Quine–Putnam indispensability argument

Quine–Putnam indispensability argument.
The Quine–Putnam indispensability argument reasons that we should believe in abstract mathematical objects such as numbers and sets because mathematics is indispensable to science. One of the most important ideas in the philosophy of mathematics, it is credited to W. V. Quine and Hilary Putnam (pictured). The roots of the argument can be traced back to thinkers such as Gottlob Frege and Kurt Gödel, but Quine introduced its key components, including naturalism and confirmational holism. Putnam gave Quine's argument its first detailed formulation, although he later expressed disagreement with some aspects of the argument. Many counterarguments have been raised against the idea. An influential argument by Hartry Field holds that mathematical entities are dispensable to science. Other philosophers, such as Penelope Maddy, have argued that we do not need to believe in all of the entities that are indispensable to science.