The following definition of the number 2 has been brought to my attention: "[t]he set of all pairs." Against this definition, I raise the following objections:
I. It seems viciously circular: if 2 is the set of all pairs, what are pairs other than a set of two objects? The same objection seems to hold for defining 3 as the set of all triples, etc.
II. Even if this kind of definition can be used for the natural numbers, I am at a loss to understand how numbers such as fractions, irrational roots, pi, imaginary numbers, and negative numbers are to be defined in terms of sets of objects.
III. If “pairs” is understood to mean “pairs of physical objects,” then it would seem that in a possible world with only two physical objects, numbers larger than two could not exist, since e.g. “the set of all triples” would be empty.
IV. Or again, take two possible worlds which each contain only twenty physical objects, with the stipulation that each world contains objects that are entirely different than the objects in the other. In this case, it would seem that “2” in one world would not be identical to “2” in the other, since the set defined as “the set of all pairs” would have different members in each possible world.
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I. "Pairhood" could be brute. It would avoid vicious circularity. Probably the issue in this account is to avoid existential quantification over numbers, which if you do, commits you to them. On this account, you just get committed to sets. If they feel an explanation of the predication of pairhood, is needed, they may just claim that it is a brute fact.
III. I bet that there is a way around this through possible worlds. Suppose that the definition of two is the set of all pairs, but that the universal quantifier here is taken to quantify not only this world, but over other possible worlds. This may make the most sense on Lewisian modal realism, where other worlds are concrete wholes. Then, even in a world where there are only two particulars, some sense could be made of threeness. You could also make sense of the claim that there could or might have been three objects in that world.
IV. Interpreting the quantifier as unrestricted across all worlds will solve this problem too.
This isn't a problem for the nominalist who is willing to have something like "transworld" sets or classes. It's easiest to imagine how this would work with Lewisian modal realism. Thus, the set of pairs need not only include particulars in this world, but in all other non-spatio-temporally related wholes to this.
Comment forthcoming within [the set of all pairs of pairs of pairs of trios] hours.
It should be noted that I am in no way a trained metaphysician.
It is not in fact viciously circular. There are things that exist and things that exist next to those who happen to be, in relevant ways, the same sort of thing. Thus, there's a person here, and next to that person is a thing which shares the relvant property, personhood. You could think of a pair of a derivation of the number 2, but there's no reason not to think it the other way, that the number two is a theoretic placeholder for "the amount in a pair," which in turn is understood as the relvant properties that all pairs (that is, a thing and another thing, sharing a relevant property) have in common. This gives us, in my mind, a perfect way to grant 2-ness as existing without inviting the worry that non-physical things exist.
The above analysis may give us a way of understanding natural numbers, which we understand, synthetically as corresponding to properties of natural space. These correspondences, in turn, provide the proper grounds for further synthetic information about regularities in space that correspond to number. These regularities, or the set of all physical things that instantiate these regularities, allow for positing of theoretical entities such as geometrical and physical constants (pi), as well as fractions (which are really just a shorthand for an arithmetic equation) and irrational roots. I don't remember what imaginary numbers are. Negative numbers are just a twist on a subtraction relation.
your third point confuses an ontological barrier with an epistemic one. William's point is a good one, and what's more i think that if there was a possible world with two objects, from which no world with more than two objects was visible, it would only be as counterintuitive to say "3 wouldn't exist" as it would be to imagine such a world existing in the first place.
Your fourth point misunderstands, I think, the way we ought to understand something like "the set of all pairs". For instance, if your reading is a valid criticism of my position, it would be the case that every time a baby was born, the number two changes (because the set of all pairs now includes the babies eyes, ears and brain hemispheres.) Maybe this means I'm wrong, but I think, on pain of some circularity, we could say that the set of all pairs, having been constructed because of something that all the pairs have in common (that there is a thing, and another thing, which shares a relevant property with the first thing) is the best way to understand two, and adding and removing members from this set doesn't change the fact that the members of the set all have "pairness" in common.
Also, you could challenge my definition of a pair (a thing and another thing with a relevant similarity) by pointing to "a" as parasitic on the number one, but I think this would be silly, and that if anything is an a priori primitive, "a" would be such a thing.
If I'm wildly wrong about this stuff, let it be a message to you all: don't to flaunt your intuitions as if they give you cred as a metaphysician.
By the way, I'm really glad we're commenting on eachother's blogs. I'll keep it up if you will. Note that Halloran's blog is on my blog roll. You should take note of it.
I'd love to establish a blogging community among the interested graduate students.
Woo, it's the internet battle of the metaphysical intuitions! Except for Bill, who actually knows what he's talking about. Onward:
I. My worry was that this definition traded on the fact that in English we have a word "pair" (or "pairhood") that basically means "a set of two objects" (which word is defined in terms of the other seems irrelevant at this point).
A.) If "pairhood" is a brute given, doesn't that just mean that all numbers are similarly brute, and that any attempts to define them are pointless?
B.) We don't seem to have similar words (like "pair" and "triple") for all numbers, so how would you define e.g. the number 1,024,048? If you say something like "the number n is the set of all n-tuples," we're back at A.). Which just means that we should spend our time doing something more productive, I guess :)
II. My main worry on this issue was that talk of numbers as being sets of sets seems to lead to absurdities: is pi the set of all sets that have 3.1415926...etc members? Is -5 the set of all sets that have -5 members? Incidentally, I'm not sure that subtraction is the best way to understand negative numbers, since it doesn't make sense to speak e.g. of subtracting 5 apples from 3 apples. I think understanding them in terms of locations in a coordinate system works better, and that understanding numbers as being properties of space is much more promising. But that seems to me a much different approach than talking of sets of sets. Still, that leaves the issue of imaginary numbers (numbers involving the number i, which is defined to be the square root of -1). You might try defining them as a certain relation between the square root operator and the number -1, but that seems problematic to me given what the square root operator usually does.
Aside:
It's funny that you bring this up, Dan, because I'm actually reading about this kind of thing in my Meta class this semester.
The Quinean metaontology is really important in these cases. Quine's big dictum went something like this: if you have a sentence that you are committed to the truth of, anything that is a bound variable under the existential quantifier is something that you're committed to the existence of. He was thinking that it was just the sentences of our best scientific theories that we should be worried about applying this to, but I figure it works for any sentence you take to be true. This seems right - the existential quantifier simply states that some x exists, and that's what ontological commitment is all about. Unlike the material conditional, it seems to me that the existential quantifier leaves little room for argument that it's just an artifact of the logician's fevered mind.
Consider the following sentence:
1) ∃x Fx
The sentence commits you to the existence of x's, but not F's, on the Quinean metaontology, because the predicate isn't what the existential quantifier's quantifying over. Now, I know that there are second-order logics that DO, but a) I don't know much about 2nd order logics and b) there are problems that I DO know of with such logics - like I dont' think they are complete and things like that. And Quine might just be able to deny such logics without too much trouble.
So, if you think that something like this is true:
2) ∃x (x is a dog)
You are committed to the existence of a dog, but not to doghood.
But there are cases that appear to be more troublesome, I believe that it was Frank Jackson who raised these kinds of issues:
3) Red resembles orange more than red resembles blue.
4) Courage is a virtue, but cruelty is a vice.
On the Quinean meta-ontology, you're committed to red, orange, blue, courage, and cruelty.
Because of the Quinean ontology, the nominalist can beg off of any demand by the realist to give an explanation of predicates. He simply will reply that the only things he's committed to are the things the existential quantifier quantifies over.
DM Armstrong called this 'Ostrich Nominalism' because the Quinean refuses to give an account of how predicates predicate over multiple objects: 'Instead, he thrusts his head back into his desert landscape.'
On Quinean Meta-Ontology:
Devitt, Michael, "'Ostrich Nominalism' or 'Mirage Realism'" Pacific Philosophical Quarterly, 61 1980 433-9
-Devitt defends Quine against DM Armstrong.
Armstrong, D.M., "Against 'Ostrich' Nominalism: A Reply to Michael Devitt", Pacific Philosophical Quarterly, 1980, 440-9.
-Armstrong defends his theory of unviersals against Devitt.
van Inwagen, Peter, 'Meta-Ontology' , Ertkenntnis 48 1998.
-PVI is not a nominalist, but defends the meta-ontology.
Aside:
It's funny that you bring this up, Dan, because I'm actually reading about this kind of thing in my Meta class this semester.
The Quinean metaontology is really important in these cases. Quine's big dictum went something like this: if you have a sentence that you are committed to the truth of, anything that is a bound variable under the existential quantifier is something that you're committed to the existence of. He was thinking that it was just the sentences of our best scientific theories that we should be worried about applying this to, but I figure it works for any sentence you take to be true. This seems right - the existential quantifier simply states that some x exists, and that's what ontological commitment is all about. Unlike the material conditional, it seems to me that the existential quantifier leaves little room for argument that it's just an artifact of the logician's fevered mind.
Consider the following sentence:
1) ∃x Fx
The sentence commits you to the existence of x's, but not F's, on the Quinean metaontology, because the predicate isn't what the existential quantifier's quantifying over. Now, I know that there are second-order logics that DO, but a) I don't know much about 2nd order logics and b) there are problems that I DO know of with such logics - like I dont' think they are complete and things like that. And Quine might just be able to deny such logics without too much trouble.
So, if you think that something like this is true:
2) ∃x (x is a dog)
You are committed to the existence of a dog, but not to doghood.
But there are cases that appear to be more troublesome, I believe that it was Frank Jackson who raised these kinds of issues:
3) Red resembles orange more than red resembles blue.
4) Courage is a virtue, but cruelty is a vice.
On the Quinean meta-ontology, you're committed to red, orange, blue, courage, and cruelty.
Because of the Quinean ontology, the nominalist can beg off of any demand by the realist to give an explanation of predicates. He simply will reply that the only things he's committed to are the things the existential quantifier quantifies over.
DM Armstrong called this 'Ostrich Nominalism' because the Quinean refuses to give an account of how predicates predicate over multiple objects: 'Instead, he thrusts his head back into his desert landscape.'
Slick,
"Thus, there's a person here, and next to that person is a thing which shares the relvant property, personhood."
I don't know if you're interested in defending some form of nominalism or another, but if you are, the wording above is uncautious. If two things literally share a relevant property, then there is a universal. But you probably meant something more like this, right?
"There is some x and some y such that x is a person and y is a person."
That's normally the sort of account that you would want to give.
I'm not going to try and say much more on the issue of numbers though. I'm not sure I understand it well enough not to be a jackass.
Dan's point in 3 is good (or something in the neighborhood is), without modal realism. Even if you allow yourself the metaphysician's paradise (as Lewis would have it) of modal realism, you still end up with difficulties in differentiating necessarily coinstantiated properties, like trilaterality and triangularity and the like. If the nominalist needs to make an accounting of predicates like these, and he tries to do it via sets of objects across possible worlds, then there is an issue here. (pace the Quinean metaontology)
If the issue is also to get rid of 'occult entities' I'm inclined to think you'll also want to purge any talk of sets of particulars, because sets end up being ontologically pretty wonky from a hard-core physicalist perspective. But, I've no idea whether you want to go that hard core. I don't think Quine ever even managed to rid himself of sets, though I'm not positive.
Aren't there supposed to problems with getting back classical math with multiple orders of infinity on nominalism? I seem to remember Quine wondering about whether or not he could get on without it. Again, I don't know enough about philosophy of math - hell, Dan will probably vouch for me that I'm just plain bad at math.
My name is Raleigh. I won't comment as slick anymore after this..it's pretty dumb...way back in the day I always wanted slick to be my poker game, but I could never get it to stick. Now I take the opportunity to be I
It's undoubtedly true that I was uncautious, and i have no doubt that I was at many important points flat wrong.
Perhaps if I back up to provide my motivation: This all started when I made the off the cuff comment that I had "become a physicalist" in a post on my own blog. Now, this itself is only true in a loose sense. First, it's only something I've even thought about with respect to PhilMind, so if there's any reason to posit non-mind non-physical entities, I'd simply be guilty of ignorance. With respect to the mind, however, I have dualist intuitions that I had been trying to make work for a long time, and when I found myself, last month, seriously entertaining ephiphenomenalism, I came to the conclusion that I needed to just let go of dualism.
So Dan shoots back with "where in the physical world to numbers exist?" now, of course, this doesn't necessarily speak to the question as I was concerned with it, but it didn't seem hard to provide a straightforward physical account of how we get to the idea of two, namely by encountering pairs in the world. Furthermore, the idea of sharing a property seems awfully basic as well, which surely speaks to my own ignorance. It seems, though, that if we wanted to account for the concept of "pair" without circularly involving the number two, we'd want to talk, for instance like this:
"If x and y are a pair, than x does not equal y, x and y share a property which is relevant to the current discourse."
I always wanted slick to be my poker nickname that is...not game...I need to start rereading things...
Ah, I see your ontological motivation. I have a few old papers on Kim's "Mind in a Physical World" that I'll have to dig up and clean up for posting. One of Kim's critiques of Churchland-style eliminativism is basically that qualia seem to pose a unique problem, since they don't seem to be reducible or functionalizable. So, IIRC he ends up endorsing a heavily qualified version of physicalism, since dualism is a dark spooky cave (his words), and while I don't fully endorse his views I do find his line of argumentation interesting.
Numbers have usually been a go-to topic for me whenever physicalism comes up; when you spend a lot of time thinking about mathematical entities it's hard to think of them in other than Platonic terms. It still seems to me that thinking of them in terms of the properties of space is the most promising line for the physicalist, but I lack the metaphysical training to pursue this further.
Here's why I'm unmoved by III.
Suppose that everything has a singleton set. And suppose that there is a physical object o.
On just these two assumptions, there are at least as many things as there are natural numbers: once we've o, we've got {o} and {{o}} and {{{o}}}, and so on. Further, there are various sets whose elements are these set-theoretical constructions out of o; the set of all such sets that have exactly three members would, on this definition, be the number three.
Do you still see a problem here in the vicinity of III?
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