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Laplace’s Demon

if there could be any mortal who could observe with his mind the interconnection of all causes, nothing indeed would escape him. For he who knows the causes of things that are to be necessarily knows all the things that are going to be. (…) For the things which are going to be do not come into existence suddenly, but the passage of time is like the unwinding of a rope, producing nothing new but unfolding what was there at first.
– Cicero, de Divinatione, 44 BC

Summary

A deterministic universe consists of causal chains that link past, present, and future in an unbreakable bond; the fact that you are reading these words right now is an inevitable consequence of domino-like cause-and-effect relationships that date back all the way to the Big Bang. If this is true, Laplace argued, then complete knowledge of the universe at any particular time allows one to perfectly predict the future and flawlessly retrace the past. The hypothetical intelligence that would possess such complete knowledge has become known as ‘Laplace’s demon’.

 


 

Pierre-Simon Laplace (1749–1827) is one of the most brilliant and influential scientists of all time. Known as ‘the French Newton’, Laplace contributed to astronomy, mathematics, and physics. His five-volume magnum opus Mécanique Céleste (Celestial Mechanics) (1799–1825) pioneered the application of calculus to the movement of the planets and the stars.

Laplace also independently discovered, applied, and promoted Bayes’ theorem, the mathematical formula that allows one to use the rules of probability theory to quantify the extent to which new data should update one’s knowledge.1 Laplace wrote two seminal works on probability theory: first, in 1812, the relatively technical Théorie analytique des probabilités, and then, in 1814, the more accessible Essai Philosophique sur les Probabilités. After discussing the foundation of probability theory and its application in fields such as physics, law, and insurance, Laplace concludes the Essai in style:

“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 exactitute 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 its 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 offtimes 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.” (Laplace 1902, p. 196)

Napoleon Bonaparte and the Celestial Mechanics

Napoleon Bonaparte and Laplace knew each other from the time that Napoleon attended the Ecole Militaire in Paris, when Laplace had been his examiner. Quickly after Napoleon came to power in 1799, he sought to increase the legitimacy of his government by appointing Laplace as Minister of the Interior. Laplace’s stint in politics lasted only six weeks, and in conversations with general Gourgaud, Napoleon described the administrative capacity of Laplace as “detestable” (Latimer 1904, p. 276).

Nevertheless, Napoleon and Laplace remained on relatively good terms, as is evident from the following fragment:

“Bonaparte recognized the splendor which the great intellect Laplace shed upon his administration. On October 19, 1801, received a volume of the Mécanique Céleste, he wrote to the author: ‘The first six months at my disposal will be employed on your beautiful work’ ” (Lovering 1889, p. 189)

Until recently, I always took Napoleon’s response for a joke, a thinly veiled way of saying that he would never have the opportunity even to browse the book – when in the business of conquering Europe, very little time would seem to remain at one’s disposal, let alone a full six months. Surely, it seemed to me, the ‘Nightmare of Europe’, the ‘Devil’s Favorite’, had matters to attend too –crushing the Prussians, obliterating the Russians, annihilating the Austrians– of a more earthly and pressing nature than those that are the topic of the Mécanique Céleste.

However, Napoleon appears to have been entirely serious:

“On November 26, 1802, after reading some chapters of a new volume dedicated to himself, he [Napoleon] refers to ‘the new occasion for regret’ that the force of circumstances had directed him to a career which led him away from that of science. At least, he added, ‘I desire ardently that future generations, reading the Mécanique Céleste, should not forget the esteem and friendship I have borne to the author.’ ” (Lovering 1889, p. 189, quotation marks added for clarity)

In Laplace’s philosophy, chance is ‘but an expression of man’s ignorance’ – a deterministic view that Laplace shared with almost all of the philosophers from antiquity2, as well as with many later scientists such as Jevons and Einstein. And if chance is but an expression of ignorance concerning the hidden causes that ultimately govern both the universe at large and our own behavior in particular, then free will is revealed to be only an ‘illusion of the mind’:

“Present events are connected with preceding ones by a tie based upon the evident principle that a thing cannot occur without a cause which produces it. This axiom, known by the name of the principle of sufficient reason, extends even to actions which are considered indifferent; the freest will is unable without a determinative motive to give them birth; if we assume two positions with exactly similar circumstances and find that the will is active in the one and inactive in the other, we say that its choice is an effect without a cause. It is then, says Leibnitz, the blind chance of the Epicureans. The contrary opinion is an illusion of the mind, which, losing sight of the evasive reasons of the choice of the will in indifferent things, believes that choice is determined of itself and without motives.” (Laplace 1902, pp. 3-4)

Laplace then continues and argues that for a hypothetical intelligence with perfect knowledge, chance ceases to exist – for that intelligence, “nothing would be uncertain and the future, as the past, would be present to its eyes.” (see Figure 6.1). The ‘intelligence’ that Laplace postulated became known as ‘Laplace’s demon’, and it is a powerful symbol of a deterministic world view.

Figure 1: Viktor Beekman’s rendition of Laplace’s demon: “We ought then to regard the present state of the universe as the effect of its anterior state and as the cause of the one which is to follow. 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; for it, nothing would be uncertain and the future, as the past, would be present to its eyes.” (Laplace 1902, p. 4)

The Greek Demon

It is sometimes believed that Laplace was the first to postulate the ‘demon’ – this is false. An alternative view is the demon was first introduced by several scientists that lived in the same era as Laplace – this is also false. As an example of the latter misconception, the 2018 Wikipedia entry on Laplace’s demon states:

“Apparently, Laplace was not the first to evoke one such demon and strikingly similar passages can be found decades before Laplace’s Essai philosophique in the work of scholars such as Nicolas de Condorcet and Baron D’Holbach (van Strien 2014). However, it seems that the first who offered the image of a super-powerful calculating intelligence was Roger Joseph Boscovich, whose formulation of the principle of determinism in his 1758 Theoria philosophiae naturalis turns out not only to be temporally prior to Laplace’s but also—being founded on fewer metaphysical principles and more rooted in and elaborated by physical assumptions—to be more precise, complete and comprehensive than Laplace’s somewhat parenthetical statement of the doctrine (Kožnjak 2015).” [references inserted for clarity]

In truth, the idea of Laplace’s demon was known already to the Greek philosophers from antiquity, who by and large believed in a deterministic cause-and-effect universe; as stated in the epigraph “the passage of time is like the unwinding of a rope, producing nothing new but unfolding what was there at first”.3 The point of departure is the principle of sufficient reason – ‘the evident principle that a thing cannot occur without a cause which produces it’. Once this principle is accepted, the demon practically suggests itself.

The principle of sufficient reason goes back at least to the world’s first atomist, Leucippus, who (if he existed!) predated the better- known atomist Democritus (c. 460 – c. 370 BC). The only surviving fragment from Leucippus is “οὐδὲν χρῆμα μάτηῳ γίνεται, ἀλλὰ πάντα ἐκ λόγου τε καὶ ὑπ’ ἀνάγκης” (“Nothing occurs at random, but everything for a reason and by necessity.”; Kirk and Raven 1957, p. 413; see also Landsman 2018).

When Laplace proposed his ‘demon’ he did not claim originality. In his Essai, Laplace cites Cicero at length, and also discusses one of Cicero’s anecdotes (without attribution). Laplace therefore appears to have been well aware of Greek philosophy and the principle of sufficient reason that pervades it. Nevertheless, Laplace’s demon does come with an unusual twist, namely the explicit statement that –based on perfect knowledge of the present– the demon can not only predict the future but can also unravel the past.4

Some philosophers (e.g., Earman 1986) believe it is important to make a distinction between ontological determinism (‘the universe is governed by domino-like cause-and-effect relationships’, that is, the principle of sufficient reason) and epistemic determinism (‘because the universe is deterministic, we can perfectly prefect the future and unravel the past, at least in principle’, that is, Laplace’s demon). One may perhaps accept the ontological version but reject the epistemic one.5 In one of the upcoming posts we will discuss the current status of Laplace’s demon; as we will see, depending on whom you ask, the demon is either healthy, struggling, comatose, or dead.

Belief in God

Laplace’s demon is omniscient and god-like: no mortal could ever hope to have the kind of perfect knowledge that Laplace alludes to. But the demon is entirely hypothetical, and my own reading leads me to believe that Laplace was an atheist. As proof I tender an anecdote, a testimony by Napoleon, and a fragment from Laplace’s own writing.

First, the anecdote concerns Laplace’s famous interaction with Napoleon:

“Someone had told Napoleon that the book [Mécanique Céleste] 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 for that hypothesis]; Napoleon, greatly amused, told this reply to Lagrange, who exclaimed, ‘Ah! c’est une belle hypothèse; ça explique beaucoup de choses.’ [Ah! It is a nice hypothesis; it explains many things] ” (Rouse Ball 1893, p. 423)

Figure 2: The Emperor Napoleon in His Study at the Tuileries. Painting by Jacques-Louis David.

There is some debate about the veracity of this anecdote, so let us proceed to the testimony of Napoleon. When exiled on Saint Helena, Napoleon’s conversations with General Gaspard Gourgaud repeatedly featured claims of Laplace’s atheism. For instance: “I often asked Laplace what he thought of God. He owned that he was an atheist.” (Latimer 1904, p. 276). Another example is the following:

“Gazing up at the starry heavens, Gourgaud says, ‘They make me feel I am so small, and God so great.’ Napoleon replies: ‘How comes it, then, that Laplace was an atheist? At the Institute neither he nor Monge, nor Berthollet, nor Lagrange believed in God. But they did not like to say so.’ ” (Latimer 1904, p. 274)

The third suggestion can be found in Laplace’s own writings. The Essai features a chapter titled “Illusions in the estimation of probabilities”; in it, Laplace described how Leibniz, a devout Christian and genius scientist, spectacularly overinterpreted a simple mathematical series:

“It was thus that Liebnitz [sic] believed he saw the image of creation in his binary arithmetic where he employed only the two characters, unity and zero. He imagined, since God can be represented by unity and nothing by zero, that the Supreme Being had drawn from nothing all beings, as unity with zero expresses all the numbers in this system of arithmetic. This idea was so pleasing to Liebnitz that he communicated it to the Jesuit Grimaldi, president of the tribunal of mathematics in China, in the hope that this emblem of creation would convert to Christianity the emperor there who particularly loved the sciences. I report this incident only to show to what extent the prejudices of infancy can mislead the greatest men. [emphasis mine]”

Final Thoughts

Prior to 1790, measurement in France was a mess. “It has been estimated that on the eve of the Revolution in 1789, the eight hundred, or so, units of measure in use in France had up to a quarter of a million different definitions because the quantity associated with each unit could differ from town to town, and even from trade to trade.” (Wikipedia; see also Alder 2007). To bring order to this measurement jungle, a panel of five French scientist (including Laplace) proposed a new system of weights and measures, which ultimately resulted in the metric system including the mètre and the gramme. Napoleon was not a fan of the metric system, and in 1812 he partially restored the old ‘system’. Only after Napoleon’s fall in 1837 was the metric system reinstated. Napoleon clarified his position in a conversation with General Gaspard Gourgaud:

“I never have approved the system of weights and measures adopted by the Directory, and invented by Laplace. It is all based on the mètre and conveys no ideas to my mind. I can understand the twelfth part of an inch, but not the thousandth part of a mètre [mil- limètre]. The system created much dissatisfaction with the Directory. Laplace himself assured me that if, before its adoption, all the objections I made to it had been pointed out to him, he would have recognized its defects and have given it up.” (Napoleon Bonaparte, as cited in Latimer 1904, 86–87)

This anecdote attests to Laplace’s infamous political flexibility, as his agreement with Napoleon is almost certainly insincere.


 

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Footnotes

1 Historically it would be defensible to speak of ‘Laplacian’ inference instead of ‘Bayesian’ inference. We need not feel sorry for Laplace, however, as enough of his contributions carry his name.

2 A notable exception is Epicurus, who sought to keep alive the concept of free will by assuming that the ‘atoms’ he hypothesized had occasional unpredictable ‘swerves’. It is remarkable how Epicurus’ speculation conceptually anticipates the Copenhagen interpreta- tion of quantum theory, which holds that chance is an inherent property of nature’s smallest elements.

3 The quotation is from Cicero, but this Roman orator expertly summarized the contributions of the different Greek philosophical schools. For more history on Laplace’s demon see the overview on http://www. informationphilosopher.com/freedom/ laplaces_demon.3.en.html.

4 See also Leibniz (2017, I, 36).

5 One may also accept the epistemic version where the demon can predict the future while rejecting the version where the demon can retrace the past.

Want to Know More?

Kožnjak, B. (2015). Who let the demon out? Laplace and Boscovich on
determinism. Studies in History and Philosophy of Science, 51, 42-52.

Laplace, P.–S. (1829/1902). A Philosophical Essay on Probabilities. Lon-
don: Chapman & Hall.

Latimer, E. W. (1904). Talks of Napoleon at St. Helena with General Baron
Gourgaud
(2nd ed.). Chicago: A. C. McClurg & Co.

van Strien, M. (2014). On the origins and foundations of Laplacian determinism. Studies in History and Philosophy of Science, 45, 24-31.

About The Author

Eric-Jan Wagenmakers

Eric-Jan (EJ) Wagenmakers is professor at the Psychological Methods Group at the University of Amsterdam.

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