What role do hypothetical situations play in philosophy? For example; most of us consider it to be a moral axiom that paedophilia is never morally justified. But we can think of a hypothetical situation, for example person X being forced to engage in acts of paedophilia by a demented individual who threatens to kill a child if person X does not engage in lascivious acts with the child. Now this hypothetical situation is wildly speculative and extremely unlikely to ever occur in the real world. So does it disprove the axiom that paedophilia is never morally justified or not?

I think the answer depends very much upon what one thinks one is doing philosophically. But the important point here is that moral claims in particular, and many of the philosophical claims that get evaluated using these invented examples, are meant to be more than just true as things actually are. So, for example, the claim that it is wrong to torture babies just for fun is meant to mean not just that all the actual baby torturing that is done just for fun is wrong, but that any baby torturing that is done just for fun would be wrong. Counterexamples to that claim, therefore, do not have to involve actual cases. As the philosopher Timothy Williamson has pointed out, moreover, many of the sorts of hypothetical counterexamples philosophers use either do have real-world instances or else such instances can easily be created. There are, for example, some very famous examples concerning knowledge known as "Gettier cases", and, at the start of one of his papers, Williamson cleverly sets up a situation ...

Last night my 4 year old son asked me, "where was I before I was born (or in your tummy?) was I alone?" What should I tell him?

Kids do ask some amazing questions. I am no expert on child psychology. I am just a philosopher who is also a parent. So please do not take what I will say the wrong way. I do not really mean to be giving parenting advice here. To some extent, what you should tell your son depends upon your religious beliefs. Some traditions would hold that your son was with God, waiting to be embodied. Some would hold that your son may have had a prior life, about which you would not know very much. But I am guessing that none of these traditions is yours, since otherwise the answer to the question would be clear enough. So I will answer assuming that you believe that, prior to your son's birth, he did not exist. (Note that many religious traditions would hold this view. So this is not a religious vs non-religious issue.) So, telling your son the truth would mean saying something like this: We can make cookies, but before we make the cookies, there aren't any cookies. There is flour and butter and...

Could it ever be rational to come to a belief on the basis of evidence which is only accessible to oneself? I have in mind, for instance, people who claim to have arrived at a belief in god by way of some critically personal spiritual experience.

I think the answer to this has to be "yes". Suppose I have a sharp pain in my foot. On the basis of that experience, I form the belief that I have stepped on a nail. This belief seems to be justified (and so rational), and it is partly based on evidence available only to me (the felt quality of the pain). The idea that religious belief might be justified on the basis of religious experience has a long history, and some---Swinburne, possibly, but I don't remember for sure---have explicitly drawn the analogy to the role experience plays in the formation of belief.

Is it possible that there exist types or methods of argument/reasoning that have not been discovered or employed before? (I do not mean specific arguments for specific problems, but Forms of arguments, so to speak.)

Sure, why not? Historically, there certainly have been. An example would be mathematical induction, which was known, in some form, to Euclid, but, so far as I know, is not present in earlier Greek mathematics. That seems to me to count if anything does. Now that was a long time ago, but why shouldn't there be such argument-forms that are unknown now even to us? Discovering one would be a very good thing, then!

How does Godel's incompleteness theorem affect the way that mathematicians understand and see mathematics as well as the world (if at all)? I'm not even close to a mathematician, but even a slight dose of the idea and theorem were enough to affect me so I suppose that I'm curious.

This depends in part upon what you mean by "mathematicians". Ordinary mathematicians, by which I mean mathematicians who aren't particularly or specially interested in logic, have generally, as a group, been utterly uninterested in Gödel's theorems. Reactions vary from case to case, and some are based on ignorance. But we do know, generally, that a huge proportion of "ordinary mathematics" can be done in what are, by the standards of set theorists, very weak theories. So the incompleteness of these theories tends not to be an issue. We've hardly exploited the strength they have. Another way to put this point is that, by and large, we know of very few "interesting" mathematical claims---claims that would be interesting to an "ordinary mathematician"--- that can be shown to be independent of these same, quite weak theories, let alone independent of Zermelo-Frankel set theory plus the axiom of choice, which is what most logicians would regard as sufficient to formalize the principles used in ordinary...

I want to start to study logic on my own to the level that I can understand a book like Enderton's and Michael Potter's Set Theory and Its Philosophy, also be able to study mathematical logic like recursive functions and model theory and Understand Logicist Reduction and why It failed because of Godel. Suppose that I've read nothing on logic and I'm not fair in math, but I want to be good at it. I'll be gleeful if you introduce me a series of books on a) Introductory books on logic followed by more professional ones, b) Mathematics of logic, c) books on Logicist Reduction and Godel or what is necessary to understand them. I have a plenty of time to study all these and will face all the troubles. Thank You Very Much Pouria From Turkish part of Iran (I think you guessed why I have to study on my own)

OK, so we're going to start with an introductory logic text. There are lots and lots of these, and people have very different views about what is best. But you could definitely do worse than to get a copy of our own Peter Smith's book . That has the advantage that you might then also read his book on intermediate logic , which covers Gödel's theorems, and flows naturally from the first. This book may be especially good for you, if, as I understood you to be saying, you aren't a math genius. While still remaining fairly rigorous, Peter keeps the technicalities to a minimum, so the book is accessible to more people than most such books. If you want something a little more technical, as well as wider ranging, then you might try Boolos, Burgess, and Jeffreys, Computability and Logic , which really gives a very solid introduction to basic recursion theory. Enderton's Introduction to Mathematical Logic covers the same sort of ground, but in a much more "mathematical" way. It is often used...

I am reading a book that explains Gödel's proofs of incompletenss, and I found something that disturbs me. There is a hidden premise that says something like "all statements of number theory can be expressed as Gödel numbers". How exactly do we know that? Can that be proved? The book did give few examples of translations of such kind (for example, how to turn statement "there is no largest prime number" into statement of formal system that resembles PM, and then how to turn that into Gödel number). So the question is: how do we know that every normal-language number theory statement has its equivalent in formal system such as PM? (it does seem intuitive, but what if there's a hole somewhere?)

Peter's explanation is as good as it could be, but let me elaborate on something. You will note that he keeps referring to the "standard proof" of the first incompleteness theorem. There are other proofs, and some of them do not rely upon a coding of this kind. Here's one nice way that stays very close to the "standard" proof. (There are also other, totally different sorts of proofs.) First, prove incompleteness for a simple theory of syntax, that is, a theory that is actually about symbols: sentences and the like. We can do this by looking at a theory that talks about its own syntax. (Quine sketches such a proof at the end of his book Mathematical Logic .) This part of the proof will look almost exactly like the standard proof, but minus the coding. The reason no coding is needed is that the theory really is one about symbols,and we can define things like sequences and take proofs to be sequencesof formulas, etc, etc. Second, establish the classical result that any theory that ...

We rarely, if ever, see headlines such as "A team oh philosophers in Berlin finally solves the Is-Ought Dilemma!". Of course, philosophy in general rarely makes headlines, but even within philosophy itself, it seems rare for philosophical ideas to be expounded or developed by *teams* of people, like scientists are doing more and more often. One would think that working in teams would increase the speed of an idea's development magnitudes, considering one would always have others off of whom to bounce ideas, and weaknesses could be worked over far more quickly, in live dialog rather than over months or years of exchanging arguments in academic publications, or books. Yet it seems that most philosophers choose to go it alone; what are the reasons for this?

I don't know what the reasons are, but I think co-authored papers and books are becoming more common, especially in the more technical parts of philosophy (language, epistemology, etc). I could be wrong about that, as I haven't done an extensive study, but that's my sense. Part of the reason may just have to do with technology. Working on some philosophical problem is a very ill-defined process much of the time. Just writing a paper can be a very long process. When things had to be snail-mailed back and forth, it was difficult to work with anyone not down the hall. Now, of course, collaboration is much easier, both in the developmental phases and in the writing phases, and so, as I said, we are seeing more of it. People can have a quick conversation at a conference, hit on an idea, and then develop it over email, the phone (which is much cheaper than it used to be), Skype, or whatever, and then write the paper without ever having to be physically in the same place again.

Has philosophy adequately dealt with the mind-body problem? I am looking for a serious answer from a person who is genuinely passionate about philosophy and not mere deferrals of the question through cliche stances so abundantly available amongst hobbyist-philosophers. Not to worry I am not out to justify some sort of theological stance, I am merely curious if professional philosophers are still concerned by this question or its derivatives. I would be very grateful for a response.

I'm not sure what's meant by "adequately dealt with", but if it means something like, "Come up with an answer that satisfies a fairly large group of people", then no, I don't think so. But to the other question, whether philosophers today still care about the mind-body problem, the answer is undoubtedly that they are. You might start here: http://plato.stanford.edu/entries/physicalism/ . The problem isn't that no-one has any good ideas what to say about mind and body, it's rather that too many people have too many good ideas, and the problem is fantastically hard. So hard that some philosophers, such as Colin McGinn, have argued that human beings are cognitively incapable of solving it (just as, say, dogs are cognitively incapable of even fairly basic mathematics). I don't say McGinn is right, just that one shouldn't assume the contrary.