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*To*: "FOM" <fom@math.psu.edu>*Subject*: Re: FOM: Re: defining ``mathematics''*From*: "Karlis Podnieks" <podnieks@cclu.lv>*Date*: Fri, 24 Dec 1999 09:50:50 +0200*Sender*: owner-fom@math.psu.edu

-----Original Message----- From: Kanovei <kanovei@wmwap1.math.uni-wuppertal.de> Date: 1999. 23. dec. 4:43 Subject: Re: FOM: Re: defining ``mathematics'' KP> About Hegel, quantity and mathematics. Many thanks to Prof.Kanovei for indicating my mistake. I was too fascinated by Prof.Simpson's returning to the seemingly "old-fashioned and outmoded" definition of mathematics as the science of quantity. The official marxist definition of mathematics was formulated in 1878 by Engels in his book "Herrn Eugen Duehrings Umwaelzung der Wissenschaft": "Die reine Mathematik hat zum Gegenstand die Raumformen und Quantitaetsverhaeltnisse der wirklichen Welt..." Thus, Engels says that mathematics is the science of space forms and quantitative relations in the real world. Of course, this sentence was just another form of the trivial "equation": mathematics = geometry + arithmetic + Calculus. Perhaps, Marx and Engels did not know about first ideas of modern algebra, set theory (invented by Cantor in 1873), non-Euclidian geometry etc. Still, for marxists of Stalin's era, the exact wording of the (for 1930s) extremely narrow Engels's definition was mandatory to survive. To save the wording, they were forced ... to modify (extend) the meaning the notion of quantity... Surprisingly, this "problem" was "solved" already in 1812 by Hegel in his book "Wissenschaft der Logik" (see the very beginning of Zweiter Abschnitt des ersten Buchs). Hegel defines quantity as the opposite of the notion of quality: "Die Qualitaet ist die erste, unmittelbare Bestimmtheit, die Quantitaet die Bestimmtheit die dem Sein gleichgueltig geworden ... ist." Thus, Hegel defines quality as "first, direct definiteness", and quantity - as "definiteness that has become indifferent to the Existing" (sorry, I'm not very strong neither in German, nor in English). For me, "definiteness that has become indifferent to the Existing" is just the notion of a model that has become distracted from its "original", and hence, can be investigated independently of this "original". In other words, this is the notion of self-contained models (models that can be used by robots). I am fascinated by this equivalence of quantity and self-contained models since the beginning of 1970s when I read Hegel for the first time. I'm glad to see outstanding people approaching this position from various angles... Could we agree on defining mathematics as the science of self-contained models? Let us call it Hegel's definition, not Podnieks's. Could this help? Merry Christmas, greetings from a former marxist, Karlis Podnieks University of Latvia Institute of Mathematics and Computer Science

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