# An Argument Against Naturalism from Abstract Objects

Some naturalists, like Quine, feel compelled to admit abstract objects, like numbers, sets, and propositions, into their ontology. But I’ve always had the sense that abstract objects are incompatible with naturalism. Here I will lay out some premises about abstract objects and naturalism that appear fairly intuitive to me. I will then represent those premises in logical notation, and demonstrate that they do, indeed, serve as a defeater for naturalism.

1. If a thing1 is natural then it is possibly not the case that there exists a thing2 where thing2 is natural and identical to thing1.

In other words, if a think is natural, then it possibly doesn’t exist.

2. There exists some proposition such that necessarily that proposition is true.

For example, the mathematical proposition ‘2 + 3 = 5’ is necessarily true and cannot be false.

3. For all propositions, necessarily, if a proposition is true, then there exists some thing1 such that thing1 is abstract and thing1 is identical to that proposition.

This is just to say that it is necessary that if a proposition is true, then it is an existing abstract object. After all, it would seem odd to predicate a truth-value to a proposition, but deny that said proposition doesn’t exist.

4. If there exists a proposition that is necessarily true, and everything is natural, then for all proposition, necessarily, if there exists a thing1 that is an abstract object identical to that proposition, then there exists a natural thing2 identical to that proposition.

I defend this premise on the grounds that, if a proposition is natural, then in every world where it obtains, it obtains as a natural proposition. That is, if ‘natural’ is predicated of a proposition, it is essentially predicated of it, which is to say that in every world where that proposition exists, it exists as a natural object.

From these four premise, we can conclude that naturalism, which I take to be the claim that everything is natural, is false.

The deduction is as follows:

Let
Nx – x is natural
Tp – p is true
Ax – x is an abstract object

1. (∀x){Nx ⊃ ♢~(∃y)[Ny & (y=x)]} (premise)
2. (∃p)☐Tp (premise)
3. (∀p)☐[(Tp ⊃ (∃x)(Ax & (x=p))] (premise)
4. (∃p)(☐Tp & (∀x)Nx) ⊃ (∀p)☐{(∃x)[Ax & (x=p)] ⊃ (∃y)[Ny & (y=p)]} (premise)
5. (∀p)☐{(∃x)[Ax & (x=p)] ⊃ (∃y)[Ny & (y=p)]} (IP)
6. ☐Tu (2 EI)
7. ☐[(Tu ⊃ (∃x)(Ax & (x=u))] (3 UI)
8. ☐(∃x)(Ax & (x=u) (6,7 MMP)
9. ☐{(∃x)[Ax & (x=u)] ⊃ (∃y)[Ny & (y=u)]} (5, UI)
10. ☐(∃y)[Ny & (y=u)] (8,9 MMP)
11. (∃y)[Ny & (y=u)] (10 NE)
12. Nv & (v=u) (11 EI)
13. Nv ⊃ ♢~(∃y)[Ny & (y=v)] (1 UI)
14. Nv (12 Simp)
15. ♢~(∃y)[Ny & (y=v)] (13,14 MP)
16. (v=u) (12 Simp)
17. ♢~(∃y)[Ny & (y=u)] (15,16 ID)
18. ~☐(∃y)[Ny & (y=u)] (17 MN)
19. ☐(∃y)[Ny & (y=u)] & ~☐(∃y)[Ny & (y=u)] (10,18 Conj)
20. ~(∀p)☐{(∃x)[Ax & (x=p)] ⊃ (∃y)[Ny & (y=p)]} (IP 5-19)
21. ~(∃p)(☐Tp & (∀x)Nx) (4,20 MT)
22. (∀p)~(☐Tp & (∀x)Nx) (21, QN)
23. ☐Tu (2 EI)
24. ~(☐Tu & (∀x)Nx) (22 UI)
25. ~☐Tu ∨ ~(∀x)Nx (24 DeM)
26. ~~☐Tu (23 DN)
27. ~(∀x)Nx (25,26 DS)
28. (∃x)~Nx (27 QN)

Line 28 is our conclusion, namely that something exists that is not natural. I take this to be incompatible with naturalism. Therefore, I take the existence of abstract objects, like propositions, to be a defeater for naturalism. I suspect that the naturalist will take issue with one or more of the premises, but at a cost. Likely (4) will require the most defending. Again, (4) says that, given naturalism and the existence of necessary truths, it is necessarily the case that if a proposition exists as an abstract object, it will exist as a natural object. If one denies this, then it would seem possible that an abstract object be natural and possibly not natural. But then in what way is it the same sort of thing? It seems odd to me that a proposition is a natural thing in this world, but a non-natural thing in other possible worlds. For it seems to me that the property of being natural is an essential property. If something is natural, it is necessary that it is natural. Thus, if naturalism is true, then all abstract objects are natural and essentially natural. But our argument shows, by indirect proof, that it is possible for there to exist an abstract object that is not natural. Giving up on (4) entails that ‘natural’ is non-essential to some things, and I find that to be implausible.

(My thanks to Skepticism First on Twitter, who dialogued with me on this argument and pushed me in some new directions)

Posted on March 26, 2014, in Naturalism and tagged , , , . Bookmark the permalink. 5 Comments.

• ### Comments 5

1. I have no doubt that some naturalists will say that some naturally existing things necessarily exist, and so abstract objects could be those things. Of course, “natural” is a nearly vacuous term at this point. If naturalism hopes, at the end of the day, to be monistic, then admitting that some natural things are contingent while others are necessary is not a step in the right direction. Let me put it this way, if you think every part of the multiverse could logically fail to exist, but that there would still be “things”, then you’re a funny sort of naturalist.

Again, some naturalists are platonists, and others are some variety of non-reductive dualist (e.g. epiphenomenalists and panpsychists). They all seem to want to grant that the mental is ontological dependent on the physical. But most of them also deny the principle of sufficient reason, and so admit of brute facts. One wonders why the universe could be an inexplicable brute fact, but an ontologically independent mind an absurdity, but that is neither here, nor there. I suspect that, as some have said, the only unifying feature among the family of resemblances that is “naturalism” is the denial of a transcendent personal God (you cannot say the denial of God simpliciter, since some have deified nature).

2. Hi there,

This is an interesting post. Ultimately I disagree, but I do think that abstract objects pose a unique problem for naturalism, and that anyone seriously committed to that view (as I am) should think long and hard about how to make sense of them within the scope of a naturalistic ontology.

In your comment you raise some issues regarding the meaning of naturalism. You mention, for example, that not all naturalists would necessarily consider themselves monists, and the philosophy of mind is a land of plenty for these swine. Another I think worth mentioning is whether naturalism is taken as a methodological or metaphysical thesis. Quine’s argument for the existence of numbers, for example, might run like so: our best theories of physical science make heavy use of mathematics, and so we should be committed to the existence of the entities referred to in mathematics. In this sense the metaphysics Quine is doing is perfectly naturalised (slogan: what exists is whatever science tells us exists) but questions concerning the immateriality of such entities is left unaddressed. As you mention, if this is all we mean by naturalism then obviously we’re casting the net very wide. For this reason I see the worry as relevant specifically for (explicitly monistic) materialism, and this is the spirit in which I have taken your post.

Coming to your argument, then, while I do think it might be possible to quibble whether material objects exist contingently, that’s certainly not one I’d be inclined to make – so let’s accept premise 1. Premise 2 – that some propositions are necessary – also seems clearly true (I’m referring to the informal premises, here, of course).

I find premise 3 extremely puzzling. It’s not at all clear to me that a proposition being necessarily true implies there’s any abstract anything. If nurgles are defined as a kind of gurgle, and all gurgles by definition have burbles, then it’s a necessary truth that all nurgles have burbles. That’s a proposition, but I don’t think I’ve just brought any abstract objects into existence (or discovered any previously unknown ones). Clearly we can invent all sorts of trite examples like this, so it seems to me like accepting premise 3 would force us to implausibly swell our ontology with useless entities. If 3+2=5 is an abstract object, then how about 3+1+2-1=5? Maybe they’re the same abstract object? If so, how does the other abstract object 1-1=0 mediate this identity? Perhaps I’m misreading you here, so my apologies if that is the case. Either way I see very little reason to accept premise 3, and I find it hard to imagine that someone like Quine would either. While I know that Quine thought we should be committed to the existence of sets and numbers, I’d be very interested to hear where exactly he wrote about propositions in this way.

The thrust of your argument, if I’ve understood correctly, is to show that the existence of necessary truths implies the existence of necessarily existing entities, which refutes naturalism if naturalism entails that all existents exist contingently. While I have my doubts about premise 3, if accepted it would give us an abstract object. Moreover, if this abstract object is a necessary truth, the idea is that this would truth would hold in all possible worlds and thus give us the existence of the abstract object in all possible world, i.e. its existence would be necessary. Please tell me if I’ve got that wrong.

Here’s how I think about the problem of abstract objects. The problem emerges not because propositions exist and are abstract (this seems to me a fairly crazy to say), but from the fact that certain propositions which are very useful to us appear to refer to entities which do not appear in the world. That certain truths are necessary doesn’t of itself suggest anything about the existence of abstract objects, because many necessary truths are non-referring. Utility matters, because if some truths which appear to reference entities which don’t appear in the world are useful, then the thesis that referents are real becomes a candidate for inference to the best explanation. This is, I believe, how Quine approached matters in the case of numbers and sets.

But once we see that the issue is really a question of how we underwrite statements that allegedly reference immaterial entities, we can see that other options appear for the materialist: notably fictionalism or nominalism. At this point of course the game is on – the nominalist can argue that the realist is left with all sorts of metaphysical weirdness (and the interaction problems that are part and parcel of this), while the realist can point out that how nominalism works is actually pretty hard to spell out, especially in the case of something like imaginary or irrational numbers. (Though hey, isn’t this just more points for fictionalism?) The necessity of mathematical truths remains a mystery for everyone, and as such is a distinct problem from that of the ontology of abstract entities. What we can gather, though, is one thing that the materialist absolutely cannot be is a naive realist about abstract objects. It’s a tough cookie, but in the end I think that’s a skin which has to be shed.

Sam

• Hi Sam,

Thanks for your response. You bring up several important criticisms and points of clarification, and I appreciate it. Here are some of my thoughts on what you’ve said:

Quine’s argument for the existence of numbers, for example, might run like so: our best theories of physical science make heavy use of mathematics, and so we should be committed to the existence of the entities referred to in mathematics. In this sense the metaphysics Quine is doing is perfectly naturalised (slogan: what exists is whatever science tells us exists) but questions concerning the immateriality of such entities is left unaddressed. As you mention, if this is all we mean by naturalism then obviously we’re casting the net very wide. For this reason I see the worry as relevant specifically for (explicitly monistic) materialism, and this is the spirit in which I have taken your post.

Indeed, I would agree with your assessment here. I think many naturalists are willing to grant the existence of mathematical entities simply because they are needed for our best scientific theories. And because those entities are needed for our theories of nature, they are “naturalized.” I do wonder, suppose it turns out that a transcendent personal necessarily existing explanation is needed to account for our theories, would that “naturalize” the explanation? At the end of the day, it seems like something is “naturalized” if a self-professed naturalist can’t rationally exclude its existence.

That being said, it seems odd to me that if numbers, sets, propositions, and other such entities are necessary, that at least some natural objects exist in every possible world. In other words, one would have to say that there is no possible in which nothing is natural, or that somethings are only accidentally natural (because they are non-natural in other possible worlds). And again, it seems odd to me that some natural things are concrete, contingent, and physical while other things, things needed to explain the concrete, contingent, and physical, can be abstract, necessary, and immaterial. But who am I to place limits on what “natural” means.

I find premise 3 extremely puzzling. It’s not at all clear to me that a proposition being necessarily true implies there’s any abstract anything. If nurgles are defined as a kind of gurgle, and all gurgles by definition have burbles, then it’s a necessary truth that all nurgles have burbles. That’s a proposition, but I don’t think I’ve just brought any abstract objects into existence (or discovered any previously unknown ones).

The point of the argument is to point out a tension between naturalism and abstract objects. I picked propositions as an example just because I thought it would be easy to represent in LPC. This may have added to the controversial nature of the argument, since some people accept that abstract objects exist, bu that propositions are not abstract objects. The idea is that there are certain abstract objects that exist in all possible worlds. If you reject the existence of abstract objects or of necessarily existing abstract objects, the force of the argument is avoided of course. But, if an abstract object can be predicated with some property in w1, it seems to me that it should exist in that world. Perhaps that is a controversial claim. Some think that things can have properties and also not exist. I can’t really wrap my mind around something like that, nor am I motivated to try to wrap my mind around it.

Clearly we can invent all sorts of trite examples like this, so it seems to me like accepting premise 3 would force us to implausibly swell our ontology with useless entities. If 3+2=5 is an abstract object, then how about 3+1+2-1=5? Maybe they’re the same abstract object? If so, how does the other abstract object 1-1=0 mediate this identity? Perhaps I’m misreading you here, so my apologies if that is the case. Either way I see very little reason to accept premise 3, and I find it hard to imagine that someone like Quine would either. While I know that Quine thought we should be committed to the existence of sets and numbers, I’d be very interested to hear where exactly he wrote about propositions in this way.

I don’t think you are misreading me. Frankly, I don’t see the problem or implausibility of “swelling” our ontology to the point of admitting an infinity of abstract objects. Furthermore, I don’t think we would be inventing them so much as discovering them. They already exist. As for whether the entity has immediate use for me, I don’t see how that is relevant to the existence of the entity. It seems to me that if the existence of the entity is entailed by some other admissions that are necessary for me to grant, then I need to grant those entities. For example, I might grant the existence of some set of natural numbers in order to apply some formula on a physical phenomenon. Am I going to only permit the existence of just those numbers needed for the formula, and wait to admit other numbers until they are needed? No, I think once you’ve admitted some natural numbers in, you’ve got to grant all of them, and so your ontology is already fairly swollen. What does it matter if you then start admitting other abstract objects? Is the realty in “Platonic Heaven” at a premium? Just to be clear, I don’t really see this as an issue of parsimony, since I think such entities are, in some way, needed insofar as entities within the same domain are needed to explain other things. And once those entities are granted, the other entities are necessitated by those that are explicitly admitted.

The thrust of your argument, if I’ve understood correctly, is to show that the existence of necessary truths implies the existence of necessarily existing entities, which refutes naturalism if naturalism entails that all existents exist contingently. While I have my doubts about premise 3, if accepted it would give us an abstract object. Moreover, if this abstract object is a necessary truth, the idea is that this would truth would hold in all possible worlds and thus give us the existence of the abstract object in all possible world, i.e. its existence would be necessary. Please tell me if I’ve got that wrong.

Yes, (2) and (3) together imply the necessary existence of some abstract objects. I think (3) could be reformulated to drop out the question of whether propositions are abstract. For instance, I could say that there exists some number, n, such that n is necessarily odd. Then the question is whether it makes sense to say “there is a number, n, that is odd and n refers to nothing that exists”. I don’t think you can.

Here’s how I think about the problem of abstract objects. The problem emerges not because propositions exist and are abstract (this seems to me a fairly crazy to say), but from the fact that certain propositions which are very useful to us appear to refer to entities which do not appear in the world. That certain truths are necessary doesn’t of itself suggest anything about the existence of abstract objects, because many necessary truths are non-referring. Utility matters, because if some truths which appear to reference entities which don’t appear in the world are useful, then the thesis that referents are real becomes a candidate for inference to the best explanation. This is, I believe, how Quine approached matters in the case of numbers and sets.

Yes, this is the more classical way of describing the problem of abstract objects. To be clear, my argument isn’t saying that the problem has to do with the existence of abstract objects, but with certain properties that abstract objects seem to have and that other accepted natural objects clearly lack. The problem you’ve hit on is that abstract objects, like numbers and sets, are used to understand our best scientific theories about natural phenomena, but the numbers and sets don’t appear to refer to entities within the world. I might put it this way: the sort of entities to which our concepts of numbers and sets refer are not the same sort things as other entities to which our best scientific theories refer. We can point to an iguana, detect an isotope, and feel inertia. Numbers and sets don’t seem to refer to anything like those things. I hesitate to say that they don’t refer to entities in this world though. Perhaps my argument converges with this broader observation insofar as I am looking at a particular property that some abstract objects seem to enjoy, i.e. necessary existence, and noting that some natural objects are essentially different from them in those regards. Ultimately, I would agree that nominalism or fictionalism are safer options for the naturalist. Of course, I find those views highly problematic in their own right, but I won’t get into that here.

Thanks for interacting with me on this point, and holding my feet to the fire a bit about Quine. I mentioned him as an example of a “naturalist” who is a realist for at least some abstract objects. But it is clear that he was not a platonist, and that he rejected the existence of propositions and possibilia. Quine would likely recoil at the mention of worlds, necessary existence, etc. Nonetheless, I think he made it acceptable for “naturalists” to be realists about all sorts of abstracta.

Cheers.

• Hey, thanks for your response.

Yes, I can see now that the argument you’re making isn’t supposed to rely specifically on propositions – this was something I was a bit unsure of at first, so thanks for clearing that up. I think it may be helpful, then, if I flesh out my objection purely in terms of mathematics and numbers.

But first let me just reiterate a point we agree on. You seem to have worries that naturalism means whatever naturalists want it to mean. I also have this worry, and think finding a useful, non-arbitrary way to characterise it is in everyone’s interests. When you focus on the existence of necessary entities I think you’ve picked up on the heart of this. When you say “suppose it turns out that a transcendent personal necessarily existing explanation is needed to account for our theories”, my reply would be that whenever this might be the case, a contingent transcendent existing explanation would do just as well. Physical theories are theories of the contingent physical world, and if this is what’s dictating your metaphysics then the best you’re ever going to be able to justify are contingent entities. Denying the existence of any necessary entities could well be a decent definition of naturalism (it certainly seems necessary, sufficiency I’m not so sure about.) In virtue of this, the existence of a necessary entity would indeed constitute a defeater for naturalism, in my view.

The reason to pick something like numbers is, I guess, the fact that we have necessary truths about them:

1. necessarily, 2 + 3 = 5

What’s not clear to me is how we get from necessary truths to necessarily existing entities. Sure, 1 is necessary, but maybe this necessity is just a kind of de dicto necessity or analyticity – it just follows from the definitions of ‘2’, ‘3’, ‘+’, ‘=’, and ‘5’ as we’ve defined them, and has no metaphysical content whatsoever. What we really need to make the move to the metaphysical world of abstract objects is a way of ensuring that the modality in 1 is a de re or metaphysical necessity – that 1 expresses a necessary relation of real entities. In short, we need another premise:

2. necessarily, numerals in true statements of arithmetic refer to real abstract entities.

With this we can make the deduction we need: Let M be a possible world. Then by 1., 2 + 3 = 5 is true in M. So by 2., the numeral ‘2’ refers to a real abstract object in M. So we get:

3. necessarily, the abstract object named by the numeral ‘2’ exists.

(Of course, I’m assuming that “2 + 3 = 5” carries the same semantics in M, so we don’t have to worry about ‘2’ having the same referent in M as it does in our common usage – otherwise it would be a different statement.)

The important thing about the above is that it depends on the necessity of 2. This is what I don’t think is defensible. In our conversation so far, two ways of justifying something like 2 have come up: (i) inference to best explanation of utility in physical theory, and (ii) what I’ll call the predication argument, which you express when you say “if an abstract object can be predicated with some property in w1, it seems to me that it should exist in that world”. I’ll now try to argue that while (i) can arguably justify the existence of abstract entities, it doesn’t give you that vital necessary existence, and also that (ii) is false in the sense it needs to be taken to justify 2.

First (ii). Now, while I agree that if a thing has a property then it must exist, I deny that if a linguistic term can be the argument of a linguistic predicate this is sufficient for us to conclude that it refers to some real entity – i.e. what we would need to get something like 2. This would seem to make metaphysics dependent on grammar, and that seems to me to be enough to call reductio immediately. The only way it could be independent of grammar is if it were the case that there was already an entity to correspond to every possible naming term in every possible language in every possible grammatical variation. This is what I was getting at when talking about ‘implausibly swelling the ontology’ – the issue is not parsimony so much (although there is an element of that) as all the problems that come with this – interaction problems in particular, but also fictional entities, etc. The dilemmas and paradoxes opened up by this line of thinking are, I think, sufficient to write it off as hopelessly unlikely to be true.

(i) is a much better strategy for arguing for the existence of abstract objects, but because it appeals to physical contingencies it can’t justifying the necessary existence of abstract objects, for the same reasons I mentioned in the preamble about naturalism. Note that the contingent existence of numbers is not in any way at odds with necessary truths of arithmetic, if you understand this necessity as a de dicto necessity. In a possible world where the arithmetic of natural numbers had no use in describing its physics “2 + 3 = 5” would still be true, but it would be the empty truth of nurgles and burbles. Also, I disagree when you say “I think once you’ve admitted some natural numbers in, you’ve got to grant all of them”. I just don’t see why. I really do think there’s a much better case to be made for the existence of the number 7 than there is for, say, Graham’s number to the power of itself (or higher-order cardinals for that matter). The annals of mathematical research is littered with entities that were defined but never went anywhere. But even if we did grant this, where is the necessary existence coming from?

So… that’s a long one again, sorry about that! My actual view is a kind of nominalism, so I think mathematical statements do refer (at least where useful in physical theory) but not to abstract entities – but without going into that I’ve tried to argue that even if we adopt realism about abstract objects we can do so without saying anything about the necessity of their existence, even when the truths that refer to them are necessary truths, because we don’t have justification for taking this as anything more than a de dicto necessity.

Sam

3. Great delivery. Outstanding arguments. Keep up the great
work.