The Quine-Putnam Indispensability Argument

Hello! Welcome to the tenth lesson in our course.

In our last lesson, we explored mathematical structuralism as a response to Benacerraf's Dilemma. We saw that both ante rem and eliminative structuralism attempt to resolve the tension between providing a plausible semantics for mathematics and a viable epistemology. Ante rem structuralism preserved a simple semantics at the cost of reintroducing an epistemological mystery, while eliminative structuralism solved the epistemology problem but required a complex and non-standard semantics.

Today, we turn to a completely different, and very influential, argument for mathematical realism (Platonism). Instead of focusing on the internal structure of mathematical theories, this argument looks outward, to the role mathematics plays in the empirical sciences. This lesson will focus on the following learning outcome: Explain the Quine-Putnam indispensability argument for the existence of mathematical objects.

This argument provides a powerful response to Benacerraf's epistemological challenge by suggesting that our knowledge of mathematical objects is justified in the same way as our knowledge of other unobservable scientific entities, like electrons or quarks.

1. The Core Argument: No Double Standards

At its heart, the indispensability argument is a challenge to be consistent. If you are a scientific realist—that is, you believe our best scientific theories are true and the entities they posit (like electrons, fields, and black holes) actually exist—then you should also believe in the mathematical entities (like numbers, sets, and functions) that are essential to those very same theories.

The argument, championed by philosophers Willard Van Orman Quine and Hilary Putnam, can be stated quite simply:

(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.

(P2) Mathematical entities are indispensable to our best scientific theories.

(C) Therefore, we ought to have ontological commitment to mathematical entities.

This is a valid argument, so if the premises are true, the conclusion must follow. To deny the existence of numbers while accepting the existence of electrons would be, in Putnam's words, "intellectual dishonesty." It would be applying a double standard.

To get a strong intuitive grasp of this argument from a leading proponent, let's watch a clip featuring philosopher of mathematics Mark Colyvan.

As Dr. Colyvan explains, the argument forces a choice: either accept the existence of mathematical objects or reject the scientific realism that underpins our belief in much of modern physics.

2. The Philosophical Bedrock: Naturalism and Holism

The argument's power comes from its first premise, (P1), which rests on two important philosophical doctrines: Naturalism and Confirmation Holism.

Naturalism

Naturalism is the view that science is our best and only guide to what exists. There is no higher authority or "first philosophy" that can stand in judgment over the findings of science. If we want to know what the world is made of, we should look to our best scientific theories (physics, chemistry, biology, etc.), not to intuition, revelation, or philosophical speculation alone. Naturalism supports the "only" part of P1: we should only be committed to entities posited by science.

Confirmation Holism

Confirmation holism, often associated with Quine, is the idea that scientific theories are not tested sentence by sentence, but as a whole "web of belief." When scientists conduct an experiment, the resulting evidence doesn't just confirm or deny a single hypothesis in isolation; it tests the entire network of theories, background assumptions, and mathematical tools used to make the prediction.

This means that the evidence confirming a theory (e.g., observing a particle track in a cloud chamber) is not just evidence for electrons, but also for the mathematical equations that describe the electron's behavior. The mathematical and the empirical are confirmed together as a single package. Holism supports the "all" part of P1: we are committed to all indispensable parts of the theory.

The following video explains these two supporting pillars and how they function in the argument.

Given your background in mathematics and logic, you'll appreciate that Quine formalized this idea with his famous criterion of ontological commitment: "To be is to be the value of a variable." This means a theory is committed to the existence of whatever must be in the domain of its variables for the theory's sentences to be true. Since the laws of physics are full of variables ranging over numbers, functions, and sets, a naturalistic and holistic reading of science commits us to their existence.

To see this in a more formal light, let's turn to a text that breaks down Quine's specific procedure.

3. Challenges to the Argument

The indispensability argument is powerful, but it's not without its critics. Objections have been raised against both of its premises.

Challenging Premise 1: Do Scientists Believe Everything in Their Theories?

Philosopher Penelope Maddy challenges P1 by appealing to the very naturalism that Quine champions. If we are naturalists, we should observe the actual practice of scientists. Maddy points out that scientists often use idealizations and simplifications they know are false. For example, fluid dynamics treats water as a continuous substance, even though we know it's made of discrete molecules.

If scientists don't believe in every entity indispensable to their daily work, it undermines the holistic claim that we must be committed to all parts of our theories. This suggests we might be able to treat mathematics as a useful idealization, rather than a literal truth.

You can see this objection explained clearly in the video from earlier.

Challenging Premise 2: Is Mathematics Really Indispensable?

The second premise—that mathematics is indispensable—seems obvious on the surface. Can you imagine doing modern physics without it? However, this has been the target of the most sustained objection.

Philosopher Hartry Field argued that mathematics, while incredibly convenient, is ultimately dispensable. He claimed that for any scientific theory that uses mathematics, we could (in principle) state a "nominalistic" version of that theory—one that doesn't quantify over mathematical objects. According to Field, the role of mathematics is to be a "conservative" tool: it helps us derive conclusions about the physical world more easily, but it doesn't add any new physical conclusions we couldn't have derived without it.

If Field is right, then P2 is false, and the argument fails. We will explore his view, known as mathematical fictionalism, in detail in our next lesson.

Conclusion

Today we've examined the Quine-Putnam indispensability argument, a cornerstone of modern philosophy of mathematics. It represents a powerful attempt to ground our belief in abstract mathematical objects on the solid footing of empirical science, thus providing an empiricist-friendly answer to Benacerraf's epistemological challenge.

Key Takeaways:

• Core Claim: We should believe in mathematical objects for the same reason we believe in other theoretical entities of science (like electrons): they are indispensable to our best theories.
• Philosophical Foundations: The argument rests on Naturalism (science is the arbiter of reality) and Confirmation Holism (theories are confirmed as a whole).
• Ontological Commitment: Quine's criterion, "To be is to be the value of a variable," formally links a theory's commitments to the objects it must quantify over to be true.
• Main Objections: The argument can be challenged by questioning whether we must believe in all parts of our theories (Maddy's critique) or by questioning whether mathematics is truly indispensable in the first place (Field's project).

Preview of the Next Lesson:
The indispensability argument sets the stage for one of the most interesting contemporary positions in the philosophy of mathematics: mathematical fictionalism. In our next lesson, we will dive deep into the project started by Hartry Field, exploring his attempt to show that mathematics is a "useful fiction" and that P2 of the indispensability argument is false. We'll analyze his strategy for accounting for the applicability of mathematics without committing to the existence of mathematical objects.