FOM: "Mathematical" versus "Mathematics"

Karlis Podnieks podnieks at
Mon Jan 5 02:02:48 EST 1998

I propose to distinguish between the more general concept of the
"mathematical" ("mathematical method", or "mathematical model",
or "mathematical theory") and the more specific term
"mathematics". It seems to me, that the meaning of the
"mathematical" can be defined precisely (I quote from my home
page paper):

Fixed (i.e. self-contained) system of basic principles is the
distinguishing property of mathematical theories. A mathematical
model of some natural process or technical device is essentially
a fixed model which can be investigated independently of its
"original" (and, thus, the similarity of the model and the
"original" is only a limited one). Only such models can be
investigated by mathematicians. Any attempt to refine a model
(to change its definition in order to obtain more similarity
with the "original") leads to a new model, which must remain
fixed again, to enable a mathematical investigation of it.

Working with fixed models mathematicians have learned to draw
maximum of conclusions from a minimum of premises. This is why
mathematical modelling is so efficient.

After this, one can try to define the (somewhat social and/or
organizational) term "mathematics". This approach to the problem
was never widely accepted. Is the fixed (self-contained)
character of models really not the distinguishing property
between the "mathematical" and the "non-mathematical"?

>From the above "definition" of the "mathematical" I derive my
very enthusiastic attitude to the mathematical platonism: it
seems to me, that platonism is (for a human being) the only way
to work with fixed (self-contained) models effectively:

Platonist attitude to objects of investigation is inevitable for
a mathematician: during his everyday work he is used to treat
numbers, points, lines etc. as the "last reality", as a specific
"world". This sort of platonism is an essential aspect of the
mathematical method.

Best wishes,
podnieks at
University of Latvia
Institute of Mathematics and Computer Science
Rainis boulevard 29, Riga, LV-1459, Latvia

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