FOM: Does Mathematics Need New Axioms?

V. Sazonov V.Sazonov at
Mon May 22 15:41:58 EDT 2000

Of course mathematics needs new axioms such as
those of group theory, geometry, etc. But the question
is of course about the axioms of a different, more
foundational kind.

It seems to me that the real need in foundations and
therefore in corresponding axioms arises in mathematics
when mathematicians feel a lack of rigor, as it was in
the case of infinitesimals. New concepts, probably arising
in connection with applications, whose formalization is urgent,
but seems problematic, may be a new serious crisis in mathematics
are "necessary". (However, after the experience we already got
in foundations of mathematics the crisis, if any, may be probably
avoided easily.)

Thus, I would reformulate the question as follows:

Does mathematics need radically new concepts?
Or, should we confine mathematics to what can be done in
extensions of ZFC and look only for these extensions?

Vladimir Sazonov,
Manchester Metropolitan University,
Senior Researcher,
Scientific interests: Logic in Computer Science, Foundations

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