MoM Ep13: Pierre-Simon Laplace

This podcast episode of Men of Mathematics discusses the life and work of Pierre-Simon Laplace, a French mathematician and physicist who made significant contributions to celestial mechanics, probability theory, and mathematical physics.

Key Topics:

• Celestial Mechanics
• Probability Theory
• Laplace Transform
• Determinism
• Laplace's Equation

Summary:

Laplace, born in Normandy in 1749, quickly rose to prominence in the French scientific community. Patronage secured due to his mathematical abilities allowed him to move to Paris where he was soon presenting papers to the Academy of Sciences. D'Alembert, recognizing his genius, helped to launch his career. He skillfully navigated the tumultuous periods of the French Revolution, Napoleon's Empire, and the Bourbon Restoration, demonstrating political adaptability. Laplace briefly served as Minister of the Interior under Napoleon, who later quipped that he brought the spirit of infinitesimals into government, which was not intended as a compliment.

Laplace made groundbreaking contributions to celestial mechanics. Addressing a question left open by Newton, he proved the stability of the solar system, demonstrating that planetary irregularities are periodic and bounded. His five-volume "Mécanique Céleste," published between 1799 and 1825, systematized all known knowledge about planetary motion. He also proposed the nebular hypothesis, suggesting that the solar system formed from a rotating cloud of gas.

Laplace also founded modern probability theory. He derived a formula for estimating success probability based on prior trials, expressed as (s+1)/(n+2), where 's' is the number of successes and 'n' is the number of trials. This Bayesian approach provides smoothed estimates, accounting for uncertainty, especially with limited observations. A good example is the sunrise problem. Even after a million days of the sun rising, Laplace's formula doesn't give a 100% probability for the next sunrise, reflecting a residual degree of uncertainty.

In mathematical physics, Laplace's equation, which states that the sum of the second partial derivatives of a potential function equals zero, appears ubiquitously in fields such as electrostatics, fluid flow, and heat conduction. Laplace also developed the Laplace transform, a powerful technique that converts differential equations into algebraic equations, simplifying their solution and proving essential in engineering and physics.

Laplace is known for articulating a deterministic worldview. He proposed the existence of an intelligence (Laplace's demon) that, knowing the position and momentum of every particle in the universe at a given moment, could predict the entire future and reconstruct the entire past. This concept embodies classical determinism, where randomness is seen as merely a result of ignorance. Although 20th-century physics, with quantum mechanics, Heisenberg's uncertainty principle, and chaos theory, challenged this deterministic vision, understanding the limitations of determinism required centuries of scientific progress. Furthermore, Laplace anticipated the concept of black holes by calculating that a star with a diameter 250 times that of the sun but with the same density would trap its own light. Laplace also developed the mathematics of functions on spheres, which is crucial for geophysics, quantum mechanics, and modern computer graphics. Despite criticisms that he sometimes failed to credit the work of others, Laplace's achievements are undeniable. His final words, "What we know is not much. What we do not know is immense," encapsulates his perspective as a mathematical physicist.