I'm really glad you enjoyed the Gödel book!

Suppose that S is Goldbach's conjecture. And suppose theory T is your favourite arithmetic (which includes Robinson Arithmetic). Then Theorem 9.3 applies to S. So if not-S is not logically deducible from T, then S must be true.

So if we had a proof that S is a "naturally" arising Gödel sentence -- i.e. a demonstration that T proves neither S nor not-S -- we'd ipso facto have a proof that S is true.

As to the "odds": my hunch is that GC is true, and can be proved in PA -- but I wouldn't bet even a decent meal out on it!!

Sure. For one thing, nature doesn't care about our arbitrary units. Suppose we have a plank of wood that''s exactly a foot long. Now I define a new unit: a schfoot. Anything one foot long is exactly pi schfeet long. Is there any mystery about things being pi schfeet long?

Also -- since we're setting aside issues about quanta and, I assume, the possibility that space is granular, can't we make sense of something changing length continuously? A twig that's a foot long and growing will pass through an uncountable number of irrational lengths on its way from being a foot long to being two feet long.

You need to distinguish the claim that mathematics is grounded on logic, and the claim that mathematics uses logic.

The weaker second claim is evidently true, at least in this sense. Mathematical reasoning is a paradigm of good deductive reasoning. And standard systems of logic explicitly aim to codify, more or less directly, the kinds of good deductive reasonings that mathematicians use. (And computers might be used to echo some such reasonings too.)

But the fact that mathematics uses logical reasoning doesn't show that mathematics is grounded on logic if that is the much stronger thesis that at least arithmetic, maybe the whole of classical analysis, just follows from pure logic plus definitions of mathematical notions in logical terms. (I take it that it is this logicist thesis which you are thinking of, when you say that there are "serious problems" about the idea that mathematics is grounded on logic).

For example, you might think that in set theory we use logic to deduce what follows from the set-theoretic axioms. But you might suppose that is not a matter of logic that the axioms are true -- that depends on the nature of sets!

We need to keep two questions straight here: (i) why is it necessary that 2+2=4; (ii) why should we believe it's necessary that 2+2=4.

The first question assumes that this is, in fact, a necessary truth, and asks what grounds the necessity. The second asks how we know.

On the first: why is it necessary that 2+2=4? Part of the problem is to decide what would count as an answer. I'll leave it to wiser heads than mine to offer a thought. But I think your worry actually lies with (ii).

Now of course, in some sense of "could," it could be that we're all utterly deluded, and in fact 2+2 isn't equal to 4 at all. All that this means is that I can't rule out beyond any possible doubt that we're utterly addled in ways we can't even imagine. But to quote my old colleague Dudley Shapere for the mty-nth time, the possibility of a doubt isn't a reason for doubt. Translation: the mere fact that we can dimly imagine that we're utterly and totally confused about even the most basic things doesn't mean we should quake in our epistemic lizard-skin boots.

Which is a good thing. Because once we put into doubt whether we know that 2+2=4, it's hard to see what would be left with to base any reasoning on. After all, if I'm wrong in thinking that 2+2=4, maybe I'm wrong when I think that from "If P then Q" and "P", it follows that "Q" is true. But in that case, the game is up and we have no reason to take any of our reasoning -- even about the frailty of our reasoning -- seriously.

Which mathematics manages to describe the physical world? Mathematicians offer us, e.g., Euclidean and non-Euclidean geometries of spaces of various dimensions (and the non-Euclidean geometries come in different brands). They can't all correctly describe the world, since they say different things even about such simple matters as the sum of the angles of a three-dimensional triangle. But we hope that one such geometry does indeed describe the sort of structure exemplified by physical space (or better, physical space-time).

That's pretty typical. Mathematicians explore all kinds of different possible structures. Only some of them are physically exemplified. For example, group theories explore patterns of symmetries; some of the patterns are to be found in the world -- but I guess that no one thinks e.g. that the Monster Group is physically instantiated. Mathematical physicists tell us which kinds of structures are to be found in the physical world and then use the appropriate mathematics to deduce more facts about the relevant structures.

Perhaps though it is different with elementary arithmetic. Perhaps, whenever we talk about objects at all, we talk about things that in principle can be counted and to which arithmetic can be applied. 'Logicists' like Frege hold that arithmetic thus has the same kind of necessary applicability as logic. But if so, it is a rather special case.

I don't myself have a view on whether 2+2=4 is absolutely certain. I suppose it's as certain as anything is or could be. But the question here is different. It's whether that certainty is undermined by doubts about what happens empirically.

As Gottlob Frege would quickly have pointed out, however, the mathematical truth that 2+2=4 has nothing particular to do with what happens empirically. It might have been, for example, that whenver you tried to put two things together with two other things, one of them disappeared. (Or perhaps they were like rabbits, and another one appeared!) But mathematics says nothing of this. That 2+2=4 does not tell you what will happen when you put things together. It only tells you that, if there are two of these things and two of those things, and if none of these is one of those, then there are four things that are among these and those. It's hard to see how one's theory of identity could affect that.

So I asked my brother about this, and he tells me that this kind of question is much discussed in the literature on mathematics education. Here's what he had to say:

"Good thing to think about. A related idea I've been considering for some time---and maybe the difference is just a matter of cognitive development---is whether solving an equation algebraically is a proof.

"Another spin on the idea---which is what got me thinking about it in the first place---is whether solving equations ought to be taught as proof, since every step one takes in algebraic solutions can be mathematically/logically justified through some equivalence that leads to the solution set. What most kids end up learning to do is to conduct the procedures of solving without any real understanding (or caring) of why what they are doing is mathematically justified. I have a sense that if solving were taught as proof, then it would be more natural for kids to pay attention to why certain steps that seem OK actually introduce other solutions, or botch everything because they might be dividing by zero, etc."

An important paper here, he says, is Andreas J. Stylianides, "Proof and Proving in School Mathematics", Journal for Research in Mathematics Education 38 (2007), 289-321.

Well, there are a lot of questions there. I won't try to answer the first one: That's a topic for a book, not an internet posting. And I'm not sure I understand the second one. But regarding the next two, yes, of course there are infinitely many numbers between 17 and 18. Here are some of them: 17.1; 17.11; 17.111; 17.1111; etc. And between any two of those, there are infinitely many more. But if you want something that will really make your head spin: Consider all the rational numbers, that is, all the numbers that can be written as fractions m/n. There are no more of those numbers than there are "natural numbers", like 0, 1, 2, etc., and that is true even though there are infinitely many rational numbers in between any two natural numbers. The proof of this is not terribly difficult. The point is that we can order all the rational numbers like this: 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, .... (What's the pattern?) Some rational numbers occur more than once, of course, but they can be weeded out. But once we have them in that order, we can "corrleate" them one-to-one with usual natural numbers. So there are just as many of the one as of the other.

Why is there no smallest number? Well, pretty much for the reason just given. (I'm assuming here that we mean no smallest positive number.) Between any two numbers, there is another. So if there were a smallest postive number, then there would be a number between zero and it, and that would be smaller. So there can't be a smallest number.

Why is there no largest number? (I'll assume here that we're talking about finite positive numbers.) Just because, given any number, there's always the number one greater than it. So if n were the greatest number, then we'd also have n+1, but then that would be greater. So there's no greatest number.

Now, of course, you might ask me why there's always a number that's one greater. And that's a perfectly reasonable question to ask. I've spent a good deal of time thinking about it myself. What is it about the numbers that makes this true? But that too is a topic for a book, not a posting.

Perhaps the first thing to say here is that we need to distinguish the question whether it is necessary that 2+2=4 from the question whether the sentence "2+2=4" is necessarily true. It seems to me that no sentence is necessarily true. Any sentence might have been false, simply because that sentence might have meant something other than what it in fact means. For example, "2+2=4" might have meant that 3+3=4, and then it would have been false. And that is what it would have meant had "2" meant 3 rather than what it does mean, namely, 2. So I agree absolutely the whether "2+2=4" is true depends upon how we what "2" and "+" and "4" and "=" all mean, not to mention the grammatical rules that govern the significance of combining them in a certain way. And if you want to put that by saying that the truth of this sentence depends upon how we "define" the numeral "2", I won't object. Not too strongly, anyway.

But it is an entirely different question whether it is necessary that 2+2=4. That is not at all a question about what our words mean. It is a question about addition, numbers, and the like. The general view is indeed that 2+2 could not have been other than 4, and the reason is pretty simple: It's very difficult to conceive how 2+2 could have been other than 4.