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Reuben Goodstein - Wikipedia

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Reuben Goodstein
Born	(1912-12-15)15 December 1912 London, England
Died	8 March 1985(1985-03-08) (aged 72) Leicester, England
Alma mater	Magdalene College, Cambridge (MA) Birkbeck, University of London (PhD)
Known for	Goodstein's theorem Primitive recursive arithmetic
Scientific career
Institutions	University of Leicester University of Cambridge
Thesis	An axiom-free equation calculus (1946)
Academic advisors	Ludwig Wittgenstein^[1]^[2]
Doctoral students	Alan Bundy S. Barry Cooper Martin Löb

Reuben Louis Goodstein (15 December 1912 – 8 March 1985) was an English mathematician with an interest in the philosophy and teaching of mathematics.^[3]

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 but ultimately spent most of his academic career at the University of Leicester. He earned his PhD from the University of London in 1946^[4] while still working in Reading.

Goodstein also studied under Ludwig Wittgenstein.^[1]^[2]

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 but provable in stronger logical systems (such as second-order arithmetic). He also introduced a variant of the Ackermann function 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 Stockholm. Among his doctoral students are Martin Löb and Alan Bundy.^[5]

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^[6]
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

^ ^a ^b Nuno Venturinha, The Textual Genesis of Wittgenstein’s Philosophical Investigations, Routledge, 2013, p. 39.
^ ^a ^b In Goodstein, R. L. (1939). "Mathematical Systems". Mind. 48 (189): 58–73. doi:10.1093/mind/XLVIII.189.58., at p. 58, Goodstein refers to Wittgenstein as his former teacher.
^ O'Connor, John J.; Robertson, Edmund F., "Reuben Goodstein", MacTutor History of Mathematics Archive, University of St Andrews
^ Goodstein, R. L. (1945). "Function Theory in an Axiom-Free Equation Calculus". Proceedings of the London Mathematical Society. s2-48: 401–434. doi:10.1112/plms/s2-48.1.401.
^ Reuben Goodstein at the Mathematics Genealogy Project
^ Rogers, Hartley (1958). "Review: R. L. Goodstein, Mathematical logic". Bull. Amer. Math. Soc. 64 (1): 32–35. doi:10.1090/s0002-9904-1958-10141-x.

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