Before digging in more: what's with this analytic/synthetic distinction?

When I was reading Hintikka's "Kant on the Mathematical Method" in Posy's collection , I had a difficulty to structure a good distinction between analytic and synthetic method, where Hintikka mentions briefly in pp. 30-31. Accordingly analytic method goes back to Plato and gets stronger with Descartes where there are no constructions used and it proceeds with assuming the result and going backwards. Synthetic method tries to effect the desired result and it's based on the use of actual constructions. (cf. Posy, p. 31) As far as I understand, the latter became as a proof method in Euclidean/ synthetic geometry as well while the analytic method remained as an old tool after the development in analysis in 19th century (Here it says reductio ad absurdum is that kind of a method but I doubt it). Descartes used the analytic method in Discourse on the Method and in Meditations since he wanted to reach certain new truths. In analytic method the proposition that is used in the beginning is not like a theorem but more like an hypothesis. The premises are to be found during the process, they are not assumed in the beginning. Once premises, axioms and inference rules are found for the desired hypothesis then synthetic method can be used to prove that the general proposition hold. 

Then there is also the talk of analytic proof. Is this the same thing with the analytic method. NO. By definition, an "analytical proof is a proof of a theorem in analysis that only makes use of methods from analysis, and which does not make use of results from geometry". Synthetic proof is the same thing with the synthetic method.

Then what is analytic philosophy? Does this have anything to do with the methods or proofs used? Yes but it is not related to any of the above methods or proofs. Analytic philosophy was developed after formal logic got really strong and the clarity of arguments can be checked by formulating the methods and the arguments logically. 

Then what is analytic/ synthetic distinction for judgments and proposition.  This is whole another story. There are at least three main different definitions of analytic and synthetic belonging to Kant, Frege and Quine. 

For Kant in analytic judgments the concept of predicate is contained in the subject. These are affirmative ones. In synthetic judgments the predicate is completely different from what we think in the concept of the subject. (A7/B11) His examples are:

• "All bodies are extended"  (analytic)
• "All bodies are heavy" (synthetic)

In the first case all I need to do is to analyze the concept "body" to reach to the concept "extended". Hence the concept extended is contained in the concept body but this is not the case in the second example. These examples use mere syllogistic logic, no strict object and concept distinction, no quantifiers...

How does Frege define analytic./ synthetic distinction? According to him when a proof of a true proposition is carried on by purely logical means and when the premises and definitions of the terms can be given logically then this proposition is analytic. If one has to use in the argumentations of a true proposition, rules belonging to some special science apart from logic and/or terms that are not logically described then this proposition is synthetic.(Grundlagen, paragraph 3) Frege's definition uses two different notions to define the same concepts, a new kind of of logic: a very strong predicate logic, and propositions instead of judgments. While Frege is concerned with the justification of the propositions Kant is concerned with judging synthetically and with judgments. This last sentence will be of importance also when we discuss discrepancies in Hintikka's paper.  

And lastly the distinction of Quine, the contemporary consensus of the analytic/ synthetic: In "Two Dogmas of Empiricism" he defines analytic propositions as based on the meanings and independent from the facts. And the synthetic propositions as the propositions that are grounded on the facts. This paper of Quine is beautiful, the structure, the method, the feeling of it is really something. After arguing impressively whether there can be any distinction between analytic and synthetic propositions he concludes that " For all a priori reasonableness, a boundary between analytic and synthetic statements simply has not been drawn. That there is such a distinction to be drawn at all is an unempirical dogma of empiricists, a methapysical argument of faith" ("Two Dogmas", p. 37)

Any other uses of analytic and synthetic? Let me think a bit more...