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 <[email protected]> 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. > - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > archives: http://friam.471366.n2.nabble.com/ > FRIAM-COMIC http://friam-comic.blogspot.com/ >
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