(in Knuth's up-arrow notation) would be considered only a formal expression that does not correspond to a natural number.
Some versions of ultrafinitism are forms of constructivism, but most constructivists view the philosophy as unworkably extreme.
The logical foundation of ultrafinitism is unclear; in his comprehensive survey Constructivism in Mathematics (1988), the constructive logician A. S. Troelstra dismissed it by saying "no satisfactory development exists at present."
This was not so much a philosophical objection as it was an admission that, in a rigorous work of mathematical logic, there was simply nothing precise enough to include.
Other mathematicians who have worked in the topic include Doron Zeilberger, Edward Nelson, Rohit Jivanlal Parikh, and Jean Paul Van Bendegem.
The philosophy is also sometimes associated with the beliefs of Ludwig Wittgenstein, Robin Gandy, Petr Vopěnka, and Johannes Hjelmslev.
Shaughan Lavine has developed a form of set-theoretical ultrafinitism that is consistent with classical mathematics.