The Indispensability Argument: A Powerful Case for Mathematical Realism
1. The indispensability argument
The indispensability argument is a powerful argument for mathematical realism, put forward by W. V. Quine and Hilary Putnam. In essence, the argument goes as follows: if mathematics is essential to our best scientific theories, then it must be true in some sense; therefore, we are committed to the existence of mathematical entities (such as numbers and sets) even if we cannot observe them directly.
2. Criticism of the indispensability argument
The indispensability argument has been criticized on a number of grounds. First, some have argued that the argument conflates epistemology and ontology, i.e. it confuses what we can know about the world with what exists in the world. Second, others have argued that the indispensability argument begging the question against those who deny the need for mathematical entities in our scientific theories (such as instrumentalists or fictionalists). Finally, some have argued that the argument commit us to an overly strong form of mathematical realism, one that is at odds with common sense (for example, by entailing that fictional entities such as Sherlock Holmes must exist).
3. Implications of the indispensability argument
Despite its detractors, the indispensability argument remains a powerful argument for mathematical realism. If it is successful, then it has far-reaching implications for our understanding of the nature of mathematics and its role in our scientific theories. In particular, it suggests that mathematics is not just a convenient tool for describing the world; rather, it is an objective part of reality itself.
The indispensability argument is a powerful argument for mathematical realism. While it has been criticized on a number of grounds, its implications remain far-reaching and significant.