Introduction
The proof that 1 + 1 = 2 might seem trivial at first glance, but it has been approached in various ways throughout the history of mathematics, each with its own philosophical underpinnings and technical machinery. This document compares different modern approaches to proving this seemingly simple statement, analyzing their strengths, weaknesses, and philosophical implications.
1. Set-Theoretic Approach
Axioms of Zermelo-Fraenkel Set Theory (ZFC)
The set-theoretic approach to proving 1 + 1 = 2 is based on the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), which includes the following axioms:
- Axiom of Extensionality: Two sets are equal if and only if they have the same elements.
- Axiom of Regularity (Foundation): Every non-empty set contains an element that is disjoint from the original set.
- Axiom Schema of Specification (Separation): Given a set and a property, there exists a subset containing exactly the elements of the original set that satisfy the property.
- Axiom of Pairing: For any two sets, there exists a set containing exactly those two sets as elements.
- Axiom of Union: For any set of sets, there exists a set containing exactly the elements of those sets.
- Axiom Schema of Replacement: If a function is defined on a set, then the range of that function also forms a set.
- Axiom of Infinity: There exists an infinite set.
- Axiom of Power Set: For any set, there exists a set containing all its subsets.
- Axiom of Choice: Given a collection of non-empty sets, there exists a function that selects one element from each set.
These axioms provide the foundation for defining numbers as sets and operations like addition in terms of set operations.
Description
In modern set theory, particularly within the Zermelo-Fraenkel with Choice (ZFC) framework, numbers are typically defined as specific sets, and operations like addition are defined in terms of set operations.
The Proof
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Definition of numbers:
- 0 is defined as the empty set: 0 = ∅
- 1 is defined as the set containing only the empty set: 1 = {∅}
- 2 is defined as the set containing 0 and 1: 2 = {∅, {∅}}
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Definition of addition:
For any sets (numbers) m and n, m + n is defined as the cardinality of the disjoint union of sets representing m and n.
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Proof:
- 1 + 1 is the cardinality of the disjoint union of {∅} with itself
- This gives us {(∅,0), (∅,1)}, which has cardinality 2
- Therefore, 1 + 1 = 2
Strengths
- Provides a rigorous foundation based on set theory
- Integrates naturally with the rest of mathematics built on ZFC
- Avoids the need for separate axioms for arithmetic
Weaknesses
- Highly abstract and counterintuitive representation of numbers
- The connection to our intuitive understanding of numbers is not immediately obvious
- Requires understanding of set theory concepts
Philosophical Implications
This approach reflects the formalist and logicist traditions in mathematics, where mathematical objects are defined purely in terms of sets and logical operations. It emphasizes the view that mathematics can be reduced to set theory and logic.
Criticisms of ZFC Axioms
- Ontological Concerns: ZFC assumes the existence of abstract mathematical objects (sets) without clear connection to physical reality.
- Axiom of Choice Controversies: The Axiom of Choice leads to counterintuitive results like the Banach-Tarski paradox, which states that a solid ball can be decomposed and reassembled into two identical copies of the original ball.
- Size Limitations: ZFC cannot handle certain large collections (proper classes) directly, requiring workarounds for concepts like "the category of all sets."
- Arbitrary Nature: The specific axioms chosen may seem arbitrary rather than inevitable, raising questions about why these particular axioms should be accepted.
- Alternative Set Theories: Other set theories (like New Foundations or NBG) offer different approaches to foundations, suggesting that ZFC is not the only possible foundation.
- Constructivist Objections: Non-constructive proofs allowed by ZFC are rejected by some mathematical philosophies that require explicit constructions rather than mere existence proofs.
Response to Criticisms
Despite these criticisms, ZFC remains the standard foundation for mathematics due to its flexibility, power, and ability to formalize most of mathematics. The specific representation of numbers as sets doesn't matter as much as the structural properties that emerge, which align with our intuitive understanding of numbers and their operations.
2. Peano Axioms Approach
Peano Axioms
The Peano axioms provide a direct axiomatization of the natural numbers:
- Zero is a natural number: 0 is a natural number.
- Successor: For every natural number n, S(n) is a natural number (where S is the successor function).
- Injectivity of Successor: For all natural numbers m and n, if S(m) = S(n), then m = n.
- Non-Successor: For every natural number n, S(n) ≠ 0 (zero is not the successor of any natural number).
- Induction: If a property is true of 0, and if the property being true for a natural number implies it is true for the successor of that number, then the property is true for all natural numbers.
These axioms directly capture our intuitive understanding of counting and provide a foundation for arithmetic operations.
Description
The Peano axioms provide a formal foundation for arithmetic based on a few simple principles about natural numbers and the successor function.
The Proof
-
Definition of addition:
- a + 0 = a
- a + S(b) = S(a + b)
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Definition of numbers:
- 1 is defined as S(0)
- 2 is defined as S(1) or S(S(0))
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Proof:
- 1 + 1 = S(0) + S(0)
- By definition of addition: S(0) + S(0) = S(S(0) + 0)
- By definition of addition: S(0) + 0 = S(0)
- Therefore: 1 + 1 = S(S(0)) = 2
Strengths
- More intuitive connection to our understanding of counting
- Directly captures the essential properties of natural numbers
- Provides a clear recursive structure for arithmetic operations
Weaknesses
- Requires accepting axioms specifically about numbers
- Doesn't reduce numbers to more fundamental concepts (unlike set theory)
- First-order formulations are incomplete (by Gödel's incompleteness theorems)
Philosophical Implications
The Peano axioms approach reflects a more direct formalization of our intuitive understanding of numbers as a sequence starting from zero and proceeding through successive "next" elements. It's less reductionist than the set-theoretic approach.
Criticisms of Peano Axioms
- Primitive Notion: Takes the concept of natural numbers as primitive rather than reducing them to more fundamental concepts, which some philosophers find unsatisfying.
- First-Order Limitations: First-order formulations of Peano arithmetic are incomplete (by Gödel's incompleteness theorems), meaning there are true statements about natural numbers that cannot be proven within the system.
- Non-Categorical: First-order Peano arithmetic admits non-standard models that include elements beyond the standard natural numbers, raising questions about what the axioms truly capture.
- Philosophical Status: The induction axiom has been questioned philosophically as it makes an assertion about all properties, which some find problematic.
- Constructivist Concerns: Classical formulations don't address constructive aspects of numbers, which some mathematical philosophies require.
Response to Criticisms
The directness of the Peano axioms can be seen as a strength rather than a weakness. By capturing our intuitive understanding of counting without unnecessary abstraction, they provide a clear and accessible foundation for arithmetic. The limitations revealed by Gödel's theorems apply to any sufficiently powerful formal system, not just Peano arithmetic.
3. Category Theory Approach
Axioms of Category Theory
Category theory provides an abstract framework for mathematical structures and their relationships:
- Objects and Morphisms: A category consists of a collection of objects and a collection of morphisms (or arrows) between these objects.
- Composition: For any morphisms f: A → B and g: B → C, there exists a morphism g∘f: A → C (composition).
- Associativity: For morphisms f: A → B, g: B → C, and h: C → D, composition is associative: h∘(g∘f) = (h∘g)∘f.
- Identity: For each object A, there exists an identity morphism idₐ: A → A such that for any morphism f: A → B, f∘idₐ = f and idₐ∘g = g for any g: C → A.
These axioms provide a framework for understanding mathematical structures in terms of their relationships rather than their intrinsic nature.
Description
Category theory provides a framework for mathematical structures and their relationships. In this approach, natural numbers are characterized by their universal property as an initial algebra.
The Proof
-
Definition of natural numbers:
- Natural numbers form the initial algebra for the endofunctor F(X) = 1 + X
- This means there's a set N with a distinguished element 0: 1 → N and a successor function s: N → N
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Definition of addition:
- Addition is defined as the unique homomorphism from the algebra (N, [0,s]) to (N, [n,s])
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Proof:
- 1 + 1 is the result of applying the addition homomorphism for 1 to the element 1
- By the properties of the homomorphism, this equals s(1)
- By definition, s(1) = 2
- Therefore, 1 + 1 = 2
Strengths
- Provides a highly abstract and general framework
- Captures the essential structure without unnecessary details
- Connects naturally to other mathematical structures
Weaknesses
- Extremely abstract and requires significant mathematical background
- Less intuitive for those not familiar with category theory
- Further removed from computational implementations
Philosophical Implications
The category theory approach emphasizes structural relationships and universal properties rather than specific constructions. It reflects a structuralist philosophy of mathematics, where mathematical objects are characterized by their relationships rather than their intrinsic nature.
Criticisms of Category Theory as Foundation
- Class vs. Set Issues: Handling of proper classes and the category of all sets raises foundational concerns, as category theory itself often requires a foundation to handle size issues.
- Accessibility: Highly abstract and requires significant mathematical background, making it less accessible than other foundations.
- Formalization Challenges: Formalizing the notion of a "collection" of objects and morphisms can be problematic without appealing to another foundation like set theory.
- Equality Definition: The meaning of equality between morphisms is not always clear and may depend on the specific category.
- Ontological Status: Questions about what category-theoretic entities "really are" can be difficult to answer without reference to another foundation.
- Practical Usability: Further removed from computational implementations than other foundations, making it less practical for some applications.
Response to Criticisms
The abstraction of category theory, while challenging, reveals deep connections between seemingly different mathematical structures that might otherwise remain hidden. The focus on relationships rather than specific constructions aligns with how mathematics is often practiced, where the properties of objects matter more than their specific representations.
4. Type Theory Approach
Axioms of Type Theory
Type theory, particularly Martin-Löf Type Theory, is based on the following principles:
- Type Formation: Rules for forming new types from existing ones.
- Term Introduction: Rules for constructing terms of a given type.
- Term Elimination: Rules for using terms of a given type.
- Computation Rules: Rules governing how terms compute or reduce.
- Identity Types: For any type A and terms a,b:A, there is an identity type Id_A(a,b).
- Dependent Function Types (Π-types): If A is a type and B(x) is a type for each x:A, then Π(x:A)B(x) is a type.
- Dependent Pair Types (Σ-types): If A is a type and B(x) is a type for each x:A, then Σ(x:A)B(x) is a type.
- Inductive Types: Natural numbers, booleans, etc., defined by their constructors and induction principles.
These principles provide a foundation where proofs and programs are treated as the same kind of mathematical objects.
Description
In type theory, particularly in constructive type theories like Martin-Löf type theory or in proof assistants like Coq or Agda, numbers and proofs are treated as types and programs.
The Proof
-
Definition of natural numbers:
- Natural numbers are defined as an inductive type with constructors zero and successor
- 0 : Nat
- S : Nat → Nat
-
Definition of addition:
add : Nat → Nat → Nat
add 0 n = n
add (S m) n = S (add m n)
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Definition of numbers:
- 1 is defined as S 0
- 2 is defined as S 1 or S (S 0)
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Proof:
- 1 + 1 = add (S 0) (S 0)
- By definition of add: add (S 0) (S 0) = S (add 0 (S 0))
- By definition of add: add 0 (S 0) = S 0
- Therefore: 1 + 1 = S (S 0) = 2
Strengths
- Computationally meaningful: proofs correspond to programs
- Can be verified by computers using proof assistants
- Provides a foundation for both mathematics and computer science
- Constructive approach ensures proofs are algorithmic
Weaknesses
- Requires understanding of type theory
- Some classical mathematical techniques are not directly available
- Can be more verbose than other approaches
Philosophical Implications
Type theory reflects a constructivist philosophy of mathematics, where mathematical objects must be constructible and proofs must provide algorithms. It bridges mathematics and computer science, emphasizing the computational content of mathematical statements.
Criticisms of Type Theory
- Constructive Limitations: Constructive restrictions limit the application of classical techniques, requiring workarounds for principles like the law of excluded middle.
- Complexity: Can be more verbose and complex than set-theoretic approaches, requiring more detailed specifications.
- Multiple Variants: Many different type theories with different properties makes standardization difficult and can lead to confusion.
- Philosophical Status: Questions about the ontological status of types versus sets and what they "really are."
- Accessibility: Requires specialized knowledge and notation, making it less accessible to mathematicians trained in classical approaches.
- Historical Development: Relatively recent compared to set theory, with ongoing development and less established conventions.
Response to Criticisms
The computational content and verifiability of type theory provide stronger guarantees and practical applications, particularly in computer science. The constructive approach, while sometimes limiting, ensures that proofs have algorithmic content, making them more directly applicable to computation. The connection between proofs and programs in type theory provides insights that are not as apparent in other foundations.
5. Comparison with Russell and Whitehead's Approach
Key Differences
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Foundational Framework:
- Russell and Whitehead built their entire logical system from scratch
- Modern approaches typically start from established frameworks (ZFC, type theory, etc.)
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Notation and Formalism:
- PM used a cumbersome notation that has largely been abandoned
- Modern approaches use more streamlined and standardized notation
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Scope:
- PM attempted to derive all of mathematics from logic
- Modern approaches are more modular and focused
-
Efficiency:
- PM's proof was extremely lengthy due to building everything from first principles
- Modern proofs are much more concise, leveraging established frameworks
Why Russell and Whitehead's Proof Was So Long
The proof in Principia Mathematica wasn't just about showing 1 + 1 = 2; it was about:
- Developing a complete logical system from scratch
- Defining what numbers are in terms of pure logic
- Defining operations like addition
- Proving properties of these operations
- Finally showing that 1 + 1 = 2 within this system
As one mathematician explained: "They spend the treatise defining what was hoped to be a complete and consistent basis for all of mathematics. That means they weren't just proving that 1+1=2 (under their system of mathematics) but also defined what '1', '+', '=', and '2' meant."
6. Which Approach Is "Best"?
There is no single "best" approach, as each serves different purposes:
- For foundational rigor: Set-theoretic or type-theoretic approaches provide the most comprehensive foundations.
- For intuitive understanding: The Peano axioms approach is most accessible.
- For computational implementation: Type theory is most directly implementable.
- For revealing structural connections: Category theory excels.
The choice depends on philosophical preferences, practical needs, and the mathematical context.
Conclusion
The various approaches to proving 1 + 1 = 2 reflect different philosophical perspectives on the nature of mathematics and different technical frameworks. While Russell and Whitehead's approach in Principia Mathematica was groundbreaking for its time, modern approaches have streamlined the process considerably.
What remains fascinating is that such a seemingly simple statement can reveal so much about the foundations of mathematics and the different ways we can conceptualize numbers and their operations. The diversity of approaches demonstrates the richness of mathematical thought and the multiple valid perspectives from which we can understand even the most basic mathematical truths.
Each approach is built on different axioms, which themselves are subject to philosophical scrutiny and criticism. By understanding these axioms and their limitations, we gain deeper insight into the foundations of mathematics and the assumptions that underlie our mathematical reasoning.