Tractatus Logico-Philosophicus · Chapter
6.02 explained
Wittgenstein defines numbers through formal operations and recursive definitions, establishing the series of natural numbers as successive applications of the operation Ω to a base element x.
What happens
Wittgenstein defines numbers through formal operations and recursive definitions, establishing the series of natural numbers as successive applications of the operation Ω to a base element x.
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Narrative arc
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- Setup
- Escalation
- Breaking point
- Aftermath
- Closing
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Follow the arc while readingThemes in this chapter
- Logic and Tautology
The definition of numbers through formal operations reflects the tautological nature of mathematical propositions.
Characters to notice
- Mr. Wittgenstein
Defines numbers via formal rules and the operation Ω.
Key passages
“And thus we come to numbers: I define x = Ω 0 ′ x Def. and Ω ′ Ω ν ′ x = Ω ν + 1 ′ x Def.”
We now arrive at numbers: I define x as the result of applying operation Ω zero times, and applying Ω to the result of ν applications yields ν+1 applications.
Wittgenstein introduces the recursive definition of numbers based on iterated operations.
“Therefore I write in place of “ [ x , ξ , Ω ′ ξ ] ”, “ [ Ω 0 ′ x , Ω ν ′ x , Ω ν + 0 ′ x ] ”.”
So I replace the notation for a series generated by repeated application with one explicitly indexed by the number of applications.
Wittgenstein reformulates the general form of a series in terms of numerical indices.
