Displaying similar documents to “Mathematics and Philosophy - Panel Discussion”

What machines can and cannot do.

Luis M. Laita, Roanes-Lozano, Luis De Ledesma Otamendi (2007)

RACSAM

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In this paper, the questions of what machines cannot do and what they can do will be treated by examining the ideas and results of eminent mathematicians. Regarding the question of what machines cannot do, we will rely on the results obtained by the mathematicians Alan Turing and Kurt G¨odel. Turing machines, their purpose of defining an exact definition of computation and the relevance of Church-Turing thesis in the theory of computability will be treated in detail. The undecidability...

Leopold Kronecker’s conception of the foundations of mathematics

Jacqueline Boniface (2005)

Philosophia Scientiae

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Most of the time, Kronecker’s views on the foundations of mathematics are reduced to some scattered ideas. However, they constitute an original and coherent doctrine, justified by epistemological convictions. This doctrine appears in the article , published in the (1887) and, especially, in the last course taught by Kronecker, which took place in Berlin during the 1891 summer semester. This article would precise the principles and the insights of the Kroneckerian doctrine and then compare...

Positive Thinking. Conceptions of Negative Quantities in the Netherlands and the Reception of Lacroix’s Algebra Textbook

Danny J. Beckers (2000)

Revue d'histoire des mathématiques

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The beginning of the 19th century witnessed the emergence of several new approaches to negative numbers. New notions of rigour made the 18th century conceptions of negative quantities unacceptable. This paper discusses theories of negative numbers emerging in the Netherlands in the early 19th century. Dutch mathematicians then opted for a different approach than that of their contemporaries, in Germany or France. The Dutch translation (1821) of Lacroix’s illustrates the ‘Dutch’ notion...

The Impact of Modern Mathematics on Ancient Mathematics

Wilbur R. Knorr (2001)

Revue d'histoire des mathématiques

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In a hitherto unpublished lecture, delivered in Atlanta, 1975, W.R. Knorr reflects on historical method, its sensitivity to modern work, both in mathematics and in the philosophy of mathematics. Three examples taken from the work of Tannery, Hasse, Scholz and Becker and concerning the study of pre-euclidean geometry are discussed: the mis-described discovery of irrational ‘numbers’, the alleged foundations crisis in the 5th century B.C. and the problem of constructibility.