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Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
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Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
Gödel on computability
"... Around 1950, both Gödel and Turing wrote papers for broader audiences. 1 Gödel drew in his 1951 dramatic philosophical conclusions from the general formulation of his second incompleteness theorem. These conclusions concerned the nature of mathematics and the human mind. The general formulation of t ..."
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Around 1950, both Gödel and Turing wrote papers for broader audiences. 1 Gödel drew in his 1951 dramatic philosophical conclusions from the general formulation of his second incompleteness theorem. These conclusions concerned the nature of mathematics and the human mind. The general formulation of the second theorem was explicitly based on Turing’s 1936 reduction of finite procedures to machine computations. Turing gave in his 1954 an understated analysis of finite procedures in terms of Post production systems. This analysis, prima facie quite different from that given in 1936, served as the basis for an exposition of various unsolvable problems. Turing had addressed issues of mentality and intelligence in contemporaneous essays, the best known of which is of course Computing machinery and intelligence. Gödel’s and Turing’s considerations from this period intersect through their attempt, on the one hand, to analyze finite, mechanical procedures and, on the other hand, to approach mental phenomena in a scientific way. Neuroscience or brain science was an important component of the latter for both: Gödel’s remarks in the Gibbs Lecture as well as in his later conversations with Wang and Turing’s Intelligent Machinery can serve as clear evidence for that. 2 Both men were convinced that some mental processes are not mechanical, in the sense that Turing machines cannot mimic them. For Gödel, such processes were to be found in mathematical experience and he was led to the conclusion that mind is separate from matter. Turing simply noted that for a machine or a brain it is not enough to be converted into a universal (Turing) machine in order to become intelligent: “discipline”, the characteristic
Gödel on Intuition and on Hilbert’s finitism
"... There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the con ..."
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There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, Gödel’s writings represent a smooth evolution, with just one rather small doublereversal, of his view of finitism. He used the term “finit ” (in German) or “finitary ” or “finitistic ” primarily to refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [Gödel, 1938a] and the lecture notes for a lecture at Yale [Gödel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of firstorder number theory, P A; but starting in the Dialectica paper
Tous droits réservésDogmas and the Changing Images of Foundations
"... Le contenu de ce site relève de la législation française sur la propriété intellectuelle et est la propriété exclusive de l'éditeur. Les œuvres figurant sur ce site peuvent être consultées et reproduites sur un support papier ou numérique sous réserve qu'elles soient strictement réservées ..."
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Le contenu de ce site relève de la législation française sur la propriété intellectuelle et est la propriété exclusive de l'éditeur. Les œuvres figurant sur ce site peuvent être consultées et reproduites sur un support papier ou numérique sous réserve qu'elles soient strictement réservées à un usage soit personnel, soit scientifique ou pédagogique excluant toute exploitation commerciale. La reproduction devra obligatoirement mentionner l'éditeur, le nom de la revue, l'auteur et la référence du document. Toute autre reproduction est interdite sauf accord préalable de l'éditeur, en dehors des cas prévus par la législation en vigueur en France. Revues.org est un portail de revues en sciences humaines et sociales développé par le Cléo, Centre pour l'édition
non–standard Analysis
"... soll heute die Rede sein von Logik und Mathematik, und von den Antworten, ..."
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soll heute die Rede sein von Logik und Mathematik, und von den Antworten,
BETWEEN THE FINITARY AND THE IDEAL
"... Within contemporary philosophy of mathematics there is a trend focussing on how mathematics is done and how it evolves, rather than how it should be done or how it should evolve. This fact is somewhat contrary to the philosophy of mathematics in the 20th century, which to a large extent was dominate ..."
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Within contemporary philosophy of mathematics there is a trend focussing on how mathematics is done and how it evolves, rather than how it should be done or how it should evolve. This fact is somewhat contrary to the philosophy of mathematics in the 20th century, which to a large extent was dominated by views developed during the socalled foundational crisis in the beginning of that very century. These views have primarily focused on questions pertaining to the logical structure of mathematics and questions regarding the justication and consistency of mathematics. Mathematicians and philosophers like Gottlob Frege (1848{1925), Bertrand Russell (1872{1970), David Hilbert (1862{1943), Kurt Godel (1906{1978) and others were very successful in their development of logic from around 1890 to, say, 1940 and they had a huge impact on the philosophy of mathematics of those days. Most probably it was Hilbert's program, rise and fall, which was the most single in
uential factor of the foundational studies until 1960's. Among the results was the widespread conception that the proper { if not the only { approach to philosophy of mathematics was through
Is a Chris tian Math e mat ics pos si ble?
"... Against the back ground of the ar ti cle deal ing with “Pre lim i nary questions on the way to a Chris tian Math e mat ics ” this ar ti cle sets out to inves ti gate the im pli ca tions en tailed in the givenness of di verse trends with in math e mat ics. In or der to ar gue about the pos si bil i ..."
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Against the back ground of the ar ti cle deal ing with “Pre lim i nary questions on the way to a Chris tian Math e mat ics ” this ar ti cle sets out to inves ti gate the im pli ca tions en tailed in the givenness of di verse trends with in math e mat ics. In or der to ar gue about the pos si bil ity of a Christian math e mat ics the na ture of the schol arly en ter prise is ar tic u lated on the ba sis of a nonreductionist on tol ogy and by ac knowl edg ing – as an al ter na tive to the onesid ed ness of arithmeticism, geometricism and logicism – an other sys tem atic op tion, one in which the unique ness and irreducibility of num ber and space (the in tu ition of dis crete ness and con ti nu ity as Bernays pre fers to des ig nate these ba sic re al i ties of mathe mat ics) are taken se ri ously, while at the same time it ven tures to account for the un break able in ter con nec ted ness (mu tual co her ence) prevail ing be tween the do mains of dis crete ness and of con ti nu ity. The idea of Chris tian schol ar ship pro vides a ba sis for con sid er ing the com plex ity of an anal y sis of the mean ing of num ber and sapce. It is briefly ex
Primitive Recursive Arithmetic and its Role in the Foundations of Arithmetic: Historical and Philosophical Reflections In Honor of Per MartinLöf on the Occasion of His Retirement
"... We discuss both the historical roots of Skolem’s primitive recursive arithmetic, its essential role in the foundations of arithmetic, its relation to the finitism of Hilbert and Bernays, and its relation to Kant’s philosophy of mathematics. 1. Skolem tells us in the Concluding Remark of his seminal ..."
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We discuss both the historical roots of Skolem’s primitive recursive arithmetic, its essential role in the foundations of arithmetic, its relation to the finitism of Hilbert and Bernays, and its relation to Kant’s philosophy of mathematics. 1. Skolem tells us in the Concluding Remark of his seminal paper on primitive recursive arithmetic (PRA), “The foundations of arithmetic established by means of the recursive mode of thought, without use of apparent variables ranging over infinite domains ” [1923], that the paper was written in 1919 after he had studied Whitehead and Russell’s Principia Mathematica and in reaction to that work. His specific complaint about the foundations of arithmetic (i.e. number theory) in that work was, as implied by his title, the essential role in it of logic and in particular quantification over infinite domains, even for the understanding of the most elementary propositions of arithmetic such as polynomial equations; and he set about to eliminate these infinitary quantifications by means of the “recursive mode of thought. ” On this ground, not only polynomial equations, but all primitive recursive formulas stand on their own feet without logical underpinning. 2. Skolem’s 1923 paper did not include a formal system of arithmetic, but as he noted in his 1946 address, “The development of recursive arithmetic” [1947], formalization of the methods used in that paper results in one of the many equivalent systems we refer to as PRA. Let me stop here and briefly describe one such system. ∗Is paper is loosely based on the Skolem Lecture that I gave at the University of Oslo in June, 2010. The present paper has profited, both with respect to what it now contains and with respect to what it no longer contains, from the discussion following that lecture. 1 We admit the following finitist types1 of objects: