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<p class="MsoNormal">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.<o:p></o:p></p>
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<p class="MsoNormal"><b><span style="font-size:12.0pt;color:black">From: </span></b><span style="font-size:12.0pt;color:black">Friam <friam-bounces@redfish.com> on behalf of Frank Wimberly <wimberly3@gmail.com><br>
<b>Reply-To: </b>The Friday Morning Applied Complexity Coffee Group <friam@redfish.com><br>
<b>Date: </b>Saturday, June 20, 2020 at 9:04 AM<br>
<b>To: </b>The Friday Morning Applied Complexity Coffee Group <friam@redfish.com><br>
<b>Subject: </b>Re: [FRIAM] Thanks again Marcus<o:p></o:p></span></p>
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<p class="MsoNormal">3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679...<o:p></o:p></p>
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<p class="MsoNormal" style="margin-bottom:12.0pt">is enough.<o:p></o:p></p>
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<p class="MsoNormal">---<br>
Frank C. Wimberly<br>
140 Calle Ojo Feliz, <br>
Santa Fe, NM 87505<br>
<br>
505 670-9918<br>
Santa Fe, NM<o:p></o:p></p>
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<p class="MsoNormal">On Sat, Jun 20, 2020, 10:01 AM Frank Wimberly <<a href="mailto:wimberly3@gmail.com">wimberly3@gmail.com</a>> wrote:<o:p></o:p></p>
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<p class="MsoNormal">I understand, Jon. Do you Nick? I think (hope) he understands my explanation.<o:p></o:p></p>
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<p class="MsoNormal">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. <o:p></o:p></p>
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<p class="MsoNormal">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.<o:p></o:p></p>
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<p class="MsoNormal">Frank<o:p></o:p></p>
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<p class="MsoNormal">---<br>
Frank C. Wimberly<br>
140 Calle Ojo Feliz, <br>
Santa Fe, NM 87505<br>
<br>
505 670-9918<br>
Santa Fe, NM<o:p></o:p></p>
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<p class="MsoNormal">On Sat, Jun 20, 2020, 9:12 AM Jon Zingale <<a href="mailto:jonzingale@gmail.com" target="_blank">jonzingale@gmail.com</a>> wrote:<o:p></o:p></p>
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<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Verdana",sans-serif;color:#333333">I think that reinterpreting computability in terms of truncation<br>
<i>obfuscates</i> the <i>philosophical content</i> that may be of interest to Nick.<br>
As a thought experiment, consider the collection of all computable<br>
sequences. Each sequence will in general have many possible algorithms<br>
that produce the given sequence up to the nth digit. Those algorithms<br>
which produce the same sequence for all n can be considered the <i>same</i>.<br>
Others that diverge at some digit are simply <i>approximations</i>. Now, if I<br>
am given a number like π, I can stably select from the collection of<br>
possible algorithms.<br>
<br>
Now we can play a game. To begin, the <i>dealer</i> produces n digits of a<br>
sequence and the <i>players</i> all choose some algorithm which they think<br>
produce the <i>dealer's</i> sequence. Next, the<i> dealer</i> proceeds to expose<br>
more and more digits beginning with the n+1th digit and continuing until<br>
all but one <i>player</i>, say, is shown to have chosen an incorrect algorithm.<br>
In the case of π, one can exactly choose a winning algorithm. If the<br>
<i>dealer</i> had chosen a <i>random number</i>, a player cannot win without<br>
cheating by forever changing their algorithm.<br>
<br>
This seems to be a point of Gisin's argument, there is meaningful<br>
philosophical content in the computability claim. He is not saying<br>
that the rationals are real, he is saying that the reals are not.<br>
π is a special kind of non-algebraic number in that it <i>is</i> <i>computable</i>,<br>
and not just a matter of measurement. It is this switch away from<o:p></o:p></span></p>
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<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Verdana",sans-serif;color:#333333">measurement that distinguishes it (possibly frees it) from the kinds<o:p></o:p></span></p>
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<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Verdana",sans-serif;color:#333333">of pitfalls we see in quantum interpretations, the subjectivity with<o:p></o:p></span></p>
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<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Verdana",sans-serif;color:#333333">which we choose our truncations is irrelevant. A similar argument is<o:p></o:p></span></p>
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<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Verdana",sans-serif;color:#333333">made by Chris Isham.<o:p></o:p></span></p>
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