You've pretty much answered your own question.

There are two ways of thinking about this. On the first, the "domain" of the theory being axiomatized is taken to consist only of the natural numbers (i.e., the non-negative integers). So it is, in a way, like when the coach says to the driver, "Everyone is on the bus". She doesn't really mean that everyone is on the bus, only that everyone on the team, or whatever, is on the bus. We speak this way all the time. It's not exactly the same phenomenon, but it's close enough to get the idea.

The second way, which you mention in connection with Tarski, is to introduce an explicit restriction to the natural numbers into the axioms. So let "Nx" be a predicate letter meaning: x is a natural number. Then the idea is to "relativize" the axioms to Nx: We replace each universal quantifier (∀x) by: (∀x)(Nx → ...); each existential quantifier (∃x) by: (∃x)(Nx & ...) So the addition axioms will take the form:

(∀x)(Nx → x + 0 = x)

(∀x)(∀y)(Nx & Ny → x + Sy = S(x + y))

We'll also need axioms about N itself:

N0

(∀x)(Nx → NSx)

(∀x)(∀y)(Nx & Ny → N(x + y))

(∀x)(∀y)(Nx & Ny → N(x × y))

These assure us that the objects the original theory speaks about all fall under Nx. Induction will be reformulated as:

A(0) & (∀x)(Nx & A(x) → A(Sx)) → (∀x)(Nx → A(x))

The translation here is thus pretty straightforward. (Note that the last two axioms about N will be provable given induction, but in general we need them, as one can see by considering Robinson arithmetic, which doesn't have induction.)

Indeed, the idea here is obviously quite general, and one can give a general treatment of this kind of translation. E.g., one can show that if some formula is a theorem of the original theory, then its "relativization"—the result of replacing each quantifier by its relativization to Nx—will be a theorem of the new theory, and conversely.

So the usual way of formulating the Peano axioms is just a sort of simplification, allowing us to suppress all the Ns that would otherwise clutter the formulae. But there are cases where one wants the Ns, say, if we wanted to mix set theoretic ideas with arithmetical ones in a single theory.

The empty set is indeed defined to be that set which contains no elements. Another definition we need is that of identity of sets: we say that set A and set B are identical just in case they contain exactly the same elements, i.e., whatever is in A is also in B, and vice versa. So, with these two definitions in hand, consider empty set E₁ and empty set E₂. Well, they are equal since any element that is in the one is in the other (for the trivial reason that neither set contains any elements). So there really is only one empty set - which is what licenses our use of the definite article "the" in "the empty set".

I'm not sure I understand your last question. In set theory, you've specified a set completely when you've specified its elements. And when we say that the empty set contains nothing, we have indeed specified exactly which elements it contains (namely, none).

Why not? We can conceive a nice large space filled with moving matter, all as in our universe, except that the laws of nature vary randomly in space and time -- which is really to say that there are no laws of nature. You could still use geometry to describe the trajectories of objects, but you could not simplify these descriptions with general formulas that cover, say, the force that objects exert on one another. Nor of course could you project any descriptions into the future (predict what will happen) nor even describe with any accuracy what is happening elsewhere or what was happening in the past (because you would have no firm ground for reasoning backward from the data you have to their origins).

So it seems that we can conceive such a world. But whether a cognitive subject could have experience of such a world, could hold it together in one mind, that's another question, one that is very interestingly examined in Kant's Critique of Pure Reason.

"...[A]ll sciences are products of our minds. They are our constructions, as are most of the physical objects in our immediate worlds."

That is no doubt true, but it misses a crucial point: Scientific theories are of course human creations. But what those theories are about are (generally) not human creations. People do not make quarks, atoms, or molecules, fields, stars or galaxies, bacteria, birds, or insects, etc, etc, etc. Nor is it up to us whether the theories we invent are true. And even when physicists discuss colliding billiard balls, the fact that the balls were made by us is neither here nor there. They are external objects, and it is not up to us how they will behave.

Much the same is true of mathematical objects. I see no reason whatsoever to believe that numbers are a human creation, any more than tectonic plates are. And it is just a confusion to think that a circle is an idea. Surely whatever ideas I have are in my mind. Is mathematics supposed to be about the contents of my mind? Should we stop proving things and instead do MRIs to find out if the circle can be squared? What is, perhaps and in some sense, our creation is the concept of a circle. But in that sense, whatever it is, the concept of a molecule is also our creation. That does not make DNA a product of our minds, and the corresponding facts about mathematics do not make set products of our minds, either.

So what precisely are circles? What exactly is mathematics about? If it's true that every even number is the sum of two primes, even though we can't prove it, what makes it true? These are the questions that motivate philosophers, and it does nothing to clarify or help answer them simply to say that mathematics is "the science of quantity". The notion of quantity itself is badly in need of clarification.

Nothing I have just said is at all original. It's largely borrowed from Gottlob Frege, and I can do little more than recommend reading his Foundations of Arithmetic.

Let's start with a quick comment about string theory. My knowledge is only journalistic, but it's clear that string theory is a mathematical theory and states its hypotheses about extra dimensions using mathematics. And as your comment about additional variables already suggests, there's nothing mathematically esoteric about higher dimensions. When variables have the right sort of independence, they represent distinct mathematical dimensions in a mathematical space, though not necessarily a physical space. (Quantum theory uses abstract spaces called Hilbert spaces that can have infinitely many dimensions. But these mathematical spaces don't represent space as we usually think of it.)

Of course, it might be that getting the right physics will call for the development of new branches of math. Remember, for example, that Newtonian physics called for the invention of Calculus, and though earlier thinkers had insights that helped pave the way, Calculus was something new. Just what sort of new mathematical ideas science might lead to is something we'll have to wait to see. But you've raised another question: if sound physics calls for new math, would the math we have now "still hold true" as you put it?

An example might help. General relativity tells us that the geometry of space-time isn't Euclid's geometry. It's something more complicated called pseudo-Reimannian geometry. Does that mean that Euclidean geometry isn't true?

A good answer calls for making a distinction. As a mathematical construction, there's nothing wrong with Euclidean geometry and there are lots of true statements that go with it. From the axioms of Euclidean geometry, it follows that the square of the hypotenuse of a right triangle is the sum of the squares of the other two sides. Briefly, it's true that Euclidean triangles satisfy Pythagoras's rule. However, this is a statement within math itself, so to speak. Whether physical space fits Euclid's axioms isn't a mathematical question but an empirical one, and the answer turns out to be "No" (or at least "not always.")

Here's a way to look at it: math gives us ways of describing possible structures. (Euclid's axioms describe a very general sort of possible structure.) We can construct abstract proofs about those structures whether or not they fit anything in physics. A theory in science might say that one kind of structure rather than another (this geometry rather than that, this probability distribution rather than that, this kind of differential equation rather than that...) gives us the best model of some part of physical reality. But changing our mind about which mathematical structures are good models for the world doesn't amount to changing our minds about math itself.

Perhaps it will help to distinguish between what "7+5" refers to and how it does the referring. The expressions "7+5," "8+4,", "2x6," "36/3" and countless others all refer to the number 12. (Though not everyone agrees that there really are numbers, we'll set that issue aside here.) But they do it in different ways. Compare:

"The 44th President of the United States is Barack Obama"

This is true, and it's true because "The 42nd President of the United States" refers to the same person as "Barack Obama." Barack Obama is the same person as the 42nd President of the United States, just as the number 12 is the same number as 7+5. (Of course, the process of adding two numbers is not a number, but "7+5 = 12" doesn't say it is.)

The sense of confusion here comes from the fact that there can be more to the meaning of a referring expression than just what it refers to. The description "The 42nd President of the United States" refers to Barack Obama, as does the description "The first African American President of the United States." But the two descriptions don't have the same meaning; meaning isn't exhausted by reference.

That said, the arithmetic case raises some more complicated issues. The meaning of "The first African American President of the United States" doesn't guarantee that it refers to the same person as "The 44th President of the United States." But the meaning of "7+5" does guarantee that it refers to the same thing as "12." Just how to account for this while taking account of the fact that there is an apparent difference in meaning between the two expressions is something I'll leave to those who work on such issues. But whatever the best answer, what "7+5+ refers to is the same number that "12" refers to.

I'm not sure about the relationship to Plato's theory of ideas, but there are many connections between programming and philosophy. I'll mention just a few.

Some of the earliest investigations into natural language semantics appealed to ideas connected to the notion of compilation. Roughly, the thought was that understanding an uttered sentence might be something like compiling a program, i.e., translating it into the "machine language" of the brain. My own view, which is probably the majority view, is that this is seriously confused, but it has been attractive to many people.

The idea that "the mind is the software of the brain" has also been very attractive, since it was first articulated (though not quite in those terms) by the great British logician Alan Turing. There are many ways to implement this idea, perhaps the most familiar of them being the various forms of functionalism.

Finally, philosophers are often interested in formal languages, and software languages are certainly a variety of such languages. They are different from the usual languages we consider, because they tend to contain not assertions but rather instructions, so they are more "imperative" than "declarative". But I think they would nonetheless repay attention from philosophers of language.

Acceptable to whom? I don't think the evidence you provide would or should convince mathematicians. They justify their beliefs about conjectures like this by appeal to proofs or counter-examples. So long as neither is forthcoming, they will rightly suspend belief.

But for the rest of us, perhaps the kind of "observational" evidence you suggest might be convincing. Here it would not help much to argue that, because we have found Goldbach's conjecture to be correct up to 10^n, it is probably correct all the way up. One reason this would be unhelpful is that the as yet unexamined even numbers are infinitely more numerous than the examined ones, so we will always have examined only an infinitesimally small sample. Another reason this would be unhelpful is that the examined even numbers are not a representative sample -- rather, they are all very much on the small side, as far as numbers go.

So the probabilistic argument would have to go differently. Let's first establish that the two prime numbers we're looking for are both odd. There is only one case where this is not true -- 2+2=4 -- but we can just put this case aside as settled.

Now examining any even number involves writing it, in as many ways as is possible, as the sum of two odd numbers and then looking through all these sums to find one where both of the odd numbers are primes. Obviously, the larger even numbers get, the more different options there are of writing this number as the sum of two odd numbers. When the number in question is 2n, then we have about n/2-1 such options. I say "about", because the number of options is the same for any number divisible by 4 and the even number just below it. Thus we have 2 distinct options of writing 12 as the sum of two odd numbers (3+9, 5+7) and 2 distinct options of writing 10 as the sum of two odd numbers (3+7, 5+5). For any large even number, the number of decomposition options is about one quarter of this number. So it looks like, as we get to larger and larger even numbers, the probability that each can be written as the sum of two primes gets larger and larger, because there are more and more options. This probabilistic reasoning assumes that the probability that any given way of writing some even number as the sum of two odd numbers (E=O1+O2) is such that O1 and O2 are both primes can be estimated as the square of the probability that any odd number smaller than E is prime.

To illustrate, let the even number be 100. There are 50 odd numbers below 100 and of these 24 are prime. So the probability that any randomly selected odd number below 100 is prime is 0.48. And the probability that any two randomly selected odd numbers below 100 are both prime is then 0.48^2 or about 0.23. So the probability that any two randomly selected odd numbers below 100 fail to be both prime is about 0.77. But this substantial probability now gets whittled down by the fact that there are many options for writing 100 as the sum of two odd numbers -- in fact, there are 25 such options. And we may then surmise that the probability that none of these options involves a pair of primes is about 0.77^25, or about one in 700, or about one seventh of 1%, or about 0.0014.

Let's repeat the exercise for E=1000. There are 500 odd numbers below 1000 and of these 167 are prime. So the probability that any randomly selected odd number below 1000 is prime is about one third. And the probability that any two randomly selected odd numbers below 1000 are both prime is then about one ninth. So the probability that any two randomly selected odd numbers below 1000 fail to be both prime is about eight ninth or 0.89. But there are 250 options for writing 1000 as the sum of two odd numbers. And the surmised probability that none of these options involves a pair of primes is then about 0.89^250. This amounts to one chance in 6 trillion, or a probability of about 0.00000000000016.

We see here that, as we go from E=100 to E=1000, the probability of finding no decomposition into two primes falls off very steeply: going from 100 to 1000, the probability was cut by a factor of about 10 billion. Just to be sure, let's repeat the exercise one more time for E=10000. There are 5000 odd numbers below 10000 and of these 1228 are prime. So the probability that any randomly selected odd number below 10000 is prime is about one quarter. And the probability that any two randomly selected odd numbers below 1000 are both prime is then about 6% or 0.06. So the probability that any two randomly selected odd numbers below 1000 fail to be both prime is about 0.94. But there are 2500 options for writing 10000 as the sum of two odd numbers. And the surmised probability that none of these options involves a pair of primes is then about 0.94^2500. This amounts to one chance in 15 times 10^66, or to a probability of about 0.000000000000000000000000000000000000000000000000000000000000000000066.

Note that, as we went from E=1000 to E=10000, the probability of finding no decomposition into two primes has fallen off even much more steeply (than in the earlier move from E=100 to E=1000): going from 1000 to 10000, the probability was cut by a factor of about 2.5 times 10^54!

Now we must also consider one other factor here. Let's think of E=100, E=1000, E=10000, and so on as centers of "neighborhoods". So defined, these neighborhoods become larger as we go up. There are about 140 even numbers in the E=100 neighborhood (those between 31 and 310, roughly) and about 1400 even numbers in the E=1000 neighborhood (those between 310 and 3100, roughly), and so on. How does this factor affect the calculation? Again, estimating very roughly, if the chance that any given even number around 100 is decomposable into two primes is about 0.9986, then the chance that all 140 of them are so decomposable is about 82% or 0.82. If the chance that any given even number around 1o00 is decomposable into two primes is about 0.99999999999984, then the chance that all 1400 of them are so decomposable is about 0.99999999999984^1400, or about 99.999999977%. And if the chance that any given even number around 10000 is decomposable into two primes is about 0.999999999999999999999999999999999999999999999999999999999999999999934, then the chance that all 14000 of them are so decomposable is about 0.999999999999999999999999999999999999999999999999999999999999999999934^14000, or about 0.999999999999999999999999999999999999999999999999999999999999999.

As these very rough calculations show, the probability of finding a counterexample declines very (and increasingly) steeply as we get to higher neighborhoods. To be sure, there is a non-zero probability in each neighborhood, and there are infinitely many neighborhoods. But this still does not add up to a substantial probability -- much like the infinite series of $1 + $0.1 + $0.01 + $0.001 + ... does not add up to a fortune.

I conclude then that, if there's no counter-example to Goldbach's conjecture in the lower neighborhoods, then this conjecture is extremely likely to be correct. My support for this conclusion rest on a kind of (quick and dirty) probabilistic reasoning that perhaps fits what you have in mind when you speak of "observational evidence" (I "observed" how many prime numbers there are in this and that neighborhood, and so on). I find this reasoning compelling, so am starting herewith to believe that Goldberg's conjecture is true. But I would not be surprised at all if mathematicians were entirely unmoved by this sort of reasoning.

Suppose someone had made the analogous argument about dividing a line of 1 unit into three equal parts. She tells us that "the length of each of these parts is 1/3 which is 0.333333333333 .... It is never ending. So theoretically we cannot determine the exact length of these parts."

I think this would be a bit overblown. We know that the length of each of these three parts is exactly 1/3, and we also know that, while this leads to an infinitely long expression in the decimal system, it would not do so in the duodecimal system (which is based on the number 12 rather than the number 10).

I want to suggest that you consider a similar response to your question. Yes, there is a notation in which we cannot express the length of the hypotenuse you have in mind with a finite number of signs. But there are other notations in which this is possible -- we can just call it "sqroot(2)". So, contrary to what you are saying, we can determine the exact length of that hypotenuse.

You can refresh your problem by pointing to some physical object and then asserting that we cannot determine its precise length. No matter how many digits we may manage to add (through clever measurement) behind the decimal point, there will still remain many further such digits unknown. Leaving modern physics aside, I agree with this but still see no dilemma, no tension with you exclamation that "physically it should have a definite length". Yes, it should, and it can have a definite length even if we cannot possibly ascertain what this length is with perfect precision.

Our human understanding is limited here, but fortunately our human curiosity is limited as well: beyond a few dozen digits after the decimal point, even the nanotechies lose interest in greater precision.