# Prospectus

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bergsonian.org 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.

# Bergsonian Axioms in the context of other new approaches to physics

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To combine quantum mechanics and general relativity into one pleasing theory has been the main challenge to fundamental physics for going on a century. It has proved difficult (to say no more than that) and it has left us today with a spectrum of competing approaches, from the mainstream ones pursued by thousands to the pet projects of individual visionaries. Bergsonian axioms lie hard against the latter endpoint. But it is worth stressing the uniqueness of the axioms’ motivations, lest they be assimilated to projects quite unlike them, and so overlooked by their natural constituency — namely, opponents of the “multiverse” philosophy who suspect that its wrongness is deeper and more widespread than commonly acknowledged.

Many of today’s approaches follow a common template. A genius with solid math and physics training identifies an abstract structure, either new or hitherto neglected, that seems to fit nicely into fundamental physics, and seems exquisitely beautiful to its identifier; the identifier carries out an impressive analysis confirming that the structure does fit, more or less; and then various factors determine the larger or smaller audience that will find the approach worth pursuing. Amplituhedrons and Octonions come to mind as recent examples of such structures; maybe even the String itself is an example.

The Bergsonian approach does not follow this template. For one thing, its structures are not particularly beautiful, new, or neglected (in the fields they are drawn from). But the greatest divergence from the template is that neither Bergson himself nor the present axiomatizer ever set out to find more elegant or effective physics theories. Our motivation has instead been a deep misgiving about something behind physics theories, namely the “possible worlds” idea that seems, from one angle, to form the background to all of them.

Of course, possible worlds are the foreground of some recently popular theories — the string-theoretic “landscape,” the “multiverse.” What is insufficiently stressed is that even theories that deny or are mute about the reality of other worlds can be seen as implicitly granting their existence as unrealized possibilities. Scientists themselves may have little reason to remark on this, but philosophers of science have started drawing attention to its importance. To take one example, L. Ruetsche, in Interpreting Quantum Theories:

“The content of a [scientific] theory is given by the set of [possible] worlds of which that theory is true.” This idea is sufficiently deep-seated and widespread that I’ve called it the standard account. (p. 6)

The paradox that ensues — call it Bergson’s paradox — is that this standard account of science lays out the timelessly-existing ensemble of all possible “block universes,” and implies that we ourselves are bits encoded into one or more of these universes, which is insane, as it contradicts our immediate sense of time and freedom and consciousness. (Of course, many people find this viewpoint sane and non-contradictory; God bless them.)

Bergson realized that this paradox cannot be escaped by altering, augmenting, or otherwise jazzing up a scientific theory within the standard account. Instead, the whole framework of a timelessly-existing ensemble of all possible worlds must be scrapped, and replaced by a framework in which the world is the continual creation of genuinely new possibilities. But it was not possible to formalize this idea until set theorists developed rigorous machinery for enlarging the universe of possible structures. Now that this machinery exists, it is not particularly hard to write down axioms for a model of set theory that “constructs itself” over time, or (more narrowly) axioms for a self-constructing family of continua. Whether such a family can exist, and be structurally similar to the world we actually observe, is now the question.

Before we turn to this question we should remark that a new and more satisfying solution to another infamous paradox, Russell’s paradox of the “set of all sets,” emerges once we countenance the idea of new sets being constructed over time. The consensus solution to Russell’s paradox is to declare “Thou shalt not reason about the class of all sets as though it were a set itself,” and this works in the sense that it keeps contradictions out of mathematics. But many regard this as an ad hoc prohibition rather than a satisfying solution, and wonder, “why, if the totality of all sets has a well-defined extension, is it not a set in a more extensive totality?” (W. W. Tait, “Constructing cardinals from below,” p. 10.) The Bergsonian axioms allow the totality of all sets now to be a set in a more extensive totality that becomes possible later.

The potential charm of Bergsonian axioms, then, is that while they may have the air of a too-clever-by-half philosophical contrivance, their mathematical upshot may include constraints on physics that coincide remarkably with our current picture of the world, and yield hints on how to refine that picture. If our mathematical conjectures are borne out we may be in a position to say: “Notwithstanding the cheekiness of this solution to the Bergson and Russell paradoxes, it would resolve them in a far more satisfying way than the other solutions on offer — but only if the world were exotically structured with type III von Neumann algebras in a curved spacetime that is continuous but sprinkled with discrete ‘collapse’ points.” The punch line, of course, would be that the world manifestly is structured in this exotic way (though we do not know all the details). So our approach would — assuming we were able to back up the claim just stated — explain why it must have this exotic structure, rather than a simpler Newtonian or Maxwellian one.

# Self-Constructing Continua

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Here is the latest 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.

# The Bergsonian Axioms

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The constructive closure of an arbitrary set $X$ of real numbers, written $\mathbb{R}(X)$, is the set of all real numbers in the constructive hierarchy $L(X)$ “seeded” with $X$.

continuum is a set $X$ of real numbers that is constructively closed, meaning $\mathbb{R}(X) = X$.

A set $\mathcal{N}$ of continua self-collects into another continuum $X$ if the following hold: (i) $X \not \in \mathcal{N}$;  (ii) $\mathcal{N}$ is a directed set under the inclusion ordering; (iii) $\mathcal{N} \in L(X)$; (iv) for no $x \in X$ does $\mathbb{R}(x)$ lie between $\mathcal{N}$ and $X$ in the sense that $\bigcup \mathcal{N} \subseteq \mathbb{R}(x) \subset X$; and (v) $X = \mathbb{R}(\bigcup \mathcal{N})$.

self-constructing family $\mathcal{F}$ of continua is one that satisfies the following:

Self-Collection Ax.: $(\forall X \in L(\mathcal{F}))(X \in \mathcal{F} \iff (\exists \mathcal{N} \subseteq \mathcal{F})(\mathcal{N}$ self-collects into $X) )$;

Foundation Ax.: $(\forall x \in X \in \mathcal{F})(\exists Y \in \mathcal{F}) (x \in Y \subseteq X$ and $(\forall Z \in \mathcal{F})(Z \subset Y \Rightarrow x \not \in Z))$.

A Bergson history is a self-constructing family that also satisfies the following:

Ordinal Creation Ax.: $(\exists X,Y \in \mathcal{F})(\lambda(X) > \lambda(Y))$, where $\lambda(X)$ is defined as the smallest ordinal $\alpha$ such that $L_\alpha(X)$ satisfies $ZF^-$, that is, the axioms of Zermelo-Frankel set theory without the power-set axiom.

Brief explanation of the axioms:

The Self-Collection Axiom can be construed as saying that a continuum $X$ arises as a member of $\mathcal{F}$ if and only if $X$ is the constructive closure of the union of some smaller continua in $\mathcal{F}$ that “naturally form a collection.” This is the case when these smaller continua have already been brought together in $\mathcal{F}$ in all finite combinations (directedness), so that taking their union is just the last “step” in a process of combining, each of whose infinitely many previous steps are already complete. Note that if $X$ is the continuum of a Cohen-real extension or of a random-real extension, it cannot have an $\mathcal{N}$ that satisfies this axiom. In both cases $X$ has an unbounded increasing sequence of sub-continua; however, the sequence’s union is itself constructively closed, hence a continuum (of a model of ZF that notably does not satisfy the Axiom of Choice) that is smaller than $X$.

The Foundation Axiom is a “well-foundedness” guarantee ensuring that each real number $x$ “first arises” in an $\mathcal{F}$-member that is the constructive closure of some other continua in $\mathcal{F}$ that do not have $x$ as a member; thus $x$ does not “slip in unaccountably” during the “growth” of $\mathcal{F}$.