If I understand your epistemological argument correctly, I suppose it could be summarized like this:

1. In order for one to have knowledge of an external object, there must be a cause-and-effect relationship.
2. If the object being known is causally inefficacious, then it cannot bring about the agent’s knowledge.
3. Therefore, something else must bring about this knowledge.
4. What brings about this knowledge must be a mind and is either necessary or contingent.
5. If the mind is contingent, then it cannot cause the knowledge of the object, and so no knowledge of the object would exist.
6. But there is knowledge of the object.
7. Therefore, the mind must exist necessarily.

To use a bit of symbolic logic:
Let K=knowledge
O=object known
R=cause-and-effect relationship
I=causally inefficacious
N=necessary mind
C=contingent mind

1. (K ^ [O = I]) ^ (K –> R)
2. I –> ~R
3. O = I –> ~R
4. K –> R = N v C
5. R = ~C
6. R = N –> K ^ N

The objection I hear most concerns (5). Could a contingent being bring about knowledge of something necessary? It is obvious that a contingent being cannot be the ground of anything that exists necessarily, but I’m wondering whether this applies equally to passive knowledge. Could our minds have a copy mechanism by which it recognizes forms?

I share your intuition that abstract objects are mental concepts, however. However, I find the epistemological argument against Plantonism worth taking a deeper look at. Let me know what you think.

Here is a rough sketch of the two arguments I’m offering for (2b), in addition to one other consideration. If you recall, establishing (2b) can be achieved by refuting the prior premise (1b). So what’s wrong with (1b)?

The first and most frequent argument lodged against (1b) is known as the epistemological objection. If abstracta are independently existing realities as platonism holds and are also causally effete (as no philosopher to my knowledge denies), then it is impossible for us to have knowledge of them. To have knowledge of something independent of us entails a relation of sorts; one where information about said object of knowledge can pass from it to us. But how can this be if abstracta just exist inexplicably a se? (1b) therefore renders abstracta epistemologically inaccessible. For powerful presentations of this argument, see Benacerraf’s “What Numbers Could Not Be” Philosophical Review 74 (1965, pp. 47-73. “Mathematical Truth,” Journal of Philosophy 70 (1973), pp. 661-79. More recent and exhaustive, see Colin Cheyne’s Knowledge, Cause, and Abstract Objects: Causal Objections to Platonism (Springer, 2001).

The second and perhaps strongest (though hardly mentioned) argument against (1b) is that it commits us to the existence of an actual infinite (ℵ₀). If abstract objects exist conceptually independent, then that means each abstract object is a definite and discrete entity. But there are an infinite number of abstract objects. Think of the natural number series alone. Or if you think each natural number can be denumerated from zero, as I do, then consider sets (see the figure I posted a few comments above). Or again the infinite number of propositions expressing possible states of affairs. The problem is that an actual infinite number of things generates absurdities. So the argument could be outlined:

1) An actual infinite cannot exist
2) If Abstracta are independently existing realities, then there is an actual infinite number of abstracta
3) Therefore, abstracta are not independently existing realities

Two qualifications need to be made in order for this argument to go through. First and obviously, it needs to be shown that an actual infinite cannot exist. Second, it needs to be shown how conceptualism doesn’t commit us to an actual infinite number of mind-dependent realities. I think both of these are easy to show. On this argument see William Lane Craig, The Kalam Cosmological Argument (Wipf and Stock, 1979. Ed. 2000), pp. 88-94.

Moreover, if there are sufficiently strong arguments in favor of conceptualism, this counts as evidence against (1b). Unfortunately most philosophers see no arguments here strong enough to sufficiently eliminate an alternative view such as platonism. However, this point favoring (2b) is not entirely lost for at least two reasons. First, it can be argued that conceptualism has more scope than platonism, as it nicely accounts for precisely the desiderata platonism is at odds with. Second, it is the testimony of many philosophers that conceptualism has more intuitive support than its alternatives, or at least enough to make it rationally acceptable in the absence of compelling pros and cons. See Alvin Plantinga, Does God Have A Nature? (Marquette University Press, 2003) pp. 127-140. Quentin Smith, “The Conceptualist Argument for God’s Existence,” Faith and Philosophy 11 (1984), pp. 38-49.

I’ve found this intuition shared by laypeople also. I’ve discussed the ontological status of abstract objects with many people wholly unfamiliar with philosophical parlance. As I describe the peculiar existence of abstract objects, I often receive the eager opinion that “they’re like, in your brain/mind” before I even mention conceptualism as an option. One instance was with a Jr. High student! This doesn’t prove that conceptualism is true, but I certainly think it is a consideration in its favor, and hence out of favor with (1b). Furthermore, in a situation like this we might be able to justify our inference (to conceptualism) by appealing to the Principle of Credulity expounded by Richard Swinburne: certain beliefs with which agents find themselves are—in the absence of counter evidence—probably true; the mere fact that you have a belief is grounds for believing it. See his The Existence of God, (Oxford, 2nd Ed. 2000.), pp. 303-315.

I think these are persuasive reasons for accepting (2b). As someone familiar with Putnam and Benacerraf, what do you think, Doug?

That does strike me as pretty darned strong.

I’ll definitely check it out if I can find it, or him, or any of his stuff online.

Thanks!

A.

Here are some arguments I plan on incorporating in my (ever-growing) revised version of the argument:

Abstract objects seem necessarily existent. That is to say, abstracta exist in all possible worlds. Propositions which are broadly logically necessary, such as “whatever has a shape has a size” seem to exist and be true in all possible worlds. Even in worlds where nothing exists, it would still be true that whatever could have had a shape would also have a size. Moreover, propositions whose truth-values are subject to change also seem necessary. For example, the proposition “there are human beings” could have been false, but it is hard to see how it could have been non-existent.

Moreover, if propositions are necessary, then numbers can plausibly be taken as necessary also. Neil Tennant has persuasively argued that certain propositions, such as “There are n Fs” and “The number of Fs is n” incur ontological commitment to at least one number; namely, 0. Tennant further argues that we can deduce the entire natural number series granted only the existence of the number 0. If 0 exists, says Tennant, it must exist in all possible worlds. Hence, the entire series of natural numbers must exist in all possible worlds. Why does Tennant think 0 must exist in all possible worlds? Consider the proposition

There is no possible world such that there are no things that are not self-identical

It follows simply that 0 is the number of such things that are not self-identical. Therefore, 0 (sequentially, each natural number) exists in all possible worlds. Tennet writes:

Once the language of arithmetic has been adopted…one incurs commitments. These are, first, to the number zero, as the number of non-self-identical things; and, thereafter, to each natural number n in turn, as the number of numbers preceding n. In any world in which one uses a rich enough first-order language—with the identity predicate, the existential quantifier, negation and the numerical term-forming operator #—one has (on reflection) to acknowledge the existence of zero.

His argument can be outlined thus

1) There is no possible world such that there are no things that are not self-identical
2) 0 is the number of such things that are not self-identical.
3) Therefore, 0 exists.

(1) seems self-evidently true, so what support does Tennant give for (2)? I have not paraphrased Tennant’s support yet. Perhaps it would be best to quote him at any rate:

Why can one maintain the premiss? One can only consider the question whether 0 exists by framing the thought x(x = 0) in a language one of whose sentences can be thus regimented. Moreover, one has to be thinking of 0 as a number, that is, thinking of “0″ as a term which, if it denotes anything at all, denotes a number. One cannot be thinking of “0″ as a term which might denote some person, or physical object.

I think these are intriguing arguments, serving not only to illustrate the necessary existence of abstract objects but also reasons for embracing their existence. These are just snippets of Tennant’s rigorous article. I urge you to check it out. Tennant, Neil. 1997. “On the Necessary Existence of Numbers,” Nous, 31.

If I understand correctly, the issue comes down to whether, in order for it to maintain its eternal truthfulness or validity, Euclidian geometry and numbers must exist in the mind of a supreme being.

My earlier comments argued that knowledge by such a supreme being was not necessarily a pre-requisite to the existence of numbers or Euclidian geometry (I’m going to stick to calling them “abstract objects,” for now). As I argued, abstract objects exist when we create them in our minds for whatever purposes we find them convenient. In that sense, “two” did not exist until I decided to use that label to represent a quantity. Mathematics simply derives from the interaction of quantities, but, in both cases, our minds alone account for their existence.

The conceptualist argument, I think, assumes numbers and mathematics necessarily exist, but I don’t find an argument here in this post that establishes their necessity. I assume someone has tried to make that case elsewhere, but that’s, admittedly, my ignorance, not Chad’s! I hope he can lay that case out at some point, although I suspect he has already referred me to a source for such an argument, but I simply haven’t had the time to get to it.

Based solely on the four corners of this post, I think numbers are necessary to us for very practical reasons that did not exist until we invented them and cease to exist the moment the last human being dies. Why would the world need numbers otherwise? If they are necessary, well, then to whom and for what purpose? If the answer is god, then I think that’s putting the cart before the horse– God exists, because numbers are necessary, because God needs them, therefore God exists.

My question, that I have rather inartfully untangled for myself over my comments on this post, is “What makes numbers necessary? Why must they exist” Convincing me of the necessity of numbers, or explaining to me what, exactly, that even means, would go a long way with me as far as the conceptualist argument for god’s existence.

Maybe not ALL the way, but I think that’s my basic hang up with this proof.

In any event… HAPPY THANKSGIVING.

Thus, “numbers” are just abstractions formed by the human mind of real stuff existing out there in the universe. But I can’t say that for all of mathematics. The elementary abstractions have been extended by the human mind, to form concepts and systems that don’t have their origin in stuff outside the human mind. Like Non-Euclidean geometries, for instance. Or whole systems, like the complex numbers. Did parallel lines intersect before humans existed? Possibly not, because of the curvy nature of spacetime. Does that mean Eucliden geometry is invalid? I don’t think so. It’s a human-invented system.

“Do they only exist in the epistemological sense, as you were sympathetic to above? If so, are you prepared to defend that view, maintaining a straight face in saying that sentences such as “4 is divisible by 2”, “Jones believes that [some proposition x]”, or “these two balls have the same shape” are really truth-valueless? Or do you mean to say that abstract objects exist in the ontological/metaphysical sense, the grounding of which is some mind? If so, are you conceding a major premise in the conceptualist argument?”

The more I read that paragraph, the more I think I’m not prepared to make the “straight faced” argument or concede that abstract objects ontologically exist! I’m going to try and break up the issues as I seem them.

I think mathematical truths can be called true solely in the epistemological sense because, again, what is a number other than a name we assign to a quantity? “Four” and “two” do not exist independently of our need to convenienty convey information, as you put it. “Four is divisible by two,” and indeed, maybe all mathematics, amounts to tautologies about these concepts. One side of the equation equals another. These truths exist as derivatives of mental concepts we use to convey information.

These things exist in our mind as replications of reality that we can use and communicate to others. I don’t see how there existence is dependent on being known by anything other than our temporally existing minds. Now I am not saying that there is no truth outside of our minds– at least I don’t think I am! Obviously rocks exist. Identical cubes exist. But what our mind does with different things and quantities of things– categorize them and make abstractions– even if they amount to “truths” does that necessarily mean that they pre-existed our minds as true before we knew them?

Funny… as I thought of that, another thing popped into my head which cuts against most of what I just said. Namely, “Well, A., were they FALSE before you knew them? Weren’t they true before you knew them? Before ANYBODY knew them? And if it could only have become true when someone came to know it, then how could it have been true before I knew it unless it was already known?”

Does that make sense? Did I just make your argument for you?

Hmmm… let me chew on that for a while.

I suppose that’s my round about way of saying whatever benefit you derive from my comments are certainly reciprocal. Honestly, I think you have a great page here that engages in a far more serious debate on these issues than I have found elsewhere, and I often recommend it to people.

Thanks.

I hear you as far as the need for an argument explaining why abstract objects might exist only epistemologically. Speaking ontologically, however, what does it mean to say and abstract object is “real and distinct” as opposed to “created”?

When I say an abstract object is “real and distinct” as opposed to “created,” I mean to say that abstract objects are discovered as opposed to invented. So long as we’re talking about abstract objects only in the epistemological sense, they should be seen as abstractions or inventions deduced from other pieces of knowledge. On this account, abstract objects are merely convenient fictions we use to convey information, they don’t really exist. Even though several theists have embraced this view, I think there are severe problems with it. See my brief remark on it in the fifth comment, especially the article reference.

See, I don’t see how numbers exist independently of our minds. If they did, then I could experience “three” without ever seeing three of anything. I could directly know it. Numbers cannot be known independently of the physical universe. Why would it be impossible to know them without existing ontologically (a term I’m not sure I understand as a lay person with little training).

I think this paragraph is loaded with issues the untangling of which would go far beyond the conceptualist argument, as interesting as they are! Your first sentence seems to concede that abstract objects exist. But what do you mean “exist”? Do they only exist in the epistemological sense, as you were sympathetic to above? If so, are you prepared to defend that view, maintaining a straight face in saying that sentences such as “4 is divisible by 2”, “Jones believes that [some proposition x]”, or “these two balls have the same shape” are really truth-valueless? Or do you mean to say that abstract objects exist in the ontological/metaphysical sense, the grounding of which is some mind? If so, are you conceding a major premise in the conceptualist argument?

Your third sentence in conjunction with the first entails a strict physicalism of mind, which is incompatible with the belief that numbers don’t exist independently of our minds yet are themselves immaterial. So are there immaterial features of mind (numbers) or not?

Why would it be impossible to know them without existing ontologically? Well, if they only exist epistemologically, the issue is not that it is impossible to know them simply because there really is nothing to know at all. If they lack positive ontic status, as the epistemic-sense only view holds, they don’t exist to be had as objects of knowledge.

Again, I take great joy in reading and considering your comments–I benefit greatly from them and hope that benefit to be reciprocal.

I hear you as far as the need for an argument explaining why abstract objects might exist only epistemologically. Speaking ontologically, however, what does it mean to say and abstract object is “real and distinct” as opposed to “created”?

See, I don’t see how numbers exist independently of our minds. If they did, then I could experience “three” without ever seeing three of anything. I could directly know it. Numbers cannot be known independently of the physical universe. Why would it be impossible to know them without existing ontologically (a term I’m not sure I understand as a lay person with little training).

Feel free to direct me to other sources if we’re chewing up too much of your time.

Thanks.