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*To*: "FOM" <fom@math.psu.edu>*Subject*: FOM: Re: defining ``mathematics''*From*: "Karlis Podnieks" <podnieks@cclu.lv>*Date*: Wed, 22 Dec 1999 09:24:18 +0200*Sender*: owner-fom@math.psu.edu

-----Original Message----- From: Stephen G Simpson <simpson@math.psu.edu> Date: 1999. 22. dec. 2:26 Subject: FOM: defining ``mathematics'' ... The idea of identifying mathematics with rigor/objectivity per se, or the rigorous/objective part of our thinking, has a long pedigree going back to Descartes. Nevertheless, in my opinion, this idea is fundamentally flawed. ... It seems to me that the right way to distinguish the various sciences from each other is not in terms of methodological issues, but in terms of subject matter. Thus mathematics, like every other science, is to be defined as the study of a specific subject matter. To delimit that subject matter may be difficult, but as a first attempt let's call it ``quantity''. In other words, I am suggesting to define mathematics as the science of quantity. KP> For me as a former marxist, this should sound good. Marx and Engels defined mathematics as "the study of quantitative relations in the real world". In 1930s, Kolmogorov tried to justify this definition by extending the notion of quantity to cover graphs, groups and all the other mathematical structures. I could prove that this totally extended notion of quantity coincides with the notion of self-contained models (i.e. models that could be used by robots). In other words, I am suggesting to define mathematics as the science of self-contained models. Since self-contained models represent the rigorous part of our thinking, we arrive to the "wrong" (yet equivalent with the "right") definition of mathematics in terms of methodological issues. No problems to defend this position against arguments from biology or history of mathematics. Still, see some other arguments against it on the page "Digital mathematics and non-digital mathematics. Trying to understand anti-formalists" at http://www.ltn.lv/~podnieks/ . Karlis Podnieks University of Latvia Institute of Mathematics and Computer Science

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