FOM: some terminological questions

[Neil Tennant]

The sense of `elementary' may be different for a working mathematician
and for a mathematical logician or model-theorist. I have always understood
the logicians' sense of `elementary' to be that the ideas at hand can be
expressed at first-order. But of course if one recasts everything in set-theoretical terms, then everything presumably becomes `elementary'.

One can give an `elementary' axiomatization of geometry by taking as primitive
notions those of Point, Line and Plane, and laying down axioms about incidence
and determination (e.g. for projective geometry: a line and a plane intersect at a point; two points determine a line...). Alternatively, one could take *only
points* as the individuals in the mathematical domain, and treat lines and planes as set of points, or properties of points.

Suppose, though, that on the obvious intertranslation, the two stocks of geometrical theorems coincide. Would each approach count as `elementary' on that score?
Or would the fact that the second approach invoked higher-order entities (sets or properties) make it non-elementary?

Different topic:

Is there a term accepted among mathematicians for the kind of mathematics of a
structure that is done `directly', using terms both primitive and native to the
domain in question, and not seeking to re-conceptualize everything in terms
of sets? I'm thinking here of geometry done in the way just described; or Peano
arithmetic; or the kind of theorizing about the reals that Tarski showed to be
complete and decidable at first-order.  I need a term for this kind of
mathematics, and have been using the term `synthetic'. Does anyone on the list
know of a better term, or of any established usage for `synthetic' that would
make it inappropriate as a term for what I have in mind?

Neil Tennant