakespeare will think, feeling everything that he will feel, knowing everything that he will know, and so perceiving everything that he will perceive — consequently, occupying the same point in space and time, having the same body and the same soul: Shakespeare himself.

But I am probably insisting too much on something self-evident. All these considerations are unavoidable when you talk about a work of art. I believe we ultimately find it obvious that the artist creates possibility at the same time as reality when he carries out his work. How comes it then that we will probably hesitate to say the same of nature? Is not the world a work of art, incomparably richer than that of the greatest artist? And is there not as much if not more absurdity in saying that the future is composed in advance, that possibility preexists reality? I grant, once again, that the future states of a closed system of material points are calculable, and can consequently be seen in their present state. But I repeat that this system is extracted or abstracted from a whole which comprehends, beyond inert and unorganized matter, that which is organic. Take the concrete and complete world, with the life and consciousness that it shelters; consider the whole of nature, generating new species with forms as creative and new as any artist’s picture — then focus within these species on individuals, plants or animals, each of which has its own character — I almost said personality (for one blade of grass resembles another no more than a Raphael resembles a Rembrandt); then raise yourself, above the level of individual people, to societies, where actions and situations comparable to those of any drama unfold: how can one still speak of possibilities preceding their own realization? How can one not see that if the event is always explained, after the fact, by such and such earlier events, a totally different event would have been explained just as well, in the same circumstances, by differently chosen earlier events — what am I saying? by the same events sliced differently, distributed differently, in short seen differently by hindsight? A constant front-to-back remodeling is carried out by the present on the past, by the cause on the effect.

We don’t see this — always for the same reason, always as victims of the same illusion, always because we treat as more what is less, and as less what is more. Let us return possibility to its proper place, and evolution will become something other than the execution of a program; the doors of the future open wide; freedom is given a limitless field. The fault of those doctrines — rare indeed in the history of philosophy — that have made room for indeterminacy and for freedom in the world, is that they have failed to see what their affirmation implied. When they have spoken of indeterminacy and freedom, they have meant by indeterminacy a competition between possibilities, by freedom a choice between possibilities — as if possibility were not itself the creation of freedom! As if any other hypothesis, giving to reality an ideal existence in possibility, would not reduce newness to a mere rearrangement of old elements! As if in doing so it would not, sooner or later, have to be led to consider it calculable and foreseeable! To accept the postulate of the opposing theory is to let the enemy right into the camp. We must take a side: it is reality that makes itself possible, and not possibility that becomes real.

But the truth is that philosophy has never frankly admitted this continual creation of unforeseeable newness. The ancients abhorred it straight away because, platonists to one degree or another, they supposed Being to have been given once and for all, perfect and complete, in the changeless system of Ideas. The world unfolding before our eyes could thus add nothing to it, and was on the contrary nothing but diminution or degradation; its successive states measured the growing or shrinking gap between what is — a shadow cast in time — and what ought to be, Idea enthroned in eternity; the world’s states would record changes in a deficit, the variable form of a void. And Time would have spoiled all of this. The moderns put themselves, it’s true, in a totally different point of view. They no longer treat Time as an intrusion, a perturber of eternity; but they willingly reduce it to a mere appearance. The temporal viewed this way is just the confused form of the rational. What is perceived by us as a succession of states is conceived by our intelligence, once this fog descends, as a system of relations. The real becomes once more the eternal, with the one difference that it is the eternity of Laws in which phenomena are resolved, instead of the eternity of Ideas serving as their model. But, in one case as much as the other, we’re dealing with theories. Let us stick to the facts. Time is immediately given. That is enough for us, and, at least until someone proves its non-existence or its wrongheadedness, we will simply affirm that there is this effective surging forth of unforeseeable newness.

Philosophy would do well to find some absolute in the moving world of phenomena. But we would do well too, to feel stronger and more joyful. More joyful, because reality inventing itself before our eyes will give each of us, endlessly, certain satisfactions that art supplies occasionally to fate’s chosen few; beyond the stasis and monotony that we first see in this reality, with senses hypnotized by the constancy of our wants, reality will reveal to us ceaselessly regenerating newness, the moving creativity of things. But we will above all be stronger, for in this great work of creation that has been there from the beginning and keeps going before our eyes, we will feel ourselves to be participants, creators of ourselves. Our power to act, taking hold of itself once again, will intensify. Having been abased until then in a submissive attitude, slaves of some natural necessity, we will stand again, masters in allegiance with a greater Master. Such will be the conclusion of our study. Let us beware seeing a mere game in speculation on the relations of the possible and the real — it may well be preparation for living well.

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Notes on translation: I have rendered Bergson’s key terms dure, durée as “abide” and “abiding”, which connote more activity than “last” or “persist” or “endure”. These other options would also suggest “keep going”, or “fail to expire”, which is not what Bergson wants to get across. Finally, there is a Buddhist idea that one often sees translated as “abide”; I have no idea what the original Sanskrit or Pali is, but the idea seems close to the one Bergson tries to get across with durée.

I put “teleology” for finalité but confess to not having given much thought to this. There may be a technical philosophical meaning better served by some other word.

I think I put “surge” for élan once, but I mostly left it untranslated, since its use in English is close enough to what Bergson means by it. It doesn’t come up much in this essay.