Born in Overschie in the Netherlands, Brouwer took his first degree and doctorate at the University of Amsterdam, where he became successively Privatdozent and professor in the mathematics department. From 1903 to 1909 he did important work in topology, presenting several fundamental results, including the fixed-point theorem. This is the principle that, given a circle (or sphere) and the points inside it, then any transformation of all points to other points in the circle (or sphere) must leave at least one point unchanged. A physical example is stirring a cup of coffee – there will always be at least one particle of liquid that returns to its original position no matter how well the coffee is stirred.

Brouwer's best-known achievement was the creation of the philosophy of mathematics known as intuitionism. The central ideas of intuitionism are a rejection of the concept of the completed infinite (and hence of the transfinite set theory of Georg Cantor) and an insistence that acceptable mathematical proofs be constructive. That is, they must not merely show that a certain mathematical entity (e.g. a number or a function) exists, but must actually be able to construct it. This view leads to the rejection of large amounts of widely accepted classical mathematics and one of the three fundamental laws of logic, the law of excluded middle (either p or not-p; a proposition is either true or not true).

Brouwer was able to re-prove many classical results in an intuitionistically acceptable way, including his own fixed-point theorem.