The Conceptualist Argument
*Updated version here*
An argument which I think is begging for a more contemporary and public defense is the conceptualist argument for the existence of God.1 You’d be hard pressed to find it given a fair treatment of more than a few sentences at best or a passive mention at worst in a typical introductory apologetics text (and unfortunately even thicker books as well). The argument isn’t totally neglected, however. Greg Bahnsen used a form of this argument as his devastating weapon of choice in his lively debate with Gordon Stein.2 More recently, William Lane Craig sometimes uses it in his some of his debates.3 Interestingly enough, prominent atheist philosopher Quentin Smith has developed a version of the conceptualist argument that has become quite popular.4 The conceptualist argument is named for its relevance to the philosophical view known as conceptualism, which holds that abstract objects are metaphysically grounded in the mind of an agent. But, according to the argument, abstract objects aren’t metaphysically grounded in just any mind, but an ultimate (omniscient), divine mind. That being said, the main focus of the conceptualist argument is the (seemingly) peculiar existence of abstract entities.
Usually contrasted with concrete objects, abstract entities are things like numbers, sets (and other mathematical entities), propositions, properties (and universals), values, relations, laws, theories, etc. What all of these entities seem to have in common is the curious possession of positive ontic status without spatiotemporal extension. That is to say, an abstract object is a real entity that does not exist in space and time. Something exists in space (or time) if it has spatial (or temporal) duration or location.5 If something has spatiotemporal duration, that is to say it has a length, height, size, etc. If something has spatiotemporal location, it exists at a certain place during a certain time; we could ask of its whereabouts and when-abouts. But abstract objects seem to lack spatiotemporal duration and location. So if we take numbers as abstract entities, for example, it would be nonsensical inquire of the number 2’s dimensions or whereabouts. Or in the words of Corey Washington, “When’s the last time you bumped into the number one? When’s the last time you slipped on the concept of truth? Or saw a justice sitting by the side of the road?” In addition to being immaterial and timeless, abstract entities are generally understood as being utterly effete (the number two cannot cause anything) and metaphysically necessary (it exists in every possible world).
The conceptualist argument, in short, is that the best metaphysical grounding for the existence of abstract objects is an omniscient mind whose concepts they are.6 But unfortunately, the argument can get pretty intricate not only in its formulation, but also its defense. But to keep things clear, a tidy outline of it can be as follows:
- Abstract objects either:
a. do not exist,
b. are independently existing realities, or
c. exist as mental concepts. - Abstract objects:
a. exist and
b. are not independently existing realities. - Therefore, abstract objects exist and are mental concepts.
- If abstract objects exist and are mental concepts, they exist as mental concepts in the mind of an omniscient, metaphysically necessary being.
- Therefore, an omniscient, metaphysically necessary being exists.
Premise (1) should be uncontroversial, for there has traditionally been three main schools of thought with respect to the ontological status of abstract objects: (1a) nominalism, (1b) Platonism, and (1c) conceptualism. Premise (2) would involve a refutation of the alternatives to conceptualism: (1a) nominalism and (1b) Platonism. With that in mind, lets take a brief look at each.
2a. Abstract objects exist.
As just said, to establish this, a refutation of (1a) is in order. (1a) implies the philosophical view known as nominalism. There are varying degrees of nominalism. Most forms of nominalism amount to trying to save a naturalistic ontology.7 If we take naturalism to be the worldview that the only thing that exists is the spatiotemporal universe and nothing more, then there is immediate problem with what to do with abstract objects. Either the naturalist denies their existence outright (extreme nominalism) or he reduces them to spatiotemporal entities and/or mere linguistic or categorical referents (nominalism and moderate nominalism) that don’t have objective meaning (fictionalsim). Arguments against nominalist views are numerous and complex, but I’ll only note a few that can generally be raised against them all. First, it is extremely hard to see how we can, in theory, reduce or eliminate the entire realm of abstract objects and not have our knowledge of the physical world severely depreciated if not precluded. Abstract objects seem truly indispensable. Second, most forms of nominalism run into major difficulty with the problem of property exemplification. It seems concrete objects do have properties like shape, color, size and so forth that are not identical to one other nor to the object itself, but are real, independent nonphysical properties had by the object. The realist (or conceptualist in this case) has a straightforward and powerful account for these features. The extreme nominalist has to give his reductive analysis to show this is not the case.
2b. Abstract objects are not independently existing realities.
Again, demonstrating (2b) would be to show (1b) false. (1b) implies the philosophical view known as Platonism. In contrast to nominalism, Platonism in its basic form has us believe that while abstract entities do in fact exist, their existence is nonetheless inexplicable; they exist naturally and necessarily a se. While there are several major arguments often leveled against Platonism, I’ll briefly mention just one.8 The first problem springs from the nature of abstract objects themselves—their causal inefficacy. If abstract objects are truly effete and exist independently from minds, then how is it we have knowledge of them? If Platonism were correct, then it seems reasoning would be impossible. For Platonism would preclude our having a working knowledge of abstract entities like numbers and the laws of logic, each of which are indispensable to our epistemic life.
So it seems neither (1a) or (1b) are plausible accounts for the metaphysical grounding of abstract objects. But that leaves us with (1c); or rather
3. Therefore, abstract objects exist and are mental concepts.
With Platonism, conceptualism affirms the necessary existence of abstract objects but maintains their existence is conceptual in nature; that is, they exist not inexplicably a se but as concepts to be had by minds. But the theist usually has a different kind of conceptualism in mind than what is typically inferred by the term. A basic doctrine of conceptualism holds that abstract objects are grounded in the minds of humans as perceiving agents, as developed by Kant. The theist takes it a step further and argues human minds are an inadequate basis on which to metaphysically ground abstract objects, and so argues further that only an omniscient mind has the capacity to ground such. For our purposes, the former can be called Kantian conceptualism and the latter theistic conceptualism. This distinction leads the theist into a defense of (4), which involves a refutation of the Kantian-type conceptualism:
4. If abstract objects exist and are mental concepts, they exist as mental concepts in the mind of an omniscient, metaphysically necessary being.
Why can’t abstract objects be metaphysically grounded in human minds? Alvin Plantinga provides us with at lest one reason:9
It seems plausible to think of numbers as dependent upon or even constituted by intellectual activity. But there are too many of them to arise as a result of human intellectual activity. We should therefore think of them as… the concepts of an unlimited mind: a divine mind.
The main point is that there are far too many abstract entities that have not yet been the object of human conception. There are multitudes of numbers, sets, properties, etc. not thought of—an infinite amount, even! But if abstract objects exist and are mental concepts, but are not mental concepts of humans, then they must conceptually reside within another kind of mind—a mind that has the capacity to host such (metaphysically necessary) infinitude. In short, if not humanly conceptual on account of our ignorance, abstract objects are most plausibly divinely conceptual on account of omniscience. And so we may now accept premise (4). And this of course brings us to our conclusion:
5. Therefore, an omniscient, metaphysically necessary being exists.
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- The conceptualist argument is closely related to the transcendental argument for the existence of God.
- This debate is freely available online in both audio and transcribed formatting here.
- For example, see the Craig-Tooley debate and the Craig-Washington debate.
- Quentin Smith, ‘The Conceptualist Argument for God’s Existence’, Faith and Philosophy, 11, 1 (January 1994), pp. 38-49. In fact, the whole chapter is available online here.
- I have borrowed this wording from J. P. Moreland, Universals (McGill-Queen’s University Press: London, 2001), p. 17.
- This definition is borrowed from J. P. Moreland and William Lane Craig, Philosophical Foundations for a Christian Worldview (IL: InterVarsity Press, 2003), p. 498.
- For a good survey of the major arguments against the various forms of nominalism, see J. P. Moreland, Universals (McGill-Queen’s University Press: London, 2001), pp. 23-73. And for the problems naturalists face with abstract objects, namely, properties, see J. P. Moreland, “Naturalism and the Ontological Status of Properties,” in Naturalism: A Critical Analysis, ed. William Lane Craig and J. P. Moreland, Routledge Studies in Twentieth-Century Philosophy 5 (London: Routledge, 2000), 67-109.
- For a good survey of the major arguments against Platonism, see two articles by Paul Benacerraf, “What Numbers Could Not Be,” Philosophical Review 74 (1965): 47-73 and “Mathematical Truth,” Journal of Philosophy 70 (1973): 661-79.
- Alvin Plantinga, “Two Dozen (or so) Theistic Arguments,” lecture presented at the 33rd Annual Philosophy Conference, Wheaton College, October 23-25, 1986.
July 30th, 2006 at 8:32 pm
Great stuff, Chad! I don’t recall ever reading much about that argument before.
November 2nd, 2007 at 2:51 am
I don’t think that our capacity for abstract thought proves, as a matter of logical necessity, the existence of a god.
All abstraction starts from sensory experience. We cannot conceive of anything that we have not encountered “out there” in the physical universe. The evolution of obviously adaptive survival skills like memory and pattern recognition adequately explains our capacity to look at a maple and an oak and think, “Gee, they have a lot in common. I’ll call them both trees.”
Even things that don’t exist… things like mermaids… are derived from creatures we have observed. It exists in my mind, and I wouldn’t dare to call it as “real” as a fish or a woman. It only exists in my mind because of what I have SEEN in my life.
I don’t see how anyone has proven that an idea needs to exist in another all knowing mind in order for me to conceive of it. It actually sounds kind of weird to me to think that my mind makes some sort of cosmic withdrawl from the idea bank of god’s mind everytime I have a new idea. Where is that mental ATM machine? The pineal gland?
I honestly wonder whether god himself is nothing more than an abstraction based on our ability to conceive of a disembodied mind that knows everything. Seriously, even god could be “made up” based on what we have encountered in the universe. You might say he is the personification of the universe– an abstraction based on a relationship we have with the physical world. He exists the same way as a mermaid does– in our minds.
Sometimes I question that value of these kinds of debates, because I don’t think one side ever convinces the other. Still, you seem like a smart guy, I enjoyed your post, and I find the arguments interesting.
November 3rd, 2007 at 8:24 pm
Thanks for the compliments and the time considering the argument. I’ve actually been working on revising this argument, so perhaps the near-future version of it might be more compelling.
As for a response to the above, I think you’ve confused concepts deducible from abstraction with what philosophers call abstract objects (or, less often, abstract concepts). Perhaps this is my fault for wording it misleadingly in the post. I use “abstract concept’ and ‘abstract object’ interchangeably, designating certain specific objects such as numbers, sets, other mathematical entitles such as functions, propositions, laws, properties, values, and so on. These are definite and discrete objects. Abstract objects (well, at least some of them) are universally recognized as necessarily existing, causally impotent nonspatiotemporal entities. Abstract objects are so called because they are contrasted with concrete objects, which are contingent, can stand in causal relations, and are spatiotemporal.
So understood, the debate is not about abstraction, such as how we are able to formulate concepts, but rather the ontological status of entities called abstract objects.
November 5th, 2007 at 1:43 am
I think what you refer to as my “confusion” is that I don’t see how concepts deducible from abstraction are any different from abstract objects.
Let’s take numbers as prototypical abstract objects (with the understanding that mathematical entities are necessarily derivatives of numbers). The only reason numbers exist as abstract objects is because they are concepts we deduced through abstraction. The numbers one and two did not exist until a physical mind decided to distinguish between quantities. Our ability to do that can be adequately explained by the fact that organisms that can categorize information are that much closer to being able to use information and thereby increase their survival capacity.
Okay, they exist in a “mind world” and cannot be properly called physical entities. I agree they are impotent, but I believe once human minds no longer exist there will be no more “numbers.” What’s more, I don’t see how anyone has proven they must, by definition, be eternal or exist in the mind of an omniscient being. A number is just a label for an idea– a useful mental category for information.
In any event, I’ll keep my eye out for your next post on this subject.
Thanks for taking the time to respond.
November 5th, 2007 at 2:18 am
A,
Thanks again for the comment and that clarification. Indeed, I see you are not at all confused about the issue at hand! Your view strikes at the very heart of the argument and has been defended professionally by a host of first rate philosophers such as Hartry Field and Mark Balaguer, aptly calling it fictionalism. So you’re in good company!
Suffice it to say I do not think fictionalism provides an adequate treatment of the sort of abstracta relevant to my argument and seems to generate more problems than it was crafted to solve (see the below link). One main problem with such a view is its extreme denial of obviously true descriptive states of affairs the truth of which seem to be grounded in certain abstracta, such as “4 is divisible by 2″ or “two entities x and y can have the same shape and size” by treating them merely as useful fictions. Further, I think this view falls victim to Quine’s indispensability arguments, who was himself no friend of realist commitments. But admittedly, I’ll have to substantiate these claims with further argument in the post to come.
I hope to deal more thoroughly with this view in the future. In the mean time, check out Stanford Encyclopedia of Philosophy’s entry on the topic. Thanks again for your comment!
November 10th, 2007 at 7:08 am
Yours is the only site I could find with an understandable explanation of the conceptualist argument. Thanks for providing it. It’s always hard for people like me who have an interest in, but no real knowledge of philosophy to refute, or even raise an objection to such an argument. Please forgive me if I commit any error which might, well, make me look like a dunce.
The best way is to go point-by-point. I can accept the definition of an abstract concept. Don’t 2a and 2b follow from the definition of abstract objects? Please elaborate on this point.
The most problematic is 4, and the defense you cite of it seems to me to miss the point. Yes, there are too many numbers to be a product of human minds, if every number is a separate abstract concept . But I think it’s enough to have about 10 abstract concepts (ie, axioms of arithmetic) to come up with ALL the numbers. So this doesn’t necessarily show that the human mind is incapable of having all abstract objects as mental concepts.
November 11th, 2007 at 1:58 am
Thanks for your comment and compliments, Anonick!
I am currently working on a much clearer and researched article on the conceptualist argument, which may answer some of your queries in terms of structure and content (no doubt the current one needs work in both areas).
But to respond directly to your very good questions: To assume (2a) and (2b) follow by definition in my argument would be to commit the fallacy of begging the question. Surely nobody would accept the argument if I defined its central object of contention in my favor! The logic of the argument is valid, so if it is to be shown sound (2a) and (2b) need argumentative support for (3) to follow, as would (4). Which brings me to your second question.
To illustrate why there must be an infinite number of mathematical entities, consider sets:
What this figure shows is that if we assume the existence of just one set, we can deduce from that one set alone an infinite number of non-identical sets. Now if some entity is not identical to itself or another, then it is numerically distinct. But if there are an infinite number of numerically distinct real entities, then there must be an infinite number of corresponding numbers, repetitive as they may be. In other words, the infinitude of the natural number line can be illustrated by putting them into a one-to-one correspondence with numerically-distinct entities. If we have 22 red balls before us, it does precious little to say the number 22 doesn’t exist just because it is expressed with two twos. It is helpful to draw a distinction between the expression of a number (which are all multiples of are simply [0-9]) and the entity that some number expresses.
Keep in mind also the other feature of abstract objects which I argued a human mind couldn’t account for—their necessity. If human minds are not necessary, and abstract objects are necessary, then it follows that no human mind could ground abstract objects. I’m hoping to elaborate more on these and other points in favor of (4) in the new post.
November 12th, 2007 at 1:45 am
It may be true that we need abstract concepts to describe the universe, but why would anyone else need them? I think numbers are needed to describe the universe, but the universe doesn’t need them. We humans need the concepts for describing the universe, and they don’t exist outside human minds, outside our own attempts to analyse nature.
We need a law to describe gravity, but nature itself doesn’t, because gravity is the result of a process. I’m of the view that nature does not need any laws, and the laws we know are the results of processes, which, at the lowest level, might not follow any “laws”, but rather depend of initial conditions, or be the result of a random selection at that time.
Can’t say whether abstract concepts really exist without humans existing, unless you define how exactly an abtract concept exists without minds. Do you mean, as I say above, that when nature follows certain laws, and these laws being abstract concepts, we effectively have an abstract concept that is necessary for the universe?
November 12th, 2007 at 3:21 am
Anonick,
To ask why the universe needs abstracta is equal to asking why the universe needs space and time. Abstract objects are as necessary as space and time themselves in a space-time universe. The only difference is that abstracta are necessary conditions for all possible worlds whereas space and time are only sufficient conditions for some. In other words, space and time cannot exist without abstracta, but abstracta can exist without space and time.
You might rightly object that this does not seem like a valid parallel. True, but only insofar as we are considering “abstract object” in an epistemological sense and not in an ontological, or metaphysical sense. The former holds that an abstract object is one that is placed before the mind by an act of abstraction. The latter holds that an abstract object is not “created,” so to speak, by an act of abstraction by some mind, but is a real and distinct entity. My argument does not utilize the epistemological sense of abstract object. The claim that that’s all an abstract object is requires argument, which is precisely the nominalist’s strategy. My argument asks, “If an abstract object is a real and distinct entity, what grounds, or accounts for, its existence?”
I certainly don’t mean that “when nature follows certain laws, and these laws being abstract concepts, we effectively have an abstract concept that is necessary for the universe,” for that would render abstracta mere inductive generalizations rather than necessary objects.
Most of the questions I’ve gotten on this argument as it appears here have been aimed at the epistemological sense of abstract objects. I take this to be a flaw in my description above—one that I hope to remedy this next time around.
Thanks again for your comment! I love to hear thoughts like these. I am doing my best at considering and answering them.
November 12th, 2007 at 10:39 am
Oh, Ok… I realise I didn’t thank you last time for the clarifications. Thanks so much, I now realise what you mean by an abstract object exactly, and why 2a doesn’t follow from the definition.
So, I’m siding with the nominalists, I guess. I don’t have a background in philosophy, so I must argue only when I’m well-equipped to do so, I guess. Otherwise I’ll just keep annoying you with my silly objections.
I’ll going to read up on nominalism, and then come back to you.
November 13th, 2007 at 12:53 am
I found a parallel to steps one and two in what is called “The problem of universals”. Indeed, it seems to be the same problem.
There are not just three options between which we have to choose what abstract concepts are: there are subdivisions between them. Realism as given my Plato might seem untenable, but moderate realism seems to me to provide the nicest answer, and one that I happen to agree with.
As regards numbers, for example, it suffices to see that basic arithmetic was discovered independently in many parts of the world. While we may not bump into numbers, we know that sets are an abstraction performed by the human mind to represent a group of objects. “Numbers” are an abstraction of counting, a way to distinguish fifty-nine sheep from sixty.
Sure, modern mathematics has moved far beyond what is found in nature. We don’t have imaginary numbers existing anywhere in nature, nor can they be viewed as an abstraction of some process. So, to say that one philosophy applies to every abstract object would be to say that the human mind explores nature in just one manner. Most of modern mathematics is just mental concepts.
But that doesn’t mean the human mind is incapable of creating the infinity of the natural number line, or the “Incompleteness” of the systems it itself has created. Godel’s theorem may be viewed as a limitation which humans have discovered on systems they themselves have created.
But I’m not a mathematician, my stronghold is physics. But I hope I’m not wrong in any of the arguments above. Also, I don’t see why you say that abstract objects are “necessary”. Please elaborate on that.
November 13th, 2007 at 3:48 am
Chad, this is good stuff. I came across your post while doing some research on the Conceptualist Argument (CA). Kudos!
I have some thoughts, but since I am new to theistic conceptual realism (I am actually a Thomist), please feel free to clarify. You pointed out in the beginning of your entry that the CA is quite similar to TAG, and I agree. It seems that the notion of abstract, universal, and invariant entities has its strongest case in the laws of logic. While an infinite set of numbers may require a necessary mind, it is perhaps easier to demonstrate the necessity of logic, at least in my opinion.
Here is how I might re-formulate the argument:
1. Laws of logic either:
a. are contingent,
b. exist necessarily and mind-independently,
c. exist as necessary concepts of a necessary mind
2. Laws of logic:
a. exist necessarily and
b. do not exist mind-independently
3. Therefore, laws of logic exist as necessary concepts of a necessary mind
4. This necessary mind is God.
I combined some of the premises in the second half of the proof. In any case, you have done some great work. I look forward to digging into this deeper.
November 13th, 2007 at 7:29 pm
Thanks for the suggestion, Doug. I wavered over whether or not to present the argument similarly, though with numbers as the object. However, I decided to frame my argument as I did for a few reasons.
The main one is to allow those interested in defending a more precise statement of the argument the liberty of choosing which abstract object they think best suits their aim (just as you have).
I give more attention to numbers because they, and other mathematical objects such as sets, are usually taken to be paradigmatic examples of abstract objects by most philosophers.
I want the conceptualist argument to be distinguished from the transcendental argument rather than risk people seeing the former as a species of the latter (I think the opposite characterization is more fitting). If I paid special attention to the laws of logic, I think it less likely people would see the broader scope the conceptualist argument has.
November 13th, 2007 at 7:37 pm
Anonick,
I have taken note of and try to address your questions in the new article. I apologize for not being speedy or clear in my replies, even more so for not replying directly to your last comment. I hope you’ll stick around to see if my future post on the topic is more compelling! Feel free to keep posting, as your thoughts inform me as to what needs work for it to be a better argument.
November 13th, 2007 at 8:31 pm
Good point, Chad. I have never been mathematically inclined, and only recently have I delved into the philosophy of mathematics, so the issue of numbers has always been a bit shaky for me. Numbers are certainly a good example, though, as are sets, propositions, values, etc.
November 14th, 2007 at 2:12 pm
Chad:
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.
November 15th, 2007 at 2:24 am
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.
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.
November 15th, 2007 at 2:45 pm
You said:
“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.
November 17th, 2007 at 9:31 am
I think, A., that 4 divided by 2 equals 2 was true before anyone knew it, or even formed a concept of a number because of the same reason that one would argue if he was questioned that, even if there were no humans around, could 4 stars have formed two groups of two each? I would say they would have, because even if the four stars didn’t know they were “4″ in “number”, and the resulting binary system didn’t know they were composed of two stars each, they certainly would have grouped together as said.
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.
November 19th, 2007 at 8:10 pm
Anonick:
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.
November 20th, 2007 at 1:13 am
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
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:
His argument can be outlined thus
(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:
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.
November 20th, 2007 at 10:21 am
Wow.
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.
December 3rd, 2007 at 3:53 am
Chad, premise (2b) has always been difficult for me to support. I’m currently reading Philosophy of Mathematics by Putnam and Benacerraf, but could you formalize an argument against the “mind-independent” theory?
December 4th, 2007 at 3:57 am
Doug,
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:
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?
December 5th, 2007 at 1:05 am
I hadn’t even considered the argument against an actual infinite, but now that you mention it, it does seem like a strong argument against Platonism.
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.