There are many philosophers who take a serious interest in neuroscience. Perhaps the best known are the two who were, I believe, responsible for the term "neurophilosophy", Paul and Patricia Churchland, both of whom are at UC-San Diego. But I'm no expert in that area and wouldn't know precisely where to send you for popularized treatments of that kind of work. There is plenty in the journals. You might care to look at Brain and Behavioral Sciences , which publishes a good deal of such work. Perhaps most intersting is the fact that they tend to publish papers with a lot of commentary from people in nearby disciplines.

I'm not sure what this "often said" remark is supposed to mean. Mathematics is not a language but a subject-matter. We use language, and different people use different languages at different times, to speak about mathematics. So mathematics isn't any different, in this respect, than anything else, so far as I can see. That said, perhaps what you have in mind might be that the way aliens conceptualized the world might be so different from the way we did that we could not understand their language. I have no idea if any such speculation might be true, but it presumably would be true that, at least initially, we could no more understand them than I can understand someone who was talking Chinese. And maybe their way of dealing with the world would be so different that we were unsure if they really were intelligent, and they had the same doubt about us. But then, the thought might be, I could take out a stick and start doing the following: Tap twice; rest; tap three times; rest; tap five times;...

As Alex said, one can presumably imagine there being lots of empirical evidence for God's existence. But if so, then I'm not sure why faith would be needed there any more than it is needed in ordinary scientific inquiry. But that doesn't mean faith wouldn't be required: The mere belief that there is a God hardly exhausts the content of faith. Liberal protestants and ultra-orthodox Jews agree that God exists, but there's not obviously a whole lot else about which they agree. Obviously, there are other religious doctrines, such as the status of Jesus, on which there is similar disagreement, but what is more interesting, to my mind, are the far deeper disagreements about what God wants for humanity and creation. Indeed, I'm aware I might be in the minority here, but to my mind, the question whether God exists is really a pretty boring one. This point is really just one made long ago by Hume. Even if you can prove that God exists, via any of the standard sorts of arguments philosophers consider, you'll...

I'm inclined to think, yes, that if a law is unjust it should lose its status as a law, but it doesn't immediately follow that one has no obligation to follow that law. In the case of profoundly unjust laws, such as those regarding Jews in Nazi Germany, presumably it would, at least, be permissible not to follow the law. But there are many other laws that might be regarded as unjust that are far less oppressive. One might suppose one had an obligation to follow such laws out of a kind of respect for the law. What one ought to do instead might be to attempt to get the law changed. Of course, in some cases, publicly and openly refusing to follow the law might be a way of drawing attention to its injustice: That's civil disobedience, in its simplest form.

Well, the first thing to say is that mathematics did develop for some time without the Cartesian co-ordinate system. And there are plenty of branches of mathematics where it isn't terribly important, for example, abstract algebra. It's also worth saying that there are lots of other co-ordinate systems, for example, polar co-ordinates. What is true is that Cartesian co-ordinates brought two powerful branches of mathematics, geometry and analysis, into a close relationship they had not previously enjoyed. I don't see any reason such a relationship would have had to be discovered at some point, but it is an extremely natural one.

I wonder whether the deeper question isn't one to which Nicholas alludes: Can I have lived a good life—not if I've never been loved but—if I've never loved?

And I'll add that, yes, my views on some fairly fundamental questions have changed a good deal. I used to think, for example, that language was essentially a social phenomenon. In my case, it wasn't reading any one particular paper that effected the change. It was, rather, a result of my thinking about the nature of communication, as I worked on an ostensibly quite distant set of problems concerning reference. Over time, what I found myself wanting to say about how we communicate using demonstratives ("that", "this", etc) and indexicals ("I", "here", "you", etc) veered unexpectedly toward a very different conception. It was only then that I found myself returning to material I'd read long before and explicitly re-thinking my previous views.

Are there any websites that accurately rank (as far as that ispossible) philosophy departments in English speaking countries withstrong philosophy programs? (Sorry, I couldn't resist.) I presume the question is motivated by a desire to get some sense of where one might get a good education, in a non-English-speaking country. If so, then please contact me directly, and let me know what countries you have in mind. I can (probably) put you in touch with people in the relevant area whose judgement I trust, and they will (hopefully) be able to give you some guidance.

If I can add a little, I guess I find myself puzzled about why anyone would think there was a conflict between belief in God and evolutionary theory. Some Christians, Jews, and Muslims (and, perhaps, adherents of other faiths about which I know less) do find there to be a conflict, but that is because they read their scriptures in a very literal way and take it as a matter of revealed truth that the universe and its inhabitants were created in a particular way. Then, indeed, there is a conflict, and the issue becomes how one should read scripture. This sort of very literal approach is, in Judaism and Christianity, actually a fairly recent phenomenon, and it does not fit at all with how the authors of the relevant texts understood their own writing. There are, for example, two creation stories at the beginning of Genesis, which have quite different histories, and they contradict one another at several points. That fact does not seem to have troubled the compliers of the book, and there is no reason...

The Axiom of Choice (usually denoted "AC") is a statement of set theory rather than of basic mathematical logic, so the theories of interest are versions of set theory that reject AC. As Dan said, any theory containing the Axiom of Determinacy will imply not-AC, but one can also simply look at what is possible without AC and, similarly, what cannot be proven without AC. There is a nice guide to such results: Thomas Jech's The Axiom of Choice . There are also weaker forms of AC, such as the Axiom of Countable Choice (every countable set of non-empty sets has a choice function) and the Axiom of Dependent Choice (more complicated). There are also forms that are stronger than what is usually assumed in set theory, in particular, what is sometimes called the Axiom of Global Choice.