This log was inspired by "How to Read Wittgenstein" and "Ludwig Wittgenstein: the duty of genius" by Ray Monk. It is based on reading Tractatus Logico-Philosophicus by Ludwig Wittgenstein translated by D. F. Pears & B. F. McGuinness (Routledge and Kegan Paul:1963)

Sunday, April 6, 2008

Mathematics is a method of logic.

Using equations characterizes the essence of the mathematical method. Because this so, every proposition of mathematics must be self explanatory. Mathematics arrives at equations by the method of substitution. Equations express that one can substitute an expression for another and so, starting from a number of equations, we advance to new equations by substituting different expressions as we go along in accordance with them. Thus the proof of the proposition 2 + 2 = 4 runs as follows:
(Ω^ν)^μ'x=Ω^(ν×μ')x Def.,


Exploring logic exploring everything that is subject to regularity, so everything outside of logic is happenstance. The law of induction cannot possibly be a law of logic, norcan it be an a priori law, since it is obviously a proposition that makes sense. The law of causality is not a law but the form of a law. 'Law of causality' that is the name of a type. Just as mechanics has minimizing principles, such as the law of least action, so too does physics have causal laws, laws of the causal form. In fact, one even surmised that there must be a 'law of least action' before know ing exactly how it went. (As always, what is a priori proves to be purely logical.)

We do not believe the law of conservation a priori, but rather know a priori that such a logical form is possible. All such propositions, including the principle of sufficient reason, tile laws of continuity in nature and of least effort in nature, etc. etc., all these are a priori insights about the forms which the propositions of science can take.

Newtonian mechanics, for example, provides a unified form for describing the world. Let us imagine a white surface with irregular black spots on it. Whatever kind of image these make, one can always approximate its description as closely as one wishes by covering the surface with a sufficiently fine square grid, and then declaring every square black or white. This grid provided a unified form for describing the surface. That form is optional, since I using a triangular or hexagonal grid would have achieved the same result. It could be that a triangular grid would have been simpler; that is to say, that we could describe the surface more accurately with a coarse triangular grid than with a fine square grid (or conversely), and so on. The different grids correspond to different systems for describing the world.

Mechanics determines one particular form of description of the world by saying: All propositions describing the world must be obtained in a given way from a given set of propositions, the axioms of mechanics. It supplies the bricks for building the edifice of science, and says: 'Any building that you want to erect, it must be with these and only these components.' (Just as we must be able to write down any amount we wish using the number system, so we must be able to write down any proposition of physics that we wish using the system of mechanics.)

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