  Another interesting analysis of finitary proof, which, however, does not provide as detailed a philosophical justification, was proposed  by  Kreisel (1960).  It yields the result that exactly those functions are finitary which can be proved to be total in first-order arithmetic \(\PA\). It is based on the proof-theoretic concept of a reflection principle; see Zach (2006)  for more detail and Dean (2015) for an analysis.  Kreisel (1970, Section 3.5)  provides another analysis by focusing on what is “visualizable.” The result is the same: finitary provability turns out to be coextensive with provability in \(\PA\).