The Indispensability Argument in Philosophy of Mathematics

Overview

The indispensability argument is a key philosophical argument for mathematical realism. It suggests that our commitment to the existence of mathematical entities, such as numbers or sets, is justified by their indispensable role in our best scientific theories.

Key Concepts

The core idea is that science needs mathematics. If a theory is empirically successful, and that theory quantifies over or uses certain mathematical objects, then we have a reason to believe those mathematical objects exist.

Deep Dive

Philosophers like W.V.O. Quine and Hilary Putnam developed influential versions of this argument. They argued that we should have ontological commitments to all entities that are indispensable to our best scientific theories. If quantum mechanics or cosmology cannot be formulated without sets, then we ought to believe that sets exist.

Applications

This argument has significant implications for:

• Mathematical realism vs. nominalism.
• Understanding the relationship between mathematics and the physical world.
• Justifying mathematical knowledge.

Challenges & Misconceptions

Critics question whether mathematical entities are truly indispensable or if alternative, non-mathematical formulations are possible. Some argue that the success of a theory doesn’t automatically guarantee the existence of its theoretical posits. Nominalist responses often focus on reinterpreting mathematical language.

FAQs

What is the main claim of the indispensability argument?

That indispensable mathematical entities in science should be accepted as existing.

Who are prominent proponents?

W.V.O. Quine and Hilary Putnam.

What is a common criticism?

That mathematics might not be truly indispensable, or that scientific utility doesn’t equate to existence.