When young children perform long division or multiplication, are they constructing a proof?

What are numbers? Are they unquestionably EVERYTHING? Let's take 17 and 18 for example: Isn't there an infinite amount of numbers that exist between 17 and 18? There is no such thing as the smallest number, and there is no such thing as the largest number. WHY?!

We generally hold that a mathematical proposition such as "2 + 2 = 4" is necessarily true; it is difficult to imagine a possible world in which it is false. However, is it possible that "2 + 2 = 4" is not a statement that expresses a mathematical necessity (or an operation involving numeric values that must provide a certain result), but rather presents an inductive inference based on how people currently "define" the number "2", and the operator "+"? We could, for example, someday come to discover that "2" does not represent "2 things or ideas"; what we call 2 things may turn out to be 3 things, or 1 thing, etc. If this is possible then it would seem that "2 + 2 = 4" is an empirical, not a rational truth. Is this intelligible? I realize that this last statement, that we could discover 2 to refer to 3 things, etc., entails a theory of what a number is, i.e. a number "represents a quantity or amount of something". It seems, though, that in order to conclude that "2 + 2 = 4" is a necessary truth we must...

Are the infinitely small and the infinitely large the same thing?

Could questions in the philosophy of language in principle be answered in terms of the structures of the human brain? Might we imagine, for instance, pointing at a certain lobe and saying "Well, this shows that Russell was wrong about denotation"?

I am an American who has embraced the ideals of the Enlightenment, specifically the inherent value, perspective, and rights all humans on this planet. How do we reconcile these values with contemporary ideologies, specifically Zionism, that posits a racially, religiously unique group with "overriding" rights?

After discussing Socrates and his views on the state in Crito, a question came to mind. How would Socrates behave in 1930's Nazi Germany when the time came to join the military? Would his sense of right and wrong win out over his loyalty to the state? Or would he feel too great a responsibility to the state, as he clearly seems to in Crito, to put his personal choices and morality over it? Thanks, Jan-Erik

I have a question about Descartes' response in Med. VI to the dreaming argument. It seems to me that his knowledge that he is not dreaming any set of beliefs is based upon the knowledge that his current experiences are consistent with reality, which relies upon the knowledge that he is not dreaming his set of beliefs about reality. Would it be accurate to accuse the response of circularity?

I think that a lot of our common intuitions about punishment require that pure retribution be considered as one of the goals thereof. It is easy to conceive of scenarios in which punishment does not act (1) as a deterrent to crime or (2) to relieve the suffering of any injured parties. Often it seems that one of the main reasons we have to punish someone is simply our conviction that he somehow "deserves" to suffer. I'm sure that most people don't see this as problematic. Yet I wonder in particular how a utilitarian would address the question of retribution, since it is not obvious (at least to me) just what the utility of retribution is.

Having just read Dawkins's The God Delusion I was appalled to learn how reviled atheists are in America. In Europe a person's stance (including politician's) on religion is largely irrelevant unless they draw attention to it. What is going on in America? What should skeptics and atheist philosophers do there to point out that atheism is a reasoned and logical viewpoint that doesn't presuppose immorality, etc.? It beggars belief that all presidential aspirants have to (in some cases as Dawkins remarks) probably pretend to be Christians in order to have any chance of being elected. I know of the Atheist's Wager, acceptance of which seems braver to me than blindly accepting the religious promises of heaven as dictated by those who brought you up. And what place do 'faith-based initiatives' have in an ostensibly secular government where church and state are separate under the constitution?