I applaud your adeptness with the now-prevalent analytic philosophical writing style. You clearly have read a lot and learned well.

How familiar or well-read are you with regard to Pragmatism? Poststructuralism? Postmodernism, at-large?

I take it that these areas are not your most researched areas, although correct me if I’m wrong. I expect that there are certain philosophers who may be able to provide you with some valuable insight - your dismissiveness toward those who reject ‘necessary truths’ concerned me just after your warning against premature dismissal of ‘fictionalism’. I am not concerned because of the ultimate position you took but rather the manner in which you took your position - with reference to ‘obviously true descriptive states of affairs’ and a caricature of ‘nominalist’ theses (I use quotes because, in all honesty, the universalist/nominalist distinction, or however you want to chop up the debate, is now a relatively useless way to describe the issue in question due to the heavy historical baggage and archaic language attached to the debate). Either your characterization of ‘fictionalism’ as the most plausible form of ‘nominalism’ is seriously skewed or you were dealing with a caricatured version of ‘fictionalism’. Perhaps the key to understanding ‘fictionalism’ (I admit, I am not entirely familiar with the work of Field) is the USEFUL fiction part of your description - see Richard Rorty for details (Rorty because of the relative ease of access to his work, his status as a sort of conglomerator of many different key 20th century philosophers whose isolated ideas may have been incomplete, among other things).

Ultimately, I applaud your effort but I believe your work calls for a reevaluation of ‘truth’ - not necessarily a change of position but rather a thorough treatment of the field.

I’d love to hear what you think.

Thank you for the comment.

I suspect you must have only skimmed the argument. Read the very next couple of sentences after the quote to which you allude:

Armstrong’s and Neilson’s characterization of naturalism is clearly incompatible with (3). The defender of (N), on the other hand, can yet consistently maintain (3), though not without considerable tension (for reasons outlined above).

I agree with you, and that’s why I use Oppy’s characterization of naturalism (which is (N) in the paper), not Neilson’s. My quoting Robinson, Mackie, Armstrong, and Neilson was not to argue for strict inconsistency between (3) and (N), but to highlight tension between the two. I make this quite clear all throughout the paper, such as when I say

There is no prima facie inconsistency between (N) and the existence of abstract objects. However, there are good reasons for thinking there is at least tension between the two—a burden theism doesn’t bear. In other words, even if one could draft a realist ontology of abstracta compatible with naturalism, a better, more-at-home account can be offered by theism.

It is only after (4) is introduced does strict inconsistency result.

I am not at all convinced that,

“(3) Abstract objects exist and are mental concepts”

and

“It is the view that anything that exists is ultimately made up of physical components. … There are no purely mental realities in a naturalistic account of the world.”

actually are clearly incompatible. It seems to me very, very possible we are using different meanings of the phrase “mental reality” and “exist”. To show a contradiction it is necessary to show that the terms are actually the same, or at least the same in a relevant manner.

As to the argument using

“(P) The notions of “true” and “false” are in and of themselves merely useful fictions

which is just to say

(P’) For any proposition p, p is such that it is truth-valueless,”

it seems clear that in order to represent a useful fictionalist’s position, (P) would have to read more like, “The notions of ‘true’ and ‘false’ are in and of themselves merely useful fictions when applied to statements about abstract objects,” and (P’) would have to read more like, “For any proposition p about abstract objects, p is such that it is truth-valueless.” Then the useful fictionalist would still be OK, since (P) isn’t a statement about abstract objects but is instead a statement about how we will apply the notions of truth and falsity to statements about abstract objects.

———————-

It’s missing the point to think that if there were actual infinitudes, the operation of subtraction could be performed upon them just as they can be upon actual finitudes. One could remove infinitely many books from an infinite library, choosing which infinite collection of books to remove; and which was removed would affect whether infinitely many books or only finitely many (and if finitely many, then how many) remained; but one wouldn’t apply the operation of subtraction to infinitudes. (You use the example 3-5=-2. But your own example, if it shows anything, shows that you can’t subtract five marbles from three marbles; it doesn’t show that you can’t have three marbles or five marbles. Why, then, should a problem with “subtraction of infinitudes” mean that you can’t have infinitely many of something?) And as for the failure to define an inverse operation–well, I’m not quite sure what addition would be for infinite sets, but if there were actual infinitudes, one could presumably take the infinitely many items in group A and put them in the same collection as the infinitely many items in group B, and then remove exactly those same items that started out in group A–the inverse operation of putting them into the same collection!–and be left with exactly those items in group B.

——————–

(P’’) p is true for S and ~p is true for S*

Naturally, one may reply that P” is true for S or S* if he is a relativist, but false for S or S* if he is not. A truth-relativist cannot assert the absolute truth of any of his claims, but can only assert their truth for him–including the correctness of truth-relativism. But I don’t think he’s inconsistent in so doing.

The real problem is with why one should accept that truth is always truth for S, rather than correspondence with reality (or some such).
—————-
One Brow: I’m not sure you understand what is meant by “indispensability,” in what I take to be Chad’s usage. 0’s indispensability doesn’t mean that you can’t live without it; it doesn’t mean that you can’t have concepts without it; it doesn’t mean that you can’t do mathematics without it (although your Indian example, in which a blank space was left, showed that Indians did have the concept); what it means is that you cannot completely conceptually analyze reality without it. It means that it is indispensable to a proper understanding of the world. Specifically, it’s indispensable to a complete ontology that accurately captures all that exists.

Somewhat similarly, I don’t see why “2+2=4″ could not be viewed as ontologically a “useful fiction” without entailing that it has no truth value. One might say that it is in fact grounded in empirical reality–that when we say that two plus two equals four, what we really mean is that anytime we have two discrete objects and two other discrete objects and put them together discretely (i.e., without merging, the way drops of water merge), we wind up with four discrete objects. Or one might say that it is a truth by fiat: One might say that we make certain arithmetic statements axioms, giving them truth by fiat, and then all arithmetic theorems have that same truth by fiat. Or one might say that the axioms are chosen so as to mirror empirical reality, so that they have the same sort of truth as “Mary is watching television,” and then describe theorems’ truth as deriving from that of the axioms. At any rate, I don’t see why “2+2=4″ has to be supposed to be an existing thing in order to have a truth-value.

I’m not a Platonist (I don’t know whether I’m a nominalist or a conceptualist; I don’t think, though, that I have thoughts–i.e., I think but do not have things called “thoughts” residing in my head, in much the way in which my leg hurts [i.e., I hurt in-my-leg-ishly] but I do not have a thing called a “pain” residing in my leg), but I’m not sure that the Platonist can’t answer (2b1) pretty much as you suggest. Physicists didn’t have to have direct evidence of quarks in order to infer their existence. Similarly, the Platonist might say that he infers the existence of abstract objects from facts about the world. The inference is merely epistemological, but it is an inference *to* existence.

Against psychologism, you write, “Say some abstract object O is the concept of some human mind at time t1. Surely there must have been times ti–tj prior to t1 such that there were no human mind that had O as a concept. It would be less congenial to say that at t1 O came into existence than to say O must have existed as a concept of a necessary mind during ti–tj. This argument runs backwards as well: given their voluminous and complex nature, there must be abstract objects that have not yet nor will ever be concepts in human minds.” But why would it be “less congenial” to suppose O’s coming into existence at t1 than to say that it already existed as the concept of a necessary mind? This is where I think that reifying mental phenomena is a problem: Instead of thinking of human beings as thinking and as hurting, the realist conceptualist with respect to abstracta thinks of human beings as having thoughts and as having pains (well, OK, pains aren’t abstract objects, but perhaps they’re illustrative), and then, thinking of them as *things*, has to wonder *how they came into existence*–and then concludes that human thinking wouldn’t suffice for the task of their creation (although why it strikes one as more plausible that an unevidenced necessary, omniscient mind would do the job better, I’m not sure). Similarly, thinking of as-yet-unthought-of abstracta as *things* that are not yet “contained in” any human mind invites the question of where they are (although they could simply be thought of as nonexistent until contained in some human mind). Yet, if abstracta are not thought of as *things*–if human beings abstract but do not think of things called “abstractions,” just as we do not have pains residing in our legs (where were those pains before we hurt?)–then the problem of how they came into existence vanishes. (Of course, the big problem I, and everyone else, I think, am left with, is how it is that material bodies are able to be aware at all–David Chalmers’s “hard problem.”) Similar considerations would apply to Quentin Smith’s infinitely complex conjunction of all true propositions (if such a thing did not turn out to be bedeviled by Russell’s Paradox-type problems), of course.

The conclusion that there is a necessary, omniscient mind seems to be one that relies on thinking of abstracta (and of propositions) as *things*, and not only that, but as things whose existence cannot simply be accepted but must be accounted for in terms of an existing cause for being. It is a cosmological argument applied to abstracta (which, frankly, seems like a weaker argument than a cosmological argument applied to concreta, since it’s harder to question the existence as things of concreta than of abstracta). Yes, some propositions are eternally true, but that simply means that they will be true no matter who speaks them (or thinks them) or when or where they are spoken (or thought). Yes, some propositions are analytically true, but that just means that they will be true by virtue of their meanings no matter who speaks them (or thinks them), and so on.

On to the comments.

“(2a.4) There is no possible world such that there are no things that are not self-identical
(2a.5) 0 is the number of such things that are not self-identical
(2a.6) Therefore, 0 exists in all possible worlds”

You (Chad) acknowledge that this seems to beg the question against the nominalist, sneaking in the existence of numbers, and indeed it does. (2a.5) is like saying, “Red is the color of cardinals,” and deducing therefrom that the color red exists. But red doesn’t exist; “red” is adjectival, not nominal–it’s descriptive. Yes, we can use “red” as a noun, but that just shows how we’ve agreed to use words; it doesn’t suddenly convert the color red into a “thing.”

Similarly, the argument

(2a.7) a exists ≡ (∃y) a=y
(2a.8) (number x) F(x)=0 ≡df ¬(∃x)F(x)
(2a.9) F(x) is x≠x
(2a.10) (∃y) y=0

doesn’t establish the existence of the number zero. As in the red cardinal example, “zero”–or any count of “how many”–functions as adjectival, not nominal. But even were it nominal, having a label–having a name and specifying what object or sort of object it would be applicable to, were that object or sort of object to exist–doesn’t bring an object into existence.

(2a.8) establishes how we agree to use the symbol “0″–how we agree to use the word “zero”–so that when we find lacking any instances of something, we say there are zero of them–we say that the number of them is zero. Numbers are indeed part of our universe of discourse–”number” and “three” are nouns, all right–but that doesn’t make them part of the metaphysical universe. When we find that there are no objects unidentical to themselves, we agree to express this by saying that “the number of such objects is zero.” This is a linguistic convention; our word use does not bring the number zero–or any other number–into existence.

(D) “There does not exist an x such that Fx” is not the same as (E) “There exists an x such that x is the number of x such that Fx, and x=0.” (D) says nothing about anything’s existence; (E) says a number exists. It’s convenient linguistically to have numerical terms that are nouns, so that we can equate the two sentences; but the kind of existence in (E) is not (or need not be) metaphysical existence.

As to the argument against actual infinitudes…. I’m not at all a fan of reifying abstracta; I don’t think of abstracta as “things.” But I can’t see this argument from the impossibility of actual infinitudes (well, as actual as a Platonist might say abstracta were, anyway) as well-founded.

“Let m = the number of books in our infinite library, n = the number of odd-numbered books, and o the number of books numbered 4 or higher…
(m – n) = infinity, whereas (m – o) = 4.
But,
n = o (since both n and o are infinite)
It follows that we get inconsistent results subtracting the same number from m.19

“The conclusion is that we obviously couldn’t perform such operations in the actual world, so an actual infinite cannot exist.”

I don’t see that this conclusion follows. The idea that n=o is just wrong. The cardinality of n = the cardinality of o, true; but the operation of set subtraction is not the same as the ordinary operation of subtraction of numbers. The way the example would work in the real world, were there a real infinite library, is that removing just the even-numbered ones would leave the odd-numbered ones–an infinite “number” of them. Removing all of them after the first four would leave just the first four–a finite number of them. Neither of those is absurd. One cannot then say that n=o and therefore the same “number” of books should be left over; it simply does not follow from transfinite arithmetic.

The real issue here, stripping away the window dressing, is that a set A may have the same cardinality as set B even though the set A is a proper subset of the set B. This is only true for infinite sets; so, of course, our ordinary intuitions about finite sets won’t work with infinite sets.

If one wanted to apply the argument given, he would have to first come up with a more discriminating way of deciding a set’s “size” than its cardinality, so that, for example, the set of all even integers would have (intuitively) half the “size” of the set of all integers. We don’t have such a discriminating way of deciding set’s “sizes” now, for infinite sets, that I know of.

(I note that William Lane Craig finds it simply absurd that Hilbert’s Hotel could be full but still accommodate infinitely more customers without doubling up in rooms. His sense of absurdity stems simply from his being unused to infinite hotels; all we have are finite hotels, so he’s used to how they work. If we lived in a world with infinite hotels, he’d be used to how *they* work, too, and probably wouldn’t find their workings absurd.)

For better or worse, many people do indeed think that I lack good sense. I’ll try not to let it be too much of an issue. : )

I. Truth
You’re first main contention concerns the nature of truth as correspondence. I didn’t anticipate this being much of an issue, but I can see now that the history of philosophy up to the 19th century isn’t appreciated by everyone…

Well, I only took a couple of introductory philosophy courses, and that was 25 years ago, so I am strictly an amateur.

At any rate, your view of truth has severe problems, similar to those which plague the coherence theory. For example, in your last reply, you say

But, as I pointed out, this approach obviously doesn’t hold water. For one, it implies that since it is possible that p coheres with my model of reality and ~p coheres with your model of reality, both p and ~p could, in some absurd and contradictory way, both be true.

Not true in the sense of reflective of reality, just true in the sense of a workable proposition within that model of reality.

The only way out of this would be to bite the bullet and say that

but either (P’’) corresponds to how things are or it doesn’t. Which is it?

If you are interpreting S and S* to mean the models we have constructed to examine reality, obviously P’’ is correct in assessing the truth of the proposition within each model.

I illustrated this above by noting the problems facing your (P) and (P’), which you didn’t respond to. Either way, your view is bankrupt. So far you have refused to even give an argument for why you think “the notions of ‘true’ and ‘false’ are in and of themselves merely useful fictions”. Perhaps you can privilege me with such.

The notions of true and false are part of the model that we construct to examine reality. You can not point to a true, any more than you can point to a 2. You can classify a statement as true. From the vantage point of internal consistency, you can always create a model where a given proposition is true and a different model where that proposition is false. However, that can disregard the purpose in constructing the model, which is examining reality. For example, you can create a model where my fingers are touching a keyboard, or where they are touching an apple. Only the first is useful, because I don’t actually want to eat that thing. So, I assign the first statement to be true, and the second statement to be false.

II. Indispensability of 0
I submit that you’re yet to offer either a counterargument to the two indispensability arguments I mentioned or offer a revisionist expression of (2a.4) that doesn’t commit you to 0. You’ve mentioned reasons for thinking 0 is epistemologically dispensable, but as a debate about ontology and not epistemology, such are irrelevant to the discussion.

I could have sworn I said that I agreed 0 is useful in all possible worlds. Under that understanding, I accepted 2a.4 and 2a.5.

III. The Actual Infinite
The inconsistency of your comments on the actual infinite are painfully obvious (so much for coherence!). On the one hand you agree with me that that when it comes to mathematics, “The absurdity is the application, not the model.”

That was not all of mathematics, generally, but the specific case of negative numbers and counting apples. Negative numbers work in the model, but are meaningless in the application, so it is a bad application in that instance. There are many other applications were negative numbers work just fine, such as vectors.

On the other hand you “see no problems in the translation [application] of infinity to reality period. The problems you espouse are also present in the model.” This is textbook special pleading. “But the former comment was about your marbles illustration, and the second about the actual infinite”, you’ll say. But this misses the point because I’m arguing that both are instances where we have internal consistency at odds with external application. Why think the two are different?

Your absurdity complaints seem to be based that there is no well-defined notion of subtraction with infinite numbers. This is true in the model of numbers, so why should it be different in any theoretical infinite? When the prediction of the model carries over to any reality, this is not automatically an absurdity of the reality. Your claims strikes me no differently than a claim 0 can’t be real, because you can’t divide by 0.

One more point: you mention being at odds with external application. What external application did you have in mind? Outside of some attribute of a putative God, such as power, I can’t think of any application you would consider real.

It’s easy to lose sight of the real issues at stake in discussions like these (as opposed to, say, what you think of the kalam cosmological argument or the attribute of omnipotence), so let me summarize what I gather to be your case against the conceptualist argument as I have presented it, and then I’ll respond.

I. Truth
You’re first main contention concerns the nature of truth as correspondence. I didn’t anticipate this being much of an issue, but I can see now that the history of philosophy up to the 19th century isn’t appreciated by everyone…At any rate, your view of truth has severe problems, similar to those which plague the coherence theory. For example, in your last reply, you say

We don’t judge, or select, a model by a guarantee of correspondence to reality that can never be achieved. We judge it whether it is consistent within its own rules and useful in helping us deal with reality…We could compare our various models of reality, look at why we find them useful, and perhaps even incorporate small parts of each other’s model into one or more of our models. That could be useful.

But, as I pointed out, this approach obviously doesn’t hold water. For one, it implies that since it is possible that p coheres with my model of reality and ~p coheres with your model of reality, both p and ~p could, in some absurd and contradictory way, both be true. The only way out of this would be to bite the bullet and say that

(P’’) p is true for S and ~p is true for S*

but either (P’’) corresponds to how things are or it doesn’t. Which is it? I illustrated this above by noting the problems facing your (P) and (P’), which you didn’t respond to. Either way, your view is bankrupt. So far you have refused to even give an argument for why you think “the notions of ‘true’ and ‘false’ are in and of themselves merely useful fictions”. Perhaps you can privilege me with such.

II. Indispensability of 0
I submit that you’re yet to offer either a counterargument to the two indispensability arguments I mentioned or offer a revisionist expression of (2a.4) that doesn’t commit you to 0. You’ve mentioned reasons for thinking 0 is epistemologically dispensable, but as a debate about ontology and not epistemology, such are irrelevant to the discussion.

III. The Actual Infinite
The inconsistency of your comments on the actual infinite are painfully obvious (so much for coherence!). On the one hand you agree with me that that when it comes to mathematics, “The absurdity is the application, not the model.” On the other hand you “see no problems in the translation [application] of infinity to reality period. The problems you espouse are also present in the model.” This is textbook special pleading. “But the former comment was about your marbles illustration, and the second about the actual infinite”, you’ll say. But this misses the point because I’m arguing that both are instances where we have internal consistency at odds with external application. Why think the two are different?

Thanks again for your comments.

When my objections are actually addressed?

and deal with the real meaty issues?

I don’t mean to overlook anything. Which meaty issue would you like for me to sink my teeth into?