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Nominalism Nominalism is the thesis that there are no mathematical objects, relations and structures, or that they do not exist as abstract objects. Nominalism is roughly divided into theories that require the reformulation of mathematics (and science) to avoid commitment to mathematical objects, and theories that instead offer an account of how such theories in fact make no commitment to mathematical objects. Varieties of nominalism:??Versions of finitism according to which there are only finitely many (concrete) mathematical objects. Versions of (semi-)formalism, according to which mathematical symbols do not refer (although it is logically compatible with there still being mathematical objects) and mathematics is akin to a game. Fictionalism, according to which mathematical statements are true for merely instrumental reasons. Mathematical statements are not literally true or false, but can be 'true' or 'false' in the mathematical fiction. Modal (eliminative) structuralism, according to which mathematics is the study structures, and mathematical objects are parsed out by a suitable elimination schema. (The appeal to modality is to ensure the elimination schema is not trivial). Deflationary nominalism, which takes a deflationary approach to truth to reject Quine's criterion for ontological commitment.
We can also distinguish two nominalistic strategies:
1. Nominalising our scientific theories, and thereby showing mathematics is (theoretically) dispensable to science.
2. Drawing a distinction between belief and mere acceptance: arguing we accept mathematical objects for scientific purposes, and that mathematics is indispensable for scientific purposes, but deny this is sufficient for rational belief in its truth.
Five problems Otavio Bueno identifies five issues faced by nominalists and platonists.
1. Epistemological problem: how to explain the possibility of mathematical knowledge, given that mathematical objects do not seem to play any role in our belief-formation. Platonists seem to do worse off here - but nominalists also need to explain the difference between a mathematician (knowing lots of mathematics) and the layperson (knowing very little).
2. Application problem: how can we explain the success of mathematics in scientific theories? Platonists seem to do better here. For example, the indispensability argument, though this relies on Quine's criterion of ontological commitment. However, it is not obvious the platonist is in fact better off: the physical world seems entirely disjoint from platonic objects, so it not clear how abstract objects are relevant
to the physical world. On the other hand, to say the physical world simply instantiates a certain type of structure is not enough: many different theories are instantiated in one part of the world. For example, quantum mechanical phenomena can be characterised by the very different theories of group theory and Hilbert spaces. Nominalism, meanwhile, needs to explain the apparent indispensability of reference to mathematics in scientific theories.
3. Uniform semantics problem: Platonism allows us to adopt the same semantics for mathematical and scientific discourse, and even for mixed statements. Most nominalist theories, however, require a substantial rewriting of mathematical language.
4. Literal discourse problem: Platonism allows us to construe mathematical statements literally; as philosophers of mathematics we can therefore examine mathematical theories as they are formulated in mathematical practice. By contrast, modal structuralism introduces modal operators, and Field's fictionalism introduces fiction operators.
5. Ontological problem: how ought we specify the nature of the objects a philosophical theory of mathematics commits us to? What are our ultimate ontological commitments?
Field's nominalism Field holds there are no abstract mathematical objects, and so mathematical statements are either trivially false (existence claims) or trivially true (universal claims). Field introduces a fictional operator so that the fictionalist can reach verbal agreement with the platonist, along the lines of 'according to arithmetic, X'. Field aims to
1. Change the aim of mathematics from truth to conservativeness (something which is not weaker than truth).
2. Show that mathematics is purely instrumental. Hartry Field attempted to meet the indispensability argument - the argument he took to be the strongest in favour of platonism - by showing how mathematical physics could be nominalised. He aims to show that, for example, real analysis is dispensable in physical theories. Field's argument is motivated by a distinction he sees between mathematical and physical entities. Field identified physical entities as indispensable because they give rise to new empirical claims. By introducing electrons into a physical theory, for example, we can prove results we could not prove before. Field's claim is that mathematical entities do not play the same role: the introduction of mathematical entities introduces no new claims over and above those in a nominalistic theory. The property Field needs is therefore conservativeness. A theory M is conservative over a theory N IFF for every sentence A of N, A follows from M+N only if A follows from N.
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