[FOM] Wildberger on Foundations

[paul at personalit.net]

> Question 1.  To what extent can we develop modern analysis on the 
> basis of PA alone (or perhaps even Heyting arithmetic), without any 
> reference to infinite sets?  Can this be done in an 
> undergraduate-friendly way without a lot of extra baggage from logic?

One undergraduate-friendly approach may be to draw an analogy with the 
philosophy of (natural) science, specifically, with constructive 
empiricism as advocated by Bas van Fraassen and others, for example in 
van Fraassen's 1980 book, The Scientific Image.  The Stanford 
Encyclopedia of Philosophy has this article on "constructive empiricism":

https://plato.stanford.edu/entries/constructive-empiricism/

Wildberger claims (as I understand him) not to be interested in 
existence, but rather in computability.  This claim has more than one 
possible interpretation, but an analogy with constructive empiricism in 
the natural sciences would proceed by asserting similarity between 
computable natural numbers (or numbers having physically constructible 
notation) and observable objects, and also between non-computable 
numbers and unobservable objects.  Where constructive empiricism 
requires granting empirical adequacy (but not truth) to relevant claims 
about unobservable objects, Wildberger would be construed as granting an 
analogue of empirical adequacy (but not truth) to relevant claims about 
non-computable numbers.

Constructive empiricism differs from logical positivism by rejecting any 
verificationist criterion of meaning, so this analogy would assume that 
Wildberger also rejects verificationist limitations on meaning.

Some logical rigor, without too much baggage, may be introduced by way 
of a free logic -- or an appropriate analogue of a free logic based upon 
computability / being a number with a physically constructible notation.

Since Wildberger claims (as I understand him) that practicing 
mathematicians do not work axiomatically, but rather using natural 
deduction with special rules for computation, this analogy would neither 
require nor imply that the axioms of any theories be revised along the 
lines of free logic; only that the rules of natural deduction be so 
revised -- at least, that is my assumption. Wildberger's question for 
Joe Shipman, about developing Analysis "completely rigorously from first 
principles," was I suspect intended rhetorically, because Wildberger 
does not believe any relevant examples exist.

Also, this would abide by Timothy Chow's observation that Wildberger 
"would not approve of the above statement of Question 1," because 
developing Analysis on the basis of PA alone, etc., is presumably a red 
herring for Wildberger, as he is neither interested in that project nor 
does he think it possible -- at least, as I understand him.

If this analogy succeeds, it might be relevant to debates about 
constructive empiricism's ontological commitment to abstract objects, by 
suggesting which abstract objects constructive empiricism is committed 
to, e.g., to computable numbers / numbers having physically 
constructible notation.

Cheers,

Paul Hollander