The Analytic/Synthetic Distinction (Stanford Encyclopedia of Philosophy)
“Analytic” sentences, such as “Pediatricians are doctors,” have historically been characterized as ones that are true by virtue of the meanings of their words alone and/or can be known to be so solely by knowing those meanings. They are contrasted with more usual “synthetic” sentences, such as “Pediatricians are rich,” (knowledge of) whose truth depends also upon (knowledge of) the worldly fortunes of pediatricians. Beginning with Frege, many philosophers hoped to show that the truths of logic and mathematics and other apparently a priori domains, such as much of philosophy and the foundations of science, could be shown to be analytic by careful “conceptual analysis” of the meanings of crucial words. Analyses of philosophically important terms and concepts, such as “material object,” “cause,” “freedom,” or “knowledge” turned out, however, to be far more problematic than philosophers had anticipated, and some, particularly Quine and his followers, began to doubt the reality of the distinction. This in turn led him and others to doubt the factual determinacy of claims of meaning and translation in general, as well as, ultimately, the reality and determinacy of mental states.
Kant wants to say analytic sentences are "self-contained" - verifiable internally. This is a problem because definitions at least must come from outside.
Frege said sentences discriminate "senses" of a word - synonyms are such that one is not thinkable without the other. Analytic sentences are, in this incarnation, tautologies. But Frege's sentences are subject to infinite legal renamings, creating the problem of fecundity (which defies "thinkability" because halting).
Read Quine "Carnap and Logical Thinking" for best account of objections.