From any point of view, I think the question here ought to be: What way of teaching such material will make it most likely that students will learn it? But that's not a question a philosopher is well-placed to answer. Someone with expertise in education would be the right person to ask.

I'm not sure I understand what it is that you really want. Certainly in logic, there are well-established techniques for discussing the language of logic itself. See any good textbook on Goedel's Theorem (say, Boolos, Burgess, and Jeffrey, Computability and Logic ) for more on this. If it's natural language in which you are interested, then you might find the work of Richard Montague interesting. Montague was one of the first to try to extend logical techniques to provide a semantics for natural language. And then, of course, there is contemporary theoretical linguistics more generally.

I don't see how one could reasonably suppose there was an argument here for the existence of God. But belief in God need not be based upon any sort of argument or even be something for which one has reasons in the usual sense in which one has reasons for beliefs. A belief in God might have more in common with aesthetic judgements than with theoretical ones. If so, then perhaps the suggestion would be that such mathematical patterns are part of what constitutes the basis for the aesthetic response in question. Whether that would be "sound" is hard to say. Aesthetic judgements are not beyond criticism (unless you regard aesthetic judgements as not really judgements at all but on a par with mere expressions), but the criticism of aesthetic judgements is slippery territory. The foregoing may well require that belief in God be something very different from belief that God exists. This suggestion—or, rather, a generalization of it—is the subject of an exceptionally interesting paper...

Stewart Shapiro's Thinking About Mathematics . ( On Amazon .) It's a well-written textbook that covers most of the basics.

Your question concerns real numbers and measurement of physical phenomena using them. The question would not arise if we were talking about non-negative integers and the use of such numbers to answer "how many" questions, like: How many panelists are there on askphilosophers.org? The answer "39" is perfectly accurate. Nor would the question arise if we were talking about the use of real numbers within mathematics itself: The ratio between the circumference and the diameter of a circle in Euclidean space is π, amd that too is perfectly accurate. On the other hand, if we are, as I said, talking about real numbers and their use in measurement, the question does arise. And I think one might have to allow that, by and large, no measurement one makes is every completely precise. But scientists are quite well aware of this fact. That is why, when they are being careful, they will say that a measurement is, say, "accurate to within a millimeter". That is also what makes the concept of "significant digits" ...

Regarding your second question, whether the incompleteness theorem limits what we are capable of knowing, people disagree about this question. But the short answer is: There is no decent, short argument from the incompleteness theorem to that conclusion. If it does limit what we are capable of knowing, then it will take a very sophisticated argument to show that it does. One might think it followed from the theorem that we cannot prove that PA is consistent. But we can. I proved it yesterday, in fact, in my class on truth. The incompleteness theorem says only that we cannot prove that PA is consistent in PA , if PA is consistent. (If it's not, then we can prove in PA that PA is consistent! But that won't do us much good, since we can also prove in PA that PA is not consistent, and indeed prove absolutely everything else in PA, e.g., that 2+3 = 127.—This last remark assumes that we have classical logic at our disposal.) So when I proved that PA was consistent, I didn't do so in PA. I...

This question concerns, in effect, number-theoretic functions that grow very fast. We can say a lot about them. The operation in play here is called "superexponentiation", and is also known as "tetration". We can define it as follows: superexp(0) = 1 superexp(n+1) = 2^(superexp(n)) So superexp(4), e.g., is: 2^(2^(2^2))), and K is superexp(7). Noting that superexponentiation is just repeated exponentiation, we can now define superduperexponentiation as follows: supdupexp(0) = 1 supdupexp(n+1) = superexp(supdupexp(n)) supdupexp(4) is already an enormous number: It is superexp(2^16) = superexp(65,536) = 2^(2^...), where there are 65,536 2s in the tower. supdupexp(5) is unimaginably huge. It is 2^(2^...), where there are supdupexp(4) 2s in the tower. But there is no need to stop there. Let's rename superexponentiation 2-exponentiation, or exp 2 for short; and let's call superduperexponentiation 3-exponentiation, or exp 3 for short; and let's just write 2^x as exp ...

There are a couple different attitudes towards this kind of question. One takes the idea of variables very seriously: There really are things that are called "variables". It's not that we don't know various things about them. A variable number, for example, reallly is intrinsically not any number in particular. That, however, is a minority view nowadays, though it has a distinguished history. See Kit Fine's Arbitrary Objects for a recent defense of it. The more common view, which originates with Gottlob Frege, is that variables are like pronouns. Consider "Everyone who met someone liked her". Here, the word "her" does not stand for anyone in particular: Rather, it stands for the person each person in question met. So, if we're talking about the people on the team, and those people are Bill, Dick, and Harry, and Bill met Betty, Dick met Jane, and Harry met Sally, then what "her" refers to "v aries" as we consider the different people: It refers to Betty when we consider Bill; to Jane when we...

In a sense, the answer, strictly speaking, is "no". Infinity isn't a number. It's a property of sets. Some sets are infinite; some are not. But in a more interesting sense, the answer is "yes": There are infinite numbers. There are many ways to see this. Here's one, borrowed from Frege. What do we mean by a "number"? Well, a number is the kind of thing one can give as an answer to a question like, "How many books/dolls/cars/whatever are there?" (These are so-called "cardinal" numbers.) There are lots of such numbers, and some of them we call "finite". These are the ordinary natural numbers, zero, one, two, and so forth. Now, notice the following. Every natural number is the number of natural numbers less than it. Thus, each natural number is one less than the number of natural numbers less than or equal to it, which is to say that each natural number is strictly less than—that is, less than and not equal to—the number of numbers less than or equal to it. Now: How many natural numbers...

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;...