FOM: what is the value of reverse mathematics?

Karlis Podnieks podnieks at
Fri Aug 6 03:31:55 EDT 1999

I'd like to keep my concepts as simple as possible, and my
postings - as short as possible. For me, over-simplification is
mathematical, but making as many distinctions as possible is

If the task of the "direct mathematics" is deriving of theorems
from a fixed set of postulates, then the task of the "reverse
mathematics" is determining of the minimum set(s) of postulates
necessary to prove a fixed set of theorems.

As a rule, reverse tasks are much more difficult than the
corresponding direct tasks (a well known example from the
Calculus: integration is much more difficult than

For me, reverse mathematics was not invented in 1990s. The
invention of groups, rings, fields, modules etc. also was a kind
of reverse mathematics - determining of the minimum sets of
"axioms" necessary to prove significant sets of properties of
different mathematical structures.

Hence, reverse mathematics seems to be a normal mathematical
approach to some problems. Are you afraid that reverse
mathematics is trying to replace the "direct mathematics"
causing mass unemployment among "direct mathematicians"? I'm not
(because I'm not a mathematician).

Karlis Podnieks
University of Latvia, Institute of Mathematics and Computer

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