Paul Bernays
Encyclopedia
Paul Isaac Bernays was a Swiss
mathematician, who made significant contributions to mathematical logic
, axiomatic set theory, and the philosophy of mathematics
. He was an assistant to, and close collaborator of, David Hilbert
.
, Edmund Landau
, Ferdinand Georg Frobenius
, and Friedrich Schottky
; philosophy under Alois Riehl
, Carl Stumpf
and Ernst Cassirer
; and physics under Max Planck
. At the University of Göttingen, he studied mathematics under David Hilbert
, Edmund Landau
, Hermann Weyl
, and Felix Klein
; physics under Voigt and Max Born
; and philosophy under Leonard Nelson
.
In 1912, the University of Berlin awarded him a Ph.D. in mathematics, for a thesis, supervised by Landau, on the analytic number theory of binary quadratic form
s. That same year, the University of Zurich
awarded him the Habilitation
for a thesis on function theory and Picard's theorem. The examiner was Ernst Zermelo
. Bernays was Privatdozent at the University of Zurich, 1912-17, where he came to know George Pólya
.
Starting in 1917, David Hilbert
employed Bernays to assist him with his investigations of the foundations of arithmetic. Bernays also lectured on other areas of mathematics at the University of Göttingen. In 1919, that university awarded him a second Habilitation, for a thesis on the axiomatics of the propositional calculus
of Principia Mathematica
.
In 1922, Göttingen appointed Bernays extraordinary professor without tenure. His most successful student there was Gerhard Gentzen
. In 1933, he was dismissed from this post because of his Jewish ancestry. After working privately for Hilbert for six months, Bernays and his family moved to Switzerland
, whose nationality he had inherited from his father, and where the ETH
employed him on occasion. He also visited the Institute for Advanced Study
in Princeton, USA, and the University of Pennsylvania
.
by , discussed in Sieg and Ravaglia (2005). In seven papers, published between 1937 and 1954 in the Journal of Symbolic Logic
, republished in , Bernays set out an axiomatic set theory whose starting point was a related theory John von Neumann
had set out in the 1920s. Von Neumann's theory took the notion of function as primitive; Bernays recast Von Neumann's theory so that sets and proper classes were primitive. Bernays's theory, with some modifications by Kurt Gödel
, is now known as the Von Neumann–Bernays–Gödel set theory
.
Switzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....
mathematician, who made significant contributions to mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, axiomatic set theory, and the philosophy of mathematics
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
. He was an assistant to, and close collaborator of, David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
.
Biography
Bernays spent his childhood in Berlin. Bernays attended the Köllner Gymnasium, 1895-1907. At the University of Berlin, he studied mathematics under Issai SchurIssai Schur
Issai Schur was a mathematician who worked in Germany for most of his life. He studied at Berlin...
, Edmund Landau
Edmund Landau
Edmund Georg Hermann Landau was a German Jewish mathematician who worked in the fields of number theory and complex analysis.-Biography:...
, Ferdinand Georg Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
, and Friedrich Schottky
Friedrich Schottky
Friedrich Hermann Schottky was a German mathematician who worked on elliptic, abelian, and theta functions and invented Schottky groups. He was born in Breslau, Germany and died in Berlin....
; philosophy under Alois Riehl
Alois Riehl
The philosopher Alois Adolf Riehl was born in Bozen in Austria .The brother of Josef Riehl, he was a Neo-Kantian and worked as a professor at Graz, then Freiburg and finally in Berlin, where he commissioned Mies van der Rohe to design his house in Neubabelsberg.For Riehl, philosophy was not the...
, Carl Stumpf
Carl Stumpf
Carl Stumpf was a German philosopher and psychologist.Born in Wiesentheid, he studied with Franz Brentano and Hermann Lotze...
and Ernst Cassirer
Ernst Cassirer
Ernst Cassirer was a German philosopher. He was one of the major figures in the development of philosophical idealism in the first half of the 20th century...
; and physics under Max Planck
Max Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...
. At the University of Göttingen, he studied mathematics under David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, Edmund Landau
Edmund Landau
Edmund Georg Hermann Landau was a German Jewish mathematician who worked in the fields of number theory and complex analysis.-Biography:...
, Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
, and Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
; physics under Voigt and Max Born
Max Born
Max Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...
; and philosophy under Leonard Nelson
Leonard Nelson
Leonard Nelson was a German mathematician and philosopher. He was part of the Neo-Friesian School and a friend of the mathematician David Hilbert, and devised the Grelling–Nelson paradox with Kurt Grelling...
.
In 1912, the University of Berlin awarded him a Ph.D. in mathematics, for a thesis, supervised by Landau, on the analytic number theory of binary quadratic form
Binary quadratic form
In mathematics, a binary quadratic form is a quadratic form in two variables. More concretely, it is a homogeneous polynomial of degree 2 in two variableswhere a, b, c are the coefficients...
s. That same year, the University of Zurich
University of Zurich
The University of Zurich , located in the city of Zurich, is the largest university in Switzerland, with over 25,000 students. It was founded in 1833 from the existing colleges of theology, law, medicine and a new faculty of philosophy....
awarded him the Habilitation
Habilitation
Habilitation is the highest academic qualification a scholar can achieve by his or her own pursuit in several European and Asian countries. Earned after obtaining a research doctorate, such as a PhD, habilitation requires the candidate to write a professorial thesis based on independent...
for a thesis on function theory and Picard's theorem. The examiner was Ernst Zermelo
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.-Life:He graduated...
. Bernays was Privatdozent at the University of Zurich, 1912-17, where he came to know George Pólya
George Pólya
George Pólya 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 fundamental contributions to combinatorics, number theory, numerical analysis and probability theory...
.
Starting in 1917, David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
employed Bernays to assist him with his investigations of the foundations of arithmetic. Bernays also lectured on other areas of mathematics at the University of Göttingen. In 1919, that university awarded him a second Habilitation, for a thesis on the axiomatics of the propositional calculus
Propositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
of Principia Mathematica
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by 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 Gentzen
Gerhard Karl Erich Gentzen was a German mathematician and logician. He had his major contributions in the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus...
. In 1933, he was dismissed from this post because of his Jewish ancestry. After working privately for Hilbert for six months, Bernays and his family moved to Switzerland
Switzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....
, whose nationality he had inherited from his father, and where the ETH
Eth
Eth is a letter used in Old English, Icelandic, Faroese , and Elfdalian. It was also used in Scandinavia during the Middle Ages, but was subsequently replaced with dh and later d. The capital eth resembles a D with a line through the vertical stroke...
employed him on occasion. He also visited the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
in Princeton, USA, and the University of Pennsylvania
University of Pennsylvania
The University of Pennsylvania is a private, Ivy League university located in Philadelphia, Pennsylvania, United States. Penn is the fourth-oldest institution of higher education in the United States,Penn is the fourth-oldest using the founding dates claimed by each institution...
.
Mathematical work
Bernays's collaboration with Hilbert culminated in the two volume work Grundlagen der MathematikGrundlagen der Mathematik
Grundlagen der Mathematik is a 2-volume work by David Hilbert and Paul Bernays describing Hilbert's approach to the foundations of mathematics...
by , discussed in Sieg and Ravaglia (2005). In seven papers, published between 1937 and 1954 in the Journal of Symbolic Logic
Journal of Symbolic Logic
The Journal of Symbolic Logic is a peer-reviewed mathematics journal published quarterly by Association for Symbolic Logic.Founded in 1936, the journal publishes articles on mathematical logic....
, republished in , Bernays set out an axiomatic set theory whose starting point was a related theory John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
had set out in the 1920s. Von Neumann's theory took the notion of function as primitive; Bernays recast Von Neumann's theory so that sets and proper classes were primitive. Bernays's theory, with some modifications by Kurt Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
, is now known as the Von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory is an axiomatic set theory that is a conservative extension of the canonical axiomatic set theory ZFC. A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes...
.