Bertrand Russell and Alfred North Whitehead's monumental work, Principia Mathematica (often abbreviated as PM), published in three volumes between 1910 and 1913, represents one of the most ambitious attempts in the history of mathematics to establish a rigorous logical foundation for all of mathematics. Among its many achievements, the work is perhaps most famously known for containing a proof that 1 + 1 = 2, which appears on page 379 (Volume I, 1st edition) or page 362 (2nd edition).
This document explains the proof, its context, and its significance in a way that makes this complex mathematical achievement accessible.
Contrary to popular belief, Russell and Whitehead did not spend hundreds of pages just to prove that 1 + 1 = 2. Their goal was far more ambitious: to derive all of mathematics from pure logic, creating a complete and consistent formal system. This project was motivated by several factors:
The proof of 1 + 1 = 2 was merely one small step in this grand project, appearing relatively late in the work after establishing the necessary logical framework.
There's a common claim that Russell and Whitehead "spent 360 pages to prove 1 + 1 = 2." This is somewhat misleading for several reasons:
As one mathematician on Stack Exchange put it: "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."
To understand the proof, we need to grasp the basics of the logical system developed in PM.
PM uses a notation that differs from modern mathematical notation:
The notation uses Peano's "dots" to disambiguate precedence, where we would now use parentheses. For example, (1+2)×(3+4)×(5+6) would be written as 1+2 . 3+4 . 5+6.
To avoid paradoxes like Russell's paradox, PM introduces a "theory of types" that places grammatical restrictions on formulas. This hierarchical system prevents self-reference and ensures that sets can only contain elements of a lower type.
In PM, numbers are defined as sets of sets with a certain cardinality:
More formally, in PM notation, 1 denotes the set of all sets that have exactly one element. That is, it's the set { c : there exists a such that c = { a } }.
The actual proof that 1 + 1 = 2 appears as theorem *54.43 in PM. The theorem states:
⊢ : α, β ε 1 . ⊃ : α ∩ β = Λ . ≡ . α ∪ β ε 2
In modern notation, this would be expressed as:
"If α and β are sets that each contain exactly one element, then their union contains exactly two elements if and only if they have no elements in common (i.e., their intersection is empty)."
The proof proceeds as follows:
⊢ : α = ι'x . β = ι'y . ⊃ : α ∪ β ε 2 . ≡ . x ≠ y
This states that if α is the set containing only x and β is the set containing only y, then the union of α and β has exactly two elements if and only if x is different from y.
≡ . ι'x ∩ ι'y = Λ
This states that the intersection of a set containing only x and a set containing only y is empty if and only if x is different from y.
≡ . α ∩ β = Λ
This substitutes the sets α and β back into the formula.
⊢ : (∃x, y) . α = ι'x . β = ι'y . ⊃ : α ∪ β ε 2 . ≡ . α ∩ β = Λ
This generalizes the result to any sets α and β that each contain exactly one element.
The proof concludes with the statement: "From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2."
The authors add a comment after the proof: "From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." This indicates that the theorem *54.43 itself doesn't directly state that 1 + 1 = 2, but rather establishes a property about sets that will be used later to define addition and prove that 1 + 1 = 2.
The actual definition of addition and the final proof that 1 + 1 = 2 come later in the work, after the concept of arithmetic addition has been formally defined.
From a modern perspective, the proof in PM might seem unnecessarily complex. Today, we would approach this differently:
As one mathematician put it: "A huge amount of other machinery goes away in 2006, because of the unification of relations and sets... In 2006 we would just use the ordinary set intersection operation and talk about R ∩ (S×B) or whatever."
For a detailed comparison of Russell's approach with modern mathematical frameworks, please see our Modern Approaches page.
Russell and Whitehead's proof of 1 + 1 = 2 in Principia Mathematica represents far more than a trivial demonstration of an obvious fact. It stands as a milestone in the development of mathematical logic and the foundations of mathematics. The proof illustrates the authors' ambitious attempt to derive all mathematical truths from pure logic, requiring them to build a complex logical system from the ground up.
While their ultimate goal of creating a complete and consistent formal system for all of mathematics was later proven impossible by Gödel, their work remains a monumental achievement in the history of mathematics and philosophy. The proof of 1 + 1 = 2, appearing after hundreds of pages of logical development, serves as a powerful reminder of the depth and complexity that underlies even the most seemingly obvious mathematical truths.