What is finitistic reasoning? I have been looking at various introductions to Hilbert's program, and they all use the concept of finitistic reasoning. What is precisely finitistic reasoning, and what would be an example of non-finitistic reasoning (preferably a simple example)?
I did look at some existing answer, such as Hilbert's Program, but did not find anything that seemed a clear definition fot finitistic reasoning.
 A: The question "What precisely is finitistic reasoning?" has a long and tangled history, and a short answer would inevitably be misleading. So let me point to something a bit more substantial. A very good place to start is Richard Zach's extremely helpful encyclopedia article on Hilbert's Program, here.
Zach gives an extensive discussion of Hilbert's own characterisations of what makes for finitistically acceptable reasoning -- what is it about? what resources are allowed?
Zach goes on to explain why it is arguable (and indeed quite widely agreed that) "technical analysis yields that the finitistic numerical functions are exactly the primitive recursive ones, and the finitistic number-theoretic truths are exactly those provable in the theory of primitive recursive arithmetic.'' And what is finistically acceptable beyond that fragment of arithmetic? Whatever can be elementarily coded into primitive recursive arithmetic (e.g. syntactic facts about formal theories).
For more on why the bounds of finitistic mathematics in a Hilbertian sense are arguably set by primitive recursive arithmetic see also William Tait's "Remarks on Finitism" here.
