The Cancelling Out of Chance

Prelude

Mathematics is a strange attractor of ‘worldviews.’ From Plato to Alain Badiou, mathematics has been, and still is, a seductive vector along which various philosophical claims can be articulated. The role that certain currents of thought attribute to mathematics, and by extension to computing in some cases, can reveal a great deal about their respective worldviews, regardless of the mathematical footing these worldviews might stand on (e.g. geometry, set theory, probabilities, etc.). One can observe that these currents of thought tend to display rationalist impetuses that aim at attaining a kind of rational ‘universality’ by threading together philosophical concerns and ways of mathematical reasoning and/or rationalising. This is not to say that mathematics needs philosophy to articulate worldviews or a certain ‘universality’ in and of itself. It can, and indeed does so, quite often. As Dominik Schindler and Matthew Fuller observe, mathematics is not on the receiving end of history, nor, I would add, is it necessarily in the vanguard of it. As a singularly abstract endeavour, it can think outside our common historical constraints and at the same time act as one of their determinants.¹ There is a lot to decipher and examine from the domain of mathematics in terms of how certain types of ‘world-making’ rationalities are/were devised and formalised, as well as how some are/were rejected and derealised. Here I am not simply talking about how mathematics can be used as a ‘tool’ or ‘means’ or ‘weapon’ to assist and perhaps justify certain socio-political ‘world-making’ claims that fall outside the domain of mathematics strictly speaking; I am advancing that mathematical ideas can be, and indeed are, masks for transcendental and/or ontological ‘world-making’ ideas. These masks are more often than not left unacknowledged, perhaps simply because of what they are—masks.²

It might be worth noting that mathematics is not like physics or other sciences in that it is not concerned with the quandaries of empirical facts. Indeed, a proof or demonstration in mathematics is radically different than an experiment in physics or in any other empirical science that observes and models the world. While mathematics is mobilised and used in empirical sciences as a ‘medium’ or ‘language,’ so to speak, it does not necessarily relate to the ‘worldly facts’ of such sciences. In fact, it does not necessarily relate to empirical facts at all for that matter. To put it bluntly, taken from its milieu interieur with its battery of rules and methods of proofs, mathematics is self-sufficient and merely necessitates self-coherence to be effective. In that regard, it is a strange ‘science’ as it does not say much about the empirical world in and of itself. It is fairly ‘mute’ compared to the empirical sciences in that sense.

This muteness certainly does not mean that mathematics cannot act as an analytical matrix with which empirical facts about the world can be deciphered. The idea of deciphering the hidden ciphers of the world by means of mathematical equations is the worldview par excellence of Western scientific modernity that started with Galileo, who notoriously claimed that the book of nature is written in the language of mathematics. While Galileo’s totalising onto-epistemic claim is questionable, my point in claiming the muteness of mathematics is to emphasise that mathematics does not need to denote empirical facts to make sense and be coherent. To use Ludwig Wittgenstein’s terminology, it is instead empirical sciences’ knowledge that demands that a single type of language game—denotation—be the key modality and criterion of its phrasing for it to make sense and be coherent. This is precisely what Jean-François Lyotard argues when observing that while one can find other types of statements in scientific phrasing, ‘such as interrogatives (“How can we explain that … ?”) and prescriptives (“Take a finite series of elements…”)’, these are ‘only present as turning points in the dialectical argumentation, which must end in a denotative statement.’³ Put differently, the terminus of empirical sciences’ knowledge is a denotation: a truth-value statement that directly points to an extra-mathematical referent that warrants proofs (e.g. is this statement verifiable/falsifiable under these extra-mathematical conditions, true or false?). On the contrary, as aforementioned, mathematics does not have any other referent but itself. Rather than having for its terminus a denotation, it has a self-standing and self-referential construction. Proofs and objects in mathematics must be constructed, with their written operations followed and executed. They demand to be read and performed as such.⁴ In this sense, the key language game of mathematics has more to do with prescriptives (‘Take a finite series of elements…’) than denotatives.⁵

In line with the topic of this special issue, the reason I am marking a difference between empirical sciences’ and mathematics’ respective language games is because in what follows I examine mathematical probabilities through the so-called ‘problem of induction’ initially posed by David Hume, who severed the supposed connection between the two language games by marking a distinction between relations of ideas and matters of fact.⁶ To be clear, induction is to be understood as a type of inferential reasoning that moves from the particular to the general: if something is true for a limited number of cases, one can conclude and speculate the same thing ought to be true for an unlimited number of similar cases. Induction is usually defined as the opposite of deduction, which moves from the general to the particular. Probabilities articulate a type of induction that produces statements about the future based on past and/or present observations. Probabilities are about anticipation and predictions, as they establish a speculative bridge between past, present, and future. They relate to induction as they anticipate the general by means of past and finite observations. The theory of induction is an old riddle and has even been termed the ‘despair of philosophy’ by Alfred North Whitehead since solving it poses quite important problems in terms of epistemic demonstrability and justification, something Hume’s scepticism exposes with force.⁷

At the centre of my interrogation is the notion of chance (hazard), which I take as being the indeterminate moment between past and future that is typically harnessed by probabilities that aim to make the unknown known. Rather than looking at the various ways chance can be empirically tamed by mathematical probabilities, the main question I pose in this article is: can chance be cancelled out by mathematical probabilities?⁸ In approaching this query, I ask: What are the conditions under which probability can say that something ought to happen? What are probabilities’ theorems of certainty? Does probability have the same demonstrative import as other domains of mathematics? If so, how and why? These are all questions I address in this article to examine probability’s so-called degrees of certainty, potentially summing up to the cancelling out of chance.

Because of the unreasonable effectiveness of probabilities (to riff on Richard W. Hamming’s well-known phrase), it is not surprising to find its footprints in various fields in the sciences.⁹ The effectiveness of the mathematical instruments that are probabilities in the denotational language game of the empirical sciences is not to be underestimated as being merely a ‘technical’ concern—indeed far from this. Mathematised sciences use this effectiveness as a self-legitimising vector, that is, legitimising the ‘truthfulness’ of their own denotational and epistemic statements by means of performances. Lyotard talks about the language game of ‘performativity’ to account for this type of legitimation narrative.¹⁰ He warns that one cannot divorce the performative prowess of the ‘technical’ from sciences’ legitimising narratives, and even draws a link between a certain notion of determinism to this ‘legitimation-by-performativity’ tuple when he writes:

Determinism is the hypothesis upon which legitimation by performativity is based: since performativity is defined by an input/output ratio, there is a presupposition that the system into which the input is entered is stable; that system must follow a regular ‘path’ that it is possible to express as a continuous function possessing a derivative, so that an accurate prediction of the output can be made.¹¹

I come back to this hypothesis when discussing the cancelling out of chance by various means of interpreting probabilities. It is important to note that this hypothesis is metaphysical as it presupposes that the same input/output instrument (e.g. probability formula) and the same extra-mathematical referent cannot produce a plurality of contradictory or inconsistent proofs. If the instrument renders statements about the referent (i.e. the ‘real’) that are contradictory or inconsistent, then the instrument must be changed as the referent ought to supply non-contradictory and consistent proofs.¹²

Echoing this special issue’s call for exploring situated and pluriversal perspectives through the prism of Bayesian (probabilistic) knowledge, my approach here is not to follow the anthropological route by chronicling various non-Western cosmologies of (doing) mathematics, but instead to directly address the ‘belly of the beast’ of the West’s ‘one-world world’ rationality.¹³ Because I take the problem of induction and probabilities (Bayesian or not) as being inherently rational Western constructs, with my central interrogation on the cancelling out of chance and by staging Hume’s sceptical arguments regarding inductive reasoning and probabilities, I aim to ask: In what sense can probabilities be said to be pluriversal? What type of perspectives do they justify (and perhaps reify) in the denotational language game of the empirical sciences?

As a critical gesture, instead of aiming to render a notion of pluriversality from the conundrums of induction and probabilities I instead look elsewhere and derive one from the conceptual personages that question their self-posited legitimacy. I argue that the personages of the sceptic and the sophist can act as operators to dismantle the plinths upon which Western forms of logos (ratio, reason) stand. Along Barbara Cassin, I posit that because these figures are artefacts or effects of Western philosophy (i.e. the philosopher’s doppelgängers), they harbour a subject position that can always resist ‘one-world world’ metaphysical claims to ‘univocal universality’ and are thus potent operators to conceive of a notion of pluriversality.¹⁴

1. The Despair of Philosophy

Lecturing one hundred years ago in February 1925, Alfred North Whitehead, while explaining the differences between mathematical reasoning and inductive reasoning, made the following statement: “this process of reasoning from the sample to the whole species is Induction. The theory of Induc­tion is the despair of philosophy—and yet all our activi­ties are based upon it.”¹⁵ While this statement can be seen as blunt, Whitehead spoke of the intersections of mathematics and philosophy from a knowledgeable perspective, having had at that time published numerous treatises, most notably the colossal Principia Mathematica, a seminal treatise of logic co-authored with Bertrand Russell.¹⁶ What is striking from the quote is the word ‘despair,’ which I have emphasised in my citation from the lecture’s original transcript. Why would induction be the despair of philosophy? Why qualify induction as a despair? What kind of hope has philosophy lost (or did it ever have) regarding induction? Should we, readers from 2025, interpret Whitehead’s statement as a lament of sorts, an expression grief? If so, a grief of what exactly? What is it that philosophy despairs of that is significant enough to warrant such a statement in a lecture on ‘Mathematics as an Element in the History of Thought’?¹⁷

Whitehead’s lecture is part of a collected lecture series about the history of Western scientific culture. It is remarkable then, that when Whitehead turns to induction in some of these lectures, there is a kind of knot in the story, a certain limit that is reached, when ‘scientifico-philosophic’ reason has to rest or take a break, so to speak. In the lecture entitled ‘The Century of Genius,’ he discusses Francis Bacon’s method of inductive inference and the difficulties it poses, and promptly asserts that ‘the special difficulties raised by induction emerged in the eighteenth century, as the result of Hume’s criticism.’¹⁸

I would like to emphasise here that Whitehead states the difficulties of induction became evident specifically due to Hume’s criticism formulated in the seminal An Enquiry Concerning Human Understanding. Indeed, Hume squarely posed the problem of induction outright, but rather than being formulated from his criticism, it might be more potent to state that it resulted from his scepticism. It is well known that Hume was a sceptic, and it makes sense that a philosopher would despair of the worldview the sceptic exposes, and vice versa. In this sense, the conceptual personage of the sceptic can be viewed as the cousin of another ancient personage: the sophist. While both can give the philosopher reasons to despair, their claims also warrant certain responses. After all, didn’t Plato and Aristotle establish the base of classic philosophy out of an existential wrestle with the sophists, by devising an arsenal of discursive methods (i.e. logic, the principle of contradiction, dialectics, etc., taught at their respective academies) to counteract the sophists and their so-called sophistries? And in so doing, wasn’t their aim to draw a decisive line between those who are the real friends of wisdom (sophia) and those who are not?¹⁹ Following this line, can we not then say that philosophy’s despair stems from Hume’s scepticism? If Hume is the conceptual personage that sets limits or ties a knot in Whitehead’s story, the question that is worth asking then is what was Hume sceptical about?

Hume’s main scepticism relates to how induction can be said to be dealing with cause and effects, how the relation between cause and effects was said to be discoverable. Contra some of his contemporaries, for Hume, it is uniquely through experience and not reason that one is able to infer effects from causes, and vice versa. What does this mean? In the above section, I already discussed how induction reasons from the particular to the general, making generalising claims from limited cases to unlimited ones. Hume was not sceptical of this type of inference but warned that the conditions that enable us to make such inference must be grounded in experience (e.g. our habits, customs, etc.) and not in mere rationality (e.g. the rules of logic).

According to Hume, a proof of mathematics is grounded in mere rationality, whereas, for example, the physical laws that explain the movements of balls on a billiard table are not—they are grounded in conditions of experience. To sustain his sceptical claims, Hume marks a strict distinction in our faculty of reasoning between what he calls the relations of ideas, which fall into the domain of reason, and the matter of facts, which fall into the domain of experience. He writes:

All the objects of human reason or inquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of Fact. Of the first kind are the sciences of Geometry, Algebra, and Arithmetic; and in short, every affirmation, which is either intuitively or demonstratively certain. That the square of the hypotenuse is equal to the square of the two sides, is a proposition, which expresses a relation between these figures. That three times five is equal to the half of thirty, expresses a relation between these numbers. Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is any where existent in the universe. Though there never were a circle or triangle in nature, the truths, demonstrated by Euclid, would for ever retain their certainty and evidence.

Matters of fact, which are the second objects of human reason, are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing. The contrary of every matter of fact is still possible; because it can never imply a contradiction, and is conceived by the mind with the same facility and distinctiveness, as if ever so conformable to reality. That the sun will not rise to-morrow is no less intelligible a proposition, and implies no more contradiction, that the affirmation, that it will rise. We should in vain, therefore, attempt to demonstrate its falsehood. Were it demonstratively false, it would imply a contradiction, and could never be distinctively conceived by the mind.²⁰

All reasonings about matters of fact are grounded in the relation of cause and effects according to Hume.²¹ That the sun will rise tomorrow is a matter of fact and not a relation of ideas. That the blue billiard ball will induce some motion on the yellow one when hitting it is a matter of fact and not a relation of ideas. On the contrary, the sole end of logic and mathematics (Geometry, Algebra, and Arithmetic, etc.) is to devise and explain the principles and operations that calibrate relations of ideas.²² Logic and mathematics know nothing about factual cause and effects. On the other hand, there is no strict logical necessity in billiards, no demonstrations henceforth. Relations of ideas and matters of fact warrant very different types of reasoning, and because of this, there can be no demonstrations based on logical necessity for any conjunctions of cause and effect, according to Hume.²³

This certainly does not mean there is no knowledge that can be derived from cause and effects. Again, Hume only warns that this knowledge stems from experience. One has to experience many times the effects of billiard balls hitting one another to infer their causal relationship and to anticipate their possible behaviour; one draws conclusions from repeated past experiences regarding matters of fact. Hume writes that ‘all reasonings concerning cause and effects, are to be founded on experience, and […] all experiences are founded on the supposition, that the course of nature will continue uniformly the same, […] and that the future must be conformable to the past.’²⁴ Indeed, any statements based on inductive reasoning I might infer about the dynamics of billiard balls supposes that the table, the balls, the billiard stick, and the whole physical context of the game will be the same next time I play. If that is not the case, I will have to infer the dynamics all over again and I will not be able to anticipate. This uniformity and conformability is key for my induction to hold.

A question that is worthwhile posing is then: is it possible to demonstrate that the course of nature must continue uniformly as the same or that the future must be conformable to the past? Since there is no logical necessity that tomorrow ought to or even will conform with today, it might be hard to demonstrate following the rules of logic and mathematics. But since what is possible or probable can never be demonstrated to be false, is it possible to use probability, something like a ‘probable necessity,’ to demonstrate that the course of nature must continue uniformly the same or that the future must be conformable to the past? To these questions, Hume categorically answers ‘Nay’ and continues:

I will go further, and assert, that [one] could not so much prove by any probable arguments, that the future must be conformable to the past. All probable arguments are built on the supposition, that there is this conformity betwixt the future and the past, and therefor can never prove it. This conformity is a matter of fact, and if it must be proved, will admit no proof but from experience. But our experience in the past can be a proof of nothing for the future, but upon a supposition, that there is a resemblance betwixt them. This therefore is a point, which can admit of no proof at all, and which we take for granted without any proof. We are determined by custom alone to suppose the future conformable to the past.²⁵

So Hume’s problem of induction stipulates that inductive reasoning is based on the presupposition that nature is uniformly the same and that the future is conformable to the past. This presupposition cannot be proven a priori nor can it be with a ‘probable’ argument. Proving the future is conformable to the past with a probable argument based on the very supposition the future is conformable to the past is a circular tautology. Hume’s problem of induction asserts there is no way out of this conundrum.

About this sceptical argument, of which I have only drawn the contours here, it is not difficult to see how one can begin to despair when trying to ground the denotational language game of scientific facts on ‘infallible truths’ as ones supposedly found in logic and mathematics. And one can ‘despair’ here not only in metaphysical terms, but also in trying to mount a counter response to Hume’s problem.

In what follows, I take Hume’s scepticism to be well grounded, just as Whitehead does, and look at how Hume’s arguments can (be said to) be reverted.²⁶ I first discuss probabilities with the classic Law of Large Numbers and then offer a reading of the Bayes-Laplace formula. With this reading, I then question some of the central theorems of the subjective interpretation of the formula (which I divide into ‘orthodox’ and ‘omniscient’ interpretations) and examine how the formula is interpretated objectively. I will pay attention to how a certain interpretation of probabilities rationalise and formalise the supposition that the future is conformable to the past, and that nature is uniformly the same and thus unable to supply contradictory or inconsistent proofs. The cancelling out of chance will refer the worldviews that articulate Lyotard’s notion of ‘performative determinism’ mentioned in the introduction. The point here is to examine potential ‘cancelling’ worldviews disguised under the masks of mathematical formulas and theorems, and ponder about their respective conditions, assumptions, and limits.

2. Cancelling Out Chance by Means of Probabilities

There are two types of probabilities I would like to introduce here. The first is the most general, more ‘qualitative,’ in the sense that is it not numerical: it is probable a few black birds will fly over my house this evening. Who knows (¯\_(ツ)_/¯)? I propose to name this ‘qualitative’ meaning of probability: probable_1. With my black birds analogy, rather than simply stating that it is probable_1 that some black birds will fly over my house this evening, I could have established ratios of the amount of black birds I have previously observed in flocks that have flown over in past evenings. Typically, I would refer to these compiled ratios to sustain some inductive claims, these ratios acting as my limited cases supporting my generalising claim. I can average or manipulate the ratios numerically to anticipate the composition of the next flock of birds. I am here reckoning and generalising statistically to infer and assert what is probable. I propose to name this meaning of probability: probable_2. It differs from probable_1 in that it bases its reasoning on ratios, mobilising statistical generalisation to justify inductive statements.²⁷

There is more than meets the eye though with this type of probability. To use proper terminology, when I am counting the number of birds I observe in a flock, I am compiling relative frequencies f out of finite ratios. The aim of my inductive reasoning is to derive a ‘big picture’ of the overall proportion after a multitude of sessions of counting frequencies f. The goal is to ultimately arrive at a certain and unique probability P which I then use as a ground to anticipate the observed frequencies of my next sessions. The two elements f and P are part of the probable_2 paradigm and are central to numerical induction. If I were omniscient I could directly derive P since I would have knowledge of all the birds in the world (or my region) and would be able to anticipate with certainty the ratio of black birds I am about to observe. I cannot obviously do so, hence I revert to counting finite frequencies.

The relation between f and P is a peculiar one. The infamous Law of Large Numbers, proved by mathematician Jakob Bernoulli in the Ars Conjectandi, put these elements in correspondence.²⁸ What the law basically stipulates is: if the probability of a certain event occurring under some conditions is P, then the observed compiled frequencies f of the event occurring under the same conditions will tend towards the proportion P in the long run (i.e. after a large number of observations or ‘trials’). In other words, the law says there is a certain ‘maturity of odds,’ so to speak, in that compiled frequencies f will tend towards P in the long run. Because Bernoulli’s theorem infers from a probability P to relative frequencies f—the reasoning moves from P to f. Now while this can be of interest for whoever gambles in a game of chance, where the physical properties of the game strictly govern its possible outcomes under certain conditions, and from which, therefore, probabilities P_i for each possible outcome i can be derived a priori (e.g. 1/6 probability to get a 5 on multiple occasions with a fair dice), it does not help us in our induction quest. The Law of Large Numbers is not inductive as it moves from the already given general probability P to the particular compiled frequency f. What one does while compiling frequencies f then is to merely verify the law, or to split hairs on whether the conditions under which the observations are made match the conditions the law stipulates (e.g. a ‘fair’ dice, etc.). The Law of Large Numbers can thus be labelled a direct probability while induction requires an inverse probability, that is, an inversion of the law’s reasoning, moving from limited observations of frequencies f to general probability P.

Can such an inverse probability be derived from Bernoulli’s theorem on the Law of Large Numbers? In other words, wouldn’t finite observed frequencies f converge towards a previously unknown general probability P in the long run? Bernoulli believed this ought to be the case, yet his fellow mathematician Gottfried Wilhelm Leibniz was doubtful.²⁹. New diseases from time to time inundate the human race, as if, therefore, however many experiments you have made with regard to deaths, you have not therefore set limits to the nature of things, so that they cannot vary for the future. However, although empirically it could not be a perfect evaluation, that does not mean that an empirical evaluation would not be useful and sufficient in practice.’ Leibniz’s original Latin passage is quoted in John Maynard Keynes, A Treatise on Probability (MacMillan and Co., 1921), 402 footnote 1. The treatise is reprinted in The Collected Writings of John Maynard Keynes, vol. 8 (Cambridge University Press, 2013).] Leibniz saw that, in moving from simple cases of games of chance to ‘real life’ events such as those found in medicine or judicial practices, it is very much unclear what observed frequencies f are effectively ‘observing.’ There are incalculable amounts of causal contingencies that underpin events, observable or not, and it is virtually impossible to account for them all with finite means. While in practice, observed frequencies might be a fine heuristic that can serve as a ‘rule of the thumb’ guide for medical and judiciary decisions and the like, they are nevertheless inherently incapable to set guaranteed limits upon which to draw conclusions with certainty.³⁰ The degrees of certainty of the Law of Large Numbers are simply not the same in the case of an inverse probability.

Yet while the Law of Large Numbers might not provide the means ipso facto to establish its own inverse, the well-known and successful attempts to come up with a way to calculate inverse probabilities that account for the incalculable contingencies of observable events are found in the work and formulae of Thomas Bayes and Pierre-Simon de Laplace.³¹ Both mathematicians found ways to restrict the incalculable contingencies by imposing conditions on the observable stochastic processes of an event so as to render the incalculable calculable. To examine these conditions, consider the equation below, which is expressed by Laplace in one very long sentence:³²

$$P\left( H_{j} \middle| E \right) = \frac{P\left( E \middle| H_{j} \right)P\left( H_{j} \right)}{\sum_{i = 1}^{n}{P\left( E \middle| H_{i} \right)P\left( H_{i} \right)}}$$

In a nutshell, the equation stipulates that if an event E can be produced by n different causes H, then the probability of a single cause i given the event E—here denoted by $P\left( H_{j} \middle| E \right)$—is equal to the probability of the event given that precise single cause i—denoted by $P\left( E \middle| H_{j} \right)P\left( H_{j} \right)$—divided by the sum of all the probabilities of the event given each individual n causes—denoted by $\sum_{i = 1}^{n}{P\left( E \middle| H_{i} \right)P\left( H_{i} \right)}$.

One remarkable feature of the formula is its temporal dimension as it expresses an iterative step-by-step update.³³ It is common to talk about prior (a priori) and posterior (a posteriori) probabilities. In the formula, $P\left( H_{j} \middle| E \right)$ is the posterior and $P\left( H_{j} \right)$ is the prior. The step-by-step update means the calculated posterior at step k becomes the prior at step k+1 and so forth. At each step, the formula integrates new observed frequencies f of individual causes in the calculation of the so-called likelihood $P\left( E \middle| H_{j} \right)$ to weigh the effects of the prior on the resulting posterior. This is how the formula articulates the inductive movement from f to P.

Now, this is what the formula ‘says.’ There are obviously a myriad of ways to interpret this, though it is worth focusing on certain mathematical and theoretical caveats the formula entails.

1. Divide to conquer: It might certainly sound hypothetical to say that E can be produced by n causes considering what was said above about the incalculable amounts of contingencies an event can be determined by. One is rightly entitled to ask: How is n devised? How are we infallibly certain there is a limit n of causes to a given event E? Surely, these are all valid questions that are not easily answerable. Yet, as a cautious answer, one can advance that in order to be able to have a probability formula at all, limits need to be drawn in order to devise ratios (probable_2), and it is thus, pragmatically speaking, in this way that limit n can be justified.
2. Mathematical Coherence: This theoretical limit also implies another mathematical limit. In the formula, the denominator is the probability of the event E, which is expressed by the sum of all the probabilities of the event given each individual n causes.³⁴ For the formula to be coherent, the sum of all individual probabilities of each cause $P\left( H_{j} \right)$ must be equal to 1. This means that for each cause, a prior probability needs to be devised in such a way that when all the n causes’ respective probabilities are added up, the result is equal to 1.
3. Probability of a probability: What are the a priori $P\left( H_{j} \right)$ at step k=0? Shouldn’t these priors be unknown at this initial step? Is it even probable_1 that these probabilities exist, let alone can be determined by a numerical ratio between 0 and 1 (i.e. be probable_2)? How legitimate is it to attribute a numerical value to a probable_1 probability? If the probability of a cause is unknown prior to the step-by-step Bayesian process starting, it surely follows that the probability that it has a value between 0 and 1 is also unknown (i.e. the probability of a probability).
4. Infinite Regress: Building on the above caveat, assuming we turn these probable_1 probabilities into probable_2 ones, what initial values should these have? Consider the causes h₁, h₂, h₃, etc. having respective probability p₁, p₂, p₃, etc. If we know nothing about h₁, it is logical we will not know the value of probability p₁. Yet we must now fix the possible value of p₁ with values such as v₁, v₂, v₃, etc. having themselves respective probability q₁, q₂, q₃, etc. One can quickly see this process leads to an infinite regress.

While the above caveats relate to one another, the first two can be explained by the formula’s requirement of formal mathematical coherence. Such mathematical conditions do not preclude one from questioning the hypothetical nature of the Bayes-Laplace approach however, especially in relation to questions raised in caveat 1. That being said, the last two caveats have theoretical import as they directly speak to the likely impossibility of cancelling out chance, or at least expose the rather significant challenges trying to do so entails in terms of justification and ignorance. After all, isn’t giving a random value to an unknown probability of an unknown cause involving chance anyway?

Making distinctions between the different ways of interpreting these caveats is always going to be murky. Though since this special issue is concerned with the subjective interpretation of the Bayes-Laplace approach to probabilities, it is pertinent to discuss this interpretation along with its cousin, the objective interpretation of the approach, if even briefly.

2.1 Cancelling by the ‘Orthodox’ Subjective Interpretation?

The subjective interpretation basically treats the various unknowns expressed in caveats 3 and 4 as elements that warrant a choice that is based on a degree of belief or credence.³⁵ This approach cuts short the perplexity of the infinite regress questions of 3 and 4 by advancing the idea that giving a probability distribution to causes is a subjective choice that reflects one’s personal degree of belief in the said distribution. While someone can perhaps be knowledgeable of the nature of the event they want to model with the Bayes-Laplace formula, and thus have an ‘idea’ about a set of initial prior distribution, this does not preclude that the choice itself is subjective. That being said, does this means that there are no strict conditions/restrictions in choosing this or that distribution, and that, as a corollary, this worldview does not aim at cancelling out chance since it does not petition for a type of faultless omniscience upon which this cancelling would be effective? Yes and no. Not if one is bound to choose the distribution ‘rationally,’ that is, be coherent with the demands of the mathematical formula. Of course, no one needs to be ‘rational’ in their subjective interpretations, though not choosing a suitable distribution as initial input to the Bayes-Laplace formula will most likely lead to incoherent outputs: garbage in, garbage out goes the maxim.

There are two perspectives on the subjective interpretation I will examine in what follows: the ‘orthodox’ worldview and what I call the ’omniscient’ worldview. The latter builds on the former, so before I address the more contemporary ‘omniscient’ worldview, there are a couple of clarifications that need to be addressed to clarify what the ‘orthodox’ interpretation entails.

First, let me return to the Bayes-Laplace formula’s variables. As explained above, the posterior gets recursively produced (step-by-step) by an amalgam of both prior (i.e. old posterior in the recursive ‘loop’ at step k > 0) and likelihood (frequencies f during an observation). The recursive prior-posterior conjugate is cumulative, meaning that it encapsulates an ‘average’ of all observations that were added as frequencies f in the likelihood variable at each step. The information available from any observation at each step is entirely contained in the likelihood, which drives the recursive prior-posterior conjugate. A question that is worth asking now is: does the order of the sequence of observations matter? Quick answer: no. The observations are sequentially exchangeable since the formula is agnostic of such order as it merely cumulates likelihood observations into posterior variables which are then turned into priors, and so forth. In other words, a finite sequence of ordered observations $o_{1},\ldots,o_{n}$ cumulated in probability $P\left( o_{1},\ldots,o_{n} \right)$ is equal to cumulated probability $P\left( o_{\pi(1)},\ldots,o_{\pi(n)} \right)$ for any possible permutation $\pi$ of the subscripts $1,\ldots,n$. This exchangeability aspect of the Bayes-Laplace formula is known as ‘de Finetti’s theorem’ (named after mathematician Bruno de Finetti), which is one of subjective interpretation’s central theorems.³⁶

What does this theorem imply? In a nutshell, the theorem basically says that observation occurrences (i.e. where/when in a sequence of observations) and their content (i.e. which ‘value’) are arbitrary since the order of observation is regarded as irrelevant. It does not say that the method of observation is arbitrary but that the observation of a given relative frequency value f at step k happens arbitrarily, that is, despite all other frequencies observed at other steps.³⁷ This is equivalent to saying that making observations at step k is analogous to tossing a coin: the chance that the next occurrence falls on tails has the same probability as falling on heads. Here the ‘analogous’ adjective is important since de Finetti’s theorem does not say the underlying ‘objective process/phenomena’ needs to be independent and identically distributed (IID) to use precise jargon, but that from the perspective of the subject, their observations appears as-if the ‘objective process/phenomena’ was IID even if it might not be, objectively speaking.³⁸

I want to emphasise this as-if character here because it complicates common ideas about subjective beliefs in relation to ‘objective chance’ (i.e. chance of the ‘objective process/phenomena’). It is perhaps intuitive for many to conceive of a person’s subjective probabilities as being their subjective expectation of objective chances. After all, aren’t objective chances necessary to explain why subjective probabilities have exchangeability properties? Aren’t subjective probabilities exchangeable because objective chances are?

They are not argues de Finetti, who demonstrates that it is not objective chances that explain exchangeability but rather exchangeability that explain probabilistic and subjective beliefs—objective chances and probabilistic beliefs do not coincide in any way. One can view a certain process as being exchangeable even when denying this process is conditioned by objective chance. If one treats a process as exchangeable, because they believe the order of observations of such a process is irrelevant, then de Finetti’s theorem shows that their inductive inferences will be the same as someone else’s who thinks the process is determined by (perhaps unknown) objective chance—there is nothing that requires one to invoke objective chances when drawing probabilities since exchangeability judgments suffice.³⁹ While some might act as-if objective chances drives a process/phenomena, there are no grounds to believe it is necessary to do so according to de Finetti’s theorem: all inductive inferences that call on exchangeable subjective probabilities can be explained without chance.

The objective existence of chance and probability is a lure according to de Finetti, who writes in his preface of Theory of Probability:

PROBABILITY DOES NOT EXIST

The abandonment of superstitious beliefs about the existence of Phlogiston, the Cosmic Ether, Absolute Space and Time, …, or Fairies and Witches, was an essential step along the road to scientific thinking. Probability, too, if regarded as something endowed with some kind of objective existence, is no less a misleading misconception, an illusory attempt to exteriorize or materialize our true probabilistic beliefs.⁴⁰

In all his writing, it is clear de Finetti does not merely aim to cancel objective chance but rejects it outright. Objective chance does not exist according to his radical stance—it is a man-made chimera. Rather than mirroring chance, subjective exchangeability is the epitome of intrinsic randomness. It is randomness that presides over one’s subjective inductive credences and observations, not objective chance.

A corollary of severing the link between chance and credence and of rejecting the existence of objective chance/probability, is that the ‘experience’ of observing a process/phenomena does not hold any key in inductive reasoning. There is nothing within the experience of observing that warrants a sudden illumination about future events de Finetti writes: ‘By virtue of having observed, or having obtained the information, that some given complex A of events have occurred, what are we entitled to say about some future event E? [..] The answer is… nothing! […] The complex A (the experience) by itself determines nothing.’⁴¹ In sum, it is neither the objective process nor the forensic experience of observing it that governs so-called ‘learning from experience.’ It is rather the subjective inductive judgement, which the Bayes-Laplace formula is the ultimate expression/diagram of with its recursive reckoning process, that governs this learning. In other words, it is the coherence of the Bayes-Laplace formula and theorem that legitimates inductive ‘learning’ and not ‘experience’ per se.⁴² ‘Experience’ is in fact an awkward category if it is to stand as a surrogate proof for the existence of the objective world, a position that can be read as standing in stark contrast to phenomenological accounts of experience that have the objective world as horizon. The only legitimate influence de Finetti grants to ‘experience’ is to merely update the likelihood function of the Bayes-Laplace formula. Inductive reasoning’s coherence comes from Bayes-Laplace’s mathematics when its theorematic formula is seen as the epitome of how subjects update their beliefs inductively and not as a mere tool that models external objective reality.

Echoing the way I qualified mathematics as being ‘mute’ in the introduction, de Finetti writes:

Mathematics, logic, and geometry are now immune to the pseudo-hypothesis (so to speak) of the existence of the world, the existence of an external reality, the existence of a metaphysical reality. I cannot doubt that if all our thought were to rid itself of that embarrassing and mysterious pseudo-hypothesis, it would have everything to gain—in clarity, depth and rigor—in every field, not least in that of the formal sciences, where, by their very nature, the danger of metaphysical deviations is by far the least.⁴³

De Finetti is convinced his work on exchangeability corresponds to Hume’s ideas.⁴⁴ While it is clear that the subjective interpretation roots the problem of induction as being a subjective choice/decision problem and thus cuts short the conundrums of caveats 3 and 4 exposed above, does it readily align with Hume’s warning that knowledge about matter of facts can only derive from our habits, customs? Is the Bayes-Laplace formula the penultimate expression of our inductive habits? Considering the quote above, it is perhaps strange to talk about inductive reasoning and matters of fact and end up praising mathematics’ ‘immunity’ to the pseudo-hypothesis of ‘the existence of the world,’ a statement that resembles how Hume qualifies relations of ideas—and precisely not matters of fact. Severing the link between chance and credence is a bold move as it short-circuits the denotational language game of science by denying that the source of scientific coherence stems from external reality and our experiences of it (i.e. the world is coherent and thus our observations of it ought to be) but instead necessarily comes from internal reasoning/thinking alone (i.e. reasoning ought to be coherent since the world might not be).

De Finetti’s worldview on what probabilities are and what they are authorised to say is certainly radical and it is typically referred to as the ‘orthodox’ interpretation of the subjective interpretation. This ‘orthodoxy’ has the merit of clearly cautioning against certain types of ‘naive’ realisms I believe yet at the same time reconstructs a type of transcendental logos or rationality that can look like a conflation of Hume’s relation of ideas and matters of fact. But since lived experience is somewhat a hollow category within the orthodox worldview, what sort of ‘grounding’ does subjective rationality need to be rooted in to be effective epistemologically speaking? Below, I briefly look at contemporary worldviews that build on de Finetti’s interpretation to respond to these questions and problematise what they give as answers in terms of the cancelling of chance.

2.2 Cancelling by the ’Omniscient’ Subjective Interpretation?

One can be puzzled by how, on the one hand, the subjective interpretation advocates that no external restrictions be made on one’s subjective judgements (i.e. prior probability assignments is a subjective choice), while on the other, stipulates there ought to be internal restrictions within the process of judging for it to be legitimate, which is to say, coherent (and thus rational). Coherence restrictions and conditions are what manuals and books preaching for the subjective interpretation are mainly composed of, with de Finetti’s treatises on probability being at the ‘avant-garde’ of such writing.⁴⁵) expose such restrictions/conditions. Earlier works discuss these as well, notably ‘La Prévision: Ses Lois Logiques, Ses Sources Subjectives,’ translated in ‘Foresight: Its Logical Laws, Its Subjective Sources.’ For a more philosophical exposition of de Finetti’s ‘worldview’ on the subjective interpretation of probability see Probabilismo: Saggio Critico Sulla Teoria Delle Probabilita e Sul Valore Della Scienza (Francesco Perrella, 1931), translated in ‘Probabilism: A Critical Essay on the Theory of Probability and on the Value of Science.’]

The subjective interpretation is normative in that sense. It dictates how a subject should follow certain conditions in order to make coherent decisions. Such conditions are not only discursive but mathematical. In fact, the discursive and the mathematical coincide here since reasoning/thinking is the only vector of coherence. As hinted above, for a ‘subjectivist’ the Bayes-Laplace formula is not a mere ‘useful’ formula but the very expression or ‘diagram’ of inductive reasoning as it articulates how subjects reason inductively with their ‘updatable’ probabilistic beliefs.⁴⁶ So rather than modelling the world, Bayes-Laplace rules are here conceived as normative apparatuses that express (not ‘model’) how one thinks and acts rationally when ‘learning from experience.’ Inductive reasoning basically boils down to the Bayes-Laplace theorem as de Finetti writes: ‘the vital element in the inductive process, and the key to every constructive activity of the human mind, is then seen to be Bayes’ theorem.’⁴⁷ I have already exposed some of the coherence demands of the Bayes-Laplace formula with caveat 1 and 2, though before I delve into how these are legitimised by the subjective interpretation, it is important to highlight the premises of interpreting probabilities in a subjective key.

Historically speaking, the subjective interpretation shifted probability’s problematic from being about making assumptions about chance events themselves to ways of subjectively believing and reasoning about certain propositions about these events. The shift to evaluating propositions about events is paramount for the interpretation to hold as it shifts the ‘truth-value’ criteria from an objective/external standpoint (i.e. making ‘metaphysical’ claims) to a subjective/internal one (i.e. making ‘epistemological’ or ‘pragmatic’ ones). It is important to emphasise that probability here becomes a type of ‘writing’ whose ‘algebra’ is the logic of propositions. It is this ‘writing’ that gets evaluated by a subject.⁴⁸ As a corollary, its mode of denotation is similar to the age-old ‘assertoric syllogism’ in the way its ‘truth-value’ criteria is evaluated.⁴⁹ John M. Keynes is known to have clearly formulated this need to leave the ambiguous logic of ‘events’ behind and instead conceive of probabilities as governed by a logic of propositions.⁵⁰ This turn to the logic of propositions in the reckoning of probability thus gives it a certain epistemological coherence, one that is divorced from ambiguous and perhaps vague metaphysical claims.

Now, as discussed with the distinction between probable_1 and probable_2 statements, probabilities’ ‘truth-value’ does not simply add up to two-valued propositional judgments (‘true/false’ or ‘bet/no bet’) but to numerical ratios (i.e. bet, but how much?). Where do degrees of belief or credence fit in this distinction? To be coherent, they must fit the probable_2 side of the distinction, as Frank P. Ramsey tells us in his seminal ‘Truth and Probability.’⁵¹ Contra Keynes, Ramsey advances that there are no beliefs in probable_1 statements that are not probable_2: subjective beliefs are measurable and thus lead to numerical degrees of probabilities.⁵² More so, beliefs are coherent precisely because they take the form of probable_2 propositions and can thus uphold calculable mathematical expectations.⁵³ behaviour is governed by what is called the mathematical expectation,’ 174. This expectation relates to the pragmatic notion of ‘utility’ as discussed below.] In a similar way, de Finetti emphasises coherence conditions on a subject’s degrees of beliefs with the notion of probabilism: the principle that it is irrational to have beliefs that violate Andrey Kolmogorov’s axioms of probability.⁵⁴ These axioms directly relate to caveats 1 and 2. Probabilism is the thesis that rational and coherent degrees of belief need to have the structure of probabilities by following its axioms.⁵⁵ They thus become ‘probabilistic beliefs’ and not mere irrational beliefs.

The reason this thesis is put forth is to avoid the pitfalls of having the subjective interpretation justify subjects having contradictory beliefs and making incoherent decisions. This thesis is warranted by a demonstration of a game-like strategy that avoids the so-called ‘Dutch book problem’ of consistently betting on losing odds in a gambling context. A Dutch book is a sequence of (monetary) bets that guarantee a net loss in a gambling context, whatever the fortuitous outcomes of the game may be. Ramsey, De Finetti and Leonard J. Savage, show that when a subject (a ‘miser’) expects to maximise gains in a gambling context, if the subject’s beliefs in gambling odds violate the axioms of probability then it is sure that the gains will be less well off than if they were following the axioms.⁵⁶ In other words, ‘probabilistic beliefs’ coupled with the Bayes-Laplace update rules avoid the Dutch book problem as they can optimise subjective expected ‘utility’ or expected gains in a gambling context in the long run. So caveats 1 and 2 are not just peculiarities imposed by the numerical underpinning of the Bayes-Laplace formula but are rather normative restrictions that ensure the formula’s calculative and performative import is maximal/optimal.

Probabilism is thus vindicated because it optimises utility in a ‘pragmatic’ way. Coherence and optimisation are here intertwined. It might be strange to talk about degrees of beliefs in terms of bets in a gambling context. Beside the historical precedent between gambling and the advent of contemporary numerical probabilities, betting—as a ‘pragmatic’ act—can indeed be nicely folded under the subjective interpretation rubric, and furthermore, be conceived of as a practical ‘means of measuring [subjective] beliefs.’⁵⁷ Even an ‘orthodox’ subjectivist can adopt this pragmatic worldview since the link between objective chance and credence is not primordial, as subjective utility is not necessarily objective strictly speaking. Needless to say here is that the ‘utility’ paradigm has been (and still is) the paradigm par excellence to conceive of the so-called homo economicus in Decision Theory and Game Theory, influencing disciplines such as Economy, Finance, and the Actuarial, Behavioural and Social Sciences.⁵⁸

But what about knowledge and epistemology? Is betting on odds an act of knowing? Or inversely, is knowing an act of betting (which may be affected by luck)? I have not directly addressed the question of knowledge in the previous section. I will turn to it now.

In its denotative form, knowledge’s epistemic raison d’être is to represent a certain state of the world. A question that is worth asking here is: where does ‘belief’ stand in regards to representing such a state? When I say that I believe that Haarlem is in Noord-Holland, I regard this proposition to be the case. I thus represent the world as such, in a ‘Haarlem-is-in-Noord-Holland’ way. I likewise do so when I know for sure Haarlem is in Noord-Holland. Now, if I assign a credence of 0.63 that it will rain in Utrecht tomorrow, do I regard the proposition to be the case? Do I represent the world in a ‘0.63-chance-of-rain-in-Utrecht-tomorrow’ way? There are reasons to think I am not. What I do instead is to have an ‘attitude’ (0.63) towards a representation of the state of the world: rain in Utrecht tomorrow. I might change this attitude upon a weather report update but not the representational content of it (i.e. rain in Utrecht tomorrow).

Strictly speaking, it does not make much sense to say the ‘0.63-chance-of-rain-in-Utrecht-tomorrow’ way of representing the world is ‘true.’ It makes more sense to say it is ‘justified’ in light of given expert weather reports. The denotative or representational import of the term ‘belief’ is thus ambiguous. Consider the following statement: ‘Geert’s belief is that p’—which can mean two things: (1) the content of Geert’s belief is true; (2) Geert’s doxastic attitude towards the proposition that p (e.g. rain in Utrecht tomorrow) is justified. In the first instance, it is the content of the belief that is said to be true, while in the latter, it is the attitude (e.g. 0.63 credence) towards a proposition suggesting a certain representation of the world (e.g. rain in Utrecht) that is said to be justified. To put it bluntly as Alan Hájek and Hanti Lin do: ‘“Degrees of belief” refer univocally to an attitude, not the content of an attitude.’⁵⁹

That being said, what I want to highlight here is the abiding distinction between doxa and episteme. As it was with bets, it might still be strange to talk about standard epistemic knowledge as a mere attitude. After all, aren’t the standard norms of epistemic rationality typically implying that we should aim to accept what is true and reject what is false and thus calibrate our full belief on their content’s categorical truth-values (e.g. Haarlem is in Noord-Holland: ‘true/false’)? This is how one can be said to be right or wrong, that is: exactly ‘right’ or exactly ‘wrong.’ There is no leeway here to make a distinction between being wrong and being wrong so to speak.

This dichotomic logic is obviously not how the logic of credences operates. Instead of being categorically right or wrong, one’s credences are ‘closer’ or ‘further’ to what is the case, which is to say they are more accurate or less accurate. Being accurate here is a proxy of being right. The epistemic norms of credences thus imply that one should calibrate their credences to minimise their expected inaccuracy.⁶⁰ So in a similar way to how (rational) gamblers/misers ought to make optimal choices on bets, probabilistic subjects ought to optimise their epistemic beliefs by maximising the accuracy of their credences on certain propositions representing some states of the world. Here measures of accuracy substitute the categorical truth-values of standard epistemology, and ‘pragmatic’ concerns pertaining to the optimisation of bets become epistemic concerns pertaining to the optimisation of accuracy.⁶¹ From pragmatic utility theory, we turn to epistemic utility theory.

In his seminal article ‘A Nonpragmatic Vindication of Probabilism,’ James M. Joyce establishes this link between probabilism and epistemology along the vector of accuracy. Updating de Finetti’s work on the pragmatics of the ‘Dutch book problem,’ Joyce shows that by replacing the logic of expected gain maximisation in the context of gambling by the logic of expected accuracy maximisation in the context of knowing, it can be demonstrated that ‘any system of degrees of belief that violates the axioms of probability can be replaced by an alternative system that obeys the axioms and yet is more accurate in every possible world.’⁶² In other words, because credences are taken as estimates of truth-values and because accuracy is posited as being an analogue to ‘truth’ in standard epistemology, Joyce argues that having non-probabilistic beliefs that violate the axioms of probability lead to epistemic failings. The way these beliefs can be deemed as ‘failings’ is to show that for every string of credences that violate the axioms, there exists a unique sequence that follows them and this unique sequence is more accurate in every possible world.⁶³

But how can probabilistic beliefs be said to be accurate about representations of the states of the world if the link between objective chance and credence remains severed as it is in the orthodox worldview? How can a credence be said to be more accurate than another without being able to assess their content’s truth-values? Difficult questions indeed, and this is why Joyce, and Richard Pettigrew after him, reinstate the link between objective chance and credence through a sort of metaphysical injunction via the so-called Principal Principle.⁶⁴ Within this newfound worldview, objective Chance (with a capital ‘C’) becomes a kind of ideal ‘database-expert’ which encodes all information about a given world/system at any time and provides a ‘summary statistics that measures the “causal tendency” of the current state [of the world] to produce the effect under consideration.’⁶⁵ In other words, Chance serves as a simple summary of complicated causal patterns that have occurred in the past. It screens off information about the past and distils it as a summary at present time t. For example, if Chance assigns a probability of 1/8 to an unfair dice landing on 5, this summarises the actual sequence of values the unfair dice produced while being thrown in the past. It basically summarises that sequence and tells us that 5 appeared in an eighth of the times it was thrown. But by virtue of being a summary, Chance necessarily omits information, such as: How many throws? How many sequences of throws? Which exact sequences of success/failures were summarised? etc.

Epistemologically speaking, while one can consider representing the world in a ‘1/8-chance-5-landing-next-throw-of-unfair-dice’ way as being ‘true’ there are no guarantees this is the case. Even as an attitude, why would ‘1/8-chance-5-landing-next’ be more vindicable than ‘1/1-chance-5-landing-next’ considering a single throw of the unfair dice? Put differently, suppose the unfair dice is thrown 10 times and for each throw one has a credence in a proposition that it will land on 5. Suppose the sequence of ‘proposition’ = credences tuples is as follow: ‘dice lands 5 on throw #1’ = 1/1, ‘dice lands 5 on throw #2’ = 0/1, …, ‘dice lands 5 on throw #6’ = 0/1, …, ‘dice lands 5 on throw #10’ = 1/1. Why would the same credence of 1/8 for all these propositions be more credible than having credence 1/1 (full belief) for each proposition that is effectively true and 0/1 (disbelief) for each proposition that is effectively false? Why would a mere summary, with no information regarding the sequencing of events, be more vindicable than an ‘omniscient’ worldview that would have access to such sequencing? Joyce points out that ‘the idea that chance is a data-base expert whose access to causally relevant information about the world, though not perfect, far outstrips anything to which humans can aspire.’⁶⁶ Whereas this might very well be, metaphysical Chance nonetheless leaves open the fact that ‘chancy’ events might still occur regardless of the propensity of the world to produce the ‘correct’ event that Chance ‘imperfectly’ summarises. This is what I tried to expose with the single throw example above. Therefore, if Chance is an imperfect yard stick for measuring credences’ accuracy, why bother with it? Why not get rid of these ‘chancy’ events altogether?

These criticisms are formulated by Pettigrew.⁶⁷ To remedy the imperfection of Chance, Pettigrew constructs an ‘omniscient’ worldview of epistemic utility with the main concept of ‘ideal credence’ or ‘omniscient credence.’ This ideal credence matches the objective truth-value of a proposition for a subject. As Pettigrew writes: ‘if a proposition is true in a situation, […] the ideal credence for an agent in that situation is the maximal credence which is represented by 1’ and conversely, ‘if a proposition is false, the ideal credence in it is the minimal credence, which is represented by 0.’⁶⁸ For this construction to work, not only does the propensity of the world to produce a certain event (or state) need to be taken into consideration (as it is with Chance), but also whether the event actually occurred. Pettigrew’s omniscient credence represents both: propensity and occurrence. It is both summative and descriptive. So having declared the existence of such omniscient credence in every possible world, it follows that the accuracy of a subject’s credence is the distance to this omniscient credence in a given world/situation. Pettigrew formalises this distance measurement in an accuracy score, the so-called Brier score, that acts as an ideal mathematical yard stick for measuring credences’ accuracy and epistemic value in a given world/situation.⁶⁹ And within this newfound omniscient worldview, probabilism still holds: it is still irrational to have beliefs that violate the axioms of probability since such beliefs inevitably lead to inaccuracies.⁷⁰

It is clear that in reinstating the link between objective chance and credence via David Lewis’ Principal Principle, Joyce and Pettigrew cross the rubicon that de Finetti’s orthodox worldview warns against crossing.⁷¹ One is definitely in the world of metaphysics in this new territory. There is no more as-if character of exchangeability, as one is able to directly read the world’s objectives chances and thus its propensity to contain specific events. This reading is conducted from a kind of view from nowhere. Contra de Finetti, objective chance is here reified as what commands or ‘constrains’ rational credence. That said, instead of ‘chance’ the omniscient worldview substitutes a concept of Chance that is anything but the mark of a certain mathematical rationality. Who knows what objective chance really is, let alone if it systematically follows the axioms of probability, like clockwork? It is clear the omniscient worldview aims at cancelling out ‘chance’ with its omniscient rationality from which its notions of epistemic accuracy derives. Since omniscience knows no indeterminacies by definition, and is thus antithetic to the inherent indeterminacies of ‘chance,’ the sole reason of this omniscient perspective is precisely to cancel ‘chancy’ chance in order to determine the accuracy of credences.⁷² I believe one is still justified in advancing, with de Finetti, that such an omniscient conceptualisation is a man-made chimera, as it attempts to exteriorize or materialize a certain logos (ratio, reason) onto the world in order to justify its normative import on subjective ways of thinking and knowing.

Leaving these metaphysical concerns aside for a moment, there is another way the omniscient worldview can be justified in how it legitimises the ‘truthfulness’ of its own denotational and epistemic propositions. Coming back to Lyotard’s notion of performativity, it is clear that vindicating credences along the vector of accuracy is to ‘legitimatise-by-perfomativity’ as referred to in the introduction with the notion of determinism. While demanding that degrees of belief be coherent with the axioms of probability can be perceived as purely normative, the accuracy-first interpretation petitioned by Joyce and Pettigrew is alethic in nature as it assumes that credences can be evaluated against some kind of transcendental truth: an ideal or omniscient credence. Within this coordinate system, performativity is articulated by the distance measure (i.e. score) between one’s subjective credence and this ideal credence. The closer to the ideal the more performative and thus rational one is calibrating their credence.

Lyotard defines determinism as the hypothesis upon which legitimisation by performativity is based. I maintain that accuracy here defined is a determinism: one’s credence can be evaluated as being on a certain spectrum of rationality/irrationality precisely because of this determinism. The ideal or omniscient credence serves as the ‘referent’ that ought to supply non-contradictory and consistent accuracy measures; it provides a stable ‘path’ to strike distances from and derive rational degrees of belief. It goes without saying this ‘path’ is anything but pluriversal; it is unique and can thus be qualified as universal since all degrees of rational beliefs derive from it.

Such logos constructed out of the requirement of mathematical coherence and denotative accuracy can be framed as transcendental, even if it is taken as a normative ‘framework’ for the subjective interpretation. One can always argue that since different subjects have different perspectives, values, and beliefs about the world, they inevitably derive their respective credences in a pluriversal way (i.e. from different ‘worlds’). One can also argue that a given subject is faced with a plurality of possible/speculative worlds when making a subjective choice/decision. While these are sound arguments, my point here in stressing the necessary assumption of probabilistic beliefs and the logic of accuracy this assumption supposedly licences, is that these ‘assumptions’ are made to universally transcend all of these worlds in order to make the subject’s credences perform in the sense mentioned above. These assumptions are conditions of rationality. So if one is to keep the pluriversal claim regarding the ‘rational’ subjective interpretation intact, I believe it would be better expressed as—PLURIVERSAL = UNIVERSAL—to play on a known maxim from Gilles Deleuze and Félix Guattari.⁷³ And this goes for both the orthodox and omniscient worldviews I have exposed above which have the axioms of probability and the pragmatic and epistemic utility arguments (i.e. probabilism) as foundational first principles to justify the rationality of the subjective interpretation of probability and induction.

2.3 Cancelling by the Objective Interpretation?

I am now turning to the objective interpretation of which caveats 3 and 4, identified with the Bayes-Laplace formula at the beginning of section 2, are of prime importance. The objective interpretation aims to fix the prior distributions problem expressed in caveats 3 and 4 by formally expressing ignorance with the hope that the iterative process of the Bayes-Laplace approach will tend towards objective posteriors. Because one is indeed faced with their own ignorance when contemplating the unknowns of 3 and 4, the objective approach puts certain conditions on how to express this ignorance in terms of prior probability distribution. Assigning equal probabilities a priori to the priors of each individual cause is one of the many ways to deal with the distribution.⁷⁴ The rationale of this interpretation in terms of priors is to not let biases enter the iterative calculations from the start. Expressing ignorance formally, and objectively, prevents subjective biases from entering into the calculation of posteriors, and in so doing, presumably increases the possibility that the calculation will converge into an objective probability value in the long run: objective ignorance in, objective results out.

The objective interpretation aims at demarcating f from P in a more stringent manner than with the subjective interpretation where f can already inform the inceptive fixing of priors’ value at step k=0. This is discouraged by the objective approach.⁷⁵ I already mentioned how assigning equal probabilities a priori is one of the ways to deal with priors. This way of distributing unknown probabilities has a long genealogical lineage going back to Bernoulli and derives from the Principle of Indifference, a contemporary term coined by Keynes.⁷⁶ What the principle essentially stipulates is that if there are no reasons to assign unequal probabilities to various cases, then equal ones must be assigned to each over the range 0-1. For example, if there are 6 cases, assign each the constant 1/6. The idea with setting equal values to priors is for posteriors to converge to objective values in the long run. This idea rests on the assumption that the world has a tendency to repeat itself in such a way that a long series of repetitions of a certain state or event conditions the advent of relative frequencies f that are approximates of probability P of this given state or event. With the Bayes-Laplace formula, feeding observed frequencies f in the likelihood variable will weigh the effects of the initial ‘indifferent’ prior and ultimately shed a posterior that approximates the objective probability P. Again, this is how the formula articulates the inductive movement from f to P.

Now, the assumption of the tendency of the world to repeat itself in such a way that f will approximate P in the long run is nothing else than a cryptic way of expressing Hume’s assumption that the course of nature will continue uniformly and that the future must be conformable to the past. As Hume emphasises, these are assumptions that cannot be demonstrated logically nor empirically. Trying to prove the assumptions empirically, with for example ‘evidence’ from the Bayes-Laplace formula that f does indeed converge to P in the long run, is a circular tautology since the whole premise of the objective interpretation rests on the assumption that the world has a tendency to repeat itself in the long run. Taking this as a proof would amount to nothing but a validation of the implicit assumption the interpretation rests on—something Hume warns against.

Another way of interpreting the tendency is to think that f will match P. This means the ‘meta’ probability (i.e. probability of a probability) that relative frequencies f will match P, or be proportionate to P, increases in the long run as it tends towards a limit. When this limit is reached, what one would then be doing while compiling frequencies f is to merely verify they match P. This would serve as a proof of the cancelling out of chance as one would be certain of this matching on a specific, single case occasion and not simply in the long run. This reasoning is also flawed and circular since thinking that f ought to match P is to already assume the cancelling out of chance. What seems as though reaching the frequencies’ limit would serve as a ‘proof’ of the cancelling out of chance, is nothing but a validation of the implicit assumption the cancelling out of chance is already effective while compiling frequencies f—a petitio principii. There are no a priori guarantees f will ever match P, only assumptions it might.

While the Law of Large Numbers provides a link between probabilities and frequencies, to believe f ought to converge toward P is to see Bernoulli’s theorem’s insight not as interpreting probabilities statistically but as grounds to make ontological assumptions about the uniformity of nature. The contention here has to do with the meaning of probable_1 and probable_2. Trying to draw guarantees of success in matching f to P is to interpret a probable_1 question into a probable_2 answer (i.e. the ‘meta’ probability). This leads to a circular reasoning. Considering the proposition that something probable_2 is going to occur in a ratio of cases proportionate to its probability, and justifying the proof of this proposition on the grounds that a guaranteed success of matching f to P is not only probable_1 (an assumption it might, who knows (¯\_(ツ)_/¯)?) but probable_2 (a probability that the probability converges) is circular and leads to the same regression regarding the existence of a probability of a probability expressed in caveats 3 and 4.

There are caveats in interpreting what inverse probability (moving from f to P) insinuates, whether the formula follows the Bayes-Laplace one or not. Rather than increasing certainty, the circular probable_2 interpretation I discussed above increases doubt. Hume was aware of these circular, infinite regress culprits when writing:

Here then arises a new species of probability to correct and regulate the first, and fix its just standard and proportion. As demonstration is subject to the control of probability, so is probability liable to a new correction by a reflex act of the mind […] This is a doubt, which immediately occurs to us, and of which, if we wou’d closely pursue our reason, we cannot avoid giving a decision. But this decision, tho’ it shou’d be favourable to our preceding judgement, being found only on probability, must weaken still further our first evidence, and must itself be weaken’d by a fourth doubt of the same kind, and so on in infinitum; till at last there remain nothing of the first probability […].⁷⁷

Obviously the subjective and objective interpretations, which can be deemed worldviews, do not correspond. While both are modern worldviews on what probabilities are, what they are authorised to say, and how, they deal with the conundrums of induction in different ways. That said, both are worldviews that ‘legitimatise-by-perfomativity’ the denotational statements they produce. In the case of objective interpretations, this legitimisation follows the classic stance of having the world as a referent provides the regular ‘path’ (instead of an ideal credence) upon which the performativity of an event captured by the Bayes-Laplace formula is based. As I argued along with Hume, the regularity of the path is an assumption, as it cannot be proven a priori the path will not produce contradictory or inconsistent proofs. Elevating the rules and principles of probabilities as that which provides such proof is to uncritically substitute them to the prima facie uniformity of nature.

In this section I raised concerns about this ‘substitution’ and argued it to be based on circular reasonings, doubtful tautologies and assumptions. Some of these assumptions were formulated after Hume, notably by Laplace himself, and there are no reasons to believe these will disappear, even if we have moved from a so-called frequentist era to a Bayesian one.⁷⁸ After all, mathematical probabilities are enticing vectors along which various ontological claims have been, and will still be, articulated and justified.

‘Cancelling’ a Sceptic?

In the same chapter where the personage of Hume made an appearance, Whitehead, while still pondering on the problem of induction, rightly observes that: ‘Induction presupposes metaphysics. In other words, it rests upon an antecedent rationalism. […]; your conjectures as to the future presupposes some basis of knowledge that there is a future already sub­jected to some determinations. The difficulty is to make sense […] of these ideas. But unless you have done so, you have made nonsense of induction.’⁷⁹ It is telling that Whitehead sees that Hume’s problem cannot be inverted unless metaphysical concepts come to the rescue; the sceptic cannot easily be ‘cancelled,’ so to speak. Whitehead consequently conceived of such metaphysical concepts to seal off the ontological gap left wide-open by Hume. In this way Whitehead ‘saves’ induction from Hume’s scepticism.

What I hope to have exposed in this article is that there are metaphysical assumptions underlying certain contemporary worldviews of what mathematical probabilities are and what they are authorised to say and how, whether it is the orthodox or omniscient worldviews transcending any rational subject position in any world with their foundational probabilistic utility principles, or the objective worldview that substitutes the performative prowess of probability as ontological proof of the uniformity of nature. All these worldviews that ‘legitimatise-by-perfomativity’ rest upon antecedent rationalisms, as Whitehead puts it, in that they uncritically adopt rationalist worldviews that have been debunked before, notably by Hume himself. As I mentioned before: who knows what objective chance really is, let alone if it systematically follows the axioms of probability? Turning probable_1 propositions into probable_2 propositions sometimes entails significant consequences in terms of reasoning and justification, something I have exposed with the various caveats of the Bayes-Laplace formula I analysed.

In the guise of a conclusion, I return to this special issue’s focus on pluriversal perspectives and look at how the conceptual personages of the sceptic and sophist can be conceived of as operators that open such perspectives from within the belly of the beast of Western culture.

There is a parallel to be drawn between how the worldview analysed in this article promotes the axioms of probability as epistemic principles and how Aristotle promotes the principle of contradiction as the first epistemic principle in the book Gamma (book 4) of Metaphysics. This is where Aristotle stipulates that it is not possible for a proposition to be at once the case (i.e. true) and not the case (i.e. false). From this principle, Aristotle derives the infamous law of the ‘exclude middle’ (tertium non datur), which stipulates that there is no intermediary state between being the case and not being the case. I already mentioned that when probabilities are turned into propositions they acquire a mode of denotation that is similar to Aristotelian assertoric syllogism in the way their ‘truth-value’ criteria is evaluated.⁸⁰ I contend that the axioms of probability for probabilism are analogous to the principle of contradiction for Aristotelianism in the sense that these axioms govern how denotational propositions ought to be ‘properly’ believed in with credences that are calibrated with one another according to the axioms. Attributing credences to propositions that violate the axioms of probability is non-sense for a probabilist, just as it is non-sense to believe in a proposition that violates the principle of contradiction for an Aristotelian.

Jan Łukasiewicz offers a short yet thorough analysis of Aristotle’s principle of contradiction and shows that its articulation follows three conflated vectors: an ontological vector (a given characteristic cannot at once belong and not belong to an object); a logical vector (two contrary propositions cannot be true at the same time); a psychological vector (no one can believe the same thing can be and not be at the same time).⁸¹ I say these are ‘conflated’ because Łukasiewicz shows that although the qualification of each vector uses different words having different significations (e.g. ‘thing’, ‘proposition’, ‘belief’), the ‘logical formulation seems to be for Aristotle equivalent with the ontological formulation.’⁸² Łukasiewicz explains this conflation from Aristotle’s correlation between assertoric propositions and objective facts: the denotational language game of propositions is made to correspond to objective facts. Because of this conflation, the world seems to be structured like a language. Łukasiewicz even shows that Aristotle confuses logical succession with psychical causality. This might resonate with some of the transcendental worldviews I addressed: rational beliefs and world(s) seem to be to be structured like the ‘language’ of probabilities. This is how they make sense. They do not otherwise.

As I have argued along with Hume in the case of induction, Łukasiewicz also shows that there are no demonstrations of the principle of contradiction, no foundations for it to stand on, so to speak. The principle is only justified by its pragmatic/practical utility. Łukasiewicz writes: ‘the principle of contradiction has, to be sure, no logical worth, since it is valid only as an assumption; but as a consequence it acquires a practical-ethical values. (…) The principle of contradiction is the sole weapon against error and falsehood.’⁸³ Again, parallels can be drawn here with how some of the worldviews I addressed self-legitimise by means of performance along the vector of ‘utility.’ That being said, let’s not be fooled by Łukasiewicz’s appeal to our ethical sentiment. Aristotle did indeed use the principle as a ‘weapon’ against his various enemies: the sceptics, the sophists, the cynics, etc.—all the supposedly false friends of wisdom (sophia). Łukasiewicz signs his text by acknowledging this inceptive war against the false friends of wisdom which philosophy is born from, historically with Plato and Aristotle, writing (not without a sense of sarcasm considering his later work) that while battling: ‘[Aristotle] might well have himself felt the weakness of his argument, and so he announced his principle a final axiom, an unassailable dogma.’⁸⁴

Barbara Cassin also comments on Aristotle’s arguments in Gamma. In ‘Speak if You are a Man, or the Transcendental Exclusion,’ Cassin traces how Aristotle made the ‘ethical’ enter into language following the same tactic as exposed by Łukasiewicz: the principle of contradiction conflating the ontological, logical, and psychological; the principle of contradiction used as a weapon in the war machine against the sophists; the principle of contradiction used as first principle for governing (univocal) sense and thus as a mean of exclusion of those who do not make sense, those who fall outside the principle, who fall outside the definition of (sensible) man—‘such a person, in so far as he is such, is similar to a vegetable’ writes Aristotle.⁸⁵ Aristotle counters or ‘cancels’ the sophists with the principle and establishes a transcendental exclusion that relegates the pretended false friends of wisdom to oblivion. This act of transcendental exclusion can be seen as one of the first acts of post-socratic Western philosophy, one that is foundational to philosophy’s requirement for univocal sense/meaning. As Cassin writes: ‘ever since Aristotle, all those who refuse to submit to this requirement have been reduced to so much silence or noise, something prior to language.’⁸⁶

The war against the sophists was already in effect with Plato who, much like Aristotle after him, formulated his insurgencies by devising his theōria. Cassin writes that ‘the singularity of sophistics is effectively to already be, inasmuch as it is a historical fact [i.e. Gorgias, Protagoras, Antiphon, etc.], a structural effect […]. Fact of history, structural effect: the soldering joint that constitutes the object of sophistics, is a platonic artefact, the product of discourses. The essence of the artefact is simply to make the sophist the negative alter ego of the philosopher: his bad other.’⁸⁷ As an artefact, the sophist is the bad doppelgänger of the philosopher, the one who bastardises logos and makes use of fallacious and non-sensical arguments, who speaks just to speak, a specious reasoner in short. This is still the characterisation contemporary dictionaries give to the term ‘sophist.’

But one would be mistaken to take this characterisation as essential. The sophistical logos, in its various historical articulations, is always in one way or another brought back to what it precisely seeks to circumvent or devastate: the submission to (univocal) being, submission to the (univocal) world, to the (univocal) real. Cassin writes: ‘trying to understand sophistics, requires at least accepting to consider, beyond the oppositions between philosophy and rhetoric, sense and non-senses, its discursive performances as countless shrewd positions against ontology: sophistics as a dodging of metaphysics and an alternative, from the presocratics, to the classic lineage of philosophy.’⁸⁸ This is how the conceptual personage that is the sophist, and by analogy, the personage of the sceptic like Hume, who are, again, Western figures, can be conceived of as a resources to ‘find ways of puncturing the propensity of the North to self-sealing one-world metaphysics.’⁸⁹

In dialogue with Cassin, Badiou observes that one of the essential goals of philosophical language is to combat equivocality. Already in Parmenides, and clearly so in Plato later, mathematics becomes the most interesting philosophical language as it instantiates a norm for univocality, something that philosophy necessitates in its quest of formulating universal propositions. To this observation Cassin promptly replies:

I could introduce a couple of nuances, but I totally agree with the fact that, for sure, univocality has always been philosophy’s job, from Aristotle to Leibniz, and until today, until… you. […] All philosophies, all philosophers have wanted this, it is their job, and they do it well. Or rather, they do it well and they don’t do it: never was Leibniz’ charateristica universalis realised. Mathematics is not an everyday language. Leibniz wanted that we try together to say calculemus ’let us calculate,’ and that faulty reasoning be seen as miscalculations. We would know who is right and who is wrong like we know an addition is just or false. How simple would it be! But no, it is not that simple. It is only a dream, a strange dream and, moreover, a strange dream of domination. That is why it is not my path.⁹⁰

To defy this dream of domination is to defy worldviews that aim to univocally institute ‘what is’ (and consequently ‘what is not’), whether these worldviews are articulated along the vector of mathematics or not. And it is not so much mathematics in itself that creates worldviews but rather the transcendental language games it might be conflated with. And it is not so much language in itself that is necessarily exclusionary but rather the univocal sense that is imposed on it as a one-way street towards meaning. If univocality is what defines sense, then let plurivocality defy sense by producing ‘non-sense.’ Perhaps pluriversal paths are less about opening up ‘perspectives,’ creating more (transcendental) views or worldviews having (singular) coordinate systems, but rather about allowing plurivocal productions that defy sense and its coordinates.

Language and mathematics are performative in that they are bound up with their effects. Cassin shows this performative characteristic of language with the age-old language-operators that are the sophists, producers of effects, sophistic effects that is. Mathematics is performative in another, albeit complementary, way I believe. I mentioned how the language game of mathematics has more to do with prescriptives (‘Take a finite series of elements…’) than denotatives. It is prescriptive in the sense that it operates, asks that operations be carried out by some actions, like a recipe of sorts. As with any language game, mathematics certainly has rules. Yet these rules allow for unorthodox and plurivocal constructions to be made. I cited Łukasiewicz above. Well, Łukasiewicz came up with unorthodox ways to construct non-classical multi-valued logics, logics that do not necessarily follow the ‘sacrosanct’ principle of contraction. There are other examples. A notable one is constructive mathematics/logic, pioneered by L.E.J Brouwer and Arendt Heyting, which does away with the law of the ‘excluded middle’ and over reliance on the axiomatic approach. I believe software studies could exploit such unorthodox/‘sophistic’ constructions, perhaps to rethink anew its own relation to mathematics.

Notes