# PHILOSOPHY OF MATHEMATICS AND LOGIC

## Why did philosophers become interested in mathematics, geometry, and logic, during the nineteenth century?

Philosophers have always been interested in these subjects, but in the nineteenth century there were even more innovations in science and technology than before. Changes in the world had an invigorating effect on higher learning, and philosophers took an interest in new research in the sciences and mathematics. Logic had been a philosophical subject since Aristotle, so new forms of logic were of interest to many philosophers who were not logicians.

## What advances were made during the nineteenth century concerning the philosophy of mathematics and logic?

During the nineteenth century, a logical theory of probability was propounded, non-Euclidian geometry was discovered, the objectivity and necessary truth of scientific first principles were questioned, a new system of logical notation was devised, and the possibility that mathematics could be reduced to logic was introduced.

## Who was Pierre-Simon Laplace?

Pierre-Simon Laplace (1749-1827) was a mathematician and astronomer who explicated what was to be the classic theory of probability. He taught in Paris at different schools, such as the Ă‰cole Militaire (military school).

## What is Pierre-Simon Laplace's theory of probability?

The fact that we do not know certain things gives rise to the idea of probabilities. Because we view the world as determined in assuming that every event has a cause, the probability of an event depends on a combination of what we do know and what we do not know. Laplace's theory of

**Mathematician and astronomer Pierre-Simon Laplace was famous for his theory of probability (Art Archive).**

**Who was Pierre-Simon Laplace's most famous student?**

*The man who would later become the most famous French dictator in history, Napoleon Bonaparte, was one of Laplace's students. Laplace's definitive* Analytic Theory of Probabilities *(1812) was, in fact, dedicated to Napoleon.*

probability was that if there is no reason to believe that one of a number of events, n, will occur, then the probability of each happening is 1/n. For example, the probability that any day of the week chosen at random will be either a Tuesday or a Thursday is 2/7.

## What is non-Euclidian geometry?

Euclidian geometry depends on a number of axioms, most important of which concerns the property of parallel lines. Non-Euclidian geometry changed Euclidian axioms. It was to have application in physics, particularly Albert Einstein's theory of relativity, when it enabled a concept of "the fourth dimension."

Carl Friedrich Gauss (1777-1855) was the first to figure out the principles of non-Euclidian geometry, although because he did not publish his ideas, the credit was given to Janos Bolyai (1802-1860) and Nikolai Lobachevsky (1792-1856), who were working independently. They rejected the Euclidian assumption that could not be proved in which only one line passes through a point in a plane that is parallel to a separate coplanar line. In their new system, a line can have more than one parallel and the sum of the angles of a triangle may be less than 180 degrees.

By the middle of the nineteenth century, Bernhard Riemann (1826-1866) developed a geometry in which straight lines always meet, thereby having no parallels, and in addition allowing for the sum of the angles of a triangle to be greater than 180 degrees. (In Euclidian geometry, parallel lines never meet and the sum of the angles of a triangle is always 180 degrees.) Reimann also went on to distinguish between the unboundedness of space as part of its extent, and the infinite measure over which distance could be taken that is related to the curvature of the same space. Riemann returned to Gauss' now-published work and explained the new ideas of distance first introduced by Loybachevski and Bolyai in terms of trigonometry. The bottom line was that "arc length" could be understood as the shortest distance between two points on a surface, without reference to the geometric properties or applicable geometry of that in which the surface itself was imbedded.

In 1868, Eugenio Beltrami (1835-1899) demonstrated a model of a Bolyaitype two-dimensional space, inside a planar circle. This proved that the consistency of non-Euclidian geometry depended on the consistency of Euclidian geometry, thus reassuring skeptics that non-Euclidian geometry was valid.

**What was unusual about Carl Friedrich Gauss' personality?**

*Gauss (1777-1855) was meticulous, conservative, and did not much enjoy teaching or other disruptions of his work. He did not collaborate or help younger mathematicians. Neither did he appreciate interruptions. It is said that he was once concentrating on a problem when told that his wife was dying. He responded, "Tell her to wait a moment till I'm done."*