Finitism and primitive recursive arithmetic

[Richard Zach]

Thanks for the plug, Alasdair!

Bill Tait has written a followup to his "Finitism" paper which, with 
discussion, is reprinted in his collected essays (The Provenance of Pure 
Reason). The original (without the added discussion) is on his website:

http://home.uchicago.edu/~wwtx/finitism.pdf

He also has a more recent paper:

Tait, William. "What Hilbert and Bernays Meant by “Finitism”". 
/Philosophy of Logic and Mathematics: Proceedings of the 41st 
International Ludwig Wittgenstein Symposium/, edited by Gabriele M. 
Mras, Paul Weingartner and Bernhard Ritter, Berlin, Boston: De Gruyter, 
2019, pp. 249-262. https://doi.org/10.1515/9783110657883-015 
<https://doi-org.ezproxy.lib.ucalgary.ca/10.1515/9783110657883-015>

It's unfortunately not on his website but you might be able to look at 
it on Google Books:

https://books.google.ca/books?id=60DEDwAAQBAJ&lpg=PR1&pg=PA249#v=onepage&q&f=false

If I'm not mistaken, he discusses something very much like the idea you 
propose (the Sieg/Ravaglia argument).

FWIW I agree with Bill that the conceptually best analysis of what 
Hilbert *should have* meant by "finitism" is PRA. My "scepticism" about 
Tait's proposal only concerned the historical question of whether 
Hilbert et al. ever used methods that went beyond PRA in his proof 
theory. I think Bill agrees with me on the historical question.

-RZ

On 2021-12-08 10:21, Alasdair Urquhart wrote:
> Richard Zach has given a detailed logical and historical
> analysis of this question in his 2001 Berkeley doctoral thesis.
> It's available at Richard Zach's website.
>
> Zach expresses scepticism about Tait's proposal.  If you are interested
> in this question, I strongly recommend his dissertation.
>
>
> On Tue, 7 Dec 2021, martdowd at aol.com wrote:
>
>> FOM:
>>
>> I was just reading
>>  Takeuti's Well-Ordering Proof: Finitistically Fine?
>>   by Eamon Darnell and Aaron Thomas-Bolduc,
>>  philsci-archive.pitt.edu/15160/1/Takeuti.pdf
>> They mention Tait's thesis that PRA exhausts finitistic methods.
>>
>> It occurred to be that the terms of PRA can be augmented by adding a 
>> function,
>> which is the universal function for primitive recursive functions.  If
>> a nice extension of PRA existed allowed proving facts about this,
>> wouln't this still be finitistic?
>>
>> One possibility might be to add some version of Hoare's axioms to Meyer-
>> Ritchie loop programs.
>>
>> Martin Dowd
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