Arend Heyting. He was a Dutch logician and mathematician, specialist in the fundamentals of mathematics . Founder of a special algebra that presents models of intuitionistic logic. He established a theory that rejects the axiomatic method and is oriented towards proofs of the intuitionist type.
Summary
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- 1 Biographical synthesis
- 1 Contributions to intuitionist logic
- 2 Algebra and arithmetic
- 2 Works
- 3 Sources
Biographical synthesis
He was born in Amsterdam on May 9 , 1898 . In his first years of study he always dreamed of a career in engineering , an idea that changed when he learned and mastered mathematics, so much so that he decided on it and in 1916 he began to study it at the University of Amsterdam . At the university he was a disciple and follower of Brouwer, which greatly influenced his future work. In 1922 he left the university with a master’s degree. Heyting starts out as a high school teacher, but all his free time was spent working on his research. In 1925received his doctorate with the thesis “Axiomatics of intuitionist projective geometry” written under the supervision of Brouwer. This was the first study of the axiomatization of constructive mathematics.
In 1936 he was appointed professor at the University of Amsterdam. He held this position for twenty years until his retirement in 1968 . He died in Switzerland on June 9 , 1980 at the age of 82. He was a worthy man, who is well remembered by scholars around the world for his persistent defense of his philosophical ideals and for his inexhaustible kindness and kindness.
Contributions to intuitionist logic
Heyting’s studies are of great importance in many ways, first of all he introduced the formalisms that were later widely disseminated, intuitionist calculations of sentences and predicates, intuitionist arithmetic , often called Heyting arithmetic . In addition, an astonishing detail was revealed: classical logical calculations can be obtained from intuitionist homonyms with the simple addition of the law of the excluded third. This circumstance was a substantial stimulus for numerous investigations on the equating of classical and intuitionist formalisms.
From the point of view of the finite foundation of traditional mathematics, the interpretations, constructed later, of many important classical formal theories in their intuitionist homonyms are of particular interest. Second, it showed the very possibility of formalizing intuitionist mathematics, which raised the doubts of many. Brouwer’s constructions were formalized, such as the theories of torrents, sets and freely formed sequences. For the first time he precisely formulated a version of the continuity principle, exceptionally important for intuitionist analysis.
The purely psychological effect was immense: non-intuitionist mathematicians understood that they would have to do with the intuitionist problem and that it was not reduced to only philosophical or critical aspects, but represented an autonomous logical and mathematical interest.
Heyting made a substantial and original contribution to intuitionism, his efforts protected intuitionism from oblivion and undermining, and if intuitionism is full of life today, this is largely due to him. It is easy to imagine that, without Heyting’s efforts, the intuitionist revolution would have been extinguished and Brouwer’s ideas would have been buried in the mausoleum of mathematical history .
Algebra and arithmetic
Arend Heyting was the creator of partially ordered sets, called “Heyting algebras”. These are presented as models of intuitionist logic, a logic in which the law of the excluded third party does not hold. Complete Heyting algebras are a central object of study in pointless topology. In any Heyting algebra, the smallest and largest elements 0 and 1 are regular. Furthermore, the regular elements of any Heyting algebra constitute a Boolean algebra .
Heyting was also the first to propose the axiomatization of arithmetic following the guidelines of the intuitionist school. This is called “Heyting arithmetic” in mathematical logic. Heyting’s arithmetic adopts Peano’s axioms, but uses the inference rules of intuitionistic logic. In particular, the principle of the excluded third is not generally accepted, despite the fact that this axiom can be used to demonstrate some specific cases.
Heyting arithmetic should not be confused with Heyting algebra, which is analogously the intuitionistic equivalent of Boolean algebra.
Plays
Throughout his professional career, Heyting wrote articles and books that disseminated and supported his intuitionist principles in different fields, some of his main works are:
- 1934 Intuitionism and proof theory.
- 1956 Intuitionism: An Introduction(First Edition).
- 1965 Introduction to intuitionism.
