Non-standard methods are a thriving business since the sixties of past century; their application in practically every field of mathematics has confirmed that infinitesimals’ intuition shortens proofs and facilitates discovery, although it is questioned by a large part of the mathematical community. A different case is that of Sergeyev’s work of the last twenty years. He introduces axiomatically an infinite natural number , Grossone, which is meant to be the cardinality of natural numbers, and shows that arithmetical calculations with infinite and infinitesimal numbers give more precise and discriminating results in many areas. Even automated computing can be extended. Behind Sergeyev’s theory and techniques there lurks an unusual approach to mathematics which could qualify as an original philosophy of mathematical practice, if it could be cast in a consistent proposal. We explain how for the moment it is a mix of threads with elements of realism (in the idea of a reality observed through stronger and stronger lenses), of formalism (in his refusing to fix such reality through a definition), and of finitism (in his insistence that only a finite number of finite operations are within human capabilities). We try to trace a common source in Sergeyev’s conception of the mathematician’s work being similar to the physicist’s and in the idea of the changing strength of the available observation tools. We propose as a possible frame a Russell-type knowledge by acquaintance. We indicate a few critical points needing mending or further enlightenments.
