FOM: Connections between mathematics, physics and FOM

Andrej Bauer Andrej.Bauer at cs.cmu.edu
Thu Feb 3 02:10:05 EST 2000

```"Jeffrey John Ketland" <Jeffrey.Ketland at nottingham.ac.uk> writes:
> Replacing the reals by the ratios is primarily motivated by
> *epistemological* considerations, about what we (finite beings) can
> *experimentally measure* (ratios). But I'm very sceptical that this
> program of eliminating the reals from physics works though. (In
> fact, I think it is reactionary and anti-progressive).

I agree that suggestions like "replace reals by rationals" are a bit
simplistic and naive, and it's easy to see that would not work very
well.

Would you (or anyone else who's put some thought into it) care to
comment about replacing the reals with *intuitionistic* reals (not
constructive, just intuitionistic)? Of course, we would then also have
to replace the rest of mathematics with its intuitionistic version.

It seems to me that intuitionistic reals reflect much better the
information-theoretic nature of reality (i.e., only finite rate of
information flow is allowed). This is just a feeling based on my
intuition from denotational semantics of programming languages.
In computer science intuitionistic logic simply *works better* than
classical logic. I wonder if the same might be true for physics.

Are there any physical phenomena that would indicate, for example,
that the real numbers satisfy the classical axiom of linear order

forall x, y: (x < y) or (x = y) or (x > y)

as opposed to the intuitionistic version

forall x, y, z: (x < y) ==> (x < z) or (z < y) ?

In fact, can there be empirical evidence as to whether reality is
Boolean or Heyting? So, the widely spread belief in classical logic
might in fact be just that, a belief.

Has anyone explored what intuitionistic physics would look like? I
doubt physicists use the law of excluded middle or the axiom of choice
much in situations where that would not be valid intuitionistically.

By "intuitionistic mathematics" I mean a fairly rich topos with
natural numbers, number-number choice (so we don't have to worry about
the distinction between Cauchy reals and Dedekind reals), and perhaps
also Markov's principle, if that helps.

--
Andrej Bauer
http://andrej.com

```