[FOM] Wildberger on Foundations
paul at personalit.net
paul at personalit.net
Tue Dec 4 14:18:34 EST 2018
> 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
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