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The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program
, 2001
"... . After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the ..."
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. After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for ever stronger and more comprehensive areas of mathematics and finitistic proofs of consistency of these systems. Early advances in these areas were made by Hilbert (and Bernays) in a series of lecture courses at the University of Gttingen between 1917 and 1923, and notably in Ackermann 's dissertation of 1924. The main innovation was the invention of the ecalculus, on which Hilbert's axiom systems were based, and the development of the esubstitution method as a basis for consistency proofs. The paper traces the development of the "simultaneous development of logic and mathematics" through the enotation and provides an analysis of Ackermann's consisten...
John von Neumann and Hilbert's School of Foundations of Mathematics ∗
"... The aim of the paper is to describe main achievements of John von Neumann in the foundations of mathematics and to indicate his connections with Hilbert's School. In particular we shall discuss von Neumann's contributions to the axiomatic set theory, his proof of the consistency of a fragm ..."
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The aim of the paper is to describe main achievements of John von Neumann in the foundations of mathematics and to indicate his connections with Hilbert's School. In particular we shall discuss von Neumann's contributions to the axiomatic set theory, his proof of the consistency of a fragment of the arithmetic of natural numbers and his discovery (independent of Gödel) of the second incompleteness theorem. 1
John Von Neumann 19031957
 FORTHCOMING IN EUROPEAN MATHEMATICAL SOCIETY NEWSLETTER
"... If influence of a scientist is interpreted broadly enough to include impact on fields beyond science proper then John von Neumann was probably the most influential mathematician ever lived: not only did he contribute to almost all branches of modern mathematics and created new fields but he also cha ..."
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If influence of a scientist is interpreted broadly enough to include impact on fields beyond science proper then John von Neumann was probably the most influential mathematician ever lived: not only did he contribute to almost all branches of modern mathematics and created new fields but he also changed history after the second World War by his work in computer design and by being a soughtafter technical advisor to the postwar militarypolitical establishment of the U.S.A. To celebrate John von Neumann’s 100th birthday, the international ‘Von Neumann Centennial Conference” took place in Budapest, Hungary between October 1520, 2003. Part of this event was the “Linear operators and foundations of quantum mechanics ” conference, where von Neumann’s legacy in operator theory was reviewed and discussed by leading experts in this field. During the conference the American Mathematical Society and the János Bolyai Mathematical Society unveiled a commemorative plaque on the house in Budapest where von Neumann was born and raised. To remember von Neumann the present note sketches von Neumann’s life and career and recalls briefly some of his views on the nature of mathematics.
∗For encouragement or helpful criticism in this project thanks are
, 2007
"... responsible for all errors that remain. Unless otherwise indicated in the text, translations from German are by the author. The original German passages will be confined to footnotes wherever possible. 1 1 ..."
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responsible for all errors that remain. Unless otherwise indicated in the text, translations from German are by the author. The original German passages will be confined to footnotes wherever possible. 1 1
Book Review: Logicomix by Apostolos Doxiadis,
"... The first two parts of the review provide background on Russell’s biography and on developments in the foundations of mathematics in the early part of the twentieth century that will help the reader set Logicomix in context. The third part contains critical remarks on 1) the relationship between Log ..."
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The first two parts of the review provide background on Russell’s biography and on developments in the foundations of mathematics in the early part of the twentieth century that will help the reader set Logicomix in context. The third part contains critical remarks on 1) the relationship between Logicomix and reality; 2) the issue of faithfulness of the graphic novel to the development of ideas in philosophy of logic and the foundations of mathematics; 3) logical inaccuracies; 4) the connection to madness and tragedy.