On the Notion of Analogy and Complexity in Some Constructive Mathematical Schemata

摘要

Banach often remarked Good mathematicians see analogies between theorems and proofs; the very best ones see analogies between analogies. Mark Kac certainly belongs to the latter group. His work on problems in statistical mechanics and in number theory, disciplines so different from each other, exhibits a feeling for the role of the ideas of probability analogous in some way in these two domains which are so far apart. In what follows an elementary approach to the notion of analogy is presented and a few mathematical problems are pointed out that pose themselves once one tries to discuss this notion in a somewhat general way; namely, similarity or proximity of proofs and counting binary and unary operations at each stage with similar trees on the set of axioms. (ERA citation 07:013595)