Mike Shulman:

Mathematical theories can be classified as analytic or synthetic.

An #analytic theory is one that analyzes, or breaks down, its objects of study, revealing them as put together out of simpler things,
just as complex molecules are put together out of protons, neutrons, and electrons.

For example, analytic geometry analyzes the plane geometry of points, lines, etc. in terms of real numbers:
points are ordered pairs of real numbers, lines are sets of points, etc.

Mathematically, the basic objects of an analytic theory are defined in terms of those of some other theory.

By contrast, a #synthetic theory is one that synthesizes,
or puts together,
a conception of its basic objects based on their expected relationships and behavior.

For example, synthetic geometry is more like the geometry of Euclid:
points and lines are essentially undefined terms,
given meaning by the axioms that specify what we can do with them
(e.g. two points determine a unique line).

(Although Euclid himself attempted to define “point” and “line”,
modern mathematicians generally consider this a mistake,
and regard Euclid’s “definitions”
(like “a point is that which has no part”)
as fairly meaningless.)

Mathematically, a synthetic theory is a formal system governed by rules or axioms.

Synthetic mathematics can be regarded as analogous to foundational physics,
where a concept like the electromagnetic field is not “put together” out of anything simpler:
it just is, and behaves in a certain way.

The distinction between analytic and synthetic dates back at least to Hilbert,
who used the words “genetic” and “axiomatic” respectively.

At one level, we can say that modern mathematics is characterized by a rich interplay between analytic and synthetic
— although most mathematicians would speak instead of definitions and examples.

For instance, a modern geometer might define “a geometry” to satisfy Euclid’s axioms,
and then work synthetically with those axioms;
but she would also construct examples of such “geometries” analytically,
such as with ordered pairs of real numbers.

This approach was pioneered by Hilbert himself, who emphasized in particular that constructing an analytic example (or model) proves the consistency of the synthetic theory.

However, at a deeper level, almost all of modern mathematics is analytic, because it is all analyzed into set theory. Our modern geometer would not actually state her axioms the way that Euclid did; she would instead define a geometry to be a set
P of points together with a set
L of lines
and a subset of
P×L representing the “incidence” relation, etc.

From this perspective, the only truly undefined term in mathematics is “set”, and the only truly synthetic theory is Zermelo–Fraenkel set theory (ZFC).

This use of set theory as the common foundation for mathematics is, of course, of 20th century vintage,
and overall it has been a tremendous step forwards.

Practically, it provides a common language and a powerful basic toolset for all mathematicians.

Foundationally, it ensures that all of mathematics is consistent relative to set theory.

(Hilbert’s dream of an absolute consistency proof is generally considered to have been demolished by Gödel’s incompleteness theorem.)

And philosophically, it supplies a consistent ontology for mathematics, and a context in which to ask metamathematical questions.

However, ZFC is not the only theory that can be used in this way.
While not every synthetic theory is rich enough to allow all of mathematics to be encoded in it,
set theory is by no means unique in possessing such richness.

One possible variation is to use a different sort of set theory like ETCS,
in which the elements of a set are “featureless points” that are merely distinguished from each other,
rather than labeled individually by the elaborate hierarchical membership structures of ZFC.

Either sort of “set” suffices just as well for foundational purposes, and moreover each can be interpreted into the other.
https://golem.ph.utexas.edu/category/2015/02/introduction_to_synthetic_math.html