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Many times in mathematics there is a natural dichotomy betweendescribing some object from the inside and from the outside. Imaginealgebraic varieties for instance; they can be described from theoutside as solution sets of polynomial equations, but one can also tryto understand how it is for actual points to move around inside them,perhaps to parameterize them in some way. The concept of formalproofs has the interesting feature that it provides opportunities forboth perspectives. The inner perspective has been largely overlooked,but in fact lengths of proofs lead to new ways to measure informationcontent of mathematical objects. The disparity between minimallengths of proofs with and without ``lemmas'' provides an indicationof internal symmetry of mathematical objects and their descriptions.A principal observation of this paper is that mathematicalstructures can be embedded into spaces of logical formulae and inheritadditional structure from proofs. We shall look at finitely-generatedgroups, rational numbers and SL(2,Z), and examples fromtopology and analysis.

DOI: 10.1023/A:1005244027724

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