Black - against quidditism.pdf - Ted Sider

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Downloaded By: [New York University] At: 21:44 25 May 2010 1 O0 Against Quidditism V. Abstract modal realism is incompatible with quidditism, for we cannot have two structures which are structurally indistinguishable but nevertheless distinct. (Lewis argues this in some detail [10, pp. 158-65], though of course he takes it as a reason for rejecting abstract modal realism.) Must, conversely, the concrete modal realist be a quidditist? I think not. One could regard other possible worlds as concrete while rejecting any identity of fundamental properties across possible worlds---either by insisting that questions about the identity of properties only make sense within a world or, probably more easily, by insisting that no fundamental property is found in more than one world, the other possible worlds containing only counterparts of this-worldly properties, just as, for the anti-haecceitist, they contain only counterparts of this-worldly individuals. (Alternatively, work with tropes, and deny that there is any cross-world matching of tropes.) A concrete modal realism of this kind would work in almost exactly the same way as abstract modal realism, since on this view two possible worlds which share the same abstract structure become indiscernible. 18 Now it is (and ought to be felt as) an embarrassment for concrete modal realism, both in quidditist and antiquidditist versions, that it allows indiscernible worlds. Given a concrete possible world, how many exact copies of it are there out there in logical space? An arbitrary answer like 'seventeen' seems ruled out by the thought that there could have been (and therefore are) more but also by the same argument there could have been (and therefore are) fewer! The only two non-arbitrary answers would appear to be 'one' and 'proper-class many'. And here even these look arbitrary, for if there is only one, why couldn't there have been more, and if proper-class many, why couldn't there have been fewer? 19 If, however, we are talking about abstract structures, this problem disappears, since identity of indiscernibles applies. If we are talking not about worlds, but about types of world, then it is trivial that there is exactly one type of each type. Thus the move fi'om concrete to abstract modal realism in this respect provides a cleaner and more elegant theory. I am inclined to think that in almost all respects it provides a cleaner and more elegant theory. For example, just as the antihaecceitist can avoid having to give straight answers to unanswerable questions such as whether the Queen could have come from a different sperm and egg ([6, pp. 112-3]; cf. [10, pp. 251-2]), so the antiquidditist can avoid giving straight answers to unanswerable questions such as whether gravitational attraction is necessarily inverse-square (if we accept a force obeying Newtunian laws except that the exponent is -2.1 rather than -2 as a counterpart of gravity, then no; if we insist that any counterpart must arise as the limiting case of an Einstein-type theory, then probably yes). There is perhaps one advantage which Lewis claims for concrete modal realism and which I cannot claim for the abstract form, namely a reductive analysis of modality. I cannot claim it, because what mathematical structures there are is a matter of what is logico-mathematically possible. So logico-mathematical necessity/possibility has to be in ~8 I am using this term, as Lewis does, to deny difference, not to deny perceptible difference. 19 Lewis concedes [10, p. 224 fn.] that the question shouldn't have an arbitrary answer, but he doesn't, at least to my satisfaction, explain how any answer could fail to be arbitrary.
Downloaded By: [New York University] At: 21:44 25 May 2010 Robert Black 101 there as a primitive. But I suspect that logico-mathematical necessity has to be in there as a primitive for David Lewis as well. Certainly the necessary truth of mathematical truths can't consist in their being true at every possible world, for pure mathematical objects such as pure sets or numbers don't seem to exist at worlds at all, and anyway for Lewis mathematics can be applied across worlds--e.g, any set of n worlds has 2 n subsets. Do I need more than primitive logico-mathematical modality, however? Let us go back to my claim at the beginning: 3many W (W is a way & ~ the world is in w). The quantification here is over abstract mathematical entities. But what about the diamond? Do I need a further primitive modal notion to go from the existence of an abstract structure to the claim that the world could have exemplified it? Or is it enough for me simply to say that there are structures similar to the structure of the real world in some ways, but differing from it in others, and have I thereby said how the world 'could have been'? If the answer to this question is 'yes' (and I'll hazard that until presented with a convincing counterargument) then the abstract modal realist can claim a reduction of metaphysical necessity to logico-mathematical necessity, which I think is all, if not more than, anyone has a right to ask for. One further related problem deserves to be touched on. Among the infinity of mathematical structures, only one (or more exactly, one together with all its substructures) is exemplified in the real world. In what does this exemplification consist? Lewis raises this problem in his discussion of 'pictorial ersatzism' [10, pp. 171~], and although he classifies 'mathematical ersatzism' along with 'linguistic ersatzism', it would not surprise me if he were to classify my version as 'pictorial'. And Lewis might want to include my view among those he caricatures as belief in a large number of pictures, one of which has the additional magical ingredient 'vim' which makes it concrete rather than abstract. 'Why', he asks, 'should we think we are part of the one with vim? How could we tell and why should we care?' [10, p. 173] To this I reply that my mathematical structures are not pictures, and they are not entities of some special kind invented for this version of modal realism, they are the familiar abstract entities dealt with by mathematics. Back in t973 Lewis wrote 'I cannot believe (though I do not know why not) that our own world is a purely mathematical entity' [8, p. 90]. Well, neither can I. VI. 20 If one abandons quidditism, what becomes of the thesis of Humean supervenience? It can certainly still be stated, as the thesis that the laws of nature supervene on the pattern of instantiation of qualities in spacetime. For this we need only a primitive intramundane identification of qualities--prior to consideration of what the laws are, it has to be determinate whether or not the same quality is manifested at any two given locations. Nor is there any difficulty in stating the supervenience claim in terms of possible worlds, 20 This section expands on some points made in my [2].
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