[FOM] Harvey Friedman's "hyperaliens"

[Harvey Friedman]

On 3/13/04 12:53 PM, "Martin Davis" <martin at eipye.com> wrote:

> First, the obvious remark that none of the conceptual problems haunting the
> proponents of hypercomputation stand in the way of Harvey's idea because of
> the bound he places on the size of the Turing machines. But there are
> problems of other kinds.
> 
> I believe I could produce a Turing machine within Harvey's bounds of 100 or
> so quadruples that will halt (starting on an empty tape) if and only if the
> Goldbach conjecture is false. Suppose we put this TM into the aliens' magic
> device and the output reads: doesn't halt. Have we proved the Goldbach
> conjecture to anyone's satisfaction? Of course, if the aliens relent and
> explain how their machine works, it would be another matter.
> 

Suppose appropriate experiments are done with this device, along the lines I
indicate in

http://www.cs.nyu.edu/pipermail/fom/2004-February/007983.html

These experiments are done on the device as a "black box" with absolutely no
understanding whatsoever of its internal mechanism.

I claim that we would then obtain a "proof" that Goldbach conjecture is
false, that is not as convincing as one would hope, but substantially more
convincing than any knowledge that we presently have in the sciences and
engineering outside mathematics.

This implicit claim was one of the principal points of my posting.

In fact, it seems plausible to me that an understanding of the operation of
the machine might do nothing to increase this already very high degree of
certainty.

Harvey Friedman