The number 2 in Eliminative Structuralism- Philosophy of Mathematics:

The Problem of Multiple Representations

Mathematics faces a problem of identity. We can construct the number 2 in set theory in multiple ways. The von Neumann construction defines 2 as {∅, {∅}}. The Zermelo construction defines 2 as {{∅}}. Both constructions work for arithmetic. Both satisfy the Peano axioms. Yet these are different sets. This raises the question: which set is the number 2?

Paul Benacerraf identified this problem in 1965. If numbers are sets, then we need to choose which sets they are. But there is no mathematical reason to prefer one construction over another. This suggests that numbers are not sets at all.

Set Theory as Foundation

Set theory attempts to provide a foundation for mathematics by reducing all mathematical objects to sets. A set is a collection of objects. The empty set ∅ contains no objects. We can build other sets from the empty set.

Consider two objects with properties:

Object A has properties: {red, round, large}
Object B has properties: {blue, square, small}

In set theory, we represent these as ordered pairs:

A = (identifier_A, {red, round, large})
B = (identifier_B, {blue, square, small})

A relation between A and B is a set of ordered pairs. For example, the relation “larger than” might be represented as {(A, B)} since A has the property “large” and B has the property “small”.

Set theory builds mathematics from these basic concepts. Natural numbers become sets. Real numbers become sets of rational numbers (Dedekind cuts) or equivalence classes of sequences (Cauchy sequences). Functions become sets of ordered pairs.

Structuralism: Focus on Relationships

Structuralism shifts focus from objects to relationships. What matters about the number 2 is not what it is made of, but how it relates to other numbers. The number 2 comes after 1 and before 3. Adding 2 to 3 gives 5. These relationships define what it means to be 2.

Consider a structure with three positions and one relation:

Positions: {p₀, p₁, p₂}
Successor relation: {(p₀, p₁), (p₁, p₂)}

This structure can be instantiated by:

{0, 1, 2} with standard succession
{a, b, c} with the relation {(a,b), (b,c)}
{∅, {∅}, {{∅}}} with appropriate succession

All these instantiations are isomorphic. They have the same structure. Structuralism says mathematics studies these structures, not particular instantiations.

Non-Eliminative Structuralism

Non-eliminative structuralism treats structures as objects that exist independently. The natural number structure exists as an abstract object. This structure has positions: 0, 1, 2, and so on. These positions can be filled by various objects, but the structure itself exists separately.

Stewart Shapiro develops this view with a structural theory. His axioms specify which structures exist. For example, there exists a natural number structure satisfying the Peano axioms. There exists a real number structure satisfying the completeness axiom.

In this view, when we say “2 + 3 = 5”, we mean: in the natural number structure, the position 2 plus the position 3 equals the position 5. The statement is about positions in an abstract structure.

Eliminative Structuralism

Eliminative structuralism avoids treating structures as objects. Instead, mathematical statements are about what must be true in any system with the right relationships.

Consider the statement “2 + 3 = 5”. Eliminative structuralism interprets this as: In any system with a zero element and a successor function satisfying the Peano axioms, if x is the second successor of zero, y is the third successor of zero, and z is the fifth successor of zero, then x + y = z.

This approach uses modal logic. The statement becomes: Necessarily, for all systems S, if S satisfies the Peano axioms, then 2_S + 3_S = 5_S. Here, 2_S means the element playing the role of 2 in system S.

Set-Theoretic vs Category-Theoretic Implementation: Two Examples

We will examine two mathematical concepts to understand how set theory and category theory implement eliminative structuralism differently. These examples show why mathematicians developed both approaches.

Example 1: The Number 2

Set-Theoretic Implementation

In set theory, we start with a relational system. A relational system for arithmetic consists of:

A set N (the domain)
An element 0 ∈ N (zero)
A function S: N → N (successor)

Let us construct a specific system. Take N = {a, b, c, d, …}. Define 0 = a. Define S(a) = b, S(b) = c, S(c) = d, and so on.

In this system, the number 2 is represented by the element c, because c = S(S(0)). We reach c by applying the successor function twice to zero.

Now consider another system. Take N = {∅, {∅}, {∅,{∅}}, …}. Define 0 = ∅. Define S(x) = x ∪ {x}.

In this system, the number 2 is represented by {∅,{∅}}, because {∅,{∅}} = S(S(∅)).

Set theory identifies what is common between these systems. Both satisfy the same axioms:

0 is in N
If x is in N, then S(x) is in N
S is injective (different inputs give different outputs)
0 is not in the range of S
The induction principle holds

The statement “2 exists” means: in any system satisfying these axioms, S(S(0)) exists. The statement “2 + 2 = 4” means: in any such system, S(S(0)) + S(S(0)) = S(S(S(S(0)))).

Set theory implements elimination by quantifying over all possible systems. The number 2 is not any particular object. It is whatever plays the role of S(S(0)) in a system satisfying the axioms.

Category-Theoretic Implementation

Category theory takes a different approach. A category consists of objects and morphisms (arrows). For natural numbers, we need:

An object N
A morphism z: 1 → N (this picks out zero)
A morphism s: N → N (successor)

Here, 1 is the terminal object (an object with exactly one morphism from every object).

The number 2 is characterized by morphisms. Start with z: 1 → N. This gives us 0. Compose with s to get s∘z: 1 → N. This gives us 1. Compose again to get s∘s∘z: 1 → N. This gives us 2.

But what is 2? In category theory, we never answer this question. The number 2 is completely characterized by being the target of the morphism s∘s∘z. We know everything about 2 through its relationships:

There is a unique morphism from 1 to 2 (namely, s∘s∘z)
There is a morphism from 2 to 3 (namely, s restricted to 2)
For any object X with a morphism x: 1 → X and a morphism f: X → X, there exists a unique morphism from N to X preserving structure

This last property (the universal property) captures what makes N special. Any other object with these properties is isomorphic to N.

Category theory implements elimination more radically than set theory. Set theory still talks about elements (0 is an element of N). Category theory only talks about morphisms. The number 2 has no internal structure. It exists only through its relationships.

Example 2: The Property “Even”

Set-Theoretic Implementation

In set theory, “even” is a property that holds for some elements in our relational system. Given a system (N, 0, S) satisfying the Peano axioms, we define:

An element n ∈ N is even if one of the following holds:

n = 0, or
n = S(S(m)) for some m ∈ N where m is even

This is a recursive definition. We can make it explicit. Define a function double: N → N by:

double(0) = 0
double(S(x)) = S(S(double(x)))

Then n is even if and only if n is in the range of double.

Consider our first system where N = {a, b, c, d, e, f, …} with 0 = a and S(x) giving the next element. The even elements are {a, c, e, …}. We have:

double(a) = a (so a is even)
double(b) = S(S(double(a))) = S(S(a)) = c (so c is even)
double(c) = S(S(double(b))) = S(S(c)) = e (so e is even)

The property “even” is implemented as a subset E ⊆ N. Different systems give different subsets, but the definition remains the same: E contains exactly those elements reachable by the double function.

Set theory handles properties through subsets and membership. We ask: is element x in subset E? The elimination happens because we do not specify which particular elements are even. We only specify the condition for being even in any system.

Category-Theoretic Implementation

Category theory implements “even” through morphisms, not subsets. We start with our natural numbers object N with morphisms z: 1 → N and s: N → N.

Define a morphism double: N → N by the universal property. We want double such that:

double∘z = z (zero doubles to zero)
double∘s = s∘s∘double (successor doubles to two successors)

This morphism exists and is unique by the universal property of N.

The even numbers are characterized as the image of double. But in category theory, we do not talk about elements in the image. Instead, we use the factorization:

double: N → E → N

Here E is a subobject of N (a monomorphism E → N), and the factorization is the image factorization of double.

To test if “2 is even” in category theory, we ask: does the morphism s∘s∘z: 1 → N factor through E? That is, does there exist a morphism 1 → E making the diagram commute?

The answer is yes because s∘s∘z = double∘z, so 2 is in the image of double.

Category theory never mentions elements. Everything is expressed through morphisms and their compositions. The property “even” becomes a structural relationship between morphisms, not a set of elements.

Why We Need Both Approaches

Set theory and category theory offer different perspectives on elimination:

Set Theory’s Advantages:

Closer to standard mathematical practice
Clear notion of elements and membership
Easier to verify specific properties
Natural for discrete mathematics

Set Theory’s Limitations:

Still relies on the concept of “element.”
Requires choosing specific constructions
Less natural for continuous mathematics
Difficulty with size issues (proper classes)

Category Theory’s Advantages:

Complete elimination of internal structure
Uniform treatment across mathematics
Natural for topology and algebra
Handles size issues elegantly

Category Theory’s Limitations:

More abstract and harder to visualize
Requires learning a new language
Can obscure computational aspects
Less intuitive for basic arithmetic

The difference becomes clear in our examples. Set theory implements the number 2 as “whatever element you get by applying successor twice to zero in your chosen system.” Category theory implements 2 as “the morphism pattern characterized by s∘s∘z.”

Set theory implements “even” as “the subset of elements reachable by doubling.” Category theory implements “even” as “the image subobject of the doubling morphism.”

Both approaches achieve elimination. Neither requires saying what the number 2 really is. But category theory goes further in eliminating reference to internal structure. This makes category theory more general but also more abstract.

Conclusion

Eliminative structuralism claims that mathematical objects need not exist as entities. Mathematics studies the necessary relationships that must hold in any system with an appropriate structure.

Set theory implements this by quantifying over relational systems. Objects are eliminated in favor of positions in relations. We saw this with the number 2 as S(S(0)) in any appropriate system, and with “even” as a recursively defined subset.

Category theory implements elimination more thoroughly. Objects are characterized entirely by morphism patterns. We saw this with 2 as the morphism s∘s∘z, and with “even” as an image factorization.

Both approaches solve the problem of multiple representations. Neither requires choosing between von Neumann and Zermelo ordinals. The choice between set-theoretic and category-theoretic implementation depends on the mathematical context and the level of abstraction desired.