FOM: Re: defining ``mathematics''

Karlis Podnieks podnieks at
Fri Dec 24 02:50:50 EST 1999

-----Original Message-----
From: Kanovei <kanovei at>
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

"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

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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