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.
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Follow the arc while readingThemes 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.
