[FOM] mathematics as formal

Michael J Barany mjb245 at cornell.edu
Thu Mar 6 13:23:14 EST 2008


It's true that a lot of the work making "axiomatic system" or "structure" into
an abstract mathematical object came in the early to mid 20th century, and I'm
not so familiar with 20th century logic.

I was thinking in terms of 2 directions for "axiom systems" in the late 19th:
first, such developments as those by Lobachevsky and others of non-Euclidean
geometries, which arose out of attempts to drop Euclid's parallel postulate
and then took on a life of their own.  These can be thought of changing the
conceptual ground rules of the discipline, and expanding the range of subject
matter and mathematical objects they were willing/able to consider.  Second,
there were those in the Weierstrass and Hilbert molds who developed competing
systems of justification for the conceptual terrain of mathematics.  We can
think of these as changing the formal ground rules, and refining the range of
acceptable operations for creating new mathematical truths.

The big grand axiomatic theories-of-all-mathematics were not, I think (in
agreement with you), attempted until the very end of the 19th, and the
beginning of the 20th.

There's a related idea: that of mathematic*s*, plural.  There seem to have
been two big trends in the 20th century, paralleling those, it appears, of
physics: first, the attempt to develop mathematics out into many varied
specialities; and second, the attempt to show that it all boils down to a
single foundation, or a comprehensible system of foundations.  Before the 20th
century, it's not clear to me that people thought in terms of mathematics as a
whole, but seemed more focused on particular problems and topics, related to
each other by common methods.


> On 3/5/08 6:24 AM, "Michael J Barany" <mjb245 at cornell.edu> wrote:
>> The second half of the nineteenth century was dominated by attempts to try
>> out
>> different axiom systems and to experiment with different formal structures.
> Can you give examples? I am under the impression that the axiomatic
> approach to mathematical structures came later. For example, Dedekind
> studied algebraic extensions of Q and their rings of integers in the late
> 19th century, but his subject matter was very concrete. Later, in the
> 1920s or so, Noether developed an axiomatic theory of "Dedekind domains".
> There are other examples of concrete mathematical objects from the
> 19th century that were approached axiomatically in the early 20th century
> (culminating in the Bourbaki notion of structure in mid 20th century).
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