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

English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ide ...

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, structure, space, models, and change. History On ...

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 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 context of an educational institution. Teaching is closely re ...

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

Magdalene College, Cambridge Magdalene College ( ) is a constituent college of the University of Cambridge. The college was founded in 1428 as a Benedictine hostel, in time coming to be known as Buckingham College, before being refounded in 1542 as the College of St Mary ...

. After this, he worked at the

University of Reading The University of Reading is a public university in Reading, Berkshire, England. It was founded in 1892 as University College, Reading, a University of Oxford extension college. The institution received the power to grant its own degrees in 192 ...

but ultimately spent most of his academic career at the

University of Leicester , mottoeng = So that they may have life , established = , type = public research university , endowment = £20.0 million , budget = £326 million , chancellor = David Willetts , vice_chancellor = Nishan Canagarajah , head_labe ...

. He earned his PhD from the

University of London The University of London (UoL; abbreviated as Lond or more rarely Londin in post-nominals) is a federal public research university located in London, England, United Kingdom. The university was established by royal charter in 1836 as a degree ...

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-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He is considere ...

Research

He published many works on

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

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 In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every ''Goodstein sequence'' eventually terminates at 0. Kirby and Paris showed that it is unprovable in Pea ...

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 (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 for much, but not all, of mathematics. A precurs ...

). 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 In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common. Under the definition as rep ...

'', ''

pentation In mathematics, pentation (or hyper-5) is the next hyperoperation after tetration and before hexation. It is defined as iterated (repeated) tetration, just as tetration is iterated exponentiation. It is a binary operation defined with two numb ...

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

mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...

(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 (m ...

, and the

philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's ...

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

The Mathematical Gazette ''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...

''. In 1962 he was an

invited speaker at the International Congress of Mathematicians This is a list of International Congresses of Mathematicians Plenary and Invited Speakers. Being invited to talk at an International Congress of Mathematicians has been called "the equivalent, in this community, of an induction to a hall of fame." ...

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

Stockholm Stockholm () is the Capital city, capital and List of urban areas in Sweden by population, largest city of Sweden as well as the List of urban areas in the Nordic countries, largest urban area in Scandinavia. Approximately 980,000 people liv ...

. Among his doctoral students are

Martin Löb Martin Hugo Löb (; 31 March 1921 – 21 August 2006) was a German mathematician. He settled in the United Kingdom after the Second World War and specialised in mathematical logic. He moved to the Netherlands in the 1970s, where he remained in r ...

and

Alan Bundy Alan Richard Bundy is a professor at the School of Informatics at the University of Edinburgh,http://homepages.inf.ed.ac.uk/bundy/ Professor Alan Bundy's website known for his contributions to automated reasoning, especially to proof planning ...

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