“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.'
'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.)
