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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
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...
Construction and Schemata in Mathematics
"... scandal in philosophy is the problem of free will ” [17, p. 205]. I very much agree with Suppes that the problem of the free will is a major puzzle, which we should try to get a better understanding of by examining the deeper issues connected with the free will. This essay, however, does not treat t ..."
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scandal in philosophy is the problem of free will ” [17, p. 205]. I very much agree with Suppes that the problem of the free will is a major puzzle, which we should try to get a better understanding of by examining the deeper issues connected with the free will. This essay, however, does not treat the problem of the free will. It concerns the problems of the ontology and epistemology of mathematics. In genereal, the problems of the philosophy of mathematics are just as old and—if it makes sense to talk about solvability of such problems— perhaps just as unsolved as the problem of the free will. Mathematics is a very important ingredient of knowledge. In its most simple form mathematics plays a necessary role in our understanding of the surrounding world and is necessary for solving simple problems of ordinary life. At the other end of the simplicityscale we find the mathematics as used in science. Also here mathematics has a necessary role in our descriptions of nature and the way in which we are involved with it and each other. It truly amazes me that there seems to be very little consensus with respect to the ontology and epistemology