Find out, what is ultrafinitism? Why some people believe in it? Can you actually get math done from that perspective?
Math Overflow: Is there any formal foundation to ultrafinitism?
In the philosophy of mathematics, ultrafinitism, or ultraintuitionism, is a form of finitism.
Like other strict finitists, ultrafinitists deny the existence of the infinite set N of natural numbers, on the grounds that it can never be completed.
People associated with ultrafinitism:
Serious work on ultrafinitism has been led, since 1959, by Alexander Esenin-Volpin. Other mathematicians who have worked in the topic include Doron Zeilberger, Edward Nelson, and Rohit Jivanlal Parikh. The philosophy is also sometimes associated with the views of Ludwig Wittgenstein, Robin Gandy and J. Hjelmslev.
Fundamental Statement of Prof. Doron Zeilberger’s Ultrafinitism