Abstract
Abstract Certain mathematical objects bear the name “pathological” (or “paradoxical”). They either occur as unexpected and (temporarily) unwilling in mathematical research practice, or are constructed deliberately, for instance in order to delimit the scope of application of a theorem. I discuss examples of mathematical pathologies and the circumstances of their emergence. I focus my attention on the creative role of pathologies in the development of mathematics. Finally, I propose a few reflections concerning the degree of cognitive accessibility of mathematical objects. I believe that the problems discussed in the paper may attract the attention of philosophers interested in concept formation and the development of mathematical ideas.
Highlights
Mathematicians call an object “pathological” if its properties stay in sharp conflict with previously accepted intuitions based on the mathematical knowledge of a given epoch
In mathematics the role of pathologies is positive; they are creative in character, and they lead to new findings or even to the emergence of new mathematical domains
Philosophers are certainly familiar with the celebrated book Lakatos 1976 and classical works discussing the pitfalls of mathematical intuition, for instance Poincare 1905 and Hahn 1980, as well as the more recent work Feferman 2000, where the author discusses the role of arithmetization of analysis in the emergence of geometrical and topological pathologies and stresses the impact of selected proof methods on that process
Summary
Mathematicians call an object “pathological” (or “paradoxical”) if its properties stay in sharp conflict with previously accepted intuitions based on the mathematical knowledge of a given epoch. The necessary condition for being called “pathological” is, as already mentioned, a deep conflict with previously accepted intuitions. Certain pathological objects are mentioned so often in texts popularizing mathematics that they become known to the general public. Popular texts rarely explain in detail the circumstances of the emergence of the paradoxes in question. They recall paradoxes mainly for readers’ amusement, treating them as mathematical mysteries or curiosities. Philosophers are certainly familiar with the celebrated book Lakatos 1976 and classical works discussing the pitfalls of mathematical intuition, for instance Poincare 1905 and Hahn 1980, as well as the more recent work Feferman 2000, where the author discusses the role of arithmetization of analysis in the emergence of geometrical and topological pathologies and stresses the impact of selected proof methods on that process
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