Emotions are not involved in any direct way in our mathematical conclusions (for the conclusions are about numbers, or groups, or vector spaces, or sets, or whatever, not about human things like emotions.) And whether a purported conclusion is indeed a mathematical truth is an objective matter: again, emotions don't come into it. They are not involved in the proofs of the conclusions either (for proofs are deductions from more or less explicit premisses about numbers, or groups, or vector spaces, or sets, or whatever, to conclusions about such things, and still don't mention emotions). And whether a purported proof is indeed a mathematical proof is again objective matter. Does that mean emotions have no place in mathematics? Well, we do think of some proofs as beautiful or elegant or cute. And this, you might well think, is a matter of how we respond -- respond emotionally, in a broad sense -- to the proofs. And what makes a mathematician seek to prove a result in the first place (elegantly or...

Gödel's first theorem, with later improvements by Rosser and others, tells us that any theory of arithmetic T that is (i) consistent, (ii) decidably axiomatized (i.e. you can mechanically check that a purported proof obeys the rules of the arithmetic) and (iii) contains Robinson Arithmetic (a very weak fragment of arithmetic) is incomplete. Strengthen T by adding more axioms and the theory will still be incomplete (unless you throw in so much it becomes inconsistent or stops being decidably axiomatized). In sum, you can't "outrun" the reach of incompleteness theorem by going to a stronger theory which is still consistent and properly axiomatized. This is explained in any standard introduction to Gödel's theorems (e.g. in the first chapter of mine).

But what would an infinite decimal correspond to on the proposed mapping? It may be that every natural corresponds on that mapping to a real between 0 and 1. But you need -- and assert! -- that to every such real there corresponds a natural on this mapping, and that's quite plainly false when you think of the reals with non-terminating decimal expansions (the construction doesn't determine a natural).

1. I don't think there is any reason to suppose that learning about mathematical logic from Principia to Gödel will be any help at all in understanding what is going on with the Frankfurt School. (The only tenuous connection I can think of is that the logical positivists were influenced by developments in logic, and the Frankfurt School were concerned inter alia to give a critique of positivism. But since neither the authors of Principia nor Gödel were positivists, it would be better to read some of the positivists themselves if you want to know what the Frankfurt School were reacting against). 2. Of course, I think finding out a bit about mathematical logic is fun for its own sake: but it is mathematics and to really understand I'm afraid there is not much for it other than working through some increasingly tough books called the likes of "An Introduction to Logic" followed by "Intermediate Logic" and then "Mathematical Logic". Still, you can get a distant impression of what's going on...

On the standard account, given two finitely long lines, even of different lengths, their pointscan indeed be matched up one-to-one, e.g. by the kind of projection theteacher indicated. And the possibility of that kind of one-to-one matchingis just what we mean when we say the two lines "contain the same number ofpoints". What makes this possible, despite the different lengths? In part, the fact that there are an infinite number of points along a finite line (the issue we are dealing with here is one of those initially puzzling matters which arises when we deal with the non-finite: intuitions tutored by finite examples can lead us astray). And there being an infinite number of points along a finite line is related to the fact that the points on a line are dense -- that is to say, between any two points, however close together, there is another point . Now, consider a line with end points. Between the left hand end-point a and any other point on the line there is a further point....

I agree with Richard's and Alex's general remarks about "logicism" and what counts as "logical". It would indeed be far too quick to reject every form of logicism just because it makes the existence of an infinite number of objects a matter of "logic". Still, it is perhaps worth reiterating (as Richard indeed does) that Principia gets its infinity of objects by theft rather than honest toil: it just asserts an infinity of objects as a bald axiom rather than trying to conjure them out of some more basic logical(?) principles in a more Fregean way. So I'd still want to say that, whatever the fate of other logicisms, Russell and Whitehead 's version -- given it is based on theft! -- can't really be judged an honest implementation of the original logicist programme as e.g. described in the Principles , even prescinding from incompleteness worries. But for all that, three cheers for Principia in its centenary year!

In the Principles of Mathematics, Russell boldly asserts "All mathematics deals exclusively with concepts definable in terms of a very small number of logical concepts, and ... all its propositions are deducible from a very small number of fundamental logical principles." Principia , a decade later, is an attempt to make good on that programmatic "logicist" claim. Now, one of the axioms of Principia is an Axiom of Infinity which in effect says that there is an infinite number of things. And you might very well wonder whether that is a truth of logic . (If someone thinks the number of things in the universe is finite, are they making a logical mistake?) Another axiom is the Axiom of Reducibility, which I won't try to explain here, but which is even less obviously a logical law -- and indeed Russell himself argued that we should accept it only because it has nice mathematical consequences in the context of the rest of Principia's system. Still, there is some room for...

There's no right answer. Zermelo's original set theory allowed "urelements", i.e. entities in the universe which are members of sets but not themselves sets. Some modern writers use "ZFC" to refer to a descendant of Zermelo's theory allowing urelements. George Tourlakis is an example, in his two volume Lectures in Logic and Set Theory . Some other writers (perhaps the majority) use "ZFC" to refer to the correponding theory of "pure" sets, where there are no urelements and the members of sets are themselves always other sets. Kenneth Kunen is just one example in his modern classic Set Theory . If you are interested in set theory as a tool, then the first line is arguably the more natural one to take. If you are interested in set theory for its own sake, then for most purposes you might as well take the second line (because it seems to make no big difference to the sort of questions that most set theorists are interested in: for example ZFC-with-urelements is equiconsistent with ZFC-for...

How much do you need to know about science to begin learning about the philosophy of science? Some but not a great deal if you are interested in very general metaphysical questions about e.g. the nature of explanation, laws and causation, and in very general methodological questions about how scientific theories are confirmed and refuted. You'll need to know quite a lot more if you are interested in understanding more specific foundational questions about the interpretation of quantum mechanics or are worrying about the nature of natural selection. Similarly, how much do you need to know about mathematics to begin learning about the philosophy of mathematics? Some but not a great deal if you are interested in very general metaphysical and epistemological questions about e.g. the nature of numbers and the nature of our knowledge of such things (if they are "things"). Quite a lot more if you are bugged by more specific questions about how we are to settle axioms for set theory or to decide...

You certainly don't need to be able to do original research in maths to be able to work on the philosophy of maths. But you will need to be able to follow whatever maths is particularly relevant to your philosophical interests. How much maths that is, which topics at which levels, will depend on your philosophical projects. For example, compare and contrast the following questions (not exactly a random sample -- they all happen to interest me!): "Is our basic arithmetical knowledge in any sense grounded in intuition?" Evidently, you don't need any special mathematical knowledge to tackle that . "Can a fictionalist about mathematics explain its applicability?" Again, I guess that acquaintance with the sort of high school mathematics that indeed gets applied is probably all you really need to know to discuss this too. "Just what infinitary assumptions are we committed to if we accept applicable mathematics as true?" Here you do need to get more into the maths, and know quite a lot about...

Well, there is a logical truth in the vicinity of 1 + 1 = 2. Or perhaps better, a whole family of logical truths. Fix on a pair of properties F and G . Then it is a theorem of first-order logic that if exactly one thing is F and one thing is G and nothing is both F and G , then are exactly two things are either- F -or- G . Here the numerical quantifiers 'exactly one thing is' and 'exactly two things are' can be defined in standard ways from the ordinary first-order quantifiers and identity. And the theorem holds whatever pair of properties we choose. This elementary logical result probably captures what is driving your intuition that in some sense 1 + 1 = 2 is "reducible to formal logic". (For a bit more on this sort of thing, see my Intro to Formal Logic §33.3 -- or any other standard logic text!) But all that is quite compatible with Gödel's first incompleteness theorem. For Gödel's theorem isn't about some limitation or incompleteness in our ability to prove ...