How did Hilary Putnam and W.V.O. Quine defend the reality of mathematical objects?

Hilary Putnam and W.V.O. Quine defended the reality of mathematical objects through what is known as the indispensability thesis.1

This argument posits that because mathematical objects—such as numbers, sets, and groups—play an essential and indispensable role in our best scientific theories of the world, we are rationally committed to granting them ontological reality. By asserting that these entities are necessary for the success and explanatory power of science, Putnam and Quine argued that their existence cannot be denied without also undermining the theories that rely upon them. This perspective is a significant contribution to the philosophy of mathematics, linking the status of mathematical entities directly to their utility in describing physical reality.1