[FRIAM] Thanks again Marcus

[Marcus Daniels]

If one is talking about objects within a cutoff of a millimeter, then 8 digits might suffice to talk about the locations of things.   If one is talking about objects within Pluto, that’s another 15 digits or so.   It’s certainly not surprising that there are computational approximations to real numbers that are inadequate for some things.  That doesn’t mean that given a particular context, that there isn’t a sufficient approximation.

From: Friam <friam-bounces at redfish.com> on behalf of Frank Wimberly <wimberly3 at gmail.com>
Reply-To: The Friday Morning Applied Complexity Coffee Group <friam at redfish.com>
Date: Saturday, June 20, 2020 at 9:04 AM
To: The Friday Morning Applied Complexity Coffee Group <friam at redfish.com>
Subject: Re: [FRIAM] Thanks again Marcus

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679...

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

505 670-9918
Santa Fe, NM

On Sat, Jun 20, 2020, 10:01 AM Frank Wimberly <wimberly3 at gmail.com<mailto:wimberly3 at gmail.com>> wrote:
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<mailto: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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