Dutch mathematician and logician, a developer of intuitionistic logic and algebra.
He was the eldest son of Johannes Heyting (21 October 1865, Angerlo – 5 June 1948, Amsterdam) and Clarissa Elisabeth Kok (24 December 1868, Winschoten – 14 January 1950, Amsterdam). Both parents were school teachers; Heyting’s father was also head of school. He had two younger brothers.
Heyting first intended to become an engineer, later decided to study Mathematics (1916). But to do finance this, Heyting and his father had to earn extra money by giving private lessons. In 1922 Heyting graduated with a degree of master’s standard and after this he became a mathematics teacher in Enschede. Here he worked on his dissertation Intuitionistische axiomatieks der projektieve meetkunde (Intuitionistic axiomatics of projective geometry, 1925), the first study of axiomatisation in constructive mathematics written with the help of L.E.J. Brouwer.
The intuitionistic Brouwer and the formalist Hilbert were involved in the Grundlagenstreit involving the tertium non datur principle, the Aristotelian logical principle that something must be either true or not (Law of excluded middle). In 1923 Brouwer had written On the significance of the principle of excluded middle in mathematics, especially in function theory and on November 17, 1930, Kurt Gödel would write his famous “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme ” (On Formally Undecidable Propositions of Principia Mathematica and Related Systems) proving Gödel’s incompleteness theorems.
As a son of two teachers, Heyting was an excellent educator, presenting complex matters in a logical way. He became Privatdozent at the University of Amsterdam (UvA) in 1936 and lecturer in 1937. He spent the rest of his career at the University of Amsterdam, being professor from 1948 until his retirement in 1968. He wrote papers in Dutch, English, German and French. His books on intuitionistic algebra in 1941 and intuitionistic Hilbert spaces in the 1950’s were ground-breaking.
His book Intuitionism: an Introduction (1956, second edition 1966) presented intuitionism to both mathematicians and logicians. Gilmore begins his excellent review of this book as follows: “This is an introduction to intuitionistic mathematics for mature mathematicians. The reader is taken rapidly to the heart of several different branches of intuitionistic mathematics. The speed of development is achieved by condensing the proofs and by presuming familiarity with the classical counterparts to the theories discussed. .. The book is written as a dialogue between Class (a classical mathematician), Form (a formalist), Int (an intuitionistic mathematician), Letter (a finitistic nominalist), Prag (a pragmatist), and Sign (a significist). In the first chapter Int defends intuitionistic mathematics against the criticism of the others, asking them finally to judge for themselves. In the remaining chapters Int presents mathematics for them to judge. In these chapters Class, except for Int, is the most loquacious; he frequently compares classical results with corresponding intuitionistic results and his questions lead Int to a more detailed discussion of some points. The device of dialogue allows abbreviation of statements without loss of clarity.”
His student Anne Troelstra wrote: Heyting was retiring and modest, lacking all ostentation. His interests were very wide-ranging and varied: music, literature, linguistics, philosophy, astronomy, and botany; he also was fond of walking. As a teacher and lecturer he impressed his students and his international audiences at congresses with his exceptionally clear presentations.
Heyting was married twice and had eleven children. He died 9 July 1980 in Lugano, Switzerland
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