FOM: Maths from physics

Ketland,JJ J.J.Ketland at lse.ac.uk
Tue Sep 21 13:33:25 EDT 1999

```I'dd like to thank Joe Shipman for the info about Fubini theorems and CH. As
for the Church's Thesis discussion, is this the idea that Joe Shipman is
getting at? We have a mathematicized physical theory T. This theory T
implies the following proposition:

There exists a certain kind of "machine" M which produces a binary
output and
(i) 217th digit of M's output = 1 if ZFC is consistent and
(ii) 217th digit of M's output = 0 if ZFC is not consistent.

To find out if ZFC is consistent (or not), we build the (non-algorithmic)
"machine" and wait until the 217th digit comes out. Is that right?

What about the following kind of "machine"? Let f(n) represent a
(computable) enumeration of the sentences of the first-order language of
arithmetic. Suppose that it is a fact about the physical world that:

There exists a certain kind of "machine" M producing a binary ouput
and
(i) the nth digit of M's output = 1 if f(n) is true
(ii) the nth digit of M's output = 0 if f(n) is not true

If we could build such a (non-Turing) machine, it would provide a way of
generating new maths from physics: although M starts by telling us that such
things as 0 = 0, and ss0+s0=sss0, eventually, at some *finite* number k,
this "machine" would start generating (the truths values of) undecidables
(relative to PA, or Z_2, or ZFC, etc.).

Jeff Ketland
j.j.ketland at lse.ac.uk
Department of Philosophy
London School of Economics

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