How to take advantage of the blur between the finite and the infinite (Q1940915)

This is a historically oriented essay on some factors of development in mathematics. The author focuses on the difference between true finite as opposed to theoretical finite. He starts with the discussion of mathematical induction and the principle of Fermat being the sophisticated version of the principle of complete induction. He stresses the fact that mathematical activity is always connected with pushing the borders between what seems to be founded on firm grounds and the ``grey zone'' of phenomena not yet penetrated or not easily graspable. According to author this activity is constantly rooted in social needs and strongly connected with the developments on other fields. In particular, some examples from physics are discussed, as well as the problem of ``invisible numbers'' for which we do not have any intuition or form of convenient representation. The text is written in informal, easily accessible (even for nonmathematician) form and contains a lot of historical examples.