Paul Isaac Bernays (17 October 1888 – 18 September 1977) was a

Swiss Swiss may refer to: * the adjectival form of Switzerland * Swiss people Places * Swiss, Missouri * Swiss, North Carolina *Swiss, West Virginia * Swiss, Wisconsin Other uses *Swiss-system tournament, in various games and sports *Swiss Internation ...

mathematician who made significant contributions to

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 ...

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

atic

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

, 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 ...

. He was an assistant and close collaborator of David Hilbert.

Biography

Bernays was born into a distinguished German-Jewish family of scholars and businessmen. His great-grandfather,

Isaac ben Jacob Bernays Isaac Bernays ( , , ; 29 September 1792 – 1 May 1849) was Chief Rabbi in Hamburg. Life Bernays was born in Weisenau (now part of Mainz). He was the son of Jacob Gera, a boarding house keeper at Mainz, and an elder brother of Adolphus Bernays ...

, served as chief rabbi of Hamburg from 1821 to 1849. Bernays spent his childhood in Berlin, and attended the Köllner Gymnasium, 1895–1907. At the

University of Berlin Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative o ...

, he studied mathematics under

Issai Schur Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the University of Berlin. He obtained his doctorate in 1901, became lecturer in 1903 and, after a stay at th ...

Edmund Landau Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Biography Edmund Landau was born to a Jewish family in Berlin. His father was Leopold ...

, Ferdinand Georg Frobenius, and

Friedrich Schottky Friedrich Hermann Schottky (24 July 1851 – 12 August 1935) was a German mathematician who worked on elliptic, abelian, and theta functions and introduced Schottky groups and Schottky's theorem. He was born in Breslau, Germany (now Wrocław, ...

; philosophy under Alois Riehl,

Carl Stumpf Carl Stumpf (; 21 April 1848 – 25 December 1936) was a German philosopher, psychologist and musicologist. He is noted for founding the Berlin School of Experimental Psychology. He studied with Franz Brentano at the University of Würzburg ...

and

Ernst Cassirer Ernst Alfred Cassirer ( , ; July 28, 1874 – April 13, 1945) was a German philosopher. Trained within the Neo-Kantian Marburg School, he initially followed his mentor Hermann Cohen in attempting to supply an idealistic philosophy of science. A ...

; and physics under

Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical p ...

. At the

University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...

, he studied mathematics under David Hilbert,

Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...

, and

Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...

; physics under Voigt and Max Born; and philosophy under

Leonard Nelson Leonard Nelson (; ; 11 July 1882 – 29 October 1927), sometimes spelt Leonhard, was a German mathematician, critical philosopher, and socialist. He was part of the neo-Friesian school (named after post-Kantian philosopher Jakob Friedrich Fr ...

. In 1912, the

awarded him a Ph.D. in mathematics for a thesis, supervised by Landau, on the analytic number theory of

binary quadratic form In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables : q(x,y)=ax^2+bxy+cy^2, \, where ''a'', ''b'', ''c'' are the coefficients. When the coefficients can be arbitrary complex numbers, most results are ...

s. That same year, the

University of Zurich The University of Zürich (UZH, german: Universität Zürich) is a public research university located in the city of Zürich, Switzerland. It is the largest university in Switzerland, with its 28,000 enrolled students. It was founded in 1833 f ...

awarded him

habilitation Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including a ...

for a thesis on

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

and

Picard's theorem In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard. The theorems Little Picard Theorem: If a function f: \mathbb \to\mathbb ...

. The examiner was

Ernst Zermelo Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic se ...

. Bernays was Privatdozent at the University of Zurich, 1912–17, where he came to know

George Pólya George Pólya (; hu, Pólya György, ; December 13, 1887 – September 7, 1985) was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamenta ...

. His collected communications with Kurt Gödel span many decades. Starting in 1917, David Hilbert employed Bernays to assist him with his investigations of the foundation of arithmetic. Bernays also lectured on other areas of mathematics at the University of Göttingen. In 1918, that university awarded him a second habilitation for a thesis on the axiomatics of the

propositional calculus Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...

of ''

Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...

''. In 1922, Göttingen appointed Bernays extraordinary professor without tenure. His most successful student there was

Gerhard Gentzen Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died ...

. After Nazi Germany enacted the Law for the Restoration of the Professional Civil Service in 1933, the university fired Bernays because of his Jewish ancestry. After working privately for Hilbert for six months, Bernays and his family moved to

Switzerland ). Swiss law does not designate a ''capital'' as such, but the federal parliament and government are installed in Bern, while other federal institutions, such as the federal courts, are in other cities (Bellinzona, Lausanne, Luzern, Neuchâtel ...

, whose nationality he had inherited from his father, and where the ETH Zurich employed him on occasion. He also visited the

University of Pennsylvania The University of Pennsylvania (also known as Penn or UPenn) is a private research university in Philadelphia. It is the fourth-oldest institution of higher education in the United States and is ranked among the highest-regarded universitie ...

and was a visiting scholar at the

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

in 1935–36 and again in 1959–60.

Mathematical work

Bernays's collaboration with Hilbert culminated in the two volume work, ''

Grundlagen der Mathematik ''Grundlagen der Mathematik'' (English: ''Foundations of Mathematics'') is a two-volume work by David Hilbert and Paul Bernays. Originally published in 1934 and 1939, it presents fundamental mathematical ideas and introduced second-order arithme ...

'' (English: ''Foundations of Mathematics'') published in 1934 and 1939, which is discussed in Sieg and Ravaglia (2005). A proof in this work that a sufficiently strong consistent theory cannot contain its own reference

functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

is known as the Hilbert–Bernays paradox. In seven papers, published between 1937 and 1954 in the ''

Journal of Symbolic Logic The '' Journal of Symbolic Logic'' is a peer-reviewed mathematics journal published quarterly by Association for Symbolic Logic. It was established in 1936 and covers mathematical logic. The journal is indexed by '' Mathematical Reviews'', Zentra ...

'' (republished in Müller 1976), Bernays set out an axiomatic set theory whose starting point was a related theory

John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

had set out in the 1920s. Von Neumann's theory took the notions of

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

and

argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...

as primitive. Bernays recast von Neumann's theory so that

class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differentl ...

es and sets were primitive. Bernays's theory, with modifications by Kurt Gödel, is known as

von Neumann–Bernays–Gödel set theory In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a colle ...

Publications

* * * * *

Notes

References

* . *Kneebone, Geoffrey, 1963. ''Mathematical Logic and the Foundation of Mathematics''. Van Nostrand. Dover reprint, 2001. A gentle introduction to some of the ideas in the ''Grundlagen der Mathematik''. * * *

External links

Hilbert Bernays Project
*
''Paul Bernays: A Short Biography'' (1976)
{{DEFAULTSORT:Bernays, Paul 1888 births 1977 deaths 20th-century Swiss philosophers Institute for Advanced Study visiting scholars Jewish philosophers Jewish scientists Mathematical logicians Philosophers of mathematics Set theorists Swiss Ashkenazi Jews Swiss mathematicians Swiss philosophers ETH_Zurich_faculty