Nombres canónicos y existencia necesaria

Abstract

The last time I saw Frege, as we were waiting at the station for my train, I said to him “Don’t you ever find any difficulty in your theory that numbers are objects?” He replied “Sometimes I seem to see a difficulty —but then again I don’t see it.”1
Ludwig Wittgenstein

1. In his elegant and brilliant paper “Quantifying In”,2 David Kaplan proposes an interpretation of the de re modalities on the basis of the de dicto modalities. In furtherance of this aim he employs the Fregean analysis of the latter plus the notion of standard name and the relation of “necessarily denoting”. Generally, a standard name is for Kaplan a singular L-determined expression (Cfr. R. Carnap. Meaning and Necessity, § 17), i.e., an expression which denotes the same object in all possible worlds. The relation between the standard name and the denoted object is represented by the predicate “ΔN”; thus, “ΔN (‘9’, 9)” means: “9” necessarily denotes 9. According to Kaplan’s analysis, the sentence
(1) 9 is necessarily greater than 7,
when given a de re interpretation, i.e., when it is understood as asserting that the number nine has the property of being necessarily greater than seven, is rendered as
(2) (E α) (ΔN (α, 9)). N ┌α is greater than 9)┐.
In order to grasp the meaning of (2) we must note that a) “α” is variable of individual concepts; b) “N” stands for the “Necessary operator”, understood in the opaque, or de dicto, sense; c) the corners at the beginning and end of “α is greater than 7” work as a propositional abstractor. Taking into account these explanations, (2) says that there is an individual concept α, such that α necessarily denotes 9, and the proposition ┌α is greater than 7┐ is necessary. We must realize that if, as Kaplan supposes it to be, “9” is a standard name, then (2) can be inferred from
(3) N ┌9 is greater than 7┐
through a peculiar form of existential generalization, determined by the fact that the possible values of “α” are not numbers but individual concepts. Furthermore, such an inferential move is only admissible when at least one standard name occurs within the opaque context after the modal operator. It is convenient to add, for the sake of the following discussion, that Kaplan’s paper deals mainly with two types of standard names: numerals and quotation names.
2. After introducing the notion of a standard name, Kaplan adds an important qualification about the possibility of extending a similar trick to the case of transparent statements of belief, which would allow their formulation by means of the opaque sense of believing, in analogy to the strategy applied to modal statements:

There are however limitations on the resort to standard names. Only abstract objects can have standard names, since only they (and not all of them) lack that element of contingency which make the rest of us liable to failures of existence. Thus, Quine can have no standard name, for he might not be. (p. 196)

This statement asserts that only those objects whose existence is necessary can have standard names, and that seems to be the case of numbers and linguistic expressions.
Since in footnote 21 Kaplan refers to the discussion of this topic in the more technical frame of his Foundations of Intentional Logic, I will quote from it a more detailed formulation; we read there that although

the notion of contingent entity is quite problematic, depending, as it does, on the notion of a possible state of affairs, it is however plausible to assume that the distinction between necessary and contingent entities parallels that between abstract and concrete entities. Thus, numbers, expressions (in the sense of type, no token) [...] are capable of having standard names, whereas for any name of a physical object or sense datum we seem to be able to imagine a possible state in which such a name either would be denotationless or would name something other. (p. 57)3

To begin with, we may guess that when Kaplan says, in the first quotation, that not all abstract entities, however, can have standard names, this remark is intended to exclude such abstract entities as “the ratio of the number of centaurs to the number of unicorns”, referred to by Quine in another context; there being, as a matter of fact, neither centaurs nor unicorns, so no such abstract entity exists.4
Now Kaplan seems to be quite sure that this is not the case for numbers and expressions, in sharp contact with the situation of poor Quine, who is not necessary at all.
3. I would like first to say something about the necessary existence of expressions, because this seems to me an elephant too big to be smuggled in unceremoniously in order to explain standard names and give an account of transparent modalities. I will suggest that it is more plausible to try to assimilate the case of the existence of expressions into that of the ratio of centaurs to unicorns, and later I will dwell on the question of numbers.
What is a type-expression, for example, a type-sentence? Quine has pointed out that it would not do to take it as the class of its utterances, for in that case “all unuttered sentences would reduce to one, viz., the null class”.5 Instead he proposes to take each linguistic form as a sequence of its successive characters or phonemes, where a sequence a1, a2..., an is the class of the n pairs (a1, 1), (a2, 2),..., (an, n). But, what about the characters or phonemes? The answer is that “we can take each component phoneme a1 as a class of utterance events, there being here no risk of non-utterance”.6
This seems to me a reasonably attractive explanation. But the existence of the required utterances of phonemes or characters is a contingent affair; on the supposition that the universe does not contain such verbal events, the result would be that any such sequence would be composed of pairs whose first member is always the null class (Λ, 1), (Λ, 2),..., (Λ, n). In these conditions all sentences of the same length will be identified with the very same sequence, that is, would conflate into only one. And it would seem that to persist in calling this a sentence is rather outrageous, at least for my ontological feelings. So, if we admit Quine’s account of type-sentences, this result offers little stimulus to speak of the necessary existence of expressions; and I am inclined to say that any account of the notion of “possible world” which makes necessary the existence of language is highly suspicious. In connection with this, it is perhaps worth mentioning Benson Mates’ paper “Leibniz on Possible Worlds”,7 where, following Russell’s terminology in his book on Leibniz, it is said that “a sentence is true or false of a possible world rather than in it [...]. Thus the sentence ‘There are no sentences’ is presumably true of some possible world (though not in one)” (p. 509).
I suppose that this could have been precisely the case of our world if a catastrophe had destroyed it some time ago.
4. I turn now to the question of numbers. Kaplan says that:

so long as we hold constant our conventions of language ‘9’ will denote the same number under all possible circumstances. (p. 195)

This allows for the introduction of “ΔN”; and it is clear that for Kaplan the use of this relational predicate is not just a way of speaking, as he willingly emphasizes the contrast between the Fregean offer of “ontological insight” and Quine’s policy of “ontological security”. We have here the expression “9”, and also the object 9, and the relation ΔN between them. There is no doubt, as he adds: “I am less interested in urging an explanation of the special intimacy between ‘9’ and 9 than in noting the fact” (p. 195).
It may be that this story is a true one; after all, we are dealing with the question of the de re necessity as applied to numbers. But what is difficult to accept is the connection suggested between this and the notion of “linguistic competence”, for he goes on:

To wonder what number is named by the German ‘die Zahl Planeten’ may betray astronomical ignorance, but to wonder what number is named by the German ‘Neun’ can indicate only linguistic incompetence. (p. 195)

Now it seems to me that the “linguistic competence” relevant here, that is, the ability to correctly understand and use numerical expressions, is not necessarily linked with the relation ΔN, because it does not require that numbers be objects, nor, consequently, that numerals be names. If I manage to carry out arithmetical operations and to count the number of people in this room, I understand (at least to a reasonable extent!) numerical expressions, and cannot be accused of linguistic incompetence.
Perhaps the issue could be clarified a little more by paying attention to another of Kaplan’s remarks, connected with his defense of essentialism of numbers; according to him there is a sense in which numbers

find their essence in their ordering. Thus, names which reflect this ordering in an a priori way, as by making true statements of order analytic, capture all that is essential to these numbers. (p. 195)

In footnote 20, he comments that Benacerraf concludes thusly in “What Numbers Could Not Be”.8 But curiously enough, Benacerraf’s analysis ends with the conclusion that numbers are not objects and number-words are not names, so there would be no place for Kaplan’s relation “ΔN”. It is true that according to Benacerraf

to be the number 3 is more and no less than to be preceded by 2, 1, and possibly 0, and to be followed by 4, 5, and so forth. (p. 70),

which echoes Kaplan’s own statement. But he adds that

a sequence of number words is just that—a sequence of words or expressions with certain properties. There are not two kinds of things, numbers and number words, but just one: the words themselves [...] and any such sequence (of words) [...] will serve the purposes for which we have ours, provided it is recursive in the relevant respect [...]. The central idea is that this recursive sequence is a sort of yardstick which we use to measure sets. (p. 71)

Roughly, Benacerraf’s reasons are that we cannot identify a particular number with one of the different set-theoretical interpretations of it, because any choice would be wholly arbitrary:

I therefore argue, extending the arguments that lead to the conclusion that numbers could not be sets, that numbers could not be objects at all (p. 69).

There are many points in need of refinement, if only to do justice to Kaplan; but my aim was only to stress the fact that considerations about linguistic competence are misleading in the context of Kaplan’s discussion. That numbers are not objects (I am not too sure about this) does not imply, of course, that numerals lack meaning, but only that they do not have denotations. On Kaplan’s account, however, if Ralph understands the Arabic notation and it is true that he believes (in the opaque sense) the proposition ┌9 is greater than 7┐, then Ralph is in rapport with a certain abstract entity, for he says, speaking about the introduction of “ΔN” in order to express the de re necessity:

The same trick would work for Bel [the transparent, or relational sense of believing] if Ralph would confine his cogitations to numbers and expressions. (p. 197)

So, it seems that it is enough to understand “9” for being in rapport with the object Nine, and that a failure would betray “linguistic incompetence” on the part of Ralph.

Notas a pie de página

1 The quotation is from G.E.M. Anscombe and P.T. Geach, Three Philosophers, Basil Blackwell, Oxford, 1967, p. 130.
2 Synthèse, vol. 19, No. 1/2, December 1968, pp. 178–214.
3 I am indebted to professor Kaplan for his kindness in giving me a Xeroxed copy of his doctoral dissertation, which otherwise I would not have had the opportunity of reading.
4 W.V.O. Quine, From a Logical Point of View, New York, Harper & Row, 1963, p. 3.
5 W.V.O. Quine, Word and Object, Cambridge, Massachusetts, The MIT Press, 1960, p. 195.
6 Loc. cit.
7 In Logic, Methodology and Philosophy of Science, III, Amsterdam, North-Holland Publishing Company, 1968, pp. 507–529.
8 Paulo Benacerraf, “What Numbers Could Not Be”, Philosophical Review, vol. LXXIV, no. 1, January 1965, pp. 47–73.