Luitzen Egbertus Jan Brouwer Biography

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World of Mathematics on Luitzen Egbertus Jan Brouwer

The Dutch mathematician Luitzen Egbertus Jan Brouwer made contributions in the fields of topology and logic. He founded the school of thought known as intuitionism, which is based on the notion that the only dependable basis of mathematics consists of proofs that can actually be constructed in the real world. In addition, he developed a fixed-point theorem and demonstrated the connection between two previously distinct fields of topology--point-set topology and combinatorial topology.

Brouwer was born on February 27, 1881, in Overschie, the Netherlands. His parents were Egbert Brouwer and Henderika Poutsma. Brouwer completed high school in the town of Hoorn at the age of fourteen and attended the Haarlem Gymnasium where he satisfied the Greek and Latin requirements needed to enter a Dutch university in 1897.

At the University of Amsterdam Brouwer easily moved through the traditional mathematics curriculum and began some original studies on four-dimensional space that were published by the Royal Academy of Science in 1904. A year later he published his first book, Leven, Kunst, en Mystiek, a philosophical treatise in which he considers the role of humans in society. In 1907 Brouwer presented his doctoral thesis and was granted his doctor of science degree by the University of Amsterdam. He began teaching at Amsterdam in 1909 and spent his entire academic career at the university.

Develops the Fundamental Concepts of Intuitionism

His doctoral thesis, "On the Foundations of Mathematics," outlined a field of research that occupied Brouwer on and off for the rest of his life. In the early twentieth century the two primary schools of mathematics were logicism and formalism. Logicism is based on the premise that fundamental concepts in mathematics, such as lines and points, have an existence independent of the human mind. The job of mathematicians is to derive theorems from these concepts. Formalism is less concerned with the nature of fundamental concepts, but insists that those concepts be manipulated according to very strict rules.

Brouwer proposed a third concept of mathematics, later given the name intuitionism (also known as constructivism or finitism). The basic argument of intuitionism, according to Richard von Mises in World of Mathematics, is that "the simplest mathematical ideas are implied in the customary lines of thought of everyday life and all sciences make use of them; the mathematician is distinguished by the fact that he is conscious of these ideas, points them out clearly, and completes them. The only source of mathematical knowledge" in intuitionism, von Mises continues, is "the intuition that makes us recognize certain concepts and conclusions as absolutely evident, clear and indubitable."

Brouwer's intuitionist school was not particularly influential when it was first proposed in the 1910s. According to Victor M. Cassidy, in Thinkers of the Twentieth Century, "Brouwer made few converts during his lifetime, and Intuitionism has only a tiny number of adherents today."

Makes Contributions in the Field of Topology

The 1910s were a period of intense activity in the field of topology, the mathematical discipline concerned with geometric point sets. In 1912 Brouwer announced perhaps his most famous theorem, the fixed-point theorem, also known as Brouwer's theorem. This theorem stated that during any transformation of all points in a circle or on a sphere, at least one point must remain unchanged. Brouwer was later able to extend this theorem to figures of more than three dimensions.

Brouwer's first appointment at the University of Amsterdam in 1909 was as a tutor. Three years later, he was promoted to professor of mathematics, a position he held for thirty-nine years. In 1951 he retired and was given the title of Professor Emeritus. He died in Blaricum, the Netherlands, on December 2, 1966. Brouwer had been married in 1904 to Reinharda Bernadina Frederica Elisabeth de Holl. She predeceased Brouwer in 1959. The couple had no children.

Brouwer received honorary doctorates from the universities of Oslo (1929) and Cambridge (1955) and was awarded a knighthood in the Order of the Dutch Lion in 1932. He had also been elected to membership in the Royal Dutch Academy of Sciences (1912), the German Academy of Science (1919), the American Philosophical Society (1943), and the Royal Society of London (1948).