foundations of mathematics, philosophy of mathematics, logic, mathematical, online, web, book, Internet, tutorial, textbook, foundations, mathematics, teaching, learning, study, mathematical logic, student, Podnieks, Karlis, philosophy, free, download
Limits of Modeling (February 5, 2010, in Latvian)
My talk The Nature of Mathematics in Serbian: Daniel A. Romano. Cta je to matematika i ko su ti matematicari? Matematicki kolokvijum, University of Banja Luka, XIV(2) (2008), 43-63.
K. Podnieks. Indispensability Argument and Set Theory. The Reasoner, Vol. 2, N 11, November 2008, pp. 8-9.
Gödel's Theorem in 15 Minutes (English, Latvian, Russian)
Introduction to
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What
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Quote of the DayNiels Bohr, before 1963. "It is wrong to think the task of physics is to find out how Nature is. Physics concerns what we can say about Nature." Quoted after: Nancy Cartwright, 1983, How the Laws of Physics Lie My basic view is that fundamental equations do not govern objects in reality; they only govern objects in models. See p. 129 of: |
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What is Mathematics?Four provably equivalent definitions of
mathematics: |
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What is Mathematics? (My Main Theses) I define mathematical theories as stable self-contained (autonomous?) systems of reasoning, and formal theories - as mathematical models of such systems. Working with stable self-contained models mathematicians have developed their ability to draw a maximum of conclusions from a minimum of premises. This is why mathematical modeling is so efficient (my solution to the problem of "The Incomprehensible Effectiveness of Mathematics in the Natural Sciences" (as put by Eugene Wigner). For me, Goedel's results are the crucial evidence that stable self-contained systems of reasoning cannot be perfect (just because they are stable and self-contained). Such systems are either non-universal (i.e. they cannot express the notion of natural numbers: 0, 1, 2, 3, 4, ...), or they are universal, yet then they run inevitably either into contradictions, or into unsolvable problems. For humans, Platonist thinking is the best way of working with stable self-contained systems. Thus, a correct philosophical position of a mathematician should be: a) Platonism - on working days - when I'm doing mathematics (otherwise, my "doing" will be inefficient), b) Advanced Formalism - on weekends - when I'm thinking "about" mathematics (otherwise, I will end up in mysticism). (The initial version of this aphorism is due to Reuben Hersh / picture). Next step The idea that stable self-contained system of basic principles is the distinctive feature of mathematical theories, can be regarded only as the first step in discovering the nature of mathematics. Without the next step, we would end up by representing mathematics as an unordered heap of mathematical theories! In fact, mathematics is a complicated system of interrelated theories each representing some significant mathematical structure (natural numbers, real numbers, sets, groups, fields, algebras, all kinds of spaces, graphs, categories, computability, all kinds of logic, etc.). Thus, we should think of mathematics as a "two-dimensional" activity. Sergei Yu. Maslov could have put it as follows: most of a mathematician's working time is spent along the first dimension (working in a fixed mathematical theory, on a fixed mathematical structure), but, sometimes, he/she needs also moving along the second dimension (changing his/her theories/structures or, inventing new ones). Do we need more than this, to understand the nature of mathematics? |
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In 1974, during a Soviet army training course, I discovered a simple (almost trivial) extension of Goedel's theorem - my double incompleteness theorem. Since that time, I'm an amateur philosopher of mathematics. My education up to Ph.D. in 1979 was purely mathematical, but I was elected Professor of Information Technologies (second class computer science). However, reading all the funny things about mathematics written even by the most prominent philosophers and mathematicians, I'm feeling at least as their kind of person. Karlis Podnieks, March 25, 2007 |
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