Do pebbles actually exist? But they are composed from quarks,

I refer to this as the Reductionist Problem of Scale. "Psychology isn't real because it's all just biology. Biology isn't real because it's all just chemistry. Chemistry isn't real because it's all just Physics." I don't see this as so much of a 'minefield' as a need to recognize that "scale matters". In unaided-human-observable Newtonian space, there is no question that pebbles are "totally a thing" -- they are. You can hold one in your hand. You can touch one to another one.

Of course; if you look solely at the scale of subquarks, then this distinction becomes unintelligible.

On the other hand, when I play a computer game, do the various objects in the virtual world exist?

No. Interacting with the symbol of a thing is not interacting with the thing itself. They are, however, fully real -- just like you yourself are fully real, but do not exist (you are not your body; you are not your brain; you are not the electrons and chemicals that flow through it. You are the pattern that is so-comprised. But that pattern itself is entirely non-physical in nature; it is non-instantiable and does not itself interact with anything -- nor can it ever.)

What if I write a program to play for me and stop watching the monitor. Do they stop existing?

I... am not rightly sure how you could come to the conclusion that this is a relevant question to the definition I provided. I did not say "to exist, things must be observed" -- I said "to exist, things must interact with other things". Pebbles interacting with lakes are interacting. Regardless of whether someone watches them.

If a tree falls in a forest, the tree exists. Regardless of whether it makes a sound.

For example, would you say that software "exists"?

No. But it is real. Software is a pattern by which electrons, magnetic fields, or photochemically-active media are constrained. The software itself is never a thing you can touch, hold, see, or smell, or taste; it never at any point is ever capable of directly interacting with anything. Just like you and me; we are not our bodies; nor our brains; nor the electrons or chemicals that flow through the brains. We are patterns those things are constrained by. I am the unique pattern that, in times past, created the password to the LW account, Logos01; and you the pattern that (I presume) created Eugine_Nier. But neither of us, physically, exist. This is important to notions of substrate-independence; where goes your pattern, is you. (Remember your Ship of Theseus problem.)

Right now I am downloading onto a VM on my workstation the 12.04 release of Lubuntu. This software is being pulled over ethernet to be delivered to a virtual harddrive image where it will be configured and installed. If I say I have LibreOffice installed too, it is clear I am talking about a specific release/instance/copy. We talk about identity in terms of software "Have you tried the latest Halo? It's awesome! ^_^" -- and two people can apparently own exactly the same game. But of course these are multiple copies of the pattern. It's even possible to talk about backups and restores. This is because the only thing that matters about defining whether something is or is not the software is that pattern.

Nations?

Real but do not exist. If every last person of the US packed their bags and got onto a rocket and shot themselves to Mars, it'd still be the United States of America. Even if every last person died while on that rocket and their kids were the ones who took over for them. Substrate independence once again demonstrates this.

Boeing 747s?

Exist. It is possible to see/hear/touch/smell/taste a Boeing 747 (I hear they taste like burnt chocolate and chicken.) It is possible for two Boeing 747's to be run into one another; or for a comet to strike one. It is not possible for a factory to churn out political constructs or minds. (Though it is possible for them to assemble all the pieces that would, when activated, allow for the presence of a mind.)

While you CAN take all the individual components of a Boeing 747 apart and put them back together again to make the same object; or over time transfer pieces into / from it (Theseus's Grandfather's Axe) -- what you can't do is just "declare" a different physical object to BE that original Boeing 747. You can't have five of the same Boeing 747. That is because it is a thing which directly interacts with other things.

People?

See the above. If some temporal accident causes me to split into two, both of those people would still be ME. (Though their cohabitating the same space would cause divergence of identity over time.) Again, this is because what I physically am is irrelevant to determining my identity (and identity is the conformance to a specific pattern).

Force fields?

"In physics a force field is a vector field that describes a non-contact force acting on a particle at various positions in space." You see that word, "acting"? To "act upon" something is quite literally definitional to being said to "interact with" a thing. By the definition I have provided of 'exists', and the definition of 'force field' as found on Wikipedia, force fields definitionally exist.

it is easy to construct a formal system with which we have precisely that experience.

I included the qualification "the formal systems that you're thinking of". Were you thinking of that formal system when you wrote

There's a particular set of formal steps I can go through to convince myself that two numbers are twin primes, and a different set of steps which will convince me that some particular pair is the largest such pair, or that there isn't a largest pair.

?

By what standard are we to judge between formal systems in which 2 is provable, those in which it is disprovable, and those in which it cannot be proved either way? Do we take a vote? Is it a question of appeal to human intuition?

This is a separate question from your original one. To answer your original question, the platonist need only point to a particular formal system (e.g., PA), and say that the nonexistence of the number 2 would mean [ETA: rather, would imply] that there would be no proof of 2's existence in that particular system.

But the platonist would also find your new question interesting. For example, when you wrote "There's a particular set of formal steps ...", how did you come to settle on that particular set of formal steps?

One position would maintain that some particular formal system is implicit in the statement of the twin prime conjecture (TPC). That is, when someone asks "Are there infinitely many twin primes?", they are speaking in an abbreviated fashion, and they really mean "Does PA prove that there infinitely many twin primes?", or "Does ZFC prove that ...", or something like that. This position would claim that number-talk has meaning only when the speaker has some specific formal system in mind.

The difficulty with this position is that people seemed to be making meaningful assertions about numbers for thousands of years before settling on any formal systems of arithmetic — indeed before they even had the concept of a formal system. (Euclid's is presumably too unrigorous to count.) Ancient number-theory texts such as the Introduction to Arithmetic by Nicomachus often didn't even include anything like what we would call a proof, not even in the sense in which Euclid's arguments are called proofs. Nicomachus pretty much just flatly asserts number-theoretic propositions, perhaps bolstering his claims with a few examples. Yet somehow readers would reflect on these propositions and agree with them, even though Nicomachus didn't communicate which formal system he was working within.

Another position would maintain that people absorb some formal system implicitly from their culture, and their number-talk is automatically in terms of that formal system. Even if that culture lacks the concept of a formal system, nonetheless all its number-talk is governed by some formal system.

But it is not even known whether the TPC is decidable in any of the standard formal systems. Suppose that the TPC were proved undecidable in, say, ZFC. Would the question really then lose all meaning? Consider a physical computer that brute-force factors one odd number after the next, and prints every consecutive pair of odd numbers that have no nontrivial factorizations. Would it really become meaningless to ask whether there is a bound on how many entries the computer would print, regardless of how many physical resources it were given? Granted, this is necessarily a counterfactual, but . . .

I am not a platonist, but I still can't call myself a formalist, because I can't bring myself to declare confidently that the TPC would become meaningless if it were proved undecidable in ZFC. Indeed, even if the TPC happens to be undecidable in every formal system whose axioms would seem to us to be "intuitively correct" of numbers, it still seems to me that our concept of number may suffice to determine a truth value for the above counterfactual. Either that computer would churn out pairs forever, or it wouldn't.

Yes and the number 2 exists within the natural numbers, but not within the model your system describes.

I would argue that the same argument MinibearRex uses here to justify his belief on a reality underlying our physical observations. Specifically, if there is no real system underlying our models how do you account for their seeming consistency?