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Nominalism in the Philosophy of Mathematics (Stanford Encyclopedia of Philosophy) 9/16/13 2:21 PM Nominalism in the Philosophy of Mathematics (Stanford Encyclopedia of Philosophy) 9/16/13 2:21 PM Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia Free Nominalism in the Philosophy of Mathematics First published Mon Sep 16, 2013 Nominalism about mathematics (or mathematical nominalism) is the view according to which either mathematical objects, relations, and structures do not exist at all, or they do not exist as abstract objects (they are neither located in space-time nor do they have causal powers). In the latter case, some suitable concrete replacement for mathematical objects is provided. Broadly speaking, there are two forms of mathematical nominalism: those views that require the reformulation of mathematical (or scientific) theories in order to avoid the commitment to mathematical objects (e.g., Field 1980; Hellman 1989), and those views that do not reformulate mathematical or scientific theories and offer instead an account of how no commitment to mathematical objects is involved when these theories are used (e.g., Azzouni 2004). Both forms of nominalism are examined, and they are assessed in light of how they address five central problems in the philosophy of mathematics (namely, problems dealing with the epistemology, the ontology, and the application of mathematics as well as the use of a uniform semantics and the proviso that mathematical and scientific theories be taken literally). 1. Two views about mathematics: nominalism and platonism 2. Five Problems 2.1 The epistemological problem of mathematics 2.2 The problem of the application of mathematics 2.3 The problem of uniform semantics 2.4 The problem of taking mathematical discourse literally 2.5 The ontological problem 3. Mathematical Fictionalism 3.1 Central features of mathematical fictionalism 3.2 Metalogic and the formulation of conservativeness 3.3 Assessment: benefits and problems of mathematical fictionalism 4. Modal Structuralism http://plato.stanford.edu/entries/nominalism-mathematics/ Page 1 of 41 Nominalism in the Philosophy of Mathematics (Stanford Encyclopedia of Philosophy) 9/16/13 2:21 PM 4.1 Central features of modal structuralism 4.2 Assessment: benefits and problems of modal structuralism 5. Deflationary Nominalism 5.1 Central features of deflationary nominalism 5.2 Assessment: benefits of deflationary nominalism and a problem Bibliography Academic Tools Other Internet Resources Related Entries 1. Two views about mathematics: nominalism and platonism In ontological discussions about mathematics, two views are prominent. According to platonism, mathematical objects (as well as mathematical relations and structures) exist and are abstract; that is, they are not located in space and time and have no causal connection with us. Although this characterization of abstract objects is purely negative— indicating what such objects are not—in the context of mathematics it captures the crucial features the objects in questions are supposed to have. According to nominalism, mathematical objects (including, henceforth, mathematical relations and structures) do not exist, or at least they need not be taken to exist for us to make sense of mathematics. So, it is the nominalist's burden to show how to interpret mathematics without the commitment to the existence of mathematical objects. This is, in fact, a key feature of nominalism: those who defend the view need to show that it is possible to yield at least as much explanatory work as the platonist obtains, but invoking a meager ontology. To achieve that, nominalists in the philosophy of mathematics forge interconnections with metaphysics (whether mathematical objects do exist), epistemology (what kind of knowledge of these entities we have), and philosophy of science (how to make sense of the successful application of mathematics in science without being committed to the existence of mathematical entities). These interconnections are one of the sources of the variety of nominalist views. Despite the substantial differences between nominalism and platonism, they have at least one feature in common: both come in many forms. There are various versions of platonism in the philosophy of mathematics: standard (or object-based) platonism (Gödel 1944, 1947; Quine 1960), structuralism (Resnik 1997; Shapiro 1997), and full-blooded platonism (Balaguer 1998), among other views. Similarly, there are also several versions of nominalism: fictionalism (Field 1980, 1989), modal structuralism (Hellman 1989, 1996), constructibilism (Chihara 1990), the weaseling-away view (Melia 1995, 2000), figuralism (Yablo 2001), deflationary nominalism (Azzouni 2004), agnostic nominalism (Bueno 2008, 2009), and pretense views (Leng 2010), among others. Similarly to their platonist http://plato.stanford.edu/entries/nominalism-mathematics/ Page 2 of 41 Nominalism in the Philosophy of Mathematics (Stanford Encyclopedia of Philosophy) 9/16/13 2:21 PM counterparts, the various nominalist proposals have different motivations, and face their own difficulties. These will be explored in turn. (A critical survey of various nominalization strategies in mathematics can be found in Burgess and Rosen (1997). The authors address in detail both the technical and philosophical issues raised by nominalism in the philosophy of mathematics.) Discussions about nominalism in the philosophy of mathematics in the 20th century started roughly with the work that W. V. Quine and Nelson Goodman developed toward constructive nominalism (Goodman and Quine 1947). But, as Quine later pointed out, in the end it was indispensable to quantify over classes (Quine 1960). As will become clear below, responses to this indispensability argument have generated a significant amount of work for nominalists. And it is the focus on the indispensability argument that largely distinguishes more recent nominalist views in the philosophy of mathematics, which I will focus on, from the nominalism developed in the early part of the 20th century by the Polish school of logic (Simons 2010). Mathematical nominalism is a form of anti-realism about abstract objects. This is an independent issue from the traditional problem of nominalism about universals. A universal, according to a widespread use, is something that can be instantiated by different entities. Since abstract objects are neither spatial nor temporal, they cannot be instantiated. Thus, mathematical nominalism and nominalism about universals are independent from one another (see the entry on nominalism in metaphysics). It could be argued that certain sets encapsulate the instantiation model, since a set of concrete objects can be instantiated by such objects. But since the same set cannot be so instantiated, given that sets are individuated by their members and as long as their members are different the resulting sets are not the same, it is not clear that even these sets are instantiated. I will focus here on mathematical nominalism. 2. Five Problems In contemporary philosophy of mathematics, nominalism has been formulated in response to difficulties faced by platonism. But in developing their responses to platonism, nominalists also encounter difficulties of their own. Five problems need to be addressed in this context: 1. The epistemological problem of mathematics, 2. The problem of the application of mathematics, 3. The problem of uniform semantics, 4. The problem of taking mathematical discourse literally, and 5. The ontological problem. http://plato.stanford.edu/entries/nominalism-mathematics/ Page 3 of 41 Nominalism in the Philosophy of Mathematics (Stanford Encyclopedia of Philosophy) 9/16/13 2:21 PM Usually, problems (1) and (5) are considered as raising difficulties for platonism, whereas problems (2), (3), and (4) are often taken as yielding difficulties for nominalism. (I will discuss below to what extent such an assessment is accurate.) Each of these problems will be examined in turn. 2.1 The epistemological problem of mathematics Given that platonism postulates the existence of mathematical objects, the question arises as to how we obtain knowledge about them. The epistemological problem of mathematics is the problem of explaining the possibility of mathematical knowledge, given that mathematical objects themselves do not seem to play any role in generating our mathematical beliefs (Field 1989). This is taken to be a particular problem for platonism, since this view postulates the existence of mathematical objects, and one would expect such objects to play a role in the acquisition of mathematical knowledge. After all, on the platonist view, such knowledge is about the corresponding mathematical objects. However, despite various sophisticated attempts by platonists, there is still considerable controversy as to how exactly this process should be articulated. Should it be understood via mathematical intuition, by the introduction of suitable mathematical principles and definitions, or does it require some form of abstraction? In turn, the epistemological issue is far less problematic for nominalists, who are not committed to the existence of mathematical objects in the first place. They will have to explain other things, such as, how can the nominalist account for the difference between a mathematician, who knows a significant amount of mathematics, and a non-mathematician, who does not? This difference, according to some nominalists, is based on empirical and logical knowledge—not on mathematical knowledge (Field 1989). 2.2 The problem
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