It's hard to imagine how I could be convinced to doubt that 2+2 = 4.Whatever argument you gave me, there would surely be some assumption orstep of reasoning in it that was less obvious to me than that 2+2 = 4.Faced with the decision of denying that 2+2 = 4 or of rejecting somestep in your argument, it seems it would always be more rational for meto do the latter. Do you now want to object: "I'll grant you itwould always be more rational for me to believe that 2+2 = 4 — butstill, couldn't it be false!?"

You would have thought that we would have worked out by now whethermathematics is a human invention or not. We haven't. There is still aheated debate between those who believe that mathematics describes arealm of entities (numbers, sets, functions, etc.) that exist quiteindependently of us and those who believe that the mathematical worldis in some sense constructed as a result of human activity. This is thecontrast between platonism and constructivism that has been touched on elsewhere here.

The kind of logic that most mathematicians assume in their work is known as classical logic . Classical logic accepts the Law of the Excluded Middle, which states that for every statement P, "P or not-P" is true. Some logicians and mathematicians (though not many) work within systems of reasoning that do not assert this Law. Most constructive logics fall into this category, in particular intuitionism does. For a little more on intuitionism see Question 139 . These days, there isn't much of a debate within the mathematical community about which is "the correct logic". There has been considerable debate about this within philosophy. Especially pertinent here are the writings of the Oxford philosopher Michael Dummett.

Yes, we use language to describe the world. Yes, we need language to describe the world. Let's even assume that language is a human construction — say there was a committee a long time ago that got together and created human languages. And finally, let's accept that the theories we elaborate within those languages are also constructions of human beings. I don't understand why you think that it follows that the views we've arrived at are just products of our fantasy that bear no connection to reality. We don't just make up stories about the world, do we? We test our stories against the evidence, we drop the stories that don't help us organize and understand the evidence, and we accept and elaborate those stories that do. Yes, scientific theories are "constructions of the human mind", but not just any construction will do!

I thank the poster for this challenging question and Jyl especiallyfor her stimulating response. Her idea is a nice one: that we deemsomething to be supernatural just in case we take it to be foreverbeyond the limits of scientific understanding. I'm not sure though. Ifsomeone were like Jyl and didn't believe that there are anysupernatural objects, then it would follow that she didn't believe thatanything was in principle beyond the limits of scientificunderstanding. Now, many people do believe that it's likely that there aresome truths about the universe that are in principle beyond humanunderstanding (for instance, see here ).Does this belief commit them to the existence of the supernatural? Thatdoesn't seem right. If you want to make room for someone who believesthat, say because our minds have the particular structure they do,there are things about the world that we cannot in principle understandand yet who also holds that there are no supernatural phenomena, thenwe'll have to understand the...

Philosophy is one of the few disciplines (the only discipline?) thestudy of whose own nature is still part of that very discipline. Youcan investigate the nature of chemistry, but that's no longer part ofchemistry. You can reflect on the nature of religious belief, butthat's no longer to be entertaining religious beliefs. But to thinkabout the nature of philosophy is very much a part of philosophy.(That's why one of the Categories of this site is "Philosophy" itself.)So it shouldn't be a surprise to you that philosophers have said a lotabout what you ask and offered different views. Some believe thatphilosophical reflection does have as its end the articulation ofclaims, philosophical truths. Others have held that there are noproperly philosophical assertions: philosophy isn't a field of inquiry,like biology, that seeks to articulate truths — it's rather a methodfor clarifying the claims of, say, the natural sciences. As foryour second question, aren't many of the questions posed on this site...

There is no thing that is nothing. To say that nothing is in the room is not after all to say that something is in the room, namely nothing. That's a confusion: see Question 49 for some discussion. You're right to observe that when someone says "I'm doing nothing", they're always doing something. But the right lesson to draw from that is not that "nothing" is actually a thing or an activity, but rather that what someone intends to communicate by saying "I'm doing nothing" is that they're not doing anything that you'd care to know about. Likewise, when you're shipwrecked on an island and looking out to the horizon and your friend asks you what you see and you say "I see nothing", you shouldn't think that "nothing" really refers to the water, the sky, the puffins, etc. Rather, you're trying to get across that you see nothing that your friend would care to know about at that moment (like an approaching ship).

I don't know anything about string theory, but I assume that itemploys rich enough mathematics that, were we to articulate thatmathematics in a formal system, Gödel's 1931 Incompleteness Theoremwould apply to it to yield the result that, if the system isconsistent, then it is incomplete, that is, then there is somemathematical statement in the language of the system that is neitherprovable nor disprovable in that system. You ask whether theconsistency and (hence) incompleteness of the system would conflictwith the claim that string theory is "a theory of everything". Itdepends on what "a theory of everything" means. If it means that thetheory can answer all questions about physical phenomena ,then there need be no conflict: the undecidable statement of the formalsystem (the statement that can neither be proved nor disproved if thesystem is consistent) is one in the language of mathematics. It is notmaking a claim about the physical world. If, on the other hand, by "atheory of everything" one...

Perhaps philosophers can offer the scientist clarification of some of the concepts or claims in play in his or her theories. For the most advanced sciences, like physics, such insight typically does not lead to any change in the practice of the working scientist. (That said, some philosophers believe that philosophical illumination of the foundations of mathematics, the most advanced exact science, might lead to a revision of our mathematical practice.) For less advanced sciences, like psychology, such clarification can have far greater impact on how the subject matter is understood and the research is pursued. Philosophers can also offer scientists help in thinking about issues that cut across particular sciences, for instance questions about how to understand claims about unobservable objects, the nature of explanation, the goals of science, the rationality of science, the nature of scientific laws, and so on. I think most philosophers would agree with the conditional claim that if some...

There are a few philosophical works that express the conviction that they have solved all the important problems and that nothing new of any interest should be expected. But these are in the minority. Most philosophical works on the contrary emphasize how much work is left to be done, how much we don't know, how much room and need there is for new ideas and approaches.