axiomatics
Andrew Winkler
the_andrew_winkler at yahoo.com
Sat Apr 25 09:52:37 EDT 2020
It strikes me that it might matter in thinking about the question, to take into consideration the essential unity of mathematics.
All of the usual structures of mathematical investigation can be viewed as facets of the structure theory of a partially defined binary algebra, with a single axiom of balance. Not only are quivers, categories, monoids, groups, semigroups, etc particular examples of balanced algebras, but each arises from requiring certain canonically defined ideals and subalgebras to be trivial, in perfect analogy to the way that fields arise from rings by forcing ideals to be trivial, and simple groups and Abelian groups are carved out of groups by forcing certain normal subgroups to be trivial. Rings and modules themselves arise from a suitable tweaking of the definition of category, while topology flows directly from functors taking values in the nonnegative real numbers. Once you have topology you also have sheaves of rings, and thence analysis, algebraic and differential geometry, etc. In a rather precise sense all of mathematics can be seen to be the structure theory of balanced algebras.
How foundations fit into the picture is of course unclear, perhaps in part because of the recursion. that lets you do set theory, and build topos theory on it, or do topos theory, and. then (more or less) build set theory on it.
Of course the most interesting mathematical topics tend to have multiple axiomatic descriptions, but to me it seems significant that there is a single unified viewpoint that in some sense embraces all of it.
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