[FRIAM] Thanks again Marcus

[Frank Wimberly]

I understand, Jon.  Do you Nick?  I think (hope) he understands my
explanation.

A clarification between me and you, Jon.  A rational number isn't literally
a real number but the field of rational numbers is isomorphic to a subfield
of the field of real numbers so it makes sense to identify a rational
number with its image under that isomorphism.

Can you explain the assertion that real numbers aren't real?  Obviously the
scientists and engineers who compute the trajectory of a probe to the outer
reaches of the Solar System don't choose among algorithms to compute the
nth digit of pi and other real numbers.

Frank





---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Sat, Jun 20, 2020, 9:12 AM Jon Zingale <jonzingale at gmail.com> wrote:

> I think that reinterpreting computability in terms of truncation
> *obfuscates* the *philosophical content* that may be of interest to Nick.
> As a thought experiment, consider the collection of all computable
> sequences. Each sequence will in general have many possible algorithms
> that produce the given sequence up to the nth digit. Those algorithms
> which produce the same sequence for all n can be considered the *same*.
> Others that diverge at some digit are simply *approximations*. Now, if I
> am given a number like π, I can stably select from the collection of
> possible algorithms.
>
> Now we can play a game. To begin, the *dealer* produces n digits of a
> sequence and the *players* all choose some algorithm which they think
> produce the *dealer's* sequence. Next, the* dealer* proceeds to expose
> more and more digits beginning with the n+1th digit and continuing until
> all but one *player*, say, is shown to have chosen an incorrect algorithm.
> In the case of π, one can exactly choose a winning algorithm. If the
> *dealer* had chosen a *random number*, a player cannot win without
> cheating by forever changing their algorithm.
>
> This seems to be a point of Gisin's argument, there is meaningful
> philosophical content in the computability claim. He is not saying
> that the rationals are real, he is saying that the reals are not.
> π is a special kind of non-algebraic number in that it *is* *computable*,
> and not just a matter of measurement. It is this switch away from
> measurement that distinguishes it (possibly frees it) from the kinds
> of pitfalls we see in quantum interpretations, the subjectivity with
> which we choose our truncations is irrelevant. A similar argument is
> made by Chris Isham.
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