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Re Ashton, PRA: the identification of PRA with finitism is a long-running discussion in the philosophy of mathematics. Everyone I have read agrees that PRA is included in "finitism"; the point of contention is whether finitism goes beyond PRA or not, with some on each side. It is trivial to find papers where this is discussed using google. One influential paper is:
* "Remarks on finitism", Wiliam Tait, [http://home.uchicago.edu/~wwtx/finitism.pdf].
Some
* "Partial realizations of Hilbert's program", Stephen Simpson, [http://www.math.psu.edu/simpson/papers/hilbert.pdf]
* "Finitistic Properties of High Complexity", Dmytro Taranovsky, [http://web.mit.edu/dmytro/www/FinitismPaper.htm]
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