What Is the Philosophy of Mathematics?

A sophisticated field in philosophy that examines the relationship between math and reality, the philosophy of mathematics also looks at the underlying assumptions and implications of math. Sometimes referred to as mathematical philosophy, the term "philosophy of mathematics" is more precise, as the prior term has other meanings, such as the philosophy a particular mathematician takes in his calculations. This is not the same thing as examining the underlying philosophical foundations of math.

Philosophy of mathematics and related fields have been around for thousands of years, since Ancient Greek times at least. The followers of Pythagoras — Pythagoreans — thought deeply about mathematics and even formed a sort of cult around it. These ancient Greeks thought that math was a beautiful, self-consistent system of looking at the world, and practically magical in its predictive capacity. This view was slightly disturbed by the discovery of irrationality — that is, numerals that extend indefinitely without ever terminating, such as pi and the square root of two.

The Ancient Greeks had other peculiar qualities in their philosophy of mathematics. For instance, they doubted the existence of zero, asking, "How can nothing be something?" They even debated over the existence of 1, or whether it was a real number. It was not until the Hindu-Arabic numeral system that the modern zero was introduced, including its function as a placeholder at the end of a numeral. This was a step forward in philosophy of mathematics as well as its practical application.

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