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Formalizing mathematics

In formalizing mathematics, we must rephrase informal constructs in terms of formal ones, which is merely an extreme case of defining non-rigorous concepts rigorously. This isn't always easy, but there can be surprising successes --- for example, before Turing it would have seemed implausible that a simple and rigorous definition of 'computable' could be given. For a good historical precedent, consider how Bolzano and Weierstrass arrived at rigorous 'memberof-delta' definitions of notions like limits and continuity. We may have our own intuitive ideas of what a continuous function is: perhaps one whose graph we can draw without taking our pencil from the graph paper, or which passes through all intermediate values. But the definition of continuity which is now standard is neither of these; we say a function f:R fun R is continuous on R iff:

forallx memberof R. forallmemberof > 0. existsdelta > 0. forallx'. |x - x'| < delta implies |f(x) - f(x')| < memberof

In what sense is this or any other formal definition 'right'? Why not take the intermediate value property (called 'Darboux continuity') instead?

forallx, x', z memberof R. x < x' and (f(x) < z < f(x') or f(x') < z < f(x))
implies existsw. x < w < x' and f(w) = z

Such worries can be allayed when the various plausible-looking formalizations are all provably equivalent. For example, the various definitions of computability in terms of Turing machines, Markov algorithms, general recursive functions, untyped lambda-calculus, production systems and so forth all turned out so. But here this is not the case: continuity implies Darboux continuity but not conversely.26 The usual definition of continuity has probably won out because it is intuitively the most satisfying, leads to the tidiest theory or admits the most attractive generalization to other metric and topological structures. But note that it also led to the counterintuitive pathologies of real analysis which, as we have already remarked, caused such disquiet once. This example well illustrates how formalization can be a difficult and creative task, and that the formal version of a concept can take on an unexpected life of its own. However, since present-day mathematics is mostly quite rigorous, the final step to complete formalization shouldn't usually be so problematical.

left right up home John Harrison 96/8/13; HTML by 96/8/16