An Attempt to Diagnose the Disagreement over the Relational Explanation of Identity

Here is what I have to say in response to the schmidentity challenge as posed to the sui generis view of identity statements. (See also these two related posts from several years ago.)

OK, so we can grant that you can introduce a 'schmidentity' predicate in the way Kripke describes. We can also grant that this predicate could then get used to do what we do with identity statements. But can we, having granted these things, nonetheless deny that the meaning and function of identity statements is explained with the object-relation story?

I am strongly inclined to do all of this. Why? Because the characteristic function of informative identity statements and their denials - the way they get us to merge and separate mental files, or concepts of individuals - is passed over in this explanation. Going along with the object-relation explanation seems to render this incidental, instead of the main point. That explanation makes it look as though the main function of an 'a is b' statement is also fulfilled by the corresponding 'a is a' statement, which of course it is not.

But, someone may argue, does the object-relation explanation really create this false appearance? And here it would be easy to be dogmatic. There would be something silly about insisting that yes, this sort of explanation really does create this false appearance. After all, my opponent - the philosopher who wants to say that the object-relation story is perfectly adequate, and that there's no problem here, and that anyone who thinks there is is in a muddle - doesn't actually seem to be confused about the fact that 'a is b' statements are often useful in a way that the corresponding 'a is a' statements are not. They would happily admit that. So the difference between us seems to be in whether we are happy to leave this out in our primary explanation, so to speak, of identity statements.

And it is important that I allow that the object-relation explanation of identity statements does show something. It's not as if it is a sheer mistake. It shows that we can so to speak depict identity statements as a special case of relational statements, i.e. statements like 'John loves Mary'. I do not want to deny this, or deny that it is of philosophical interest.

There is something neat or cool about this sort of observation, too. It has a charm to it, similar to the charm possessed by clever hacks (in the sense of computer culture). I think that the philosopher who wants to defend the object-relation story lacks a proper place to put this. They feel the charm, the strikingness, of the explanation, and - not wanting this to elude them - wrongly place it in the "primary explanation" place in their thinking, instead of a place marked something like "striking and potentially instructive thing you can say". So long as we only focus on the "primary explanation" place, it looks like the defender of the object-relation story is missing something and proposing something maddeningly objectionable, but it also looks like the antagonist of the object-relation story is missing something. It is not until we consider other possibilities for the significance of the object-relation story that we are able to give both parties their due.

This, I now think, is a very important point (even though I may not have expressed it very well). I regret that I didn't manage to arrive at this point in my paper on this topic. I also think my predecessors were missing something in this regard.

So, we can grant the possibility of the schmidentity predicate, and the possibility of it coming to be used to do the characteristic work of identity statements, but nonetheless deny that the object-relation story should take pride of place in our explanation of the meaning and function of identity statements. A leftover question here is: should we also deny that the meaning and function of statements made with the 'schmidentity' predicate, if they are being used in the way we use identity statements, is explained by their stipulated semantics? And the answer, I think, is Yes. If they are being used in that way, then the object-relation story should not take pride of place in their explanation. But it is understandable that we should hesitate here, since 'schmidentity' was introduced and defined by means of the object-relation story, and this invites us to look at their use - when they are being used in the characteristic way we use identity statements - as a kind of secondary thing, a happy side-effect.

(I feel like saying something more at this point, which may be more objectionable, about what other use (schm)identity statements may have, apart from their practical use which has to do with merging and separating. A metaphysical use, so to speak. And about what attitude we take to this use, or whether it might be a kind of illusion. And this relates to one of the old posts linked to at the beginning. But I won't do more than make this hint, since these are treacherous waters and I wouldn't want to abuse the goodwill of a differently-minded reader.)

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.)