You are quite right that Gödel's (first) incompleteness theorem attracts all kinds of bizarre "interpretations". Various examples are discussed and dissected in Torkel Franzen's very nice short book, Gödel's Theorem: An Incomplete Guide to its Use and Abuse, which I warmly recommend.

My guess is that a main source for the whacky interpretations is the claim that has repeatedly been made that the theorem shows that we can't be "machines", and so -- supposedly -- we must be something more than complex biological mechanisms. You can see why that conclusion might in some quarters be found welcome (and other technical results in logic generally don't seem to have such an implication). But as Franzen explains very clearly, it doesn't follow from the theorem.

To answer this question, it may be helpful to say something about the mathematical formalism usually used in probability theory. The first step in applying probability theory to study some random process is to identify the set of all possible outcomes of the process, which is called the sample space. For example, in the case of an infinite sequence of coin flips, the sample space is the set of all infinite sequences of H's and T's (representing heads and tails). Probabilities are assigned to events, which are represented by subsets of the sample space. For example, in the case of an infinite sequence of coin flips, the set of all HT-sequences that start with H represents the event that the first coin flip was a heads, and (assuming the coin is fair) this event would have probability 1/2. The set of sequences that start with HT is a subset of the first one, and it represents the event that the first flip was heads and the second tails; it has probability 1/4.

Now, consider some infinite HT-sequence s. For any positive integer n, we can consider the set of all sequences that agree with s for the first n terms. This set contains s, and imitating the reasoning in the last paragraph we see that it represents the event that the first n coin flips come out as specified by s, which has probability 1/2ⁿ. Since {s} is a subset of every one of these sets, the event that the entire infinite sequence is exactly s must have probability less than 1/2ⁿ for every n. But that means that the event must have probability 0. So you are absolutely right that the reasoning here involves a limiting process: the probability is 0 because 1/2ⁿ approaches 0 as n approaches infinity.

With this background, it is also now easy to see the distinction between zero-probability events that are possible and those that are impossible. The event that the entire infinite sequence is s is represented by the set {s}. It has probability 0, but is possible. The event that the first flip is both a heads and also a tails is represented by the empty set (since there are no elements of the sample space that fit this description); it has probability 0 and is impossible.

I'm really glad you enjoyed the Gödel book!

I don't know if anyone has actually tried to prove that establishing Goldbach Conjecture needs additional axioms added to "regular arithmetic". But if someone had succeeded we would certainly have heard a lot about it!

Why so? Well, suppose T is your favourite theory of "regular arithmetic" (and we can assume T contains at least Robinson Arithmetic, Q). And suppose we can show that GC can't be proved by T as it stands. Then, trivially, T + not-GC is consistent. So, since GC is a Pi₁ sentence, it follows from Theorem 9.3 of my book that GC is true! In other words, proving GC can't be proved by T in fact proves Goldbach's Conjecture.

So we can put it this way: if it does need new axioms to establish GC, then proving that is so is at least as hard as proving Goldbach's Conjecture itself. Which, the evidence suggests, is very hard!

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 think your #1 should go. If you drive a sled through the snow, the two lines you draw in the snow will never meet, never get closer to each other, even if you drive on forever. If you have two intersecting lines and close the angle toward zero, then at the limit the lines will have the same direction ... but at that limit they will also coincide (be identical) and hence not be parallel.

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.