TITLE: Analytic/synthetic distinction in mathematics besides geometry?
QUESTION [7 upvotes]: In a recent answer to an old MO question, I made a distinction between a "definition" of a mathematical object in the sense of axioms that characterize it, and a "definition" that explicitly constructs the object in question.  For a concrete example, consider the "definition" of the real numbers as an ordered field with the least upper bound property, and the "definition" of the real numbers via Dedekind cuts or Cauchy sequences.
In the comments, David Roberts suggested that the words synthetic and analytic be used to describe this distinction.  He also commented that the word "analytic" in this sense came from philosophy.  That is, it is not to be confused with the use of the term "analytic" to mean involving calculus or real/complex/functional analysis.
The distinction between "synthetic geometry" and "analytic geometry" is well known, and "synthetic geometry" certainly is analogous to the development of the theory of the real numbers from the axioms for an ordered field with the l.u.b. property.  But I was a little surprised at David Roberts's suggestion that the words are used more widely in mathematics.

Are the words synthetic X and analytic X commonly used to describe this distinction in mathematics for any value of X other than geometry?

I have some background in philosophy and I am pretty sure that the terminology does not "come from philosophy" in any obvious sense.  There is a famous analytic/synthetic distinction in philosophy, but Kant was the first to make a big deal about it, and the term "analytic geometry" dates back to Descartes at least.  Moreover, the analytic/synthetic distinction in the sense of Kant (or later philosophers) does not really line up with the distinction between analytic geometry and synthetic geometry; for example, Kant thought that all of mathematics was "synthetic a priori," and Quine famously questioned whether the distinction even made sense.
Just to be clear, I am not asking for additional mathematical examples of distinctions that are analogous to the distinction between the two different "definitions" of the reals; they are easy to come up with. I'm  just asking whether the terminology is already in use, and/or would be readily understood by people, and not confused with the other notion of "analytic".
(Come to think of it, I'm not even sure that "analytic geometry" is quite analogous to Dedekind cuts; I think of analytic geometry as referring to the study of Euclidean geometry using coordinates, rather than the explicit construction of a model of Hilbert's axioms.  But never mind this quibble for now.)

REPLY [3 votes]: Based on others' responses and some followup reading I have done, I would like to summarize the situation as I understand it.
One can infer from the nLab entry on synthetic mathematics that the use of synthetic/analytic in the complete ordered field/Dedekind cut sense is common among (at least some) category theorists.  It's not clear to me, however, that this terminology is familiar to other mathematicians.  For example, I suspect that most probabilists don't commonly talk about "synthetic probability" versus "analytic probability."
As for whether the terms come from philosophy—I had forgotten that the terms analytic/synthetic were used in a less precise sense prior to Kant.  So yes, the words do "come from philosophy" in the sense that, for example, Descartes regarded himself as taking an overall "analytic" approach to philosophy and mathematics, and  François Viète's In artem analyticem isagoge (a precursor to analytic geometry as we think of it) used the word in the title.  On the other hand, Descartes did not actually use the term "géométrie analytique" in La Géométrie, and his sense of the word "analytic" is not all that closely related to "analytic" in the Dedekind cut sense; Descartes just meant that his approach was to divide problems into their simplest parts, and then solve problems by proceeding from simple to complex.  This latter notion is a very broad idea; one could even imagine someone describing the axiomatic method as being "analytic" in this sense.