The number of things

Nous

Volumn 36, Issue SUPPL., 2002, Pages 176-196

  Van Inwagen, Peter ^a  

^a UNIVERSITY OF NOTRE DAME   (United States)

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EID: 61449189878     PISSN: 00294624     EISSN: None     Source Type: Journal    
DOI: 10.1111/1468-0068.36.s1-1.8     Document Type: Review

References (23)

1. Ludwig Wittgenstein, Tractatus Logico-Philosophicus: The German text of Ludwig Wittgenstein's Logisch-Philosophische Abhandlung, with a new Translation by D. F. Pears & B. F. McGuinness, and with the Introduction by Bertrand Russell, F. R. S. (London: Routledge & Kegan Paul, second impression, 1963). In citations, I will use Wittgenstein's "proposition numbers," rather than page numbers.
2. Hilary Putnam, The Many Faces of Realism: the Paul Carus Lectures (La Salle, Illinois: Open Court, 1987). The quoted passage occurs on pp. 17-19.
3. Proposition 4. 1272 does not end at this point. See also 4. 128 and 5. 453. Proposition 4. 128 reads Logical forms are without number. Hence, there are no privileged numbers in logic, and hence there is no possibility of philosophical monism or dualism, etc. By 'monism' and 'dualism', Wittgenstein of course means the thesis that there is one object (as Spinoza held) and the thesis that there are two objects (a thesis that perhaps no one has held) - not the thesis that there is one kind of object (as materialists and idealists hold) and the thesis that there are two kinds of object (as Descartes held).
4. It might be that there were more nominalistically acceptable objects than could be numbered. This could not happen if all nominalistically acceptable objects were spatio-temporal objects and were within the same space-time. But suppose there were more space-times than could be numbered (a supposition consistent with David Lewis's modal ontology). Then there could be more objects extended in space and time - there could, in fact, be more tables - than could be numbered. In the sequel I shall simply assume that if it is impossible to number the nominalistically acceptable objects, this is not because there are too many of them. (And we assume that there are no non-existent objects of any sort: I should think that if there were non-existent tables and elephants and neutron stars, it would be entirely plausible to suppose that there were too many of them to be numbered. )
5. Material Beings (Ithaca, N. Y. : Cornell University Press, 1990), §18.
6. (I use bold-face double-quotes for "Quine corners" or "quasi-quotation-marks. ") This assumption requires us to assume that the following two theses do not hold: "For all x and for all y, if x is the same M as y and F. . . x. . . , then F. . . y. . . " "For all x and for all y, if x is the same N as y and F. . . x. . . , then F. . . y. . . ", where 'F. . . x. . . ' is a sentence in which 'y' does not occur and 'F. . . y. . . ' is the result of replacing some or all of the free occurrences of 'x' in 'F. . . x. . . ' with 'y'. Roughly speaking: if the analogue of the principle of the indiscernibility of identicals held for all "relative identities," one could not have a case in which relative identities did not "coincide. "
7. I did not have to use English phrases like 'the others' and 'one of' to say this; all that is required to say it is the apparatus of quantifier logic, the usual sentential connectives, and the predicates '1 is the same being as 2' and '1 is the same person as 2'.
8. Carnap seems to assert that the second part of the argument does not follow from the first - see The Logical Syntax of Language (London: Routledge & Kegan Paul, 1937), p. 295 - but I am not sure I understand Carnap's point.
9. Wittgenstein says that one cannot say' "There are objects', as one might say, 'There are books'. " I have no idea what the words 'as one might say' ['wie man etwa sagt'] could mean, so I will ignore them.
10. In standard notation, '∃x ∃y Fxy' is, of course, equivalent to ∃x Fxx. v ∃x ∃y (Fxy & ̃ x = y).
11. Jaakko Hintikka, "Identity, Variables, and Impredicative Definitions," The Journal of Symbolic Logic, 21 (1956), pp. 225-245.
12. 5. 534 And now we see that in a correct conceptual notation pseudo-propositions like 'a = a', 'a = b & b = c. → a = c', '∀x x = x', '∃x x = a', etc. cannot even be written down. (As before, I have replaced Wittgenstein's Principia notation with current logical notation. )
13. Suppose that the second of these sentences is uttered by someone as a parody of nautical terminology. The fourth is from Stephen Potter's Lifemanship. It is recommended for inclusion in a letter giving instructions to a visiting team on how to find the home team's playing field. It has, according to Potter, the following annoying virtue: the fact that it is meaningless will not become evident to the visiting team till they have exhausted themselves in the attempt to find the home team's playing field.
14. And assuming that 'individual' is meaningful and does not admit of borderline cases. There may well be problems about the meaning of 'individual' - but if there are, Putnam's argument does not appeal to them. (Is 'individual' perhaps ambiguous? If so, pick one of its possible senses and concentrate on that one. If Putnam is right, there can be no meaning to the question 'How many objects are there in a world containing three individuals?' when the word 'individual' is used in that sense. ) And if the word 'individual' is meaningful, it's hard to see how there could be borderline cases of individuals; at any rate, Putnam's argument does not have as a premise that 'individual' is vague.
15. The more general definition is: F(the sum of S) = df ∃!z (Fz and every member of S is a part of z and every part of z overlaps some member of S). It should be noted that we do not have to suppose that the word 'sum' can occur only within a definite description. If we wished, we could read the predicate 'every member of S is a part of z and every part of z overlaps some member of S' as 'z is a sum of S'. That would allow us to say that the members of a set had more than one mereological sum.
16. Material Beings, §8.
17. A proper part of an object is a part of that object other than the whole object - for in the formal theory of parts and wholes, it is convenient to regard every object as being by definition a part of itself.
18. As Putnam has said, some versions of mereology recognize a "null individual," the sum of the empty set. In those versions, the second axiom is that every set has a sum.
19. That is an object that is not "all in one piece": a spatial object having at least two parts that are such that every path through space that joins those two parts passes outside that object.
20. A connected object is an object that is not a scattered object: an object that is "all in one piece. " A maximally connected object is a connected object that is not a proper part of a connected object. A maximally connected part of an object x is a connected part of x that is not a proper part of any connected part of x. If there are cats and (undetached) cats' tails, then a cat's tail is a connected part of the cat, but not a maximally connected part, since there are connected parts of the cat - the cat itself if no other - of which it is a part. If a dog and a cat (spatially separated in the ordinary way; not surgically joined or anything special like that) have a mereological sum, then the cat is a maximally connected part of that sum, since there is no connected part of the sum that has the cat as a proper part. (Couldn't we simply define a maximally connected part of x as a part of x that is a maximally connected object? No: if a cat's head and tail have a sum, the tail is a maximally connected part of the sum, but is not a maximally connected object. )
21. In "Truth and Convention: On Davidson's Refutation of Conceptual Relativism," Dialectica 41 (1987), pp. 69-77
22. Putnam imagines some criticisms of the lessons he draws from the confrontation between "Carnap" and "the Polish Logician," and these criticisms bear at least some resemblance to my criticisms. The criticisms are put into the mouth of a "Professor Antipode," a figure of fun. (Professor Antipode, like most figures of fun, is not very intelligent. ) I believe that the resemblance is superficial. However this may be, I do not understand Putnam's reply to Professor Antipode. The reader must judge: either I am obtuse in the extreme, or (inclusive) the words of Putnam's reply to Professor Antipode, like the words of his original argument, cease even to seem to mean anything when they are subjected to careful analysis. In Representation and Reality (Cambridge, Mass. : MIT Press, 1988), p. 110 ff
23. Putnam makes a point that could be applied to the confrontation between Carnap and the Polish Logician in this way: the whole dispute is really about which things to apply the word 'object' to, and that dispute is to be settled by establishing a convention. (He goes on to attempt to "deconstruct" the fact/convention distinction. I am sorry to be boring about this, but I am afraid I shall have to say once more that I do not understand the words this attempt comprises. I am aware that "I don't understand" is used by many philosophers as a substitute for argument, but I really don't understand what he says. ) I have in effect replied to this argument (leaving aside the attempted deconstruction of the fact/convention distinction, which I am not in a position to say anything about) in Material Beings, pp. 6-12. The essence of my argument was this: if a thing doesn't exist, it isn't there for you to establish a convention to the effect that it shall be called an 'object' (or anything else); if it does exist, the term 'object' applies to it, since the term applies to everything.