Wittgenstein takes the successive application of an operation as the model of a proposition. His definition of the general propositional form as "[‾p,‾ξ,N(‾ξ)]" is a variation of the general form for expressing a term in a series: "[a, x, O'x]." The "‾p" is the collection of elementary propositions that a given proposition is composed of, and thus is the first term in the series of operations that generates a complex operation. The "‾ξ" is a complex proposition in this series of successive negations, and "N(‾ξ)" shows us how the next term in the series will be generated, namely by negating all the terms in "‾ξ."

Frege's search for something more certain than pure intuition to ground the concepts of number and arithmetical progression directly motivated his development of modern logic, which then served as the basis for analytic philosophy generally. Frege was largely arguing against Kant, who argued that our knowledge of mathematics is based on pure intuition. Any given number could be generated, according to Kant, by adding a certain number of ones: 4 = 1 + 1 + 1 + 1, while 98 = 1 + 1 + 1 + …. Pure intuition is necessary for the concept of "and so on" that makes it possible to add infinitely many ones together.

Frege claimed that he could make pure intuition unnecessary to mathematics by giving a definition of number based in logic that would provide a general rule more rigorous than "and so on" for adding successive ones. Frege and Russell both developed ingenious systems to prove that the laws of mathematics could be inferred from basic logical axioms. Though they were largely successful, there remained some tensions, as found in Russell's Paradox and Russell's Axiom of Infinity, which related to the conception of numbers as objects.

In defining mathematics as a "method of logic" (6.234), Wittgenstein suggests that numbers are not objects that can be constructed out of logical forms. Numbers are exponents of operations (6.021): they constitute a shorthand for expressing how many times an operation has been applied.

The curious thing about Wittgenstein's philosophy of mathematics in the Tractatus is that it relies on the concept of "and so on" (cf. 6.02) that Frege had gone to such lengths to eliminate. Wittgenstein seems not to give any rigorous account of how one number can be said to follow from the previous one. The difficulties of an expression such as "and so on" would occupy his later philosophy, but, in spite of being a careful student of Frege's works, Wittgenstein seems strangely blind to these difficulties here.

Wittgenstein also goes against Frege and Russell in claiming that the propositions of logic are tautologies that lack sense and say nothing. His conception of logic is explained in a telling metaphor at 6.124: "The propositions of logic describe the scaffolding of the world, or rather they represent it." The metaphor of scaffolding brings to light four principal aspects of Wittgenstein's conception of logic. First, scaffolding is a framework structure: it is a skeleton of joints rather than a building with walls and rooms. Similarly, logic does not consist of propositions with a sense, but only provides a framework within which propositions with a sense may fit. Second, the framework of scaffolding is used to construct a more substantial building, just as logic provides a framework within which the substantial facts about the world may fit. Third, scaffolding has points of contact with the building it is placed against, but it does not overlap with the building, nor is it a part of the building. Logic has points of contact with the world in that both logic and the world share a logical form, but the content (as opposed to the form) of facts themselves has no analogue in logic. Fourth, scaffolding is only a tool used in construction: a sturdy and complete building has no need of scaffolding. Similarly, as Wittgenstein claims at 5.5563, "all the propositions of our everyday language, just as they stand, are in perfect logical order." We do not need logic or philosophy when language is functioning normally. These tools are only needed to provide clarity when language misfires and attempts to speak nonsense.

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