I recently read Michael Friedman's Reconsidering Logical Positivism. As one might have guessed from the title of the blog, or one of my earlier posts, I think the renewed interest in positivism in recent years is a positive (sorry) development in philosophy. A short summary of Friedman's thesis in this book is that the positivists were much more Kantian than is generally recognized, and also more sophisticated than a lot of their critics give them credit for. He particularly examines the views of Carnap.
While Friedman defends Carnap from many of his critics, he apparently believes a roughly Quinean line of criticism can be successful. If he's right about that, then a lot of people need to take a close look at those arguments, as Kantian views have thrived since the collapse of positivism. If positivism was more Kantian than is generally believed, but the Kantian view doesn't work either, then that would present serious problems for many modern philosophers.
Myself, I am not convinced by Friedman's Quinean arguments. In essence, Quine complains that analyticity is impossible because analytic truths can't just be arbitrary and do the job that is required of them, but there is no way to make them non-arbitrary, except by connecting them to experience. At least, that's my brutally oversimplified version of Quine's argument.
One worry about relying on experience in any way to justify logic and mathematics is that surely we need logic and mathematics in order to make the kind of sophisticated determinations which would be necessary to figure out what our experience requires in terms of logic and mathematics. So an appeal to experience appears likely to be inevitably circular.
Certainly Quine's own theory is circular. In "Epistemology Naturalized", he suggests that we should trust our methods of inquiry to be truth-indicative, because earlier homonids who were bad at discovering truth would have been weeded out by natural selection. But the evolutionary theory Quine appeals to here was, of course, discovered via our current methods of inquiry. So our methods of inquiry tell us evolution happened, and evolutionary theory tells us our methods of inquiry can be trusted. It's all very Cartesian.
Of course, one of the reasons philosophers are still fascinated by Descartes is that the Cartesian circle is so hard to break. I do not believe Carnap thought he could break it. His analytic truths were not meant to be any kind of absolutely certain starting points (otherwise what was up with the principle of tolerance?) Rather, he thought it was useful to treat some statements as analytic in his sense, where of course determining what's useful does depend on our experience and there is a certain degree of arbitrariness and triviality in the whole procedure (his response to Quine in the Carnap volume of the Library of Living Philosophers is especially clear on this point). Certainly it is possible to draw a line between the analytic and the empirical, as Grice and Strawson emphasized in their response to Quine, and whether it is useful to do so is surely an empirical question, and one for which Carnap tries at various times to provide evidence, and against which Quine only raises a priori criticisms.