Tractatus Logico-Philosophicus · Chapter
6.241 explained
Wittgenstein provides a formal proof of the proposition 2 × 2 = 4 using his notation for operations, demonstrating how mathematical propositions are derived from definitions and operations rather than expressing substantive truths.
What happens
Wittgenstein provides a formal proof of the proposition 2 × 2 = 4 using his notation for operations, demonstrating how mathematical propositions are derived from definitions and operations rather than expressing substantive truths.
Themes in this chapter
Mathematical Propositions Express No Thoughts
The proof of 2 × 2 = 4 is shown as a tautological transformation of symbols, not a factual statement about the world.
Characters to notice
- Mr. Wittgenstein
Author of the proof and the logical notation used.
Key passages
“Thus the proof of the proposition 2 × 2 = 4 runs: ( Ω ν ) μ ′ x = Ω ν × μ ′ x Def. Ω 2 × 2 ′ x = ( Ω 2 ) 2 ′ x = ( Ω 2 ) 1 + 1 ′ x = Ω 2 ′ Ω 2 ′ x = Ω 1 + 1 ′ Ω 1 + 1 ′ x = ( Ω ′ Ω ) ′ ( Ω ′ Ω ) ′ x = Ω ′ Ω ′ Ω ′ Ω ′ x = Ω 1 + 1 + 1 + 1 ′ x = Ω 4 ′ x .”
The proof of 2 × 2 = 4 is a step-by-step manipulation of operation symbols, showing that applying the operation Ω twice, twice, is equivalent to applying it four times.
This illustrates Wittgenstein's view that mathematical proofs are sequences of tautological transformations.