A Philosophical Essay on Probabilities

by Pierre-Simon Laplace

A classic of science, this famous essay by "the Newton of France" introduces lay readers to the concepts and uses of probability theory. It is of especial interest today as an application of mathematical techniques to problems in social and biological sciences.
Generally recognized as the founder of the modern phase of probability theory, Laplace here applies the principles and general results of his theory "to the most important questions of life, which are, in effect, for the most part, problems in probability." Thus, without the use of higher mathematics, he demonstrates the application of probability to games of chance, physics, reliability. of witnesses, astronomy, insurance, democratic government and many other areas.
General readers will find it an exhilarating experience to follow Laplace's nontechnical application of mathematical techniques to the appraisal, solution and/or prediction of the outcome of many types of problems. Skilled mathematicians, too, will enjoy and benefit from seeing how one of the immortals of science expressed so many complex ideas in such simple terms.

Genres: Philosophy, Science, Mathematics, Nonfiction, Classics, Essays, Physics

224 pages, Paperback

First published December 13, 1994

About the author

People note theory of French mathematician and astronomer Marquis Pierre Simon de Laplace of a nebular origin of the solar system and his investigations into gravity and the stability of planetary motion.

His pivotal work led to the development of statistics. He summarized and extended the work of his predecessors in his five-volume Mécanique Céleste (Celestial Mechanics) (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace.

Laplace formulated Laplace's equation and pioneered the Laplace transform in many branches of mathematical physics, a field that he took a leading role in forming. People also named the Laplacian differential operator, widely used in mathematics. He restated and developed the nebular hypothesis of the origin of the solar system and was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse.

Laplace is remembered as one of the greatest scientists of all time. Sometimes referred to as the French Newton or Newton of France, he possessed a phenomenal natural mathematical faculty superior to that of any of his contemporaries.

Laplace became a count of the First French Empire in 1806 and was named a marquis in 1817, after the Bourbon Restoration.

A frequently cited interaction between Laplace and Napoleon purportedly concerns the existence of God. A typical version is provided by Rouse Ball:

Laplace went in state to Napoleon to present a copy of his work, and the following account of the interview is well authenticated, and so characteristic of all the parties concerned that I quote it in full. Someone had told Napoleon that the book contained no mention of the name of God; Napoleon, who was fond of putting embarrassing questions, received it with the remark, 'M. Laplace, they tell me you have written this large book on the system of the universe, and have never even mentioned its Creator.' Laplace, who, though the most supple of politicians, was as stiff as a martyr on every point of his philosophy, drew himself up and answered bluntly, Je n'avais pas besoin de cette hypothèse-là. ("I had no need of that hypothesis.")

Laplace's early published work in 1771 started with differential equations and finite differences but he was already starting to think about the mathematical and philosophical concepts of probability and statistics. However, before his election to the Académie in 1773, he had already drafted two papers that would establish his reputation. The first, Mémoire sur la probabilité des causes par les événements was ultimately published in 1774 while the second paper, published in 1776, further elaborated his statistical thinking and also began his systematic work on celestial mechanics and the stability of the solar system. The two disciplines would always be interlinked in his mind. "Laplace took probability as an instrument for repairing defects in knowledge." Laplace's work on probability and statistics is discussed below with his mature work on the analytic theory of probabilities.

The asteroid 4628 Laplace is named for Laplace.

His name is one of the 72 names inscribed on the Eiffel Tower.

The tentative working name of the European Space Agency Europa Jupiter System Mission is the "Laplace" space probe.

Reviews

[Jake · Rating 5 out of 5]

“Probability is a (some people would say *the*) logical calculus of uncertainty.” ~ David Braber

Often times we’re not in control of the provenance of our data. When we try to model the world from our data in some way, it should arise from our beliefs and irregularities we observe from the universe. The simplest model we often use, linear regression, is easily resolved with just two data points. When you think about Newton’s physics, it’s simply put, a simple regression model with high validation accuracy. Easily resolved with two data points, but doesn’t generalize properly when you start dealing with smaller masses and more data points. But it’s a useful model if you just care about where your starting place, ending place are and you assume masses are large

I bring up Newton because LaPlace was a colleague to Newton. LaPlace would be the one who did the grunt work of generalizing mechanics and that was used as a framework for thinking about probabilities in this book. It’s quite odd that so many people think LaPlace thought the world was deterministic. In the chapter where he brings up what would later be referred to as LaPlace’s Demon, it’s literally titled Sur Les Probabilities or “On the Probabilities.” LaPlace was setting up his demon as a straw-man so that he could later tear it down. From here on out I’ll take snippets of this chapter and talk about them.

“We out the to regard the present state of the universe as the effect of its anterior state and as the cause of one which is to follow.”

That’s Markov’s principle in action, which is to say, if a recursive algorithm cannot fail to converge, then it converges. This would lead to Church’s thesis, which connects the abstract mathematics to the real world consequences and limitations of computability. This was a conjecture by LaPlace 100 years ahead of the formal proof.

“Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it—an intelligence sufficiently vast to submit these data to analysis—it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom...”

Boltzmann would become depressed sixty years after this because no one believed him when he spoke of atoms, yet LaPlace states it as an inevitability. That formula he’s speaking of is Bayes’ rule, which really should be called LaPlaces’ Rule because he was the one responsible for its formalization and how we use it. We have a predilection for rewarding the first to discover an idea, and not the first to make it usable. This in my view is a mistake.

“... for it, nothing would be uncertain and the future, as the past, would be present to its eyes.”

This is the bit that people quote as if LaPlace were an idiot. Holding it up as the sign that he thought the world was deterministic. He would go onto later say,

“The curve described by a simple molecule of air or vapor is regulated in a manner just as certain as the planetary orbits; the only difference between them is that which comes from our ignorance...Probability is relative in part to this ignorance, and in part to our knowledge.”

Aside from this thinking being sixty years ahead of its time, he’s stating in simple terms that probability encodes our uncertainty. If there was any doubt to what LaPlace thought he would go onto say:

“In this state of indecision it is impossible for us to announce their occurrence with certainty.”

If we do not have knowledge, we cannot predict with certainty.

Now that we’re here, how do we deal with this uncertainty? He would go onto propose the use of what is now known as latent variables, although he did not use this language. Noise as it is referred to today is what we observe in terms of our data about the world that departs from what we say in our model, which we deal with with a probability distribution. We don’t know how this will happen, and this is what LaPlace talks about in terms of our ignorance. It was this thinking that would lead him to be the first to formally prove the central limit theorem which was the justification for the Gaussian density.

The real point that LaPlace was trying to make is that, effectively aleatoric uncertainty is just epistematic uncertainty. That is to say, uncertainty from lack of knowledge is the same as uncertainty arising from something which can be modeled as a stochastic system. The real difference between the two is just the depth to which our knowledge is limited by the scope and size of the problem.

“I think if it were true that P=NP or if we had no limitations on memory and computation, AI would be a piece of cake. We could just brute-force any problem. We could go "full Bayesian" on everything (no need for learning anymore. Everything becomes Bayesian marginalization). But the world is what it is.” ~ Yann LeCun

[Dan · Rating 4 out of 5]

With the scientific and materialistic worldview in full swing, Laplace is explaining and promoting the new science of probability. It started with Pascal and Fermat - as a new mathematical theory for games of chance (i.e. gambling). With new developments - especially due to Bernoulli and Moivre, and with the help of the new science of calculus that allows summations of infinite sums - Laplace develops and advocates the use of probability in Physics, Astronomy, Actuary (survival/mortality tables and insurance), Clinical Medicine (effects of vaccination on survival), accuracy of observations, and so on. He goes as far as to suggest the use of probability in building social institutions, moral and ethical norms, the probity of witnesses, and to discredit miracles (at least Laplace is not assigning probabilities to the existence or non-existence of God - as some materialists and scientists are doing these days). For Laplace, probability is defined in terms of discrete distributions (i.e. counting distributions), as highly mathematical (i.e. in terms of the calculus needed for the summation of infinite and convergent sums), and as the projection of the entire real domain (i.e. everything that it is can and should be explained in terms of causes and with the help of probabilities or physical laws). Looking back at this book, one cannot help wondering how easy some things would have been for Laplace and for some of his convoluted proofs and arguments - if only he had been aware of the already existing, trivial, simple, but extremely powerful Bayes' Theorem. Laplace's dream is still with us today - as we are still hoping to explain everything with a causal, mathematical/statistical, and unified totality (i.e. a theory of everything, General Artificial Intelligence, a comprehensive technological society, and so on).

[William Bies · Rating 3 out of 5]

The field of probability owes its origins in the seventeenth century to games of chance, the kind of thing high-school boys like to debate and may even interest themselves enough in to apply themselves to and begin to work out a systematic theory of on a quantitative basis. By the turn of the nineteenth century, however, it had achieved a certain maturity, which is why Laplace’s Essai philosophique sur les probabilités of 1819 assumes its canonical status. Here, Laplace aims to outline in simple terms for the general reader the groundwork that underlies his more formal Théorie analytique des probabilités.

Right from the start, Laplace declares his allegiance to strict determinism in a passage which contains his celebrated all-knowing spirit who can predict everything in the future (no need to reproduce it here since everyone will have seen it already; in any case, we refer to Cassirer’s criticism of its significance in a work we have reviewed here). Where then does probability come in? Here is Laplace:

All events, even those which on account of their insignificance do not seem to follow the great laws of nature, are the result of it just as necessarily as the revolutions of the sun. In ignorance of the ties which unite such events to the entire system of the universe, they have been made to depend upon final causes or upon hazard, according as they occur and are repeated with regularity, or appear without regard to order; but these imaginary causes have gradually receded with the widening bounds of knowledge and disappear entirely before sound philosophy, which sees in them only the expression of our ignorance of the true causes. [p. 3]

Hence, his definition of probability:

The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. The ratio of this number to that of all the cases possible is the measure of its probability, which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible. [pp. 6-7]

From the terms of the definition Laplace seems to assume a finite space of elementary events? His second principle [p. 11] assumes a uniform measure on the space of elementary events – what, in an infinite space can be somewhat tricky to define and justify. Laplace’s canniness as a scientific thinker of rank emerges throughout the course of these early chapters, betraying itself here and there through a thoughtful aside. For he recognizes right away that independence of events can be a subtle issue [pp. 12-13], especially when one has to do with conditional probabilities [p. 15]. From the discussion it will be apparent that Laplace, like many an overworked doctoral candidate, views everything in the world through the lens of his specialty. For instance, with the observation that

This principle [viz., Bayes’ theorem] gives the reason why we attribute regular events to a particular cause. [p. 16]

Laplace correctly perceives that the notion of the extraordinary is always relative to a Bayesian prior [pp. 16-17] – what many a reckless atheist in our day is wont to overlook. It should hardly be surprising that discriminating judgment becomes possible only when one possesses relevant experience and can impose realistic priors instead of feigning ignorance.

A handful of further highlights from Part I. The law of large numbers [p. 60ff] is applied to births of boys and girls [p. 64ff], supported by extensive statistics. Chapter nine, describing the method of least squares then at the forefront of research, is notable for exhibiting how technical Laplace’s attitude towards his subject is. See his interesting way of quoting an error bar! Regarding an estimate of the mass of Saturn as 1/3512 of the mass of the sun, Laplace has this to say:

Applying to them my formulae of probability, I find that it is a bet of 11,000 against one that the error of this result is not 1/100 of its value, or that which amounts to almost the same – that after a century of new observations added to the preceding ones, and examined in the same manner, the new result will not differ by 1/100 from that of M. Bouvard. [p. 79]

After reviewing the general principles of the calculus of probabilities in Part I, in Part II Laplace goes on to consider its applications in natural philosophy and to human affairs. He is, of course, also author of a celebrated treatise on celestial mechanics, so what he has to say here carries intrinsic interest. In the application to lunar theory [p. 84], he distinguishes a secular term in the motion of the moon due to oblateness of the earth which can be teased out of a mass of observations by statistical methods (an amazing feat for the time). Other related topics include the irregularity in Jupiter and Saturn [p. 85ff], reflecting perturbations away from their Keplerian orbital elements, and an uncomfortably extended digression on theory of tides [pp. 89-97].

But for most readers it is the potential to have something relevant to say about human affairs that is most engaging. Among the subjects covered: an interesting analysis of how to account for a testimony of a witness suspected to be false, by probabilistic arguments [p. 109ff]; a ranked voting scheme long before Pearson/Wilcox [p. 127ff]; re. criminal judgments, argues for a supermajority among the jurors [p. 132f] – still a simple analysis based on straightforward application of the binomial distribution and neglecting psychological factors, cognitive biases, the possibility of factions etc.; Bernoulli’s analysis of how much smallpox vaccination would raise the mean lifetime [p. 145f] – for want of good data, he had to employ some questionable assumptions; annuities, interest rates, insurance [p. 149ff] and lastly perceptive comments concerning illusions in the estimation of probabilities: what we could call behavioral economics avant la lettre [p. 160f].

What lesson does Laplace wish to teach in all this material? One has to be careful. Sometimes the qualitative conclusion can be intuitive but calculus shows how to quantify it [p. 114], yet other times the correct answer can be counterintuitive, as he shows with probabilities that arise in urn drawing experiments [pp. 165-166]. For the rationalist and combative atheist, polemics against astrology, divination and superstition are only to be expected [pp. 165-173]. His concluding paragraph bears reproducing here, as it sharpens his view of how the objectivity of the calculus of probabilities helps against errors in reasoning due to cognitive bias:

It is seen in this essay that the theory of probabilities is at bottom only common sense reduced to calculus; it makes us appreciate with exactitude that which exact minds feel by a sort of instinct without being able ofttimes to give a reason for it. It leaves no arbitrariness in the choice of opinions and sides to be taken; and by its use can always be determined the most advantageous choice. Thereby it supplements most happily the ignorance and the weakness of the human mind. If we consider the analytical methods to which this theory has given birth; the truth of the principles which serve as a basis; the fine and delicate logic which their employment in the solution of problems requires; the establishments of public utility which rest upon it; the extension which it has received and which it can still receive by application to the most important questions of natural philosophy and the moral science; if we consider again that, even in the things which cannot be submitted to calculus, it gives the surest hints which can guide us in our judgments, and that it teaches us to avoid the illusions which ofttimes confuse us, then we shall see that there is no science more worthy of our meditations, and that no more useful one could be incorporated in the system of public instruction. [p. 196]

Rather too optimistic, indeed – everyone knows how easy it is to lie with statistics – but in a commendable spirit of enlightened objectivity. Three stars. Large tracts of the exposition, for instance in chapter five can be hard to follow without consulting the companion volume (Théorie analytique des probabilités) because complex formulae are expressed only in words (the Dover edition has no explanatory footnotes or endnotes). In the present work, Laplace shows himself manifestly to be more a technician than a philosopher: what changed over course of nineteenth century to create the conditions for the flourishing of probability theory properly so called in the twentieth century? Most probably this can be put down to the proliferation of real-world use cases.

[ernest (Ellen) · Rating 5 out of 5]

Great essay about examples in probability (Laplace’s 10 laws) and philosophy (where they fail to meet reality). I think everyone should read the section on moral hope, ie nonlinear expectation

[Lucca Horta · Rating 3 out of 5]

"É notável que uma ciência iniciada pela consideração dos jogos tenha se elevado aos mais importantes objetos dos conhecimentos humanos."

Seria necessária a opinião de um historiador da ciência, porém poucas vezes esse livro toca em temas que consideraríamos, hoje, filosóficos, com exceção da parte inicial do primeiro capítulo.

No restante, é um livro maçante sobre probabilidade (sem nenhuma notação matemática) e sobre aplicações da probabilidade às diversas áreas do conhecimento.

[Roberto Rigolin F Lopes · Rating 4 out of 5]

We are in 1812, Laplace is educating us on probability calculus. He warms up pointing out that probability is one of the most important human developments from the 17th century. Probability is defined simply as: "The ratio of the number of favorable cases to that of all possible cases". Then he goes keenly through applications such as: decisions in assemblies and judgments, mean duration of life, illusions estimating probabilities and so on. It "probably" bursted lots of superstitions within human endeavors.

[Bastian Greshake Tzovaras · Rating 3 out of 5]

Skip the introductory parts to probability if you've ever sat in a course on the topic. The later parts on applying probabilities to all sciences and everyday life are nicer.

[Lázaro · Rating 3 out of 5]

Basto ensayo sobre la ciencia de la probabilidad y la estadística. Esta publicación de Laplace contiene en sus páginas la síntesis de las aportaciones previas al área por otro autores (por lo menos 30 autores citados, la mayoría de francia en su misma época) y le agrega toda su impronta y desarrollos propios. Se merece una puntuación alta por el intelecto que desempeña, pero le bajo el nivel porque solamente utiliza las palabras para definir expresiones matemáticas, lo cual por novedoso que sea, es poco asimilable. Más allá de esto, es un ensayo filosófico, y lo manifiesta correctamente. Es interesante ver cómo los científicos de otros tiempos pasaban de un área a otra y lo hacían mediante cuestiones de filosofía o práctica.

[Nick Black]

GT Barnes & Noble, 2009-06-08, spontaneous purchase. I was looking at the Dover Math+Science section (two shelves now, and growing! w00t!), overwhelmed by all the mathematical goodness, and suffered some crosstalk of titles; for a second, I thought I'd spied "Recreations in Cohomology Operations and Homotopy Theory". Before realizing my error, I blinked, and said aloud, "I don't want to meet the motherfucker who's like, hey, instead of going to Vail this year I'd like to do a week of recreational homotopy theory. Maybe fight a few cops and see if I can catch some recreational gonorrhea, you know, piss some recreational kidney stones, steal some fat girl's diet pills so i can stay awake working on cohomology operation theory, maybe playfully throw myself and loved ones into a chemical fire and make a game of who can drink the most white phosphorus or hell just call it a Steenrod Square Party and spike drain-o directly into my pee-hole," but then I grokked they were two different books. I sheepishly grabbed Laplace, rendered tender, and got out. You expect to run a business this way? A pox, mssrs Barnes and Nobel, on both your houses.

[Tim Clouse · Rating 3 out of 5]

One of the fundamental texts on subjective probability and Bayes' Theorem. Contains Laplace's Rule of Succession, which has bedeviled thinkers in probability for the past 200 years. Appears to be a translation from 1901, but is understandable.

[Pamela · Rating 2 out of 5]

some interesting bits written in antiquated style.. helpful.