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We introduce, develop, and apply a new approach for dealing with theintuitive notion of function, called Flow Theory. Within our framework allfunctions are monadic and none of them has any domain. Sets, proper classes,categories, functors, and even relations are special cases of functions. Inthis sense, functions in Flow are not equivalent to functions in ZFC.Nevertheless, we prove both ZFC and Category Theory are naturally immersedwithin Flow. Besides, our framework provides major advantages as a language foraxiomatization of standard mathematical and physical theories. Russell'sparadox is avoided without any equivalent to the Separation Scheme. Hierarchiesof sets are obtained without any equivalent to the Power Set Axiom. And a clearprinciple of duality emerges from Flow, in a way which was not anticipatedneither by Category Theory nor by standard set theories. Besides, there seemsto be within Flow an identification not only with the common practice of doingmathematics (which is usually quite different from the ways proposed bylogicians), but even with the common practice of teaching this formal science.

Follow the Flow: sets, relations, and categories as special cases of functions with no domain