Holism and the Nature of Mathematical Objects

The author of the article does not agree with the ontology of mathematical Platonism. Mathematical Platonism is justified by the argument that mathematics is indispensable from natural sciences. At the beginning of the article, the author gives an example of Hartry Field’s nominalistic strategy, which argues against the indispensability of mathematics. However, the main thesis of the article is that it is enough to show that the argument about the indispensability of mathematics does not actually have direct Platonic consequences. The main focus is on the Quine-Putnam Indispensability Argument in Mark Colyvan’s Platonic interpretation. The article discusses how Colyvan justifies this argument. In particular, Quine’s doctrines of epistemological naturalism and confirmational holism are considered. Colyvan’s interpretation is called into question. The author analyzes the premises of accepting the indispensability argument in the framework of Quine’s original philosophy. The author concludes that if we are trying to justify the argument by using Quine’s philosophy, we have to reject its Platonic consequences. Quine’s semantic holism and his reference theory do not assume the existence of Platonic objects that are independent of our consciousness. Quine’s assumption is that science as a whole is simultaneously dependent on both language and experience, and this duality cannot be meaningfully traced to statements taken separately. This is why we cannot assert the a priori nature of mathematical statements. This contradicts the views of mathematical Platonism. In addition, Russell’s description theory that Quine adopted allows us to restrict our ontology from any undesirable entities, including abstract mathematical objects. We postulate objects only when we involve the terms denoting them in a suitable interaction with the entire apparatus of our language. The question of whether to consider a language unit as a term is decided on the basis of the systematic effectiveness of its use as a term. Any ontological assumptions are a matter of practical usability of the conceptual schema of science. In this way, it is possible to preserve the indispensability argument, which is traditionally given in defense of mathematical Platonism. At the same time, we can reject its Platonic consequences without appealing to nominalistic interpretations of scientific theories.

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