I'm afraid this all seems terribly confused.
Do you have any evidence or argument to support your contention that disputes about whether (e.g.) real numbers, infinite sets, tables, electrons, symphonies, and justice, "exist" are best regarded as disputes about those things rather than about existence? Here, it seems that you've just asserted it repeatedly (in a way that, for me, pattern-matches to "not really as confident about this as s/he makes out").
I agree with shminux that, given the extent to which people at least think they disagree about what "existence" means and how it works, blithely describing it as "unproblematic" seems overoptimistic.
The argument against infinity that you sketch here sounds to me like utter nonsense. First you say it's "conceptually possible" that some things are distinguishable only by the fact of being different things. (Maybe so, though I'm not convinced and you offer no actual reason for thinking so.) Then you apparently go on to assume that if there are infinite sets then there are infinite sets of "brutely distinguishable" things (by no means obvious; perhaps some kinds of things can be collected into infinite sets and some can't), and that two sets of "brutely distinguishable things" can't be distinguished except by their cardinality (why? why can't they be distinguished by being brutely different things themselves?). And then you say this implies that any two such sets of equal cardinality are in fact the same thing, which makes absolutely no sense at all in conjunction with the proposal that some things are distinguishable by the fact of being different things and by nothing else.
The following paragraph ("we are able to distinguish points that exist from those that don't exist") also seems nonsensical. There are no points that don't exist; that's what "don't exist" means. There might be apparent-descriptions-of-points that turn out not to apply to any actual points, but that isn't at all the same and doesn't seem like it does the work you want the distinction between existent points and nonexistent points to do.
Having dismissed the idea that questions like "do numbers exist?" are questions about existence rather than about numbers, it seems very odd that you then say "If finitude is a condition for existence, we've learned something new about the concept of existence". Why isn't it just finitude and infinitude about which we've allegedly learned something, just as you claim that learning that numbers do or don't exist would be learning something about numbers and not about existence itself?
The last paragraph seems rather close to word salad to me.
Presumably there's some reason why you're posting this stuff -- you're working up to some big general theory, or you think many people are terribly confused about existence (though I don't know how you'd reconcile that with calling it "so unproblematic"), or something. Would you care to say a bit about the bigger picture this is meant to contribute to?
I am not sure why you insist on describing the concept of existence as "unproblematic", given how much debate it generates. Your conclusion that "The incoherence of sensationalism and of actual infinities argues for a metaphysics upholding the primacy of common-sense existence." does not follow from your premises and does not provide a useful starting point for discussing any concept based on "existence". Do you have an issue with the definition of existence (of something) as that something being an element of reality as defined, for example, by Eliezer in The Useful Idea of Truth:
I need different names for the thingies that determine my experimental predictions and the thingy that determines my experimental results. I call the former thingies 'beliefs', and the latter thingy 'reality'.
If you do, what is it? If you don't, why not adopt it?
In addition to all the other problems this has (existence is seriously not unproblematic), the central argument against infinities is simply wrong, even if we admit the possibility of "brute distinguishability".
If S is a set of these "brute distinguishables", and T=S u {x}, where x is yet another brute distinguishable (one that isn't in S), then S and T are distinguishable because T contains x and S doesn't. Whether brutely or not, the fact is that we declared that x is distinguishable, so this is valid.
Now evidently you disagree with this last part; but if we instead accept your argument, we get other problems. For instance, the number two is impossible. Why? Well, say {x,y,z,w} is a set of four distinct brute distinguishables; then {x,y} and {z,w} are different; and yet they are indistinguishable, because the only distinguishing property they have is their cardinality, and thus we have a contradiction! This is exactly the same argument you used, but with "infinity" replaced by "two". The only difference is that when you considered infinite sets, you considered S and T with S a subset of T, while here neither is a subset of the other. But this difference has no relevance to the reasoning. The assertion that "the only distinguishing property they have is their cardinality" is simply false. I could have used singletons instead of two-element sets, I was just afraid you'd miss the point and reply "But we assumed that x and y are distinguishable, thus so are {x} and {y}!" (Now wait. Have I proved that 2 doesn't exist or that 4 doesn't exist? More on that in a moment.)
Well, OK, there is one difference between the infinite case and the finite case -- a finite set of brute distinguishables could maybe be said to have a defining property, while an infinite one certainly couldn't. (Though honestly even the finite really couldn't -- it could only have a definition relative to your set/class of brute distinguishables.) But this is irrelevant; you don't need defining properties to distinguish things; any property will do. You essentially admit this yourself when you consider the cardinality as a potentially distinguishing property, even though any two infinite sets of brute distinguishables would be equally undefinable, regardless of their cardinalities. You could admit only as distinguishing properties those that don't make reference to particular brute distinguishables -- but then you're abandoning the assumption that they are brute distinguishables, because now given two brute distinguishables x and y, they have exactly the same properties. And if it was just undefinability you were after, why did you introduce this assumption of brute distinguishables in the first place? Undefinable sets exist anyway, if you want to make that the basis of your argument.
Edit Jan 1 2013: Oops! There's a mistake in the above paragraph -- I said any two such infinite sets would be equally undefinable, but (provided we accept the definabillity of finite ones, which as I detailed I don't think we should), this is incorrect, because they could be cofinite. Thus, the above paragraph should really talk about sets of brute distinguishables that are both infinite and co-infinite. This makes essentially no difference to the point.
Even if we accept your argument this far, despite all the mistakes I've pointed out, all it shows is that there can't be an infinite set of brute distinguishables, that there can be at most finitely many. (Except really it shows that there can't even be 4, by the argument above, or even 2, by the version with singletons. At which point you no longer really have brute distinguishability at all.) It's not an argument against infinite sets, just that a particular class of things must be finite.
Even if we accept your conclusion that there can't be infinite sets, this still doesn't do what you want. You aren't just trying to argue against infinite sets, you're trying to argue against "infinite quantities". What the hell are "infinite quantities"? You seem to be under the mistaken impression that "infinite quantities" is a sensible unified notion, when in fact there are many different systems of infinities we use depending on what the situation calls for and that don't fit together. Here, why don't I just point you to the discussion article I wrote on the matter some time ago.
And really, you should know better than to base an argument on the existence of something just because it's "conceptually possible", i.e. you personally find it possible to conceive of. That really has no bearing on whether it's actually possible. That is as silly as making arguments based on P-zombies being conceptually possible.
I'm not even touching the rest.
Adult readers never seriously maintain that fictitious characters exist
A) "Never" is a strong word. I imagine there are all kinds of mental disorders that can lead certain adults to confuse fiction with reality
B) "Existence" here is a cached term used for simplifying a concept to the point of being inaccurate. When a person says that, for instance, Frodo Baggins doesn't exist, he or she would be entirely incorrect to say that there is nothing in existence that matches the concept of Frodo Baggins. What the person is actually saying, is that a description of the character of Frodo calls to mind an image of something resembling an organism and that this image fits into a mental category that no actual organism can fit into. Whether or not Frodo "exists" is a pretty poor question unless fleshed out.
Existence is well-suited for the role as a primitive concept in philosophy because it is so unproblematic...
??!?!?
Everything that exists does so in some definite quantity. Existence is that property of conceptual referents such that they necessarily exist in some definite quantity.
I'm confused by this mix of referring to things that exist and referring to existence as a property of conceptual referents. Are you saying that conceptual referents are the things that exist in finite and definite quantity? Or are you saying something else?
definite quantity
I see that you are claiming that existing things are bounded in some quantifiable way, but you do not seem to account for the inherent uncertainty of determining quantities.
The identification of a definite quantity requires a quantifier. Some uncertainty comes from the implementation of this quantifier; if it is incorrect then the identified quantity would be wrong. You could handle this by verifying the implementation of the quantifier, but that only pushes the uncertainty into the context of verification. To use the quantifier you must choose to halt the regression of verification and accept the remaining uncertainty.
Additional uncertainty comes from the choice of the quantifier. The quantifier used is one choice from a large and possibly infinite set of possible quantifiers. Not all of these quantifiers would provide the same answer--or even provide a "reasonable" answer, for example by replying with "hat" instead of a number like "2".
For example: I scoop up a handful of gravel from a beach. I want to count the stones in my hand. But my hand contains all kinds of stuff; rocks and dirt from the size of dust to a couple inches across, bits of wood, shell, and other organic debris. Out of this mess which bits of the stuff are stones? It depends on how I quantify stones; is it by volume, apparent area, mass, composition, color, texture... there are many possible measurements, and combination of measurements. I choose one way of counting stones and get a quantity of 5, but it could have been 1 or 1000 or "blue hat" if I had made other choices.
Given this uncertainty, how can I know that only a "definite quantity" of stones exist in my hand?
[Crossposted; Based on Can infinite quantities exist? A philosophical approach (downvoted)
The topic is the concept of existence, not why there's something rather than nothing—not the fact of existence—but the bare concept brings its own austere delights. Philosophical problems arise from our conflicting intuitions, but “existence” is a primitive element of thought because our intuitions of it are so robust and reliable. Of course, we disagree about whether certain particulars (such as Moses) have existed and even about whether some general kinds (such as the real numbers) exist, but disputes don’t concern the concept of existence itself. If Moses’s existence poses any conceptual problem, it concerns what counts as being him, not what counts as existence. Adult readers never seriously maintain that fictitious characters exist; they disagree about whether a given character is fictitious; even the question of the existential status of numbers is a question about numbers rather than about existence. As will be seen, sometimes philosophers wrongly construe these disputes as being about existence.
When essay 19.0 asked “Can infinite quantities exist?” existence’s meaning wasn't in play—infinity’s was. Existence is well-suited for the role as a primitive concept in philosophy because it is so unproblematic, but it’s unproblematic nature can be thought of as a kind of problem, in that we want to know why this concept is uniquely unproblematic. We would at least like to be able to say something more about it than merely that it’s primitive, but in philosophy, we acquire knowledge by solving problems and existence fails to provide any but the unhelpful problem of its being unproblematic. The problem of infinity provides, in the end, some purchase on the concept of existence, which concept I assumed in dealing with infinity.
In one argument against actual infinity, I proposed as conceptually possible that separate things might be distinguishable only concerning their being separate things. Then, if we assume that infinite sets can exist, the implication is the contradiction that an infinite set and its successor—when still another point pops into existence—are the same set because you can’t distinguish them. (In technical terms, the only information that could distinguish the set and its successor, given that their members are brutely distinguishable, is their cardinality, which is the same—countably infinite—for each set.)
What’s interesting here is the role of existence, which imposes an additional constraint on concepts besides the internal consistency imposed by the mathematics of sets. Whereas we are unable to distinguish existing points, we are able—in a manner of speaking—to distinguish points that exist from those that don’t exist. While no proper subsets are possible for existing brutely distinguishable points, the distinction within the abstract set of points between “those” that exist and “those” that don’t exist allows us to extend the successor set by moving the boundary, resulting in contradiction.
If finitude is a condition for existence, we’ve learned something new about the concept of existence. Its meaning is imbued with finitude, with definite quantity. Everything that exists does so in some definite quantity. Existence is that property of conceptual referents such that they necessarily exist in some definite quantity.
Existence is primitive because almost everyone knows the term and can apply it to the extent they understand what they’re applying it to. The alternative to primitive existence is primitive sensation, as when Descartes derived his existence from his “thinking.” But sensationalism is incoherent; “experiences” inherently lacking in properties (“ineffable”) are conceived as having properties (“qualia”). So, the heirs of extreme logical empiricism, from Rudolf Carnap to David Lewis, have challenged existence’s primitiveness. Carnap defined existence by the place of concepts in a fruitful theory. Lewis applies this positivist maxim to find that all possible worlds exist. Lewis isn’t impelled by an independent theory of logical existence, such as a Platonic theory that posits actually realized idealizations. Rather, the usefulness of possible worlds in logic requires their acceptance, according to Lewis, because that’s all that we mean by “exists.” Lewis is driven by this theory of existence to require infinitely many existing possible worlds, which disqualifies it on other grounds. But the grounds aren’t separate. When you don’t apply the constraints of existence because you deny their intuitive force, you lose just that constraint imposing finitude. The incoherence of sensationalism and of actual infinities argues for a metaphysics upholding the primacy of common-sense existence.