I'm not sure if he's a finitist or ultrafinitist [1] but the uncountable reals are clearly a problem for him. I'd like to argue for their usage from a "soft" point of view as opposed to just giving axioms.
Mathematics and physics both began with the advent of astronomy: Babylonians and others were curious to trace star patterns and from there both physics and math developed in tandem and influenced each other greatly. Really, neither would have developed without the other.
Calculus was invented for calculations in physics. This gave rise to differential equations which we use to model so many nontrivial things. The differential eqns describe flow and continuity and arguably reality. Differentiation and smoothness can't be defined over finite sets in the same way. My philosophical counter-argument to finitists is that clearly we're on the right path to understanding the universe and nature when using the reals. It seems foolish to shy from this because computers have trouble computing some functions. Statements like "there are only finitely many atoms in the universe" don't improve our understanding of much but PDEs explain.
Mathematics and physics both began with the advent of astronomy: Babylonians and others were curious to trace star patterns and from there both physics and math developed in tandem and influenced each other greatly. Really, neither would have developed without the other.
Calculus was invented for calculations in physics. This gave rise to differential equations which we use to model so many nontrivial things. The differential eqns describe flow and continuity and arguably reality. Differentiation and smoothness can't be defined over finite sets in the same way. My philosophical counter-argument to finitists is that clearly we're on the right path to understanding the universe and nature when using the reals. It seems foolish to shy from this because computers have trouble computing some functions. Statements like "there are only finitely many atoms in the universe" don't improve our understanding of much but PDEs explain.
[1] https://en.wikipedia.org/wiki/Ultrafinitism