Editing Ultrafinitism (section)

== Computational complexity theory based restrictions ==
Other considerations of the possibility of avoiding unwieldy large numbers can be based on [[computational complexity theory]], as in [[András Kornai]]'s work on explicit finitism (which does not deny the existence of large numbers)<ref>[https://archive.today/20120713221118/http://kornai.com/Drafts/fathom_3.html "Relation to foundations"]</ref> and [[Vladimir Sazonov]]'s notion of [[feasible number|feasible numbers]].<!-- seems unclear whether it is really *his* concept, someone can read attached to learn more. https://link.springer.com/chapter/10.1007/3-540-60178-3_78 -->

There has also been considerable formal development on versions of ultrafinitism that are based on complexity theory, like [[Samuel Buss]]'s [[bounded arithmetic]] theories, which capture mathematics associated with various complexity classes like [[P (complexity)|P]] and [[PSPACE]]. Buss's work can be considered the continuation of [[Edward Nelson]]'s work on [[predicative arithmetic]] as bounded arithmetic theories like S12 are interpretable in [[Raphael Robinson]]'s theory [[Robinson arithmetic|Q]] and therefore are predicative in [[Edward Nelson|Nelson]]'s sense.  The power of these theories for developing mathematics is studied in bounded reverse mathematics as can be found in the works of [[Stephen A. Cook]] and [[Phuong The Nguyen]]. However these are not philosophies of mathematics but rather the study of restricted forms of reasoning similar to [[reverse mathematics]].