Reuben Louis Goodstein (15 December 1912 – 8 March 1985) was an English

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

with an interest in the

philosophy Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...

and

teaching Teaching is the practice implemented by a ''teacher'' aimed at transmitting skills (knowledge, know-how, and interpersonal skills) to a learner, a student, or any other audience in the of an educational institution. Teaching is closely related ...

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

Education

Goodstein was educated at St Paul's School in London. He received his Master's degree from Magdalene College, Cambridge. After this, he worked at the

University of Reading The University of Reading is a public research university in Reading, Berkshire, England. It was founded in 1892 as the University Extension College, Reading, an extension college of Christchurch College, Oxford, and became University College, ...

but ultimately spent most of his academic career at the

University of Leicester The University of Leicester ( ) is a public university, public research university based in Leicester, England. The main campus is south of the city centre, adjacent to Victoria Park, Leicester, Victoria Park. The university's predecessor, Univ ...

. He earned his

PhD A Doctor of Philosophy (PhD, DPhil; or ) is a terminal degree that usually denotes the highest level of academic achievement in a given discipline and is awarded following a course of graduate study and original research. The name of the deg ...

from the

University of London The University of London (UoL; abbreviated as Lond or more rarely Londin in Post-nominal letters, post-nominals) is a collegiate university, federal Public university, public research university located in London, England, United Kingdom. The ...

in 1946 while still working in Reading. Goodstein also studied under

Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. From 1929 to 1947, Witt ...

Research

He published many works on

finitism Finitism is a philosophy of mathematics that accepts the existence only of finite set, finite mathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite mathematical objects (e.g., infinite ...

and the reconstruction of

analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...

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 but provable in stronger logical systems (such as

second-order arithmetic In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation of mathematics, foundation for much, but not all, ...

). 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 function, total computable function that is not Primitive recursive function, primitive recursive. ...

that is now known as the hyperoperation sequence, together with the naming convention now used for these operations (''

tetration In mathematics, tetration (or hyper-4) is an operation (mathematics), operation based on iterated, or repeated, exponentiation. There is no standard mathematical notation, notation for tetration, though Knuth's up arrow notation \uparrow \upa ...

'', '' pentation'', ''hexation'', etc.). Besides

mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

(in which he held the first professorial chair in the U.K.),

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

, and the

philosophy of mathematics Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathem ...

, Goodstein was keenly interested in the teaching of mathematics. From 1956 to 1962 he was editor of ''

The Mathematical Gazette ''The Mathematical Gazette'' is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journ ...

''. In 1962 he was an

invited speaker at the International Congress of Mathematicians An invitation system is a method of encouraging people to join an organization, such as a Club (organization), club or a website. In regular society, it refers to any system whereby new members are chosen; they cannot simply apply. In relation to w ...

(with an address on ''A recursive lattice'') in

Stockholm Stockholm (; ) is the Capital city, capital and List of urban areas in Sweden by population, most populous city of Sweden, as well as the List of urban areas in the Nordic countries, largest urban area in the Nordic countries. Approximately ...

. 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 English mathematicians Alumni of Magdalene College, Cambridge