Featured Article Summary #51

The Quine-Putnam indispensability argument is a philosophical argument for the existence of abstract mathematical objects. It is named after philosophers Willard Quine and Hilary Putnam. Quine introduced a number of philosophical positions to the argument, while Putnam gave it its first detailed formulation. Mark Colyvan developed a standard form of the argument that differs in certain ways from Quine and Putnam's formulations. Critics, such as Hartry Field, argue that mathematical entities are not necessary for science. Other philosophers have also argued that we do not need to believe in all entities that are indispensable to science. Alan Baker and Mark Colyvan support a version of the argument that claims mathematics is indispensable to specific scientific explanations. The problem raised by Benacerraf for the philosophy of mathematics is that mathematical sentences seem to imply the existence of mathematical objects, but if these objects exist, they would be unknowable to us. The philosophy of mathematics is split into two main strands, platonism and nominalism, each of which face issues due to this problem. The indispensability argument aims to overcome the epistemological problem by providing a justification for belief in abstract mathematical objects, based on the claim that their truth is indispensable for formulating scientific theories. The Quine-Putnam indispensability argument states that we should believe in mathematical entities because they are indispensable to our best scientific theories. This argument is based on the assumptions of naturalism and confirmational holism. It is mainly aimed at nominalists who are scientific realists. The argument only argues for belief in the parts of mathematics that are indispensable to science, not the most abstract parts. Some philosophers infer from this argument that mathematical knowledge is a posteriori and contingent, which is controversial. Indispensability arguments can also be used to argue for sentence realism, which is a weaker claim than platonism. The indispensability argument claims that mathematical objects are necessary for scientific theories, but this is not the same as saying they cannot be eliminated. The attractiveness of a theory can be evaluated using theoretical virtues and if an entity is dispensable, an equivalent theory can be formulated without it. However, philosophers like Hartry Field argue against this by reformulating scientific theories to not refer to mathematical objects. Field's alternative is mathematical fictionalism, which claims that mathematical theories are false. Another approach is to reformulate mathematical theories themselves to not imply the existence of mathematical objects. Quine's naturalism argues that scientific theories should guide our beliefs about what exists, rather than philosophy. This form of naturalism rejects the idea that philosophy should dictate scientific beliefs. Some philosophers support this naturalism using track-record arguments, which highlight the success of science compared to philosophy. However, critics argue that this approach discredits philosophical methods entirely. Additionally, some critics, like Penelope Maddy, argue that mathematics does not rely on the natural sciences for its legitimacy, as mathematicians do not consider scientific applications in their work. Charles Parsons also argues that mathematical truths do not depend on scientific theories.