GameLaboratory

So long as we believe that there must be a reason why what is, is the way it is, we will continue to fuel superstition, which is to say, the belief that there is an ineffable reason underlying all things. Since we will never be able to discover or understand such a reason, all we can do is believe in it, or aspire to believe in it. So long as we construe our access to facticity in terms of thought’s discovery of its own intrinsic limits and of its inability to uncover the ultimate reason for things, our abolition of metaphysics will only have served to resuscitate religiosity in all its forms, including the most menacing ones. So long as we construe facticity as a limit for thought, we will abandon whatever lies beyond this limit to the rule of piety. Thus, in order to interrupt this see-sawing between metaphysics and fideism, we must transform our perspective on unreason, stop construing it as the form of our deficient grasp of the world and turn it into the veridical content of this world as such – we must project unreason into things themselves, and discover in our grasp of facticity the veritable intellectual intuition of the absolute. ‘Intuition’, because it is actually in what is that we discover a contingency with no limit other than itself; ‘intellectual’ because this contingency is neither visible nor perceptible in things and only thought is capable of accessing it, just as it accesses the chaos that underlies the apparent continuity of phenomena. No doubt, this is yet another bid to overturn Platonism, but it differs fundamentally from those with which we are all familiar. Unlike Nietzsche, it is not a matter of abolishing the immutable realm of ideality on behalf of the sensible becoming of all things, nor even of relinquishing traditional philosophical denunciations of phenomenal time

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and of the illusions of the senses. Rather, it is a matter of relinquishing the belief, common to Platonism and anti-Platonism, that becoming pertains to phenomena while intelligibility pertains to the immutable, and of denouncing, via intellectual intuition, the ‘stabilist’ illusion of sensible becoming – the illusion that there are invariants or immutable laws of becoming. The speculative releases us from the phenomenal stability of empirical constants by elevating us to the purely intelligible chaos that underlies every aspect of it.

We said above that so long as we continue to believe that there is a reason why things are the way they are rather than some other way, we will construe this world as a mystery, since no such reason will ever be vouchsafed to us. And yet – we come now to the issue that is going to occupy us – the requirement that there be a necessitating reason is not just a symptom of ideological delusion or of theoretical timorousness. It follows from a principled rejection of the factial based upon an apparently decisive objection against the latter; an objection which we shall have to expound and refute if we want our speculative approach to retain a minimal degree of credibility.

This objection is the following: it seems absurd to maintain that not only things but also physical laws are really contingent, for if this was the case, we would have to admit that these laws could actually change at any moment for no reason whatsoever. And indeed, one unavoidable consequence of the principle of factiality is that it asserts the actual contingency of the laws of nature. If we are seriously maintaining that everything that seems to us to have no reason to be the way it is, is actually devoid of any necessary reason to be the way it is, and could actually change for no reason, then we must seriously maintain that the laws of nature could change, not in accordance with some superior hidden law – the law of the modification of laws, which we could once more construe as the mysterious and immutable constant governing all subordinate transformations – but for no cause or reason whatsoever.

However, the objection continues, anyone who accepts the foregoing thesis would have to expect objects to behave in the most capricious fashion at every moment, and thank the heavens that this is not the case, and that things continue to conform to everyday constants. Those of us who endorse this claim would have to spend our time fearing that familiar objects could at any moment behave in the most unexpected

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ways, congratulating ourselves every evening on having made it through the day without a hitch – before worrying about what the night might hold in store. Such a conception of reality or relation-to-the-world seems so absurd that no one could sincerely maintain it. We all know the old adage according to which there is no absurdity that has not at one time or another been seriously defended by some philosopher. Our objector might acerbically remark that we have just proved this adage false, for there was one absurdity that no one had yet proclaimed, and we have just unearthed it.

Our objector will conclude that we have no choice but to admit that laws are underwritten by a necessity that rules out such capricious disorder. Since this real necessity cannot be derived from logic or mathematics alone – for we can conceive of many physical universes other than our own without contradiction – there is nothing for it but to maintain that the world is underwritten by a necessity that is not logico-mathematical but rather intrinsically physical, and for which it will always be impossible to provide a comprehensible reason. But – and here is the crux of the objection – we cannot give up the idea of such a necessity on the grounds of its enigmatic character without also giving up the manifest stability of our world. For, short of stupefying coincidence, a world without physical necessity would be riven at each instant and in each of its points by an immense multiplicity of disconnected possibilities, on account of which it would implode into a radical disorder infusing even its tiniest material particles. In other words, the objection goes, if physical laws were actually contingent, we would already have noticed it – moreover, it is quite likely that we would not even be here to notice it, since the disorder resulting from such contingency would no doubt have atomized all consciousness, as well as the world that is given to it to perceive. Thus, the fact of the stability of the laws of nature seems sufficient to refute the very idea of their possible contingency – unless, of course, we were to attribute this stability to a quite extraordinary degree of coincidence, which would have allowed us to live up until now amidst constants of quite impeccable stability – a statistical miracle which we would have good reason to marvel at, but whose imminent collapse, whether in the coming day or the coming minute, we would also have every reason to fear. But it is precisely this claim about the real contingency of physical laws that we propose to defend in all seriousness. Yet we do not give thanks at every instant that we live in a stable world. For although we maintain that the laws of nature could actually change for no reason, nevertheless,

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like everyone else, we do not expect them to change incessantly. In other words, our claim is that it is possible to sincerely maintain that objects could actually and for no reason whatsoever behave in the most erratic fashion, without having to modify our usual everyday relation to things. This is what we must now attempt to justify.

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The difficulty which we have just identified is a familiar philosophical problem – it is known as ‘Hume’s problem’. To answer the foregoing criticism of the claim that physical laws are contingent is to propose a speculative solution to Hume’s problem.

What is this problem? In its traditional version, it can be formulated as follows: is it possible to demonstrate that the same effects will always follow from the same causes ceteris paribus, i.e. all other things being equal? In other words, can one establish that in identical circumstances, future successions of phenomena will always be identical to previous successions? The question raised by Hume concerns our capacity to

demonstrate the necessity of the causal connection.1 Our account will focus on the principle of causality, also known as ‘the principle of the uniformity of nature’ by those suspicious of the term ‘cause’. Whatever name one prefers, the formulation is essentially the same: this principle assumes that given the same initial conditions, the same results invariably follow. However, we must dispel an all too frequent misunderstanding here by pointing out that the necessity of this principle has never been seriously called into question by the varieties of falsificationism inaugurated by Karl Popper. Falsificationism does not claim that the laws of nature could change for no reason in the future, but ‘only’ that the theories espoused by the sciences of nature are always susceptible to refutation by unexpected experimental results. For falsificationism, it is a matter of being forced to reconsider our belief in the permanence of physical theories because of the impossibility of attaining certain knowledge of the ‘active factors’ that may be present in nature. Thus neither Popper nor any other epistemologist claims that a theory could be subverted because of a random transformation in the structure of reality. Popper does not maintain that given identical circumstances, physical laws could some day change – he maintains only that we can never demonstrate that a determinate physical theory will always be valid, since it is impossible to rule out a priori the future possibility of an experiment behaving in the manner of an as yet un-catalogued datum that

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could invalidate the predictions of the theory in question. Physicists were driven to abandon Newtonian physics in favour of Relativity physics by new experiments, or new interpretations of old experiments, not by some sudden transformation in the structure of nature occurring sometime around 1905 (the date of the publication of the Special Theory of Relativity), and which modified the physical universe itself. Thus Popper assumes without further argument that the principle of the uniformity of nature is veridical, and never genuinely confronts Hume’s problem, which is not about the question of the future validity of our theories of nature,

but about the future stability of nature itself.2 We must also make a further clarification. The problem of causality as raised by Hume is a problem that concerns the constancy of natural laws regardless of whether these are construed deterministically or indeterministically, i.e. probabilistically. Obviously, Hume formulated his problem within the deterministic framework of the physics of his day, but the nature of the problem is actually unaffected by the question of whether or not natural laws will turn out to be probabilistic. It is a matter of knowing whether given perfectly identical circumstances, the same laws will be verified in the future, irrespective of the nature of these laws. In the case of a deterministic law, we would have to ask whether given conditions X then event Y, and only event Y, will continue to occur tomorrow in the same way it occurred today. In the case of a probabilistic law, we would have to ask whether given conditions X, event Y will have the same probability of occurring or not occurring tomorrow as it does today. Thus, the problem of causality as formulated by Hume must not be conflated with the problem of determinism – it is a more general problem concerning all laws of nature, irrespective of their eventual specificity. The same point can be put differently by saying that Hume’s problem raises the question of whether we can have any guarantee that physics as such – and not just this or that variety of physics – will continue to be possible in the future. The condition of possibility for physics is the repeat-ability of experiments, which is the fundamental guarantor of the validity of a theory. But if we found tomorrow that successive experiments carried out under exactly the same conditions yielded completely different results, if it became impossible to guarantee stable effects or stable probabilities from one day to the next for the same experimental conditions, then it is the very idea of physical science that would be undermined. Thus, Hume’s question can be formulated as follows: can we demonstrate that the experimental science which is possible today will still be possible tomorrow?

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This is the proper formulation of the problem, to which up until now, three types of responses have been envisaged: a metaphysical response, a sceptical response (the one espoused by Hume himself), and of course Kant’s transcendental response. We will briefly reiterate the essential elements of all three responses before outlining those of the speculative response.

A metaphysical response to Hume’s question would proceed by demonstrating the existence of a supreme principle governing our world. Thus, for instance, one might proceed in the manner of a disciple of Leibniz, and begin by demonstrating the necessary existence of a perfect God, before proceeding to infer from the latter the fact that such a God can only have created the best of all possible worlds, which is to say, ours. Accordingly, the eternity of our world, or at least of the principles that govern it, would be guaranteed by the eternity of divine perfection itself. This would be what could be called a direct and unconditional proof of causal necessity – I produce a positive demonstration that the existence of God is unconditionally necessary (that it proceeds from His essence alone, rather than from some external condition), before directly inferring from it that our world must be and remain the way it is.
The sceptical solution is the one that Hume proposed in response to his own question. His response can be broken down into two parts: a) Hume begins by rejecting every metaphysical solution to the problem of causality – it is impossible, he insists, to establish the future stability of natural laws by any sort of a priori reasoning. According to Hume, we have only two means at our disposal when it comes to establishing the truth of an existence or non-existence – experience and the principle of non-contradiction. But neither of these will allow us to demonstrate the necessity of the causal connection. For experience can only inform us about the present or the past, not the future. It may tell us that a law exists, or is susceptible to verification in the same way in which it has existed or been verified in the past, but it cannot establish that this law will continue to be verified in the future. As for the principle of non-contradiction, it allows us to establish a priori, and independently of any recourse to experience, that a contradictory event is impossible, that it cannot occur either today or tomorrow. But for Hume there is nothing contradictory in thinking that the same causes could produce different effects tomorrow.

Let us cite Hume’s own illustration of his argument in Section IV of the Enquiry Concerning Human Understanding:

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When I see, for instance, a Billiard-ball moving in a straight line

towards another; even suppose motion in the second ball should

by accident be suggested to me, as the result of their contact or

impulse; may I not conceive, that a hundred different events might

as well follow from that cause? May not both these balls remain

at absolute rest? May not the first ball return in a straight line, or

leap off from the second in any line or direction? All these supposi—

tions are consistent and conceivable. Why then should we give the

preference to one, which is no more consistent or conceivable than

the rest? All our reasonings a priori will never be able to shew us any

foundation for this preference. (Hume 1957: 44)

It is not just causal necessity, but what Leibniz called ‘the principle of reason’ that is being called into question here. We have seen how, according to such a principle, there must be a reason why everything is as it is rather than otherwise. But what Hume tells us is that such a reason is entirely inaccessible to thought, for since we cannot demonstrate that the laws must remain as they are, we cannot demonstrate the necessity of any fact – on the contrary, it would be perfectly compatible with the requirements of logic and experience for everything to become other than it is, whether natural processes, things, or events. There is no reason for anything to be or to remain self-identical.

b) But Hume is not content with disqualifying every possibility of a response to his question – he modifies the question itself in such a way as to replace it with a problem which does admit of a solution. Since we cannot demonstrate the necessity of the causal connection, he argues, we should stop asking ourselves why the laws are necessary and ask instead about the origin of our belief in their necessity. This amounts to a relocation of the problem that replaces a question about the nature of things with a question about our relation to things – one no longer asks why the laws are necessary, but why we are convinced that they are. Hume’s answer to this question can be summed up in a single word: habit, or custom. When a fact recurs, it engenders in us a spontaneous feeling of habituation which gives rise to the certainty that the same thing will re-occur in the future. It is this propensity to believe that what has already recurred will invariably recur in the same way in the future that governs our entire relation to nature.

Lastly, the third type of response to Hume’s problem is Kant’s transcendental response – more precisely, the objective deduction of

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the categories as elaborated in the Critique of Pure Reason’s ‘Analytic of Concepts’. This deduction is certainly one of the most difficult sections of the Critique, but its basic principle is easy enough to grasp. We will briefly recapitulate it here, because we shall have occasion to return to it. The transcendental solution differs from the classical metaphysical solution to the problem by substituting an indirect and conditional proof for a direct and unconditional one. We saw above how a metaphysical or dogmatic response to Hume’s problem operated by producing a positive demonstration that there exists an absolutely necessary principle, then deriving the necessity of our world from it. By way of contrast, the transcendental enquiry produces an indirect proof of causal necessity, which is to say, a proof by reductio ad absurdum. It proceeds as follows: we begin by assuming that there is no causal necessity, and then we examine what ensues. But what ensues, according to Kant, is the complete destruction of every form of representation, for the resulting disorder among phenomena would be such as to preclude the lasting subsistence of any sort of objectivity and even of any sort of consciousness. Consequently, Kant considers the hypothesis of the contingency of the laws of nature to be refuted by the mere fact of representation. This is the regard in which Kant’s response is conditional – he does not say that it is absolutely impossible that causality could cease to govern the world in the future; he merely says that it would be impossible for such an occurrence to manifest itself to consciousness – and this because if causality ceased to govern the world, everything would become devoid of consistency, and hence nothing would be representable. This is why Hume’s imaginary scenario of the billiard-balls is impossible – for in this scenario, it is only the billiard-balls that escape causality, not the table upon which they roll, or the hall containing the table. It is precisely because the context remains stable that we are still able to represent something to ourselves when we imagine the fantastical possibilities suggested by Hume. However, says Kant, if representation were no longer structured by causality (along with the other categories of the understanding), it would no longer structure any aspect of the phenomenon at all, and nothing would remain, whether in the subject or the object, capable of sustaining our contemplation of it as spectators. Thus, causal necessity is a necessary condition for the existence of consciousness and the world it experiences. In other words, it is not absolutely necessary that causality governs all things, but if consciousness exists, then this can only be because there is a causality that necessarily governs phenomena.

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For all their apparent differences, all three of these responses to Hume’s problem share a common assumption. What they all have in common is the fact that none of them ever calls into question the truth of the causal necessity. In all three cases, the question is never whether causal necessity actually exists or not but rather whether or not it is possible to furnish a reason for its necessity. The self-evidence of this necessity is never called into question. This is obvious in the case of the metaphysical and the transcendental solutions, since they both proceed by trying to demonstrate its truth, but what is less obvious is that Hume too never really doubts causal necessity – he merely doubts our capacity to ground the latter through reasoning. For Hume, the ‘ultimate cause of any natural

operation’3 must remain unknown to us – we may well be able to reduce the principles that underlie natural phenomena to a few general causes; nevertheless, Hume insists, ‘as to the causes of these general causes, we should in vain attempt their discovery [. . .] These ultimate springs and

principles are totally shut up from human curiosity and enquiry.’4 In other words, we may well be able to uncover the basic laws that govern the universe – but the cause that underlies those laws themselves, and which endows them with necessity, will remain inaccessible to us. This is to concede that physical processes are indeed possessed of ultimate necessity. And it is precisely because Hume concedes this that he can characterize his own position as sceptical – for to be a sceptic is to concede that reason is incapable of providing a basis for our adherence to a necessity we assume to be real.

Our own speculative position refuses this assumption which all three of the aforementioned solutions share, in order to finally take seriously what the Humean – not Kantian – a priori teaches us about the world, viz., that the same cause may actually bring about ‘a hundred different events’ (and even many more). What Hume tells us is that a priori, which is to say from a purely logical point of view, any cause may actually produce any effect whatsoever, provided the latter is not contradictory. There can be no doubt that this is the evident lesson of reason, which is to say, of the thinking whose only fealty is to the requirements of logical intelligibility – reason informs us of the possibility that our billiard-balls might frolic about in a thousand different ways (and many more) on the billiard-table, without there being either a cause or a reason for this behaviour. For if reason knows of no a prioris other than that of non-contradiction, then it is perfectly compatible with reason for any consistent possibility to arise,

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without there being a discriminatory principle that would favour one possibility over another. Accordingly, it seems strange to begin by recusing the viewpoint of reason as obviously illusory, and then to attempt to provide a rational foundation for this recusal, or to conclude that the latter cannot be rationally supported. How could reason, for which the obvious falsity of causal necessity is blindingly evident, work against itself by demonstrating the truth of such a necessity? It is our senses that impose this belief in causality upon us, not thought. Thus, it would seem that a more judicious approach to the problem of the causal connection would begin on the basis of the evident falsity of this connection, rather than on the basis of its supposed truth. In any case, it is astonishing to note how in this matter, philosophers, who are generally the partisans of thought rather than of the senses, have opted overwhelmingly to trust their habitual perceptions, rather than the luminous clarity of intellection.

The sceptical position is the most paradoxical, for on the one hand it seeks to show how the principle of reason is incapable of founding its ontological pretensions, yet on the other, it continues to believe in the necessity – the real, physical necessity – that this principle has injected into the world. Although Hume no longer believes in metaphysics, he continues to believe in the necessity that metaphysics has extrapolated into things. Coupled with the unthinking propensity to believe in whatever recurs, this incomplete rejection of metaphysics results in a merely organic adherence to the phantasmatic world of metaphysics. Hume believes blindly in the world that metaphysicians thought they could prove. Thus, we should not be surprised at the ease with which such scepticism turns into superstition, for to assert and believe that there is an unfathomable necessity to the way of the world is to be prepared to believe in a great deal of providence. It seems to us that it would be wiser to believe in reason, and thereby to purge reality of the hinter-world of causal necessity.

We can now see in what regard the speculative position eliminates the contradictions usually attending Hume’s problem. From our point of view, if the necessity of the causal connection cannot be demonstrated, then this is simply because the causal connection is devoid of necessity. But this is not to say that the speculative position eliminates every difficulty. For in fact we are going to reformulate Hume’s problem in such a way as to shift its difficulty elsewhere. This reformulation can be stated as follows: instead of asking how we might demonstrate the supposedly genuine necessity of physical laws, we must ask how we are to explain the

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manifest stability of physical laws given that we take these to be contingent. Once reformulated, Hume’s question is in fact the one we raised earlier: if laws are contingent, and not necessary, then how is it that their contingency does not manifest itself in sudden and continual transformations? How could laws for which there is no permanent foundation give rise to a stable world? Our wager is that this formulation of the problem, unlike its canonical version, allows of a satisfactory solution which requires no limitation of the capacities of rationality.

For the reader who finds it decidedly difficult to concede that laws could actually be contingent, perhaps the following analogy will help. We are all familiar with the ‘adventure’ that gave birth to non-Euclidean geometries: in order to prove Euclid’s postulate that there is exactly one parallel to a right angle from a given point, Lobachevsky assumed that this thesis is false, and that it is possible to draw infinitely many lines through the point that are all parallel to the original line. He did so assuming it would lead to a contradiction, which would result in a proof by reductio of Euclid’s postulate. But instead of such a proof, Lobachevsky discovered a new geometry, which is just as consistent as that of Euclid, but differs from it. Perhaps those who are reluctant to accept the thesis we are proposing might be willing to countenance the following suggestion: let us suppose we are convinced of the necessity of the causal connection, but do not believe in the possibility of providing a metaphysical demonstration of its necessity – could we not attempt a proof by reductio of the actuality of the causal connection? Could we not cancel this necessity in thought and see if it does indeed lead us into absurdity? If it does, we will have proved apagogically what we tried in vain to prove by invoking a metaphysical principle of uniformity. But our wager is that what happened to those geometers in the case of Euclid’s postulate will also happen to us – little by little, we will discover that the acausal universe is just as consistent and just as capable of accounting for our actual experience as the causal universe. But we will also discover that the former is a universe devoid of all those enigmas that are part and parcel of the belief in physical necessity. In other words, we have nothing to lose by moving from a causal to an acausal universe – nothing except enigmas. But it is immediately apparent that such an approach is going to conflict fundamentally with the transcendental solution to Hume’s problem. For as we saw above, the transcendental deduction uses an argument by reductio in order to infer from the absence of causal necessity the destruction of representation. Our claim, on the contrary, is that

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the hypothetical suspension of causal necessity does not necessarily entail consequences that are incompatible with the conditions of representation. Accordingly, our problem can be rendered still more precise: in order to establish the validity of our speculative solution to Hume’s problem, we must expose the nature of the logical fallacy inherent in the transcendental deduction, so as to show, contrary to what the latter maintains, that the constancy of the phenomenal world does not amount to a refutation of the contingency of physical laws. In other words, we must show why it is a mistake to infer, as Kant does, the destruction of representation from the non-necessity of laws.

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For Kant, if our representations of the world were not governed by necessary connections – which he calls the ‘categories’, among which is the principle of causality – the world would be nothing but a disordered mass of confused perceptions, incapable of yielding the experience of a unified consciousness. Thus, according to Kant, the very idea of consciousness and experience requires a structuring of representation capable of making our world into something other than a purely arbitrary sequence of disconnected impressions. This is the central thesis of the so-called ‘objective’ deduction of the categories, the aim of which is to legitimate the application of the categories to experience (the categories being understood as those universal connections presupposed by physics in particular). There can be no consciousness without the possibility of a science of phenomena, because the very idea of consciousness presupposes the idea of a representation that is unified in time.5 But if the world was not governed by necessary laws, it would break up into experiences without coherence or succession, from whence no consciousness in the strict sense of the word could issue. Accordingly, the necessity of laws becomes an incontrovertible fact once one has construed it as the very condition for consciousness.

No doubt, we can only acknowledge the seemingly irrefutable character of this ‘conditioning’ argument – but this acknowledgement must be immediately qualified with the observation that it is only irrefutable insofar as it applies to the notion of stability, rather than to that of necessity. For the only incontrovertible – but also tautological – claim that can be unreservedly accorded to Kant is that the stability of phenomena provides the condition for consciousness as well as for a science of nature. Both the fact of this stability and its status as condition for science are obviously incontestable. But the same cannot be said of the inference –

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which we will henceforth call ‘the necessitarian inference’ – underlying the claim that the stability of laws presupposes, as its imperative condition, the necessity of those laws. What we must now examine are the structure and assumptions underlying this inference.

Why does the incontestable fact of the stability of the laws of nature, which is to say, the exceptionless stability of nature’s principle of uniformity, lead us to infer the necessity of this uniformity? By what reasoning do we move from the fact of stability – which is indeed general insofar as it has yet to be undermined – to an ontological necessity? The ‘necessitarian’ inference that underlies this reasoning can be broken down into three steps:

If the laws could actually change without reason – i.e. if they were not

necessary – they would frequently change for no reason. 2. But the laws do not frequently change for no reason. 3. Consequently, the laws cannot change for no reason – in other words,

they are necessary.

No one can contest proposition 2, which simply states the fact of the (manifest) stability of nature. Accordingly, the brunt of our evaluation of this inference must bear upon an assessment of implication 1, for if this implication turned out to be ‘falsifiable’, then the inference itself would collapse as a result of the disqualification of one of its premises. This implication proceeds from the contingency of laws – which is to say, from the possibility of their alteration – to the actual frequency of their alteration. This is why we will refer to it as the ‘frequentialist implication’ – for only if one accepts the validity of the implication that proceeds from the possibility of change to its frequency, can one accept the validity of the ‘necessitarian’ inference. In order to expose the invalidity of this latter inference, it is both necessary and sufficient to establish in what regard, and under what precise conditions, it becomes possible to reject the former implication.

First of all, it is important to note that the frequentialist implication not only underlies the Kantian argument in favour of causal necessity, but also the common belief in this necessity. To assert, as common opinion does, that if the laws of nature were contingent, ‘we would have noticed’, or to assert, as Kant does, that we would have noticed it to such an extent that we would know nothing, is in either case to maintain that the contingency of laws would entail an alteration of natural laws frequent enough

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to be manifested in experience, or even to destroy the very possibility of experience. Both theses, the common thesis and the Kantian thesis, rely on the same argument – viz., that contingency implies frequent transformation – differing only with regard to the intensity of the frequency which they infer from contingency (for the common thesis, the frequency would be low but manifest, while for the Kantian thesis, the frequency would be so high as to destroy representation). We must pause and take a closer look at this implication, and try to understand why it seems so obvious that Kant executes it without even bothering to justify it – something that could also be said of common sense and of the majority of the partisans of physical necessity.

We are aided in this task by Jean-René Vernes’ Critique of Aleatory

Reason, which provides us with a valuable resource.6The chief merit of this brief essay, written with a concision worthy of the philosophers of the seventeenth century, lies in the way in which it exposes the reasoning upon which both Hume and Kant implicitly rely when they assume that the necessity of laws is obvious. But it is important to point out right away that for Vernes this reasoning is legitimate – thus, he holds the belief in the necessity of physical laws to be well-founded. Vernes’ project consists in rendering explicit the reasoning that is implicit in Hume and Kant, the better to underline the sense in which it is true. By way of contrast, the relevance of Vernes’ thesis for us is that by excavating this reasoning and exposing its true nature, he allows us to discover its weak-point. Vernes’ thesis is the following: the inference whereby we proceed implicitly from the stability of laws to their necessity consists in a piece of probabilistic reasoning – ‘probabilistic’ in the mathematical sense of the term. Recall Hume’s text about the billiard-balls, which we cited above. According to Vernes, this text harbours both the (explicit) root of the problem of causality, and the (unnoticed) key to its rational solution. What lies at the root of the problem? The fact that there is a massive difference between the a priori or ‘imaginary’ possibilities – more generally, all those possibilities that are conceivable (non-contradictory) – and those possibilities that are actually experienced. Contrary to Kant’s identification of the a priori with necessity and of the empirical realm with contingency, for Vernes it is the a priori that presents us with contingency, while it is experience that presents us with necessity. The enigma resides in the fact that a priori ‘a hundred different events’ as Hume says – or even,

as Vernes puts it, ‘a practically infinite number’7 of different events – could arise from the same series of causes, ceteris paribus. Thus, a priori, the balls

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might behave in thousands upon thousands of different ways, whereas in experience, only one of those possibilities is effectuated each time – the possibility that conforms to the physical law of impacts. But by what right then do we conclude from this difference between the a priori and the experiential that it is the a priori that is deceptive, and not experience that is illusory? What is it that allows us to claim that the constancy of experience opens onto a genuine necessity, whereas the a priori does not open onto a veritable contingency? The answer is that our assumption in this case is exactly the same as that which would lead a gambler to suspect (at the very least) that a die that always lands the same face up is very probably loaded. Let us take a set of events configured in such a way that there is a priori no reason why any one of them should occur rather than any other. This would be the case for a set of dice, both of which we assume to be perfectly homogeneous and symmetrical, and about which we can assume a priori that there is no reason for any one of their faces to turn up more frequently than any other given an unbiased throw. When, armed with this hypothesis, we attempt to calculate the likelihood of one event occurring (understood as this or that outcome of throwing the dice), we implicitly assume the following a priori principle: whatever is equally thinkable is equally possible. It is precisely this quantitative equality between the thinkable and the possible that allows us to work out the probability or frequency of an event when we play a game of chance – given that, in the kinds of circumstances outlined above, one of two events has no more (this ‘more’ being strictly mathematical) reason to occur than the other (whether as the result of throwing the dice, or of tossing a coin, or of spinning a roulette wheel, etc.), then we must assume that each has as much chance of actually occurring as the other, and use this assumption to arrive at calculable estimations as to the probability of complex events entirely composed of this initial equi-probability (such as the chance of the dice turning up a double-six, or of the roulette wheel turning up 0 three times in row, etc.).

But now let us suppose that the dice with which we have been playing for an hour have continually been landing with the same face up. We would say to ourselves, invoking the principle according to which events that are equally possible are equally probable, that there is only a very small likelihood that this is the result of genuine chance. There must be a cause (perhaps a lead ball hidden in the dice) necessitating this single result . But let us now suppose that the dice we are playing with have been landing with the same face up not just for an hour, but throughout our

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entire lives, and even as far back as human memory stretches. And let us also suppose that these dice are not just six-sided, but possess millions and millions of sides. We now find ourselves in the same situation as Hume pondering his billiard-balls – for each event given in experience, we can conceive a priori of a great many different empirical outcomes (indeed, a number so immense it seems vain to try to determine it), all of which appear to us to be equally possible. Yet we continue to obtain the same outcome, which is to say, the same effects from the same causes. Thus, when Hume or Kant assume that the necessity of laws is self-evident, their reasoning is exactly the same as that of the gambler faced with loaded dice, because they implicitly assume the validity of the inference unearthed by Vernes, viz., that if physical laws were actually contingent, it would be contrary to the laws that govern chance if this contingency had never manifested itself. Consequently, there must be a necessitating reason, albeit hidden – just as there must be a lead ball imbedded in the dice – that explains the invariance in the result.

The implicit principle governing the necessitarian inference now becomes clear: the latter proceeds by extending the probabilistic reasoning which the gambler applied to an event that is internal to our universe (the throw of the dice and its result), to the universe as such. This reasoning can be reconstructed as follows: I construe our own physical universe as one among an immense number of conceivable (i.e. non-contradictory) universes each governed by different sets of physical laws; universes in which the impact of two billiard-balls does not conform to the laws that govern our own universe but results rather in both balls flying off into the air, or fusing together, or turning into two immaculate but rather grumpy mares, or into two maroon but rather affable lilies, etc. Thus, I mentally construct a ‘dice-universe’ which I identify with the Universe of all universes, bound globally by the principle of non-contradiction alone, each face of which constitutes a single universe governed by a determinate set of physical laws. Then, for any situation given in experience, I roll these dice in my mind (I envisage all the conceivable consequences of this event), yet in the end, I find that the same result (given the same circumstances) always occurs: the dice-universe always lands with the face representing ‘my’ universe up, and the laws of impact are never violated. Every time it is thrown, this dice-universe invariably results in the same physical universe – mine, the one I have always been able to observe on a daily basis. No doubt, theoretical physics will always be able to teach me new and expected things about the particular face-universe I inhabit – but

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this will simply amount to a deeper knowledge of my universe, rather than to an aleatory transformation of the universe itself. The latter has never infringed the principle of uniformity; it has always presented me with the same result given the same initial conditions. The improbability of this stability in the outcome seems so aberrant that I do not even pause to consider the possibility that it might be solely the result of chance. Consequently, I infer from it – via an inference which is generally executed too quickly even to be noticed – the existence of a necessary reason, but a necessity that is extra-logical as well as extra-mathematical. This necessity provides the ineliminable supplement to the argumentative necessity of logico-mathematical reasoning, since the latter presents me with a dice-universe that is homogeneous – in other words, each face of which is equally conceivable. Thus, I redouble the necessity proper to logico-mathematical notation with a second type of necessity, a real or physical necessity, which alone – like the lead ball imbedded in the dice – is capable of furnishing the reason for the obvious ‘fixing’ of the outcome. It is then entirely up to me whether to call the source of this second necessity ‘matter’ (as Vernes does) or ‘providence’ – whatever I choose to call it, it will remain a primordial and enigmatic fact.

To sum up: the Humean-Kantian inference is an instance of probabilistic reasoning applied not to an event in our universe, but rather to our universe itself considered as merely one among a totality of possible universes. The nub of the argument consists in registering the immense numerical gap between those possibilities that are conceivable and those that are actually experienced, in such a way as to derive from this gap the following probabilistic aberration (which provides the source for the frequentialist implication): if physical laws could actually change for no reason, it would be extraordinarily improbable if they did not change frequently, not to say frenetically. Indeed, they would change so frequently that we would have to say – and here we move from Hume to Kant – not just that we would have noticed it already, but that we would never have been here to notice it in the first place, since the ensuing chaos would have precluded the minimal degree of order and continuity required for the correlation between consciousness and world. Thus, necessity is proven by a fact of immensely improbable stability, viz., the permanence of the laws of nature, and by the subjective obverse of this permanence, which is the consciousness of a subject capable of science. Such is the logic of the necessitarian argument, and more particularly, of the frequentialist implication that underlies it.

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***

Before we proceed to outline the speculative refutation of the necessitarian argument, it is necessary to note that there is a well known rejoinder to such an argument, and it consists in demonstrating that the continuing existence of our world can in fact be explained by chance alone. It proceeds in the same way as the Epicureans explained the apparently purposeful existence of living beings – one compares the emergence of the most complex organisms to another, suitably improbable outcome (such as, for example, writing the Iliad by randomly throwing letters at a given surface), before pointing out that even this outcome ceases to be anomalous given a sufficiently gigantic number of attempts. Similarly, one response to the aforementioned probabilistic objection would be to point out that our own highly structured world could have been the result of a gigantic number of chaotic emergences that finally stabilized themselves in the configuration of our universe.

Yet this response to the necessitarian argument cannot satisfy us; and this for the very simple reason that it continues to presuppose the necessity of physical laws. Thus, it is important to notice that the very notion of chance is only conceivable on condition that there are unalterable physical laws. This is precisely what the example of the dice-throw shows: an aleatory sequence can only be generated on condition that the dice preserve their structure from one throw to the next, and that the laws allowing the throw to be carried out not change from one cast to the next. If from one throw to the next the dice imploded, or became flat or spherical, or if gravity ceased to operate and they flew off into the air, or on the contrary, were projected underground, etc., then there would be no aleatory sequence, and it would be impossible to establish a calculus of probabilities. Thus chance always presupposes some form of physical invariance – far from permitting us to think the contingency of physical laws, chance itself is nothing other than a certain type of physical law – one that is ‘indeterministic’. In Epicurus for instance, it is clear that the clinamen, the tiny aleatory deviation in the trajectory of atoms, presupposes the immutability of physical laws: the specific shape of atoms (smooth, hooked, etc.), the number of different kinds of atoms, the indivisible character of these elementary physical units, the existence of the void, etc. – none of these can ever be modified by the clinamen itself, since they provide the conditions for its effectuation. But our response to the necessitarian objection must allow us to conceive of a world devoid of any physical necessity that would still be

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compatible with the fact of the stability of laws. Accordingly, we must deploy an argument that does not at any point redouble the purely logical necessity of non-contradiction (which we derived from the principle of factiality, and which is the only necessity that is recognized as true a priori by Hume) with a real necessity, which is to say, with a necessity that would install a principle of preference between equally conceivable options. Yet it is precisely such a principle of preference that the aleatory response to the necessitarian objection introduces, since there is no contradiction involved in thinking a modification in the determinate conditions that allow a random process to be carried out (for example: a modification in the form of the dice, or in their number of faces, or in the laws that govern their throws, etc.). Moreover, such a response would not convince anyone, for it concedes to the objector the notion that we have up until now benefited from a favourable draw, and hence from a luck that might well turn, incessantly producing different outcomes. We would then be plunged into that relation to the world which we earlier qualified as absurd, and which would incite us to fear at every moment the random behaviour of reality.

Thus, the refutation of the frequential implication cannot consist in demonstrating that the stability of the world conforms to the laws of chance – rather, it should demonstrate that the contingency of natural laws remains inaccessible to aleatory reasoning. Contrary to responses of the Epicurean type, we should not grant the legitimacy of our opponent’s reasoning, and then try to render it compatible with actual experience. Instead, we must disqualify this reasoning as such by showing that it makes inappropriate use of the categories of chance and probability, outside their legitimate field of application. In other words, we must show that one cannot apply these categories to the laws of the physical world as such, and that the use of probabilistic reasoning in such a context is meaningless. We will then have succeeded in showing how the stability of laws is compatible with their absolute contingency, and we will have done so apparently contrary to all ‘healthy probability’. We will also have done so by disqualifying the absurd fear of continual disorder, since this fear relies precisely on the aleatory conception of physical laws that encourages us to interpret the actual stability of representations as an extraordinary piece of luck. In other words, we must elaborate a concept of the contingency of laws that is fundamentally distinct from the concept of chance. Yet it would certainly be possible to establish this difference between the two notions solely with the help of the preceding remark about chance,

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which is to say, by maintaining that the contingency of laws should not be confused with chance, since the effectuation of the latter already presupposes a pre-existing set of laws. We could then maintain that the contingency of laws cannot be subjected to the categories of chance, since such contingency is capable of affecting the very conditions that allow chance events to occur and exist. We would thereby disqualify the frequentialist implication that reasons about laws as if they were the result of an aleatory throw, failing to see that it is precisely such laws that provide the condition for the throw. However, even if it is not fundamentally incorrect, such a response on our part would be disappointing. It would confine itself to safeguarding contingency from a specific objection, without attempting to deepen its concept. We would confine ourselves to saying what contingency is not (i.e. chance), without taking advantage of the necessitarian objection as an opportunity for stating more precisely what it is. But factial ontology is not intended to be a ‘negative ontology’ – we do not wish to confine ourselves to maintaining that contingency as we understand it is not accessible to this or that type of reasoning; rather, our aim is to elaborate an ever more determinate, ever richer concept of contingency. Accordingly, every difficulty encountered by factial speculation should be converted into a means for identifying a determinate condition of chaos capable of allowing us to overcome the obstacle. This is the basic principle of a reason that has been emancipated from the principle of reason – its progressive unfolding demands that we exhibit the positive and differen-tiated properties of the absence of reason. Thus, a genuinely satisfactory speculative resolution of Hume’s problem should explain what could constitute a precise condition for the manifest stability of chaos. This condition would allow us to penetrate more deeply into the nature of a temporality delivered from real necessity. But as we are about to see, such a condition exists, and it is mathematical in nature: it is the transfinite.

***

In order to counter the frequentialist implication, we must begin by identifying its fundamental ontological presupposition. For in fact, this implication holds only in a very particular case, and on condition of a singularly strong ontological hypothesis, because it conjoins the being of the possible and the being of the Whole. Thus, this probabilistic reasoning is only valid on condition that what is a priori possible be thinkable in terms of a numerical totality.

Granted, the frequentialist implication does not have to provide a precise determination of the cardinality of conceivable possibilities in

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order to ‘work’ and to remain legitimate – the greater the number by which the conceivable possibilities exceed the number of actually experienced possibilities, the greater the extent of the probabilistic anomaly, and it is obvious that there are in any case immensely more conceivable possibilities than experienced ones. It does not even really matter whether this totality of possibilities is finite or infinite, since the infinite is no obstacle to the application of probabilities. Any object, even one directly given in experience, can provide me with the opportunity for a probabilistic calculation on the infinite. This infinity, which is tied to the continuity of the object under consideration, does not preclude the possibility of carrying out a positive evaluation of the sought-for event. Take for example a homogeneous rope of determinate length, both of whose extremities are being subjected to equal tension. I can calculate the positive probability that it will break at one of its points, even though these ‘breaking points’ are theoretically infinite in number along the length of the rope, since they are ‘dimensionless’. Moreover, I can do so despite the apparent paradox, according to which, since the probability of any one of these ‘dimensionless’ points being the breaking point is one over infinity, the rope is incapable of breaking at any of its points. Yet all that is necessary to obviate this ‘paradox’ is to select a segment of rope, however small, and probability once more becomes applicable to the latter as efficiently as in those cases where the number of possibilities

is a natural whole number.8

Thus, all that is required for the frequentialist implication to work is the assumption that there does indeed exist a totality of non-contradictory conceivable possibilities, and the stipulation that this totality, whatever its cardinality, is immensely larger than the set of physically possible events. But this is to say that in order for the inference to be legitimate, there is one uncircumventable condition that must be satisfied, viz., that there is a totality of conceivable possibilities. It is necessary to assume that a set of possible worlds (the ‘dice-universe’ we invoked earlier) is actually conceivable, if not intuitable, within which we can then carry out our extension of probabilistic reasoning from objects that are internal to our universe (the dice, the rope) to the universe as such. Thus, probabilistic reasoning is conceivable on condition that it be possible to conceive a totality of cases within which one can then calculate frequencies by determining the ratio of the number of favourable cases to the number of possible cases. If you revoke the notion of a set of cases and the idea of a total-universe from whence the events submitted to analysis are drawn, then aleatory reasoning becomes meaningless.

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Accordingly, aleatory reasoning – which is to say, the very idea of chance insofar as the latter is subject to a calculus of frequency – presupposes the notion of numerical totality. When totality is internal to our universe, then it is either directly given to us in experience (as in the case of the number of faces of the dice, or the number of segments of the cord), or indirectly, via an observation of the frequency of a given phenomenon. But when I attempt to apply probabilistic reasoning to our universe as a whole, I assume – without there being anything in experience that could validate this assumption – that it is legitimate to consider the conceivable as another instance of a totality of cases. Thus, I subject the conceivable to a mathematical hypothesis: I turn it into a set, however large. I turn it into the set of possible worlds because I consider it a priori legitimate to think the possible as a Whole.

But it is precisely this totalization of the thinkable which can no longer be guaranteed a priori. For we now know – indeed, we have known it at least since Cantor’s revolutionary set-theory – that we have no grounds for maintaining that the conceivable is necessarily totalizable. For one of the fundamental components of this revolution was the detotalization of number, a detotalization also known as ‘the transfinite’. On this particular point, it is the signal work of Alain Badiou – and

primarily Being and Event9– that has been decisive for us. Among Badiou’s fundamental theses is the one whereby he maintains – through his own prescriptions – the ontological pertinence of Cantor’s theorem, in such a way as to reveal the mathematical conceivability of the detotalization of being-qua-being. Although our interpretation of the ontological scope of this detotalization is not the same as Badiou’s, it is thanks to his singular project that we have been able to discover the means through which to extricate ourselves from the ontological conditions inherent in the necessitarian inference. For one of the most forceful aspects of Being and Event resides in the way in which it uses mathematics itself to effect a liberation from the limits of calculatory reason, a gesture altogether more powerful than any external critique of calculation in the name of some supposedly superior register of philosophical thought. In this regard, as well as through many other aspects of his thought, Alain Badiou has carried out a profound reinvestment of the inaugural decisions of philosophy as such, for there is no fundamental episode in philosophy since Plato that has not proceeded via a re-interpretation of its originary alliance with mathematics. Moreover, he has also, we believe, summoned each of us to try to grasp anew the meaning of this privileged link between these two discourses.

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For our part, and to begin with, we have attempted to remain faithful to this gesture through the following thesis: there is a mathematical way of rigorously distinguishing contingency from chance, and it is provided by the transfinite.

In order to be as concise and as clear as possible about the issue of the

transfinite, we can put things in the following way:10 one of the most remarkable aspects of the standard axiomatization of set-theory (known as ‘ZF’, for ‘Zermelo-Fraenkel’), progressively elaborated during the first half of the twentieth century on the basis of Cantor’s work, consists in its unencompassable pluralization of infinite quantities. ‘Cantor’s theorem’, as it is known, can be intuitively glossed as follows: take any set, count its elements, then compare this number to the number of possible groupings of these elements (by two, by three – but there are also groupings ‘by one’, or ‘by all’, which is identical with the whole set). You will always obtain the same result: the set B of possible groupings (or parts) of a set

A is always bigger than A – even if A is infinite.11It is possible to construct an unlimited succession of infinite sets, each of which is of a quantity superior to that of the set whose parts it collects together. This succession is known as the series of alephs, or the series of transfinite cardinals. But this series itself cannot be totalized, in other words, it cannot be collected together into some ‘ultimate’ quantity. For it is clear that were such a quantitative totalization to exist, then it would also have to allow itself to be surpassed in accordance with the procedure of the grouping of parts. Thus, the set T (for Totality) of all quantities cannot ‘contain’ the quantity obtained by the set of the parts of T. Consequently, this ‘quantity of all quantities’ is not construed as being ‘too big’ to be grasped by thought – it is simply construed as not existing. Within the standard set-theoretical axiomatic, that which is quantifiable, and even more generally, that which is thinkable – which is to say, sets in general, or whatever can be constructed or demonstrated in accordance with the requirement of consistency – does not constitute a totality. For this totality of the thinkable is itself logically inconceivable, since it gives rise to a contradiction. We will retain the following translation of Cantor’s transfinite: the (quantifiable) totality of the thinkable is unthinkable. Accordingly, the strategy for resolving Hume’s problem can now be stated in the following way.

We are not claiming that the non-totalizing axiomatic is the only possible (i.e. thinkable) one. Consequently, we are not claiming that the possible is always untotalizable, even if this is the case in the standard

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axiomatic of sets. For we cannot deny a priori the possibility that it could also be thinkable that the thinkable constitutes a totality. The fact that the untotalizable is thinkable within a given axiomatic does not prevent anyone from choosing another axiomatic, in which the frequentialist implication would still be valid. There is always more than one axiomatic, and despite its eminence, that of standard set-theory is only one among many. Thus, we cannot rule out a priori the possibility of selecting an axiomatic in which the realm of possible worlds would constitute an ultimate and determinate numerical totality. But this at least must be accorded to us: we have at our disposal one axiomatic capable of providing us with the resources for thinking that the possible is untotalizable. However, the mere fact that we are able to assume the truth of this axiomatic enables us to disqualify the necessitarian inference, and with it every reason for continuing to believe in the existence of the necessity of physical laws – a necessity that is mysteriously superimposed onto the fact of the stability of these same laws.

What the set-theoretical axiomatic demonstrates is at the very least a fundamental uncertainty regarding the totalizability of the possible. But this uncertainty alone enables us to carry out a decisive critique of the necessitarian inference by destroying one of the latter’s fundamental postulates: we can only move immediately from the stability of laws to their necessity so long as we do not question the notion that the possible is a priori totalizable. But since this totalization is at best only operative within certain axiomatics, rather than in all of them, we can no longer continue to claim that the frequentialist implication is absolutely valid. We have no way of knowing whether the possible can be totalized in the same way as the faces of a set of dice can be totalized. This ignorance suffices to expose the illegitimacy of extending aleatory reasoning beyond a totality that is already given in experience. Since we cannot decide a priori (i.e. through the use of logical-mathematical procedures alone) whether or not a totality of the possible exists, then we should restrict the claims of aleatory reasoning solely to objects of experience, rather than extending it – as Kant implicitly does in his objective deduction – to the very laws that govern our universe, as if we knew that the latter necessarily belongs to some greater Whole. Since both theses (viz., that the possible is numerically totalizable, and that it is not) are a priori conceivable, only experience can provide us with an assurance as to the validity of aleatory reasoning, by acting as guarantor for the actual existence of the totality which is required in order for that reasoning to work – whether in the form of a

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direct experience of a supposedly homogeneous object (e.g. the dice or the rope), or in the form of a statistical analysis (determination of average values and of appropriate frequencies for an identifiable phenomenon). Accordingly, the only totalities available to us that are capable of legitimating this type of aleatory reasoning must be given to us within our universe – which is to say, experientially. Kant’s belief in the necessity of laws is thereby revoked as an instance of aleatory reason’s unwarranted pretension to reach beyond the limits of experience.

For it is in fact this illegitimate application of probabilities beyond the limits of our experience that allows Kant, at several junctures in the 1st Critique, to derive from the hypothesis of the contingency of laws, the necessity of their frequent modification. Thus, in the following passage, the inference from the possibility of contingency to the necessary frequency of its effectuation is particularly evident:

Unity of synthesis according to empirical concepts [which in this

context means primarily the application of the relation of causality

to phenomena – QM] would be altogether contingent, if these latter

were not based on a transcendental ground of unity. Otherwise,

it would be possible for appearances to crowd in upon the soul, and yet

to be such as would never allow of experience. Since connection in

accordance with universal and necessary laws would be lacking,

all relation of knowledge to objects would fall away. The appearances might indeed constitute intuition without thought, but not

knowledge, and consequently would be for us as good as nothing.

(Kant 1929: 138, A111 – my emphasis)12

Kant infers from the assumption of the actual contingency of phenomenal laws modifications of reality so extreme that they would necessarily entail the destruction of the very possibility of knowledge, and even of consciousness. But how is Kant able to determine the actual frequency of the modification of laws, assuming the latter to be contingent? How does he know that this frequency would be so extraordinarily significant as to destroy the very possibility of science, and even of consciousness? By what right does he rule out a priori the possibility that contingent laws might only very rarely change – so rarely indeed that no one would ever have had the opportunity to witness such a modification? It can only be by the right which he derives from applying the calculus of probability to our world as a whole, rather than to any phenomenon

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given within the world, and hence from an a priori totalization of the possible, which we know, since Cantor, can no longer lay claim to any logical or mathematical necessity – which is to say, to any sort of a priori

necessity.13

***

What has been our approach to Hume’s problem, and to what extent can we claim to have provided a solution to it?

We began by reformulating the problem: instead of assuming that Hume’s imaginary hypothesis concerning ‘a hundred different events’ resulting from the same causal sequence was a chimera which had to be refuted, we sought to uncover what it was exactly that prevented us from believing in the truth of such a hypothesis, given that reason, on the contrary, issued an emphatic invitation to accept it. We then noticed that at the root of this presupposition lay an instance of probabilistic reasoning applied to the laws of nature themselves; a piece of reasoning which there was no reason to accept once its condition – the claim that conceivable possibilities constitute a totality – was revealed to be no more than a hypothesis, as opposed to an indubitable truth.

In doing so, although we have not positively demonstrated that the possible is untotalizable, we have identified an alternative between two options – viz., the possible either does or does not constitute a totality – with regard to which we have every reason to opt for the second – every reason, since it is precisely the second option that allows us to follow what reason indicates – viz., that there is no necessity to physical laws – without wasting further energy trying to resolve the enigmas inherent in the first option. For whoever totalizes the possible legitimates the frequential implication, and thereby the source of the belief in real necessity, the reason for which no one will ever be able to understand – thus, whoever does so must maintain both that physical laws are necessary and that no one can know why it is these laws, rather than others, which necessarily exist. But on the contrary, whoever detotalizes the possible is able to think the stability of laws without having to redouble them with an enigmatic physical necessity. Consequently, we can apply Ockham’s razor to the hypothesis of real necessity: since the latter is an ‘entity’ that is superfluous for explaining the world, we can safely dispense with it without depriving ourselves of anything except for a mystery. Thus, the resolution of Hume’s problem allows us to remove a fundamental obstacle to the complete embrace of the principle of factiality, because it rules out, as a henceforth familiar sophism, the transcendental objection which

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invariably infers an actual, random disorder of representations from the contingency of laws.

***

We know that the terms ‘chance’ (from the Vulgar Latin: cadentia) and ‘aleatory’ (from the Latin: alea) both refer back to related etymologies: ‘to fall’, and ‘falling’ in the case of the former; ‘dice’, ‘dice-throw’, or ‘game of dice’ in the case of the latter. Thus, these terms bring together notions that, far from being opposed to one another, are actually inseparable – the notions of play and of calculation, and of the calculation of chance which is inherent in every game of dice. Every thinking in which the identification of being with chance is dominant foregrounds the theme of the dice-totality (which is to say, of the unalterable enclosure of the number of the possible), of the apparent gratuity of the game (the play of life and of a world whose superior artificiality is acknowledged), but also that of the cool calculation of frequencies (the world of life insurance and evaluable risks). The ontology of the enclosure of possibilities inevitably situates us within a world whose aversion to gravity is but the obverse of the fact that it only takes counting techniques seriously. By way of contrast, the term ‘contingency’ refers back to the Latin contingere, meaning ‘to touch, to befall’, which is to say, that which happens, but which happens enough to happen to us. The contingent, in a word, is something that finally happens – something other, something which, in its irreducibility to all pre-registered possibilities, puts an end to the vanity of a game wherein everything, even the improbable, is predictable. When something happens to us, when novelty grabs us by the throat, then no more calculation and no more play – it is time to be serious. But what is most fundamental in all this – and this was already one of the guiding intuitions of Being and Event – is the idea that the most powerful conception of the incalculable and unpredictable event is provided by a thinking that continues to be mathematical – rather than one which is artistic, poetic, or religious. It is by way of mathematics that we will finally succeed in thinking that which, through its power and beauty, vanquishes quantities and sounds the end of play.

***

In light of the foregoing, we can now grasp the general tenor of the factial approach. Our project can in fact be formulated as follows: our aim is to supplant the contemporary dissolution of metaphysical problems by a non-metaphysical precipitation of these same problems. Let us quickly explain what this supplanting entails.

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What will be the reaction of many contemporary philosophers (albeit fewer and fewer) when confronted by Hume’s problem, or by the question as to why there is something rather than nothing? Generally speaking, they will try to find the easiest way to shrug their shoulders. They will try to demonstrate to you that there is nothing enigmatic about your question, because it does not even need to be raised any more. Thus, they will endeavour, in a spirit of charity – tirelessly repeating the Duchampian-Wittgensteinian gesture – to make you understand that there is no enigma, because there is no problem. These philosophers will claim to have dissolved your ‘naïve’ problem – ‘naïve’ because metaphysical, dogmatic, etc. – by unveiling the (linguistic, historical) source of this vain questioning. Ultimately, what they are really interested in is finding out how it was possible (and how it is still possible – you, the questioner, being proof of it) to be perplexed by such ‘pseudo-problems’. The end of metaphysics is still largely identified with this type of dissolvent approach – it is no longer a matter of asking oneself metaphysical questions, since the latter are mere semblances of questions, or questions that are now irremediably obsolete, but rather of raising questions on, or about metaphysics. But we now know that the contemporary belief in the insolubility of metaphysical questions is merely the consequence of the continuing belief in the principle of reason – for only someone who continues to believe that to speculate is to seek out the ultimate reason for things being thus and so, also believes that there is no hope of resolving metaphysical questions. Only someone who believes that the essence of the answer to a metaphysical question lies in discovering a cause or a necessary reason can estimate, rightly, that these problems will never be resolved. This is a discourse of the limits of thought, which we now know to be a consequence of the perpetuating denial of metaphysics. Accordingly, we can now see that the veritable end of metaphysics is materialized by the attempt to extract from this dissolution the precipitate of these venerable questions, now restored to their sovereign legitimacy. For it is by resolving the questions of metaphysics that we become able to understand that the essence of the latter consisted in generating problems that it could not answer so long as it retained its fundamental postulate – only by abandoning the principle of reason can we begin to make sense of these problems.

Thus, the factial stance proceeds by abandoning the dissolvent approach to metaphysics as a procedure that has itself become obsolete. For the postulate of dissolution – viz., that metaphysical problems are not

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really problems, but false problems, pseudo-questions constituted in such a way that it makes no sense to assume they could ever admit of a solution – this postulate begins to crumble once we learn to renounce the principle of reason. As a result, metaphysical problems are revealed always to have been genuine problems, since they do admit of a solution. But their resolution depends on one precise and highly constraining condition – that we begin to understand that in reply to those metaphysical questions that ask why the world is thus and not otherwise, the response ‘for no reason’ is a genuine answer. Instead of laughing or smiling at questions like ‘Where do we come from?’, ‘Why do we exist?’, we should ponder instead the remarkable fact that the replies ‘From nothing. For nothing’ really are answers, thereby realizing that these really were questions – and excellent ones at that. There is no longer a mystery, not because there is no longer a problem, but because there is no longer a reason.

***

However, we must return to our proposed resolution of Hume’s problem, for the former cannot wholly satisfy us. As we saw, this resolution is fundamentally non-Kantian insofar as it aims to establish the conceivability of the actual contingency of the laws of nature. Yet it would not be quite right to assert that our resolution itself is unequivocally speculative, just because it is anti-transcendental in intent. For although the thesis we advanced was ontological, and did indeed assert something about the in-itself rather than phenomena in maintaining the detotalization of the possible, nevertheless, it was only advanced as an ontological hypothesis. We have not established whether this non-totalization actually obtains, we have merely supposed it, and drawn the consequences of the fact that such a supposition is possible. In other words, although our proposed resolution of Hume’s problem gives us grounds for not immediately giving up on the idea of factial speculation, it has not itself been engendered as a truth by means of speculative reasoning. For a properly factial resolution of Hume’s problem would require that we derive the non-totalization of the possible from the principle of factiality itself. In order to elaborate such a solution, we would have to derive the non-whole as a figure of factiality, just as we sketched the derivation of consistency and of the ‘there is’. This would entail absolutizing the transfinite in the same way in which we absolutized consistency – which is to say that we would have to think the former as an explicit condition of contingent-being, rather than merely construing it as a mathematically formulated hypothesis that can be advantageously supported by the speculative. But it is clear that

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such a resolution of the problem would require that we be in a position to do for mathematical necessity what we tried to do for logical necessity. We would have to be able to rediscover an in-itself that is Cartesian, and no longer just Kantian – in other words, we would have to be able to legitimate the absolute bearing of the mathematical – rather than merely logical – restitution of a reality that is construed as independent of the existence of thought. It would be a question of establishing that the possibilities of which chaos – which is the only in-itself – is actually capable cannot be measured by any number, whether finite or infinite, and that it is precisely this super-immensity of the chaotic virtual that allows the impeccable stability of the visible world.

But it is clear that such a derivation would not only have to be far more complex, but also more adventurous than that of consistency, since it would have to demonstrate how a specific mathematical theorem, and not just a general rule of the logos, is one of the absolute conditions of contingency. Accordingly, it might seem wiser to confine ourselves to our hypothetical resolution of Hume’s problem, since the latter seems sufficient to vanquish the objection from physical stability, which provided the only ‘rational’ motive for not simply abandoning every variant of the principle of reason. However, there is another problem that rules out such caution, and it is precisely the problem of ancestrality. For as we saw, the resolution of the latter demanded an unequivocal demonstration of the absoluteness of mathematical discourse. We begin to see then, albeit dimly, that there now seem to be two problems tied to the issue of the absolute scope of mathematics – the problem of the arche-fossil and the problem of Hume. It remains for us to connect them in such a way as to provide a precise formulation of the task for non-metaphysical speculation.

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