Analytic/synthetic distinction in mathematics besides geometry?

Question

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.)

Answer 1

The answer is here:

https://ncatlab.org/nlab/show/synthetic+mathematics

As you can see, there are several flavors available, such as synthetic topology, probability, domain theory, etc. This effort is by no mean new, but it is true that the categorical approach has once again emphasized the synthetic over the analytical

---

SOME THOUGHTS AND A BIT OF BACKGROUND

As usual your questions cut through to the very bone. I shall try to articulate an answer (or at least a sketch thereof), knowing all too well that this is impossible in full.

The two terms, Analysis and Synthesis, go back well before Kant, to the very beginning of western thought ( they somehow appeared in Aristoteles, for instance in his Analytica Priora, ie the first formalization of logic, but he probably incorporated previous knowledge from various sources).

It is worth visiting etimological dictionary:

Analysis, circa 1580s, "resolution of anything complex into simple elements" (opposite of synthesis), from Medieval Latin analysis (15c.), from Greek analysis "solution of a problem by analysis," literally "a breaking up, a loosening, releasing," noun of action from analyein "unloose, release, set free; to loose a ship from its moorings," in Aristotle, "to analyze," from ana "up, back, throughout" (see ana-) + lysis "a loosening," from lyein "to unfasten" (from PIE root *leu- "to loosen, divide, cut apart").

**So, Analysis basically means to solve some concept into its simpler
constituents, whereas Synthesis is the opposite direction.** 

In modern day we could say that math is like an acrobat balancing itself between two (yet entwined) sides, the analytic one, and the synthetic (in Lawvere's terminology the second is called CONCEPTUAL).

One would be tempted to replace them with constructive vs platonic, and perhaps one would not go too far off, but with many caveats.

The examples abound: for instance, one can define a smooth manifold intrinsically, or present it with explicit charts. A group is an instance of an abstract group, but Combinatorial Group Theory studies concrete groups generated by a set of elements and its relations. You can pick your field, and you almost invariably see that it straddles between an analytical approach and a synthetic one (though the mixture varies...)

Now, what I find extremely interesting is the interplay between these two sides: how to pass from the synthetic to the analytic viewpoint and conversely?

I believe the method is universal: suppose you have several analytic expressions and you surmise they are in fact the same, then IF you can prove explicitly that this is indeed the case, you form an equivalence class, and you say that these two versions are different instances of presentation of one "conceptual" object.

PS Descartes is the official father of analytic geometry, even though some Greek (forgot his name) anticipated him.

What is intriguing is that Descartes is in a way the father of both modern philosophy and mathematics. In his Discourse of the Method he elaborated his universal way of reasoning, which distinguished 4 phases, 2 of them being Analysis and Synthesis. It is not too much of a stretch to think that his analytic geometry is an application of his own philosophical method (of which he was very proud, and rightly so) . But, not knowing enough of this story, I could be wrong: it could be that the mathematician arrived before the philosopher. Either way, there is no doubt that this question carries a great relevance in both philosophy and mathematics.

PPS it is not by chance that Bill Lawvere has written (with Schanuel) a book called Conceptual Mathematics. Category Theory (including higher cats) is so far perhaps the best tool we have to treat Synthetic Math in full generality, though by no means the only one. That is both its greatness and its inherent limit: it captures the invariant side of math, but of course it hides the other one (think of groups for instance: to study the cat of Groups tells you all you need to know of general groups, but if I want to study say SU(5) I need to calculate, there is no way round it....

Answer 2

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.