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# Pierre-Simon Laplace

Pierre-Simon Laplace (March 23 1749March 5 1827) was a French mathematician and astronomer, the discoverer of the Laplace transform and Laplace's equation. He was a believer in causal determinism. The Laplacian differential operator, much relied-upon in applied mathematics, is named after him.

In his Essai philosophique sur les probabilités, Laplace set out a mathematical system of inductive reasoning based on probability, which we would today recognise as Bayesian. One well-known formula arising from his system is the rule of succession. Suppose that some trial has only two possible outcomes, labeled "success" and "failure". Under the assumption that little or nothing is known a priori about the relative plausibilities of the outcomes, Laplace derived a formula for the probability that the next trial will be a success.

$\Pr(\mbox{next outcome is success}) = \frac{s+1}{n+2}$,

where s is the number of previously observed successes and n is the total number of observed trials. It is still used as an estimator for the probability of an event if we know the event space, but only have a small number of samples.

The rule of succession has been subject to much criticism, partly due to the example which Laplace chose to illustrate it. He calculated that the probability that the sun will rise tomorrow, given that it has never failed to in the past, was

$\Pr(\mbox{sun will rise tomorrow}) = \frac{d+1}{d+2},$

where d is the number of times the sun has risen in the past. This result has been derided as absurd, and some authors have concluded that all applications of the Rule of Succession are absurd by extension. However, Laplace was fully aware of the absurdity of the result; immediately following the example, he wrote, "But this number [i.e., the probability that the sun will rise tomorrow] is far greater for him who, seeing in the totality of phenomena the principle regulating the days and seasons, realizes that nothing at the present moment can arrest the course of it."

Laplace strongly believed in causal determinism, which is expressed in the following quote from the introduction to the Essai:

"We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes."

This intellect is often referred to as Laplace's demon (in the same vein as Maxwell's demon). Note that this concept of identifying the hypothetical intellect described above by Laplace with whatever sort of demon does not come from Laplace, but from later theorists: Laplace rather saw himself as an atheist scientist that hoped that humanity would progress in a better scientific understanding of the world, which, if and when eventually completed, would still need a tremendous calculating power to compute it all in a single instant. While Laplace saw foremost practical problems for mankind to reach this ultimate stage of knowledge and computation, later interpretations of quantum mechanics, which were adopted by philosophers defending the existence of free will, also leave the theoretical possibility of such an "intellect" contested: for a further discussion of this issue, see also: determinism.

There has recently been proposed a limit on the computational power of the universe, ie the ability of Laplaces Demon to process an infinite amount of information. The limit is based on the maximum entropy of the universe, the speed of light, and the minimum amount of time taken to move information across the Planck length, and the figure turns out to be 2130 bits. Accordingly, anything that requires more than this amount of data cannot be computed in the amount of time that has lapsed so far in the universe. Predicting the rise of life in the universe, for example, requires vastly more data than this, and so, according to the theory, is computationally infeasable to predict.

## Quotes

• What we know is not much. What we do not know is immense.
• I have no need of that hypothesis. ("Je n'ai pas besoin de cette hypothèse", as a reply to Napoleon I, who had asked why he hadn't mentioned God in his book on astronomy)