[FOM] Remarks on Church's Thesis

Frank Waaldijk fwaaldijk at gmail.com
Mon Sep 5 09:33:17 EDT 2016

```A (somewhat) late reaction on Harvey Friedman's post in which he writes:

> Similarly, any invoking of physical processes in connection with the
> usual infinitary Church's Thesis is also highly problematic, as there
> is no reason whatsoever to attribute any kind of infinity to the
> current actual physical world.
>

There is actually a physical experiment that I think merits attention, to
test one-sidedly whether the so-called Strong Physical Church-Turing Thesis
(PCTT+) holds. PCTT+ states more or less: `every real number that we
measure in Nature is a computable real number' (a position also known as
digital physics).

This experiment is based on the peculiar trait of recursive mathematics,
that for each n there is a constructive-recursive covering R_n of the
recursive unit interval which has measure less than 2^(-n) from a classical
point of view.

So we can start constructing such a cover R_40 say, with measure smaller
than 2^(-40), by enumerating in a recursive way a sequence of rational
subintervals of [0,1].

At the same time we flip a coin an indefinite number of times, and we
interpret heads as 1 and tails as 0, creating a binary sequence which we
interpret as a binary expansion of a real number y between 0 and 1.

If the physical world `corresponds' to classical mathematics, the odds that
at any given time y is seen to lie in one of the intervals of R_40 that we
have enumerated thus far, is less than 2^(-40). But if PCTT+ holds, then y
will surely fall into one of those intervals...if we wait long enough.

If by some strange design we see y belonging to one of the intervals in
R_40, then perhaps our probability models may need some revision... I have
detailed this idea in the article `On the foundations of constructive
mathematics -- especially in relation to the theory of continuous functions
<http://www.fwaaldijk.nl/foundations%20of%20constructive%20mathematics.pdf>'
(2005, preprint 2001). I also wrote some more on my math & science blog
<https://fwaaldijk.wordpress.com/2014/01/14/experiment-to-disprove-strong-physical-church-turing-thesis/>
(2014).

So far I have received positive feedback on this idea, but I also would
really welcome it if someone could point out one or more flaws.

I am still pondering ways to determine how long one should wait before
concluding: `if y does not belong to any of the intervals enumerated thus
far, then we have to reject PCTT+'. Some of these thoughts are related to
`drawing a natural number at random' (see my blog), with an interesting
connection to Benford's law.

Best wishes to all,

Frank Waaldijk
www.fwaaldijk.nl/mathematics.html
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