[FOM] reply to eray ozkural re "The provenance of pure reason (II)"
gstolzen at math.bu.edu
Sun Jun 4 14:22:28 EDT 2006
In reply to my "The provenance of pure reason (II)," Eray Ozkural
On 5/29/06, Gabriel Stolzenberg <gstolzen at math.bu.edu> wrote:
> > 1. Bill's argument that the basic concepts of constructive math
> > "belong to" classical math seems to work equally well with the law
> > of excluded middle replaced by a version of Church's thesis that
> > implies the negation of the law of excluded middle. Does this mean
> > that the basic concepts of constructive math belong both to classical
> > math and to this other system that is inconsistent with it?
> I would like to know more about this version of Church's thesis.
> How does this implication occur?
You're able to get counterexamples to certain classical theorems.
See, for example, "A uniformly continuous function on [0,1] that is
everywhere different from its infimum," by William Julian and Fred
Richman, Pacific Journal of Mathematics, 111, no. 2 (1984), 333-340.
It can be downloaded from the web.
> I think that the view that classical mathematics is a restricted
> subset of constructive mathematics, in analogy with non-Euclidian
> geometry and Euclidian geometry, may be more relevant from
> a logical point of view.
You may be right.
> Of course, I say that only because I think that
> the semantic notions of "existence" and "function" hardly matter.
> (When you say that empty set exists, how many people take that
> literally?) What seems to matter more is that abstract concepts are
> defined formally and communicated without error from individual
> to individual (the idea of a friend named Bhupinder S. Anandh). [*]
> After one observes this logical relation between the two schools,
> one may go further and observe how a mechanical explanation is
> sufficient to give semantics to constructive math, a situation
> which I believe to be fulfilling from a physicalist point of view.
I think I understand your point, except that I would hesitate to
say that the semantic notions of "existence" and "function" hardly
matter. To me, they seem to matter greatly for mathematical practice
(because they give rise to certain intuitions and habits of thought).
With best regards,
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