Summary
The general form of a proposition is "[‾p, ‾ξ, N(‾ξ)]" (6). That is, every proposition is built from an initial set of elementary propositions (‾p) that are then transformed into a more complex proposition through successive applications of the negating operation, "N(‾ξ)." Thus, propositions generally are produced through successive applications of an operation.
Mathematics is also founded in the successive application of operations. If we take the expression "1/2'x" to signify the operation "1/2" applied to x, we can define a number series in terms of how many times 1/2 is applied to x. For instance, x can be defined as 1/2(^0)'x, 1/2'x as 1/2(^1)'x, 1/2'1/2'x as 1/2(^2)'x, and so on: "A number is the exponent of an operation" (6.021). The general concept of number is simply the form that all numbers share in common.
The propositions of logic are tautologies (6.1), and hence say nothing (6.11). Any attempt to give content to logical propositions is misguided. That they are true shows itself in their structure, and this structure helps us to understand the formal properties of language and the world (6.12). We cannot express anything by means of logical propositions.
Because the truths of logic are all the same (in that they all say nothing), there is no real need to "prove" them. What we call "proof" with regard to logical propositions is only necessary in complicated cases where a proposition's being a tautology is not immediately evident (6.1262). This kind of proof, however, is of an entirely different kind from the proofs by which we can establish the truth of a proposition with a sense. To prove the truth of a proposition with a sense, we must show that it follows from something else that we already know to be true. A proposition of logic, however, does not need to be deduced from other propositions. Rather, we could say, the propositions of logic give us the form of logical proof (6.1264): for example, the tautology "((p ⊃ q).p) ⊃ q" shows us that, given the non-tautologous propositions "p ⊃ q" and "p" we can prove another non-tautologous proposition, "q."
"Mathematics is a logical method" (6.2): as we have seen, numbers can be derived from the successive application of operations, this application of operations being a method of logic. The propositions of mathematics are all equations, where we say that one expression is the equivalent of another (e.g. "7 + 5 = twelve"). As Wittgenstein has already discussed, (5.53–5.5352) the sign for identity is superfluous, since the equivalence of two propositions should be evident from their form. It thus follows that the propositions of mathematics are all pseudo- propositions: they do not tell us anything, but simply express an equivalence of form. As logical pseudo-propositions, the propositions of mathematics cannot themselves express thoughts. Rather, they are abstractions that help us to infer propositions about the world (6.211).
Analysis
A series is a mathematical entity that consists of a number of terms arranged in a particular order, e.g. the series of square numbers, [1, 4, 9, 16, ...]. In 5.2522, Wittgenstein gives a general form for expressing a term in a particular series as "[a, x, O'x]," where "a" stands for the first term in the series, "x" stands for an arbitrarily selected term, and "O'x" stands for the term that immediately follows "x." The "O'" is the operation by which a term in the series is generated out of another. So, for instance, we could express the series of square numbers as [1, x, (sqr(x) + one)^2].
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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