8 1. Logic and foundations for i = 1,...,n, and so by telescoping all of these identities together we obtain A1 ...An − B1 ...Bn = n i=1 B1 ...Bi−1(Ai − Bi)Ai+1 ...An which, when combined with tools such as the triangle inequality, gives a variety of stability results of the desired form (even in situations in which the A’s and B’s do not commute). Note that this identity is also the discrete form of the product rule (A1 ...An) = n i=1 A1 ...Ai−1AiAi+1 ...An and in fact easily supplies a proof of that rule. 1.5. Notational conventions Like any other human language, mathematical notation has a number of implicit conventions which are usually not made explicit in the formal de- scriptions of the language. These conventions serve a useful purpose by conveying additional contextual data beyond the formal logical content of the mathematical sentences. For instance, while in principle any symbol can be used for any type of variable, in practice individual symbols have pre-existing connotations that make it more natural to assign them to specific variable types. For instance, one usually uses x to denote a real number, z to denote a complex number, and n to denote a natural number a mathematical argument involving a complex number x, a natural number z, and a real number n would read very strangely. For similar reasons, x ∈ X reads a lot better than X ∈ x sets or classes tend to “want” to be represented by upper case letters (in Roman or Greek) or script letters, while objects should be lower case or upper case letters only. The most famous example of such “typecasting” is of course the epsilon symbol in analysis an analytical argument involving a quantity epsilon which was very large or negative would cause a lot of unnecessary cognitive dissonance. In contrast, by sticking to the conventional roles that each symbol plays, the notational structure of the argument is reinforced and made easier to remember a reader who has temporarily forgotten the definition of, say, “z” in an argument can at least guess that it should be a complex number, which can assist in recalling what that definition is. As another example from analysis, when stating an inequality such as X Y or X Y , it is customary that the left-hand side represents an “unknown” that one wishes to control, and the right-hand side represents a more “known” quantity that one is better able to control thus, for instance,

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