Reuben Louis Goodstein (15 December 1912 – 8 March 1985) was an English mathematician with a strong interest in the

philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...

and teaching of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

Education

Goodstein was educated at St Paul's School in London. He received his Master's degree from

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. After this, he worked at the University of Reading but ultimately spent most of his academic career at the University of Leicester. He earned his

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from the University of London in 1946 while still working in Reading. Goodstein also studied under Ludwig Wittgenstein.

Research

He published many works on finitism and the reconstruction of analysis from a finitistic viewpoint, for example "Constructive Formalism. Essays on the foundations of mathematics." Goodstein's theorem was among the earliest examples of theorems found to be unprovable in

Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly u ...

but provable in stronger

logical systems Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science investigating h ...

(such as second-order arithmetic). He also introduced a variant of the

Ackermann function In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total ...

that is now known as the hyperoperation sequence, together with the naming convention now used for these operations ('' tetration'', '' pentation'', ''hexation'', etc.). Besides mathematical logic (in which he held the first professorial chair in the U.K.), mathematical analysis, and the philosophy of mathematics, Goodstein was keenly interested in the teaching of mathematics. From 1956 to 1962 he was editor of '' The Mathematical Gazette''. In 1962 he was an invited speaker at the International Congress of Mathematicians (with an address on ''A recursive lattice'') in

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. Among his doctoral students are Martin Löb and Alan Bundy.

Publications

* Fundamental concepts of mathematics, Pergamon Press, 1962, 2nd edn. 1979 * Essays in the philosophy of mathematics, Leicester University Press 1965 * Recursive Analysis, North Holland 1961, Dover 2010 * Mathematical Logic, Leicester University Press 1957 * Development of mathematical logic, London, Logos Press 1971 * Complex functions, McGraw Hill 1965 * Boolean Algebra, Pergamon Press 1963, Dover 2007 * Recursive number theory - a development of recursive arithmetic in a logic-free equation calculus, North Holland 1957 * Constructive formalism - essays on the foundations of mathematics, Leicester University College 1951 * with E. J. F. Primrose: Axiomatic projective geometry, Leicester University College 1953

References

{{DEFAULTSORT:Goodstein, Reuben Louis English mathematicians 1912 births 1985 deaths People educated at St Paul's School, London Alumni of the University of London Academics of the University of Reading Academics of the University of Leicester 20th-century British mathematicians Alumni of Magdalene College, Cambridge