The Liberal Logic Of Uncertainty From 1320 To 1974

things which had been uncertain forever have recently become improbable

Probability is a strange alchemy, a science of ignorance. Given the amount of ignorance involved in getting anything done, it’s for the best that we have at least an alchemy of it now. Perhaps Poincaré put the mystery best:

“Vous me demandez de vous prédire les phénomènes qui vont se produire. Si, par malheur, je connaissais les lois de ces phénomènes, je ne pourrais y arriver que par des calculs inextricables et je devrais renoncer à vous répondre ; mais, comme j’ai la chance de les ignorer, je vais vous répondre tout de suite. Et, ce qu’il y a de plus extraordinaire, c’est que ma réponse sera juste.”

(Science et méthode, Livre premier, § IV: Le hasard)

The importance of spinning this straw into gold cannot be easily overemphasized.The probabilistic interpretation of counts collected by the state (so-called “statistics”) is the inner consciousness of the modern state. This has been strongly emphasized over the last fifty years by respected authorities such as Michel Foucault (Society Must Be Defended), James C Scott (Seeing Like A State) and most powerfully so by Ian Hacking (The Emergence Of Probability/The Taming Of Chance).

That said, there is something of a cottage industry in proclaiming the alchemical status of this so-called science. To pick an example “at random,” consider Nassim Taleb, an essayist and managerial scientist who has published extensively on the question of whether this process could truly be as simple as the textbooks present it.

There may be something better to do than to publicly name the Rumplestiltskins of economics. Until then, it is worth our time to trace the flow of “calculus albus” through modern Europe. We will see that critiques of the statistical project can be drawn back into far earlier conceptions of probability. In fact, a particular research project from the fin de siècle throws a special light on the subject - the project of John Maynard Keynes. 

Without further ado, let us see how man got so ignorant that he had to develop a science - or at least an alchemy - of it? 

From Dawn To Decadence

Part 1: Dawn

Often we are told that Descartes is the dawn of a new era called “Modern Philosophy”, an era which ends the period of “medieval philosophy” and stretches from Descartes to before Nietzsche. Descartes himself is, characteristically, more clear on this matter: his originality was in reorganization. Descartes restructured the questions of philosophy to be independent of patristics of Sic Et Non - the founding text of Medieval Scholasticism - not “even as” but most especially as he relied on Scholastic insights.

In few places is this more highlighted in the discussion of ignorance. The dominant theory of un-knowledge was Aristotle’s fallacy theory from “On Sophistical Refutations”. This tedious theory should be familiar to anyone who has been accused of a “nice ad hom” online.

The greatest medieval philosopher - Dante - is even more remarkable for his lack of analysis of ignorance, not even Aristotle’s theory is given. Dante’s reason is shown in his most direct discussion of ignorance: that of limbo, the first circle of hell. Dante believes ignorance to be the original state of mankind, thus unbaptized infants appear in Limbo. More textually, he has Vergil say

“Or vo' che sappi, innanzi che più andi,
ch’ei non peccaro; e s’elli hanno mercedi,
non basta, perché non ebber battesmo,
ch’è porta de la fede che tu credi;

e s’e’ furon dinanzi al cristianesmo,
non adorar debitamente a Dio:
e di questi cotai son io medesmo.
Per tai difetti, non per altro rio,

semo perduti, e sol di tanto offesi
che sanza speme vivemo in disio”

(Inferno, Canto 4)

Vergil, ignorant as he was of Christianity, owes “no [sinful] wages” - “non peccaro” - but merely exists immortally in the natural state of mankind: “without hope, but living in desire” - “sanza speme vivemo in disio”. Ignorance, being the original state of humanity, can be explained by causes so disappointing and ‘mere’ as simple location: “living in the rear of christianity” - “furon dinanzi al cristianesmo”.

Descartes rearranges the situation extensively, emphasizing the question of which causes are necessary rather than those sufficient. This makes sense because, where Dante’s Commedia is written in dialectic with the original state of mankind, Descartes’ Meditationes de Prima Philosophia is written in dialectic with his earlier book Discours de la Méthode - which is to say, with natural science. Philosophy is contrasted not with the emptiness of the original state, but with the fullness of natural science. The question becomes, not “What unsatisfying reasons are there for this or that bit of ignorance?” but “For what satisying reason did He make Man so stupid generally?”

Nam si, quo peritior est artifex, eo perfectiora opera ab illo proficiscantur, quid potest a summo illo rerum omnium conditore factum esse, quod non sit omnibus numeris absolutum?

(Meditationes de prima philosophia, Adam et Tannery Editio, Meditatio IV, Paragrapho 5.)

The answer is simple: because God made us individuals! In a very important sense we are born without ideas - this is part of being an individual - and thus our ideas are in a very important sense made by us and thus are imperfect. Ignorance thus takes on a dual nature, with a single kind of cause which can be investigated and - in theory - eliminated.

At an even higher level of abstraction, Spinoza further developed the essence of the theory:

“At res aliqua nulla alia de causa contingens dicitur nisi respectu defectus nostræ cognitionis.”

(Ethica, De Deo, prop XXXIII, sch I)

Spinoza’s defectus theory - “No thing is said to be contingent on any other cause except with respect to our deficit of knowledge” - is not idle speculation. He has here, in fact, a fairly complete - if highly abstract - description of how error was practically conceived in observational science. This can be illustrated by a parable.

Suppose Spinoza and Van Leeuwenhoek wanted to learn the principles of biological generation, say, was the body contained entirely in the reticulation of semen? (Historically, this question occurred long after their deaths, though Spinoza was one of the first to be aware of spermatozoa at all). To answer, Spinoza needed to look at a drop of semen through a microscope. Specifically, he used what is now called a Leeuwenhoek microscope: a very small sphere of Venice glass held within an eye plate (the sample would be on screws in front of the eye plate). The high quality Leeuwenhoek microscopes made by Spinoza were extremely powerful: they had a resolving power of one micron, small enough to see bacteria. Yet they were also so tedious to make that no one outside of Spinoza’s social circle would see a bacterium for another hundred years.

The difficulties Spinoza had with the microscopes could be quantitative: 

1. Spinoza only knew how to create (grind or blow) tiny lenses so perfectly, 

2. He only understood the dynamics of light so precisely, and 

3. He only understood the refractive index of air and glass so exactly. 

They could also be qualitative, i.e. Spinoza’s speculations as to embryology were limited by his imagination. But in all cases, the difficulty is not the mere smallness of the spermatozoa - it is mainly a personal ignorance.

Let’s say the specific problem Spinoza is having is that his image is fuzzy. This could be due to one or two of two things: a scratch in the lens or spherical aberration. The defectus in cogitatione in this case is our personal inability to interpret the distribution of fuzzyness around the edge of the image in a way that shows which cause created the effect.

Applying the defectus theory in this way makes the strange dual nature of probability clear: Probability is due to ignorance, but also has an objective effect. Probability is due to ignorance of causes in particular: given what the causes are, the effects are objective. 

Pascal makes the next decisive advance by developing a plan for a method of choosing how to act despite ignorance, by analogy to gambling. 

“Examinons donc ce point, et disons : « Dieu est, ou il n’est pas. » Mais de quel côté pencherons-nous ? La raison n’y peut rien déterminer. Il y a un chaos infini qui nous sépare. Il se joue un jeu, à l’extrémité de cette distance infinie, où il arrivera croix ou pile. Que gagerez-vous ? Par raison, vous ne pouvez faire ni l’un ni l’autre ; par raison, vous ne pouvez défendre nul des deux. … Oui, mais il faut parier : cela n’est pas volontaire, vous êtes embarqué. … Puisqu’il faut choisir, voyons ce qui vous intéresse le moins. Vous avez deux choses à perdre, le vrai et le bien… Voilà un point vidé ; mais votre béatitude ? Pesons le gain et la perte, en prenant croix, que Dieu est. Estimons ces deux cas : si vous gagnez, vous gagnez tout ; si vous perdez, vous ne perdez rien.”

(Pensees, Article X, Hachette, 1871, tome I)

Pascal’s “point vidé” is a radical new form of subjectivism: the introduction of individual caprice, a dependence on “your bliss” - “votre béatitude,” whatever you’d like. Notice, though, how caprice must be limited: “Oui, mais il faut parier : cela n’est pas volontaire, vous êtes embarqué.” If the gambler were allowed to decline a gamble, the whole system falls apart. This is both quantitatively and qualitatively the origin of the notion of an expected value.

Let’s return to our parable of Spinoza and the microscope one last time. Suppose one sees too much blur at the edge of an image. This could be caused by a scratch or by spherical aberration. In the first case, the solution would be to make more spherical lenses more perfectly. In the latter, a possible solution would be to make an aspherical lens.

Descartes advocated for the latter for bio-mimetic reasons, which Spinoza strongly criticized. His criticism was so virulent that some have suggested Spinoza didn't understand Snell’s law, but this is due to a particular misinterpretation that I will not distract us with here. I suspect Spinoza’s attitude was attached at some level to a Platonic admiration for the perfection of the sphere along with the practical fact that spherical lenses are much easier to make. Do we buy Descartes’ biomemetic reasons or Spinoza’s Platono-practical argument? There is no way to decide a priori: it is a point vidé.

The Pascalian ideal was, more or less, how the concept of probability was used throughout the era of so-called “Modern Philosophy” with mostly technical innovations in the intervening 1.5-2 centuries. Isaac Toddhunter had no difficulty in writing a smooth narrative from Pascal’s argument above to Laplace’s mathematics in A History of the Mathematical Theory of Probability from the Time of Pascal to that of Laplace.

The only technical innovation that remains relevant is that which Bernoulli crowned the “Principle Of Insufficient Reason.” As an approach, it attempted to replace Pascal’s point vidé with an egalitarian principle: in the absence of contrary evidence, all unknown causes are equally likely.

Part 2: Decadence

Often we are told that “La décadence est venue à la fin du siècle,” and the field of probability was not spared. In the late 19th century - high romanticism - one sees an explosion of new criticisms and theories.

The most spectacular criticism came from mathematician Joseph Bertrand, a ferocious critic of pseudo-mathematical social theories, like economics. In 1889, Bertrand demonstrated the mystery within a seemingly simple problem: in a unit circle, what are the odds that a random chord (line segment with both ends on the circle) is longer than the square root of three?

This problem has a simple solution. Each chord is determined by two endpoints. Choose the first point arbitrarily, by the homogeneity of the circle this makes no difference. Now imagine the equilateral triangle with a corner at this point and all corners on the circle. The chord has a length greater than the square root of three only if the second point is such that the chord passes through this triangle. By the principle of insufficient reason, the second point may be regarded as anywhere on the circle. These last two sentences show a probability of one third.

This problem has a simple solution. Except for diameters, every chord is determined by its midpoint (all diameters have the center as a midpoint). Imagine an equilateral triangle with corners on the unit circle and a small inscribed circle tangent to its lines. By rotation, one can see that a chord with its midpoint inside the inner circle must be longer than the square root of three. This inscribed circle has radius one half, so takes up one fourth of the area. By the principle of insufficient reason, any point is equiprobable. Therefore probability is one fourth.

Some visualizations of the issue

Bertrand has shown that the principle of indifference is hopelessly muddled, even internally contradictory when dealing with continuity. There is no possible a priori meaning to the term “each chord is equiprobable”, and the claim there is is subject to Bertrand’s Paradox.

Given the logical force of Bertrand’s Paradox, it is a testament to the point vidé’s power over men’s minds that Bertrand does not respond by abandoning the principle of insufficient reason. Rather, he adopts the methodological principle that probability strictly only applies to the finite and each infinite case must be handled by a careful logic of limits. This methodological principle is, of course, a part of the general late 19th century project of The Arithmetization Of Analysis, where intuitive but unclear equations between infinitesimals were replaced with convoluted but logically transparent inequalities between rational numbers.

Before moving on, I would like to say a word about early statistical physics or “gas theory” as it was known at the time. Though incredibly revolutionary, in the question of the foundations of probability theory it was mostly notable for its conservatism. The principle of insufficient reason was used without real objection until Boltzmann’s H-Theorem, when equiprobability became a physical claim. There was little doubt that in principle that “probability” always meant equiprobability. Thus Josiah Royce, an American Hegelian, was able to absorb statistical mechanics wholesale without much difficulty.

A New Hope

Part One: Hopes Betrayed 

In 1908, a 25 year old grad student submitted a thesis on the foundations of probability. The book contains the first proof - in a forbidding Boolean notation - of the fundamental principle of inverse probability (which we now call Bayes Theorem), arguing it was a simple application of a generalization of the Boolean product (which remains the modern understanding). He intends this thesis to be the technical core of a treatise for general consumption and begins work expanding it for publication. It will take 13 years and he will publish three books and one book length pamphlet in the interim, but he did eventually do it.

Keynes’ Treatise On Probability attempted to give form to the field made chaotic in the fin de siècle. It is a deeply hopeful work, with none of Bertrand or Poincaré’s despair of being bound by the principle of insufficient reason.