# [FOM] The Provenance of Pure Reason (II)

Charles Silver silver_1 at mindspring.com
Wed May 31 11:50:43 EDT 2006

```  Gabriel Stolzenberg wrote:

>    As I read the quote below from Bill Tait's "The Provenance of
> Pure Reason," Bill is talking from within and about what, in my
> previous message, I called, "classical constructive mathematics."
> On this view, constructive math consists in seeing what one can
> do without the law of excluded middle.  Period.  There are no
> distinctively constructive concepts of 'existence' or 'function.'

This seems to suggest that Bill may have ignored "distinctively
constructive concepts of 'existence' [and]  'function'." Could you
please provide a "distinctively constructive" definition of
'function' and explain how exactly it differs from the classical
definition?
Just to get things started, let's use this classical definition of
'function': "a function from set X to set Y is a relation f such
that   the domain of f is X and for each x in X there is exactly one
y in Y such that <x,y> belongs to f."  With respect to this classical
definition of 'function', could you please provide the constructive
definition?  (If 'set' must first be explained constructively, please
begin by defining 'set'.)

>    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.

Just to have this laid out for us with specificity and clarity,
could you please produce the version of Church's thesis you have in
mind that implies the negation of LEM?

Thank you.

Charlie Silver

```