FOM: Effective Bounds in Core Mathematics
Martin Davis
martin at eipye.com
Thu Jun 29 11:37:33 EDT 2000
At 03:55 PM 6/27/00 -0400, Fred Richman wrote:
>An analysis of a classical proof of P
>might reveal a constructive proof of a classically equivalent theorem
>Q. The constructivist would maintain that Q is what had been proved,
>not P. ... the constructivist ... is simply interested in finer
>distinctions
>than the classical mathematician wishes to make.
As stated this is surely not true. The non-constructivist makes the very
same distinction, but expresses it in different words. Instead of saying:
P has not been proved, the proof only shows Q
the non-constructivist would say
P has been proved, but non-constructively
Where is there greater fineness of distinction?
If the constructivist is of the Bishop variety, the tendency will be to
replace P by P* that can be proved constructively because a stronger
hypothesis has been built in, e.g., classical continuity replaced by
uniform effective continuity. With the redefinition, P* written in English
may even use exactly the same words as P. Is this making finer distinctions
or a form of obfuscation?
Martin Davis
Visiting Scholar UC Berkeley
Professor Emeritus, NYU
martin at eipye.com
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