FOM: Connections between mathematics, physics and FOM
Ayan Mahalanobis
ama78 at student.canterbury.ac.nz
Thu Feb 3 18:36:14 EST 2000
Andrej Bauer wrote:
>
> 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.
Why do you think ,"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)", is that your interpretation of
Brouwer's continuity principle? In that case it is not in the
Intuitionistic logic you need to assume it on top of Intuitionistic
Logic.
> In fact, can there be empirical evidence as to whether reality is
> Boolean or Heyting?
Well I myself find Kripke's semantics interesting, it is in some form
formalisation of Brouwer's concept of creating subject an idealised
mathematician. Why you are not thinking about it, am I missing
something?
> 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.
>
This is very interesting I always thought a computer scientist must have
some dislike towards choice (countable or full) you seem to think
otherwise. Don't you think that it is bit odd to assume even countable
choice, there is no means that you can generate the choice function or
am I missing something.
I myself work in Bishop's mathematics which is seen as mathematics with
Intuitionistic logic, i.e. mathematics without the law of excluded
middle, continuity axiom, Church's thesis and fan theorem. Ofcourse full
axiom of choice is excluded but people are often happy with the
countable axiom of choice. Though I believe that it is a compromise
between convenience to work and philosophical conviction.
Cheers
Ayan Mahalanobis
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