Forthcoming in Philosophia

My paper 'A Problem for Hofweber's Ontological Project' is forthcoming in Philosophia. It grew out of this blog post. The final draft is available at PhilPapers.

My preoccupation with arguing against Hofweber goes back further than that blog post, and he has always been very gracious and encouraging about it.

Illusory Explanatory Benefits in Philosophy

This post was prompted by a recent blog post of Wolfgang Schwarz's. See also a recent post of Alexander Pruss's.

“The difficult thing here is not, to dig down to the ground; no, it is to recognize that the ground that lies before us is the ground” - Wittgenstein (Remarks on the Foundations of Mathematics, VI 31, p.333).

I think we are often dogged, when doing philosophy, by a tendency to give credence to false theories on the grounds that they provide an explanation of something, when really the explanation is a pseudo-explanation, and where nothing of the kind is required if we see things aright.

In such situations, the false theory gives us something to say about some fact which resembles a real explanation, but gets us nowhere, and charmed by the idea that here we have an explanation where before there was none, we think better of the false theory. But having an explanation where before there was none is only a virtue if the explanation is a real one - if it actually helps us understand something, and does not merely have the form of an explanation.

A recent blog post by Wolfgang Schwarz called 'Magic, worlds, numbers and sets' contains an interesting example of this. It begins as follows:

'In On the Plurality or Worlds, Lewis argues that any account of what possible worlds are should explain why possible worlds represent what they represent. I am never quite sure what to make of this point. On the one hand, I have sympathy for the response that possible worlds are ways things might be; they are not things that somehow need to encode or represent how things might be. On the other hand, I can (dimly) see Lewis's point: if we have in our ontology an entity called 'the possibility that there are talking donkeys', surely the entity must have certain features that make it deserve that name. In other words, there should be an answer to the question why this particular entity X, rather than that other entity Y, is the possibility that there are talking donkeys.

It might be useful to consider parallel questions about mathematical entities.'

The example I want to concentrate on here is the one about mathematical entities, coming right after this passage. The post goes on to explore all kinds of weird stuff about Lewis, and I am not responding to it as a whole - I am just helping myself to something which occurs early on in the post, and using that as a vivid illustration of the particular failure mode in philosophy that I am trying to isolate and warn against.

(Before that, some sidenotes on the possible worlds case. The case is difficult, in part because there are various different ways of understanding 'possible worlds' in philosophy. We have some on which they really exist, some - perhaps for this reason closer to ordinary language - on which, apart from the actual world, they do not. We have some on which they are all the same sort of thing as the real, actual world, and some on which they are not. But on a lot of these, I too have sympathy for the idea that there is nothing here to explain. However, I think putting the point in terms of an emphatic identification of possible worlds with ways things might be missing the mark - for there are reasons due to Stalnaker in 'Possible Worlds' and elaborated on by Yablo in 'How in the World' for thinking that possible worlds are not to be identified with ways at all.

Secondly, regarding the point about an entity called 'the possibility that there are talking donkeys' (which of course need not be thought of as maximal or world-like) having to have certain features in virtue of which is deserves that name: perhaps that isn't so wrongheaded, but why can't the answer be along the following lines?: yes, one such feature is that, in this possilibity, there are donkeys. Another is that, in this possibility, they talk - or at least some of them do.)

To continue quoting:

'Mars has two moons, Phobos and Deimos. So here is a fact about the number 2: it is the number of moons of Mars. Following Lewis, one might argue that any account of numbers should explain in virtue of what the number 2 has this property. If we have numbers in our ontology, surely it can't be a brute fact that precisely this one is the number of moons of Mars.

The von Neumann construction of numbers gives a plausible answer to the Lewisian challenge. Here the number 2 is identified with the set { {}, { {} } }. This set has two members. The set of moons of Mars also has two members. And that is why 2, i.e. { {}, { {} } }, is the number of moons of Mars. In general, a von Neumann cardinal n is the number of Xs iff there is a one-one map between the members of n and the Xs.

By contrast, consider a primitive platonism about numbers on which the numbers are irreducible extra entities, distinct from sets, sticks, Roman emperors, and everything else. I do think the Lewisian objection has some bite here. One of the Platonic entities, call it X, is supposed to be the number 2. But what makes it the case that X, rather than Y, is the number 2, and thereby the successor of 1, and the number of moons of Mars? How come our label '2' picks out X rather than Y?

There seems to be an argument here for reducing numbers to sets.'

I want to criticize the line of thought indicated here. Firstly, regarding the idea that an account of numbers must explain in virtue of what the number 2 has the property of being the number of moons of Mars: aren't we being misled here by a phrasing which puts the focus on 2 instead of Mars? Intuitively, it is not an intrinsic feature of 2 that it is the number of moons of Mars, but an extrinsic one.

It is indeed plausible that it can't be a brute fact that the number 2 is the number of moons of Mars, but it doesn't follow from this that an account of numbers is the place to look for the explanation. Rather, the explanation we feel the lack of is that of, as we would more naturally say, the fact that Mars has two moons. And so I suggest, the apparent non-bruteness of the fact in question lies in its being explicable in astronomical terms. It seems like there is, whether or not we are able to figure it out, a story to tell about the formation of the planets and their moons which explains why Mars has two of them. I don't think there are any good reasons to believe that, with such an explanation on board, we would have further explaining to do as to why the number 2 is the number of moons on Mars. (Indeed, from a practical standpoint the idea seems ridiculous. But perhaps a practical standpoint isn't everything.)

I say 'I don't think there are any good reasons' above - but is that really the point? What force does my argument have? I am doing two things: firstly, I am suggesting that there is potentially a kind of bait and switch going on in our getting to the point of feeling that we need an account of numbers that explains why the number two is the number of moons of Mars: the fact calls for astronomical explanation, but if we consider the matter abstractly, we may just feel that it needs some explanation, and then the weird explanation involving the von Neumann construction is wheeled in.

Secondly, I am proposing that there is no explanatory gap between 'There are 2 Fs' and '2 is the number of Fs'. But at this point my arguments give out. Indeed, I think the best approach at this point is to stop arguing for the correct viewpoint, and switch to trying to trace the origin of the incorrect viewpoint. And I think in this case it lies in our misunderstanding expressions like '2 is the number of Fs', due to their superficial resemblance to expressions which work in a different way. (This is, it must be noted, is not at all to say that there is no such thing as the number two, or that it doesn't really have such properties as being the number of moons of Mars.)

A Fallacy in Hofweber's Arguments in Ontology

[This is a draft of a paper.]

Hofweber's ontological project crucially involves inferring negative existential statements from statements of non-reference, i.e. statements that say that some term or terms do not refer. Here, after explaining the context of this move, I want to show that it is fallacious, and that this vitiates Hofweber's ontological project.

Thomas Hofweber has for several years been developing a distinctive approach to ontological and metaontological questions.

One of his starting points is the way some ontological questions in philosophy can apparently be settled with trivial arguments - for example, since mathematics has established that there are infinitely many prime numbers, it follows that there are numbers, and so there is no room for a special philosophical discipline of ontology (if it is to respect mathematics) to deal with this as a substantial question, the way ontologists of mathematics seem to try to do. Call this the puzzle about ontology.

Hofweber attempts to solve the puzzle about ontology by independently motivating a distinction between two different readings of quantifiers, or two sorts of quantification: internal and external. Internal quantificational statements, unlike external ones, do not work by placing conditions on a domain of objects. (To see that we might need something like this, consider the quantifier in 'Santa Claus doesn't exist, therefore there is something that doesn't exist'.) He then argues that the trivial arguments go through, but only when the quantifiers are given an internal reading. Give the quantifiers an external reading, and it is not clear that their premises have been established - in the case of 'There are infinitely many prime numbers', for instance, it might be that mathematics has established this on its internal reading, but not on its external reading.

Hofweber doesn't just want to solve the puzzle about ontology with his internal/external distinction, however. He also wants to use it to establish answers to certain (external) ontological questions - negative answers. This is what I call 'Hofweber's ontological project'.

Taking the number case, the project goes roughly like this. Hofweber argues that, if we can establish internalism about number-talk, including arithmetic (i.e. if we can establish that the quantifiers involved in number-talk, including arithmetic, are internal ones which do not place conditions on a domain of objects), we can show that the external question of whether numbers exist is left open by this talk, and is thus free for the taking by ontology.

Next, Hofweber argues that numerals, number words like 'four', and terms like 'the number 2' are not referring terms. I.e., that they are not in the business of referring to things. They sometimes assume the superficial grammatical position of referring terms for sophisticated linguistic reasons involving the notion of a 'focus construction' (and other considerations, depending on the kind of occurrence).

Then, on the basis that number terms don't refer, Hofweber concludes (via a principle designed to enable one to infer non-existence of things from statements of non-reference) that numbers don't exist, i.e. that there are no numbers (the quantifier here being intended externally), thus answering one of ontology's fundamental questions. I will good-naturedly call this last step 'the Howler'.

A couple of years ago, I inconclusively argued that Hofweber's distinction between internal and external quantification is ill-motivated. Here, I want to grant that distinction, and even grant that it enables Hofweber to explain the validity of the trivial arguments.

I want to make it clear that the Howler is a fallacious move, and that this vitiates Hofweber's project for answering certain ontological questions (e.g. about numbers, properties and propositions) in the negative. I will not be concerned here with whether Hofweber succeeds in establishing internalism about number-talk - my point is only that his argument from internalism to negative answers in ontology contains a fallacy.

The Howler appears in Hofweber's contribution to the influential 2009 anthology, Metametaphysics: New Essays on the Foundations of Ontology. The contribution is called 'Ambitious, Yet Modest, Metaphysics'. (I include other relevant papers in the bibliography, to help readers piece together a more detailed view of Hofweber's overall project, but he gives a good sense of it in the paper just mentioned.)

I think my criticism will be most effective if I quote the Howler along with the argument in which it appears, rather than reconstructing it and insisting that that is what Hofweber was doing. Here is the argument:

Let’s briefly reflect on what seems to be a central thesis about reference or denotation: 

(REF) If Fred exists then ‘Fred’ refers to Fred. 

Of course, I am assuming that ‘Fred’ is unambiguous, or at least used in the same way throughout. (REF) is uncontroversial, I take it, and probably a conceptual truth. Note that it implies the following: 

(REF∗) If ‘Fred’ doesn’t refer to Fred then Fred doesn’t exist. 

There are two ways for an expression not to refer. One is to aim to refer, but not to succeed. A classic case of this are empty names. Although the details of any example one might try to give of this are controversial, let’s nonetheless take ‘Sherlock’ to be an empty name of this kind. That is, suppose Sherlock is a name and thus has the semantic function of picking out an object. But it fails in carrying out that function. It thus doesn’t succeed in referring, and thus doesn’t refer. Thus Sherlock does not exist. Nothing in the world is Sherlock, no matter what in general the world contains. There could be all kinds of people, with all kinds of professions, but no matter how general properties are instantiated in the world, nothing in it is Sherlock. And nothing could be. If ‘Sherlock’ does not refer then Sherlock does not exist. This is all fairly trivial, but I go over it to make it vivid for our next case. 

Names aim to refer, but they can fail to succeed in what they aim for. The second way in which an expression might not refer is when it does not even aim to refer. Non-referential expressions, like ‘very’, don’t refer since they don’t even aim to refer. If internalism is correct about talk about numbers, properties, and propositions, then the relevant singular terms are non-referential. They do not aim to refer, and thus they do not refer. According to the above version of internalism ‘two’ is just like ‘most’. But since it doesn’t refer we know that there is no such thing as the number two. Since ‘two’ and ‘the number two’ are non-referring expressions nothing out there is (or can be) the number two. There can be all kinds of objects, abstract or concrete, they can have all kinds of properties and relations to each other. Nonetheless, none of them is (or can be) the number two. Or any of the other numbers. Internalism thus answers the ontological question.

Note first that Hofweber says that '[a]ccording to the above version of internalism "two" is just like "most"'. But what do we get if we substitute 'most' for 'Fred' in Hofweber's (REF*) principle?:

(REF*-Most) If 'most' doesn't refer to most then most doesn't exist.

But this seems like ungrammatical nonsense. Furthermore, it doesn't seem that 'Does most exist?' or 'Is there such a thing as most?' are substantial, sensible questions. It may be argued that 'Does most exist? No.' is not complete gibberish, if it is construed as a kind of metalinguistic point - it's not true to say 'Most exists'. This question-and-answer does not appear to be about whether the domain of our external quantifiers meets certain genuine conditions (and not simply metalinguistic conditions such as 'being referred to by the word "most"').

So if 'two' really is just like 'most' in all relevant respects, Hofweber has a problem. There is, of course, an important difference. Consider:

(REF*-Two) If 'the number two' doesn't refer to the number two then the number two doesn't exist.

Unlike (REF*-Most), (REF*-Two) is superficially grammatical. It even appears not to be nonsense (if we consider it independently of Hofweber's views). Do either of these two differences help?

Superficial grammaticality doesn't help; consider the nonsensical but superficially grammatical question 'Is there a rock of eggs?'. This doesn't seem to turn on whether a domain of objects meets some genuine condition, and 'Is there a rock of eggs? No.', like the 'most' case, seems to be a metalinguistic point at best.

The appearance of sense doesn't help either, for Hofweber has no way of explaining it except in terms of internalism; he explains occurrences of number-expressions always in terms of their being non-referring terms that appear in the syntactic guise of referring terms (for sophisticated linguistic reasons). And it is not at all clear what these could possibly be doing in an external quantificational context.

In general, the point might be captured by the following principle: non-existence of something only follows by semantic descent from non-reference when the non-referring term plays the semantic role of referring. Otherwise you can't semantically descend to a well-formed, sensical proposition.

(This is a necessary condition for the non-existence of something following by semantic descent from non-reference, but it may not be sufficient. I say 'follows by semantic descent' rather than simply 'follows' because '"X" does not refer' may be argued to always imply 'The referent of "X" does not exist' - but there is no semantic descent there, as there is in Hofweber's arguments.)

This seems like the natural view, in lieu of some special story, and Hofweber hasn't given any such story.

In a forthcoming post, I will argue that Hofweber's ontological project is impossible, for a different (though related) reason: internalism at the strength he requires it is inconsistent with the thesis that there is a substantial ontological question about numbers left open by arithmetic and other non-metaphysical number-talk, since such a question would constitute a counterexample to internalism. This mistake is, I think, more profound than the one exposed here - making it involves a kind of sawing-off of the branch one is sitting on.

Bibliography

Main reference: 

Hofweber, T. 2009. 'Ambitious, yet modest, metaphysics', in David John Chalmers, David Manley & Ryan Wasserman (eds.), Metametaphysics: New Essays on the Foundations of Ontology. Oxford University Press.

Background reading for Hofweber's project:

Hofweber, T. 2005a. 'Number Determiners, Numbers, and Arithmetic', The Philosophical Review 114:2.

Hofweber, T. 2005b. 'A Puzzle about Ontology', Nôus 39:2.

Hofweber, T. 2007. 'Innocent Statements and their Metaphysically Loaded Counterparts', Philosophers' Imprint 7:1, <www.philosophersimprint.org/007001/>.

These papers are available on Hofweber's homepage:

http://web.mac.com/hofweber/Thomas_Hofwebers_homepage/Papers.html