Arend Heyting

Born: 9 May 1898 in Amsterdam, Netherlands
Died: 9 July 1980 in Lugano, Switzerland

Arend Heyting's father was Johannes Heyting and his mother was Clarissa Kok. Both Arend's parents were school teachers and Johannes Heyting was particularly successful in his profession being appointed as head of a secondary school. Arend spent his school years with the intention that he would make a career in engineering. Only near the end of his schooling did his love and ability in mathematics mean that the course of his career changed and he went to university to study mathematics.

Although Heyting's father was a successful school teacher, the family were still in financial problems when Heyting began his studies in 1916 at the University of Amsterdam. Both Heyting and his father earned the extra money necessary to pay for his studies by taking on private tutoring work. At the University of Amsterdam Heyting was taught by Brouwer who had a large influence on his future work. In 1922 Heyting graduated with a degree of master's standard.

At this point in his career Heyting began to follow the same road as his parents by beginning a career as a secondary school teacher. He taught in two schools in the town of Enschede. In the Overijssel province in eastern Netherlands, standing on the Twente Canal near the German border, this industrial town with its cotton-textile industry was not an ideal place for an academic to be living. Heyting was not well placed to make contact with colleagues in the universities, yet he spent all his free time working on his research.

He received his doctorate in 1925 for a thesis written under Brouwer's supervision. His dissertation Intuitionistische axiomatieks der projektieve meetkunde was the first study of axiomatisation in constructive mathematics. When the Dutch Mathematical Association announced a prize question in 1927 they gave Heyting an ideal topic on which to compete. They asked for a formalisation of Brouwer's intuitionist theories and Heyting's outstanding essay was awarded the prize in 1928. This essay was then polished and expanded by Heyting and published in 1930. It made Heyting's name well known among those interested in the philosophy of mathematics.

This work had another beneficial effect as far as Heyting was concerned for it brought him to the attention of Heinrich Scholz who held the chair of mathematical logic in Münster. Scholz made his extensive library available to Heyting, fortunate since Münster was relatively close to Enschede, and a lifelong friendship arose between the two. Heyting's academic isolation in Enschede no longer seemed the problem that it might have been.

By this time Heyting had married Johanne Friederieke Nijenhuis. The couple were married in 1929 and had eleven children. After 31 years of marriage they divorced in 1960.

Heyting attended the Erkenntnis Symposium at Königsberg in September 1930. There he represented intuitionism while Carnap and von Neumann represented logicism and formalism respectively. Each argued their own case and against that of the other two. Although Heyting's version of intuitionist logic differed somewhat from that of Brouwer, it is clear that one of his main aims was to make Brouwer's ideas more accessible and better known. Brouwer had presented his ideas in a deliberately non-formal, and very personal, way.

There were others interested in intuitionist logic working on similar problems of formalisation at the same time as Heyting. One was Kolmogorov who corresponded with Heyting. The article [7] ([8] is the English translation) reproduces three letters which Kolmogorov sent to Heyting, the first in 1931 questions the distinction between a proposition P and the statement "P is provable".

In 1934 Heyting published Intuitionism and Proof Theory [1]:-

... a concise and well-written survey in which the viewpoints of intuitionism and formalism are clearly described and contrasted.
Heyting was appointed as a Privatdozent at the University of Amsterdam in 1936 and in the following year he was appointed as a lecturer. He spent the rest of his career at the University of Amsterdam, being promoted to professor in 1948. He held this position for twenty years until he retired in 1968.

Heyting published a paper on intuitionistic algebra in 1941 and intuitionistic Hilbert spaces in the 1950's. These were ground-breaking works. Another major treatise which has presented intuitionism to both mathematicians and logicians was Intuitionism: an Introduction (1956, second edition 1966). 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.

The article [4] shows the major influence that Heyting has had on the study of the foundations of mathematics and in so doing shows the importance of Heyting's contributions. Franchella argues that Heyting has been the cause of two major changes of direction. Firstly, at least partly because of him, the topic has moved away from trying to answer the big problems such as "what is mathematics?". Heyting moved away from these big problems, concentrating on trying to identify formal, intuitive, and logical concepts in the study of mathematics. The second change which Franchella argues that Heyting brought about was a realisation that there exist degrees of evidence in mathematics. This is a particularly important aspect of mathematics today when computer programs are being used to verify mathematical proofs:-
What was specific to intuitionism, however, was the thesis that mathematics is an activity, a process of becoming, the exhaustive description of which is impossible, just as it is impossible to define once and for all its elementary concepts.
We should end this biography by giving an indication of Heyting's personality. Troelstra writes in [1]:-
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.
In [2] he is described as follows:-
A handsome, dignified man, he is well remembered by scholars around the world for his quiet yet persistent advocacy of his philosophical ideals and for his unfailing politeness and kindness.

Article by: J J O'Connor and E F Robertson

October 2003
MacTutor History of Mathematics