Mathematical Structuralism: An Alternative to Platonism

Hello! Welcome back to our exploration of contemporary philosophy of mathematics.

In our last lesson, we confronted Benacerraf's Dilemma. We saw how Platonism, the view that mathematical objects are real, abstract, and independent, provides a wonderfully simple account of what makes mathematical statements true (semantics). However, this came at the cost of a profound mystery: if these objects are outside of spacetime and causally inert, how could we possibly know anything about them (epistemology)?

Today, we'll investigate one of the most significant and influential responses to this dilemma: mathematical structuralism. This lesson directly addresses the learning outcome: Describe mathematical structuralism as an alternative to Platonism. We will see that structuralism is not a single, monolithic view, but a family of theories united by a core insight about the nature of mathematics. It attempts to retain the objectivity Platonism provides while sidestepping its most difficult epistemological problems.

1. The Core Insight: "Positions in a Structure"

Structuralism's starting point is another famous argument from Paul Benacerraf, published in his 1965 paper, "What Numbers Could Not Be." In the previous lesson, we saw how Platonism takes mathematical terms like '3' to refer to a specific, unique abstract object. Benacerraf challenges this very idea.

As you know from your mathematics background, we can construct the natural numbers within set theory in different ways. For example:

• Zermelo's ordinals:
  • $0=\varnothing$
  • $1=\{0\}=\{\varnothing \}$
  • $2=\{1\}=\{\{\varnothing \}\}$
  • ...and in general, the successor of $n$ is $\{n\}$.
• Von Neumann's ordinals:
  • $0=\varnothing$
  • $1=\{0\}=\{\varnothing \}$
  • $2=\{0,1\}=\{\varnothing ,\{\varnothing \}\}$
  • ...and in general, the successor of $n$ is $n\cup \{n\}$.

Both of these sequences perfectly satisfy the Peano axioms and do everything we need the natural numbers to do. Yet, as sets, they are different. For instance, in the Zermelo construction, $3=\{\{\{\varnothing \}\}\}$, which has one element. In the von Neumann construction, $3=\{\varnothing ,\{\varnothing \},\{\varnothing ,\{\varnothing \}\}\}$, which has three elements.

Benacerraf's question is: If numbers are unique abstract objects (as the Platonist claims), which of these set-theoretic objects is the number 2? Is it $\{\{\varnothing \}\}$ or is it $\{\varnothing ,\{\varnothing \}\}$? There seems to be no non-arbitrary reason to choose one over the other.

From this, Benacerraf concludes that numbers are not objects at all in the traditional sense. Rather, what matters are the relations they bear to one another. The important thing about '2' is not its intrinsic "object-hood," but the fact that it is the successor of '1' and the predecessor of '3' in an infinite progression.

This is the central idea of structuralism: mathematics is the science of structure, not of objects.

To get a feel for this idea, let's read a clear, high-level introduction.

As you've just read, a mathematical object, for the structuralist, is a position in a structure. The natural number '3' is simply the third position in the natural number structure, a structure defined by relations like 'is the successor of'. Any system of objects—be it Zermelo's ordinals, von Neumann's ordinals, or a series of beer mugs—can instantiate this structure.

This insight provides a powerful alternative to Platonism. But it immediately raises a new metaphysical question: what exactly is a structure? The different answers to this question give rise to the different versions of structuralism. We will now examine the two main families of views.

2. Ante Rem Structuralism: Structures as Real Objects

The first approach, known as ante rem structuralism (from the Latin for "before the thing"), is most famously defended by philosopher Stewart Shapiro. This view is the closest to Platonism.

Ante rem structuralism claims that structures themselves are real, abstract objects that exist independently of any systems that might instantiate them. The natural number structure, the real number structure, and the structure of a vector space are all unique, existing entities in their own right.

On this view:

• The subject matter of arithmetic is the natural number structure.
• The numbers themselves (0, 1, 2...) are places or positions within that structure.

This approach cleverly uses a distinction between two perspectives:

• Places-are-objects: When we do pure mathematics, we treat the places as objects in their own right. The statement "2+3=5" is a factual claim about the places within the natural number structure.
• Places-are-offices: When we apply mathematics, we see which objects in a given system fill the "offices" of the structure. We might say that the von Neumann ordinal $\{\varnothing ,\{\varnothing \}\}$ fills the 2-office in a particular set-theoretic system.

This is a subtle but powerful idea. The following reading explains it in more detail.

How does ante rem structuralism fare against Benacerraf's Dilemma?

• Semantics: It does very well. Like Platonism, it provides a face-value semantics for mathematical statements. The term '2' refers to a unique object: the 2-place in the natural number structure. This resolves the Zermelo vs. von Neumann issue because neither of them is the number 2; they are just different objects that can occupy the office of 2.
• Epistemology: Here, it runs into trouble. Ante rem structuralism is still a form of realism about abstract objects. Structures, on this view, exist in a causally inert, non-spatiotemporal realm. So, the original epistemological challenge returns: How do we, physical beings, gain knowledge of these abstract structures? The problem has been shifted from knowing about individual numbers to knowing about entire structures, but it has not been solved.

Test your understanding!

A traditional Platonist claims, "The number 2 is a unique, abstract object."
An ante rem structuralist claims, "The natural number structure is a unique, abstract object, and 2 is a place within it."

Explain why the structuralist's claim is better equipped to handle Benacerraf's "What Numbers Could Not Be" argument (about Zermelo vs. von Neumann ordinals) than the Platonist's claim.

Show answer

The Platonist has trouble with Benacerraf's argument because they are committed to '2' referring to one specific object. When presented with two different but equally functional set-theoretic objects (Zermelo's 2 and von Neumann's 2), they have no good reason to identify the 'real' number 2 with one over the other. Any choice seems arbitrary.

The ante rem structuralist, however, avoids this problem elegantly. They agree that '2' refers to a unique object, but that object is the 2-place in the natural number structure. The Zermelo and von Neumann sets are not the number 2 itself; they are merely different candidates for instantiating the structure. The structuralist can happily admit that both systems are perfectly good models of the number structure and that in one system, $\{\{\varnothing \}\}$ fills the 2-office, while in the other, $\{\varnothing ,\{\varnothing \}\}$ fills it. The identity of the number 2 itself is stable and unique (it's the place), while the identity of its potential occupants can vary.

3. Eliminative Structuralism: "Structuralism without Structures"

If ante rem structuralism seems like it inherits too much of Platonism's metaphysical baggage, there is a very different approach. Eliminative structuralism tries to get the benefits of a structuralist view without committing to the existence of abstract structures at all.

This view, associated with philosophers like Geoffrey Hellman, proposes that we shouldn't take mathematical statements at face value. Instead, we should eliminate talk of mathematical objects and structures by reinterpreting our statements.

A statement like "There are infinitely many primes" is not about a single structure. Instead, it is a general claim about any system that has the right structure. It should be translated as:

"In any system S that satisfies the Peano Axioms, it holds that for any element n in S, there is a prime element p in S that is greater than n."

This is a form of if-then-ism: if a system has a certain structure, then it will have certain properties.

How does eliminative structuralism fare against Benacerraf's Dilemma?

• Epistemology: It does wonderfully. We no longer need to explain a mysterious connection to an abstract realm. Mathematical knowledge is simply knowledge of logical implication—knowing what follows from a set of axioms. Since this is a matter of proof and deduction, which are concrete activities, the epistemological mystery dissolves.
• Semantics: This is its weak point. It forces us to abandon a face-value reading of mathematical sentences. "2+2=4" does not appear to be a complex "if-then" statement about all possible systems; it looks like a simple statement about specific objects. This view requires a significant and complex re-translation of our entire mathematical language.

Furthermore, this view faces the problem of vacuity. What if there are no systems in the actual world that satisfy the Peano Axioms? (For example, if the universe contains only a finite number of things). In that case, the "if" part of the statement is always false, making every mathematical theorem vacuously true. "2+2=5" would be just as true as "2+2=4".

To solve this, Hellman proposed modal structuralism. We don't need such systems to actually exist, only for them to be possible. The statement becomes:

"Necessarily, for any system S that satisfies the Peano Axioms,..."

And we add the claim that:

"Possibly, a system satisfying the Peano Axioms exists."

This avoids commitment to a vast ontology of actual objects, but it introduces a reliance on modal notions of possibility and necessity, which carry their own philosophical complexities.

Conclusion

Today we've seen how mathematical structuralism emerges as a direct response to the problems facing Platonism. Rather than being a single view, it's a diverse family of theories attempting to chart a course between the horns of Benacerraf's Dilemma.

Key Takeaways:

• Core Idea: The subject matter of mathematics is the abstract structure of systems of objects, not the intrinsic nature of the objects themselves. Mathematical objects are best understood as positions in a structure.
• Ante Rem Structuralism: A realist view that posits structures as independently existing abstract objects. It offers a strong, face-value semantics but inherits the epistemological problems of Platonism.
• Eliminative Structuralism: An anti-realist view that denies the existence of abstract structures. It translates mathematical statements into general claims about any system (or any possible system) that has the right structure. It offers a clear epistemology but requires a complex, non-face-value semantics.
• The Dilemma Revisited: Structuralism doesn't so much solve Benacerraf's Dilemma as it offers different ways to navigate it. Ante rem structuralism prioritizes the semantic horn, while eliminative structuralism prioritizes the epistemological horn.

This exploration reveals the deep and persistent trade-offs in the philosophy of mathematics between providing a simple account of truth and a plausible account of knowledge.

Preview of the Next Lesson:
In our next lesson, we will step back to examine another powerful argument for mathematical realism, one that ties the existence of mathematical objects not to semantics, but to their role in empirical science. We will explore the Quine-Putnam indispensability argument, which claims we ought to be committed to the existence of mathematical entities because they are indispensable for our best scientific theories.