The Analytic-Synthetic Distinction Is

Analytic truths are self-consistent without reliance on something besides the internal logic for truth. Synthetic propositions involve something outside the statement. They are somewhat more like algebra with variables that can make a statement true or false depending on what they are. So the logic of math may be implicitly true (if it isn’t false), yet mathematical statements may be analytically false. True math statements require verification, or something beyond themselves- for axioms or whatever, at some point need to be accepted as true for the rest of the math to follow. That premise is why Quine thought the synthetic-analytic distinction wasn’t valid, so far as I get from reading the article on the analytic-synthetic distinction. A synthetic proposition that everyone knows is true such as “the Earth has been around a long time” is like a math axiom in just being accepted as true with some verification from experience. The epistemological criterion of the analytic-synthetic distinction is unavoidably challenging like Descartes’ paradigm from the Meditations on a Method- verification of anything including experience originates inwardly and is an association of ideas and percepts with thought. Self-standing truth is God- everything else requires verification through association. 

https://plato.stanford.edu/entries/analytic-synthetic/