Mathematical objects as figures of speech: On Yablo’s figuralism Ebba Gullberg November 2007 1 Introduction This paper is intended as a preliminary study for my dissertation, which is cen- tered around Georg Kreisel’s famous dictum: “The problem is not the existence of mathematical objects, but the objectivity of mathematical statements.” Kreisel’s dictum brings up the question whether it is possible to give an account of math- ematics that preserves its objectivity without postulating a platonic domain of abstract mathematical objects. I want to explore the relationship between the assumption that mathematics is objective, and the assumption of mathematical objects. In mathematics, it is common to make statements that assert the existence of mathematical objects such as numbers, sets, and functions. From a mathematical point of view it seems obvious that these objects exist. Philosophers, on the other hand, have questioned the existence of numbers. One may ask whether the math- ematician and the philosopher are talking about the same thing when the former claims that numbers exist and the latter denies this. If they do, the philosopher’s denial of the existence of numbers comes into conflict with well-established math- ematical results. In this context, Carnap [7] introduced a distinction between internal and ex- ternal statements. The mathematical statement that numbers exist is an internal existence statement within mathematics. Without contesting this mathematical statement, the philosopher asks whether numbers really exist. Should she claim that this is the case, she will have made an external existence statement about mathematical objects. One way of interpreting Carnap’s distinction, which has recently been pro- posed by Stephen Yablo, is to equate the external statements with those that are intended to be literally true and the internal ones with those that are in some sense fictional. Yablo’s approach, figuralism, is the topic of this paper. 1 1.1 Nominalism The main thesis of nominalism can be formulated in the following way: There are no abstract objects. In the case of mathematics this can be narrowed down to the more specific claim that there are no abstract mathematical objects, such as numbers, functions, or sets.1 One of the attractions of a view like this can be seen by considering what is perhaps the most obvious objection against the opposite position, realism or platonism. The objection has to do with our possibilities of gaining mathematical knowledge. Benacerraf [2, p. 414] writes: “If, for example, numbers are the kinds of entities they are normally taken to be, then the connection between the truth conditions for the statements of number theory and any relevant events connected with the people who are supposed to have mathematical knowl- edge cannot be made out.” The worry that is expressed here is that the abstractness of the numbers and in particular the impossibility to causally interact with them seem to prevent us from giving a reasonable account of how we can come to know things about them. Since mathematics and mathematical thinking play a huge role in both science and everyday life, the nominalist’s suggestion is that we abandon the idea of abstract objects and try to give an account of mathematics in terms of something else. 1.2 Mathematical realism and ontological commitment However, it is far from clear that nominalism is a winning strategy. Let us for a moment ignore the epistemological difficulties with realism and look at what kind of argument that can be given in favor of the existence of abstract mathematical entities, and hence against nominalism. Suppose that T is some mathematical theory. A very influential line of reason- ing is the following: 1. The theorems of T are literally true. 2. If the theorems of T are literally true, objects of kind P exist. 3. Hence, there exist objects of kind P. We can get a better picture of the idea of this argument if we look at an example. Let T be the theory of arithmetic and let the objects of kind P be natural numbers. Now take a sentence such as (*) There are infinitely many prime numbers. 1This definition of mathematical nominalism presupposes the thesis that if numbers etc. exist, then they are abstract, i.e. not located anywhere in space or time. For some other possible views, see e.g. Balaguer [1]. 2 If we agree to treat this sentence no differently than we would treat any other natu- ral language sentence, it does not seem far-fetched to interpret (*) as asserting the existence of a certain kind of entity, namely prime numbers (and, more generally, natural numbers). Another assumption that is unlikely to appear controversial is that this interpretation is the literal interpretation of (*). Combine this with our knowledge that (*) is a theorem of arithmetic and we have accepted Premise 2 of the above argument. All that remains in order for us to reach the conclusion that there actually are numbers is to find an argument convincing us that the literal interpretation is the right one and that, on this interpretation, arithmetic is true. Mathematics, and not only arithmetic, is full of sentences like (*) that, on the face of it, assert the existence of various abstract entities. Some of them can be seen on closer examination to be expressible in terms of other; for instance, talk about numbers can be translated into talk about sets, but at the end of the day there always seem to be some left. We say that mathematics is ontologically committed to abstract objects. At this point it could be objected that our natural language formulations of mathematical theories are misleading (despite the fact that this is how they are usually formulated). Quine [13, p. 242] writes: “The trouble is that at best there is no simple correlation between the outward forms of ordinary affirmations and the existences implied.” Quine’s own solution to this problem is to introduce a canonical notation in the language of first order logic, and a criterion by which to decide what entities exist: “Insofar as we adhere to this notation, the objects we are to be understood to admit are precisely the objects which we reckon to the universe of values over which the bound variables of quantification are to be considered to range.” As it turns out, even with canonical notation and the more formal criterion, the ontological commitment to abstract entities in mathematics apparently cannot be made to disappear. 1.3 Indispensability and truth The realist will still need something to support Premise 1 in the argument given in the previous section, since if mathematics is not true, then it hardly matters what its ontological commitments are. Such an argument is given as part of the Quine- Putnam indispensability argument for the existence of mathematical objects. The indispensability argument has been considered by many philosophers, both real- ists and nominalists, to be the best one available. It is based on one of the most conspicuous features of mathematics, namely its vast applicability in science. If we were deprived of the possibility to use mathematical language and mathemat- ical theories when formulating and practicing science, the argument goes, our scientific theories would be irreparably damaged and impoverished. Furthermore, since the scientific theories that we have now are the best we have come up with 3 so far (even though they are not flawless) it can be argued that we are justified in believing that they are true, at least to some extent. So, if mathematics is an indispensable part of these theories, it seems that we are justified in believing that mathematics is true too. We can summarize this in the following way: 1. We are justified in believing that our best scientific theories are true. 2. Mathematics is indispensable for our best scientific theories. That is, math- ematics is necessary both for the formulation of scientific theories and for the practice of science. 3. If T and T ′ are theories, T ′ is indispensable for T , and we are justified in believing that T is true, then we are justified in believing that T ′ is true. 4. Hence, we are justified in believing that mathematics is true. By accepting this argument we get half of Premise 1 in the argument for the ex- istence of mathematical objects. The part about literalness seems to follow if we join for instance Quine [13] and Burgess [5] in seeing it as somehow dishonest or insincere to use one language when doing science and another when talking about the same thing in philosophy. The rest of the indispensability argument runs along essentially the same lines as the argument given in the previous sec- tion; literal truth and ontological commitment yield the conclusion that there exist mathematical objects. 1.4 Revolutionary and hermeneutic nominalism In order to avoid acceptance of the realist’s account of mathematical existence, there are a few different options for the nominalist to explore. Burgess [5] intro- duces a distinction between two nominalistic strategies: revolutionary and herme- neutic nominalism. The same distinction, with some slight qualifications, is used in Rosen & Burgess [15]. Revolutionary nominalism involves a normative claim about mathematics. The revolutionary nominalist agrees with the realist that our actual mathemati- cal theories are ontologically committed to abstract entities, but holds that the practice or the interpretation of mathematics can and should be revised to make the commitments go away. Contrary to the adherents of the indispensability ar- gument, the revolutionary nominalist believes that mathematics as it is now is not indispensable for science. Clearly, mathematics is an extremely useful and conve- nient tool for formulating attractive and efficient scientific theories, but, at least in principle, we could manage without it and have theories with the same explanatory power as the ones we have now but that do not refer to abstract objects.