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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).
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].
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