An interesting point but I'm not sure that it does. The ultra-finitist says that numbers above a certain limit don't exist. In my view numbers don't "exist" in

An interesting point but I'm not sure that it does. The ultra-finitist says that numbers above a certain limit don't exist. In my view numbers don't "exist" in the first place. The theory of numbers is a tool for making predictions. The statement that numbers above a certain limit don't exist turns into the statement that examples to test the theory don't exist above a certain limit, because you won't be able to find that big of an example in our universe. IMO this is a much more sensible statement. An ultra-finitist may worry about an intermediate number in a calculation not existing, even though the start and end number may be small (for instance consider log_2(2^n) = n). This is silly in my view, since the theory of numbers is a prediction tool as a whole, and there is no need to physically realize intermediate numbers. As long as you're using the rules of the theory you're good (for the same reason you don't need to be afraid of using infinite sets in the middle of a deduction). One might also worry that the theory of numbers may fail to make correct predictions for very large numbers, but this is really a different question about generalising from examples. It could already fail for very large examples that we can test in our universe. The possibility of it failing for numbers larger than we can realize in our universe is precisely what you don't need to worry about.