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Formalized mathematics
 TURKU CENTRE FOR COMPUTER SCIENCE
, 1996
"... It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In c ..."
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It is generally accepted that in principle it’s possible to formalize completely almost all of presentday mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.
Hilbert’s twentyfourth problem
 American Mathematical Monthly
, 2001
"... 1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Cong ..."
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1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of
The Ignorance of Bourbaki
, 1990
"... this article writes: "Which half of his brains did Bourbaki use ? My impression is, the left half. Perhaps I am projecting. The Bourbachistes were uncomfortable with the rightbrain mathematics of the Italian geometers, and for good reason: significant portions were suspect and might, if one takes ` ..."
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this article writes: "Which half of his brains did Bourbaki use ? My impression is, the left half. Perhaps I am projecting. The Bourbachistes were uncomfortable with the rightbrain mathematics of the Italian geometers, and for good reason: significant portions were suspect and might, if one takes `true' and `false' to be leftbrain notions and `right' and `wrong' to be rightbrain ones, be justifiably described as right, but false.
Debreu’s apologies for mathematical economics after 1983
"... Abstract: When reassessing the role of Debreu’s axiomatic method in economics, one has to explain both its success and unpopularity; one has to explain the “bright shadow ” Debreu cast on the discipline: sheltering, threatening, and difficult to pin down. Debreu himself did not expect to have such a ..."
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Abstract: When reassessing the role of Debreu’s axiomatic method in economics, one has to explain both its success and unpopularity; one has to explain the “bright shadow ” Debreu cast on the discipline: sheltering, threatening, and difficult to pin down. Debreu himself did not expect to have such an influence. Before he received the Bank of Sweden Prize in 1983 he had never openly engaged with the methodology or politics of mathematical economics. When in several speeches he later rigorously distinguished mathematical form from economic content and claimed this as the virtue of mathematical economics, he did both: he defended mathematical reasoning against the theoretical innovations since the 1970s and expressed remorse for having promised too much because it cannot support claims about economic content. The analysis of this twofold role of Debreu’s axiomatic method raises issues of the social and political responsibility of economists over and above standard epistemic issues.