<div dir="ltr"><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><span style="font-family:arial,sans-serif">On Tue, Jul 11, 2017 at 1:58 PM, spike </span><span dir="ltr" style="font-family:arial,sans-serif"><<a href="mailto:spike66@att.net" target="_blank">spike66@att.net</a>></span><span style="font-family:arial,sans-serif"> wrote:</span><br></div><div class="gmail_extra"><div class="gmail_quote"><div lang="EN-US"><div class="gmail-m_7430322852833511186WordSection1"><p class="MsoNormal"> <br></p></div></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-color:rgb(204,204,204);border-left-style:solid;padding-left:1ex"><div lang="EN-US"><div class="gmail-m_7430322852833511186WordSection1"><div><p class="MsoNormal"><u></u><u></u></p></div><div><div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-color:rgb(204,204,204);border-left-style:solid;padding-left:1ex"><span style="font-family:Arial,sans-serif">​><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​> ​</div>…</span><span style="font-size:13.5pt;font-family:Arial,sans-serif">I know this is a bit heretical but perhaps irrational numbers really do have a last digit. ​If the computational resources of the entire universe is insufficient to calculate the 10^100^100^100 digit of PI, and given that there are only about 10^81</span><span style="font-family:Arial,sans-serif"> </span><span style="font-size:13.5pt;font-family:Arial,sans-serif">atoms in the observable universe that seems like a reasonable assumption, could the ​</span><span style="font-size:13.5pt;font-family:Arial,sans-serif">10^100^100^100​ digit of PI even be said to exist?​   </span></blockquote></div><p class="MsoNormal"><span style="font-size:13.5pt;font-family:Arial,sans-serif"><u></u> </span>  </p><p class="MsoNormal"><u></u></p><p class="MsoNormal"></p><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​> ​</div>Sure can John.  We don’t even need to resort to the definition of a rational number to show that pi is irrational, with a definition: if there is a last digit then you can write it as a fraction, which makes it rational.<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>The Taylor series expansion of pi doesn’t know or care how many atoms there are:<p></p><p class="MsoNormal"><u></u></p><p class="MsoNormal"><u></u>  </p><p class="MsoNormal"><u></u></p><p class="MsoNormal"><img width="367" height="59" style="width: 3.8229in; height: 0.6145in;" id="gmail-m_7430322852833511186Picture_x0020_3" src="cid:image001.jpg@01D2FA34.97B52440"></p></div></div></div></blockquote><div><br></div><div><br></div><font size="4">The key issue are<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>those 3 little dots,<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>the "..."; if the physical universe says the series can't continue forever is it really correct to say it nevertheless does?<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>I know<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>the above is<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>the definition but<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>definitions are easy, they<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>don't<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>imply that the thing defined actually exists; I could define "glixclog" as the prime<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>number<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>that is greater than 11 but less than 13.<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>You could argue I'm not playing fair because a prime number between 11 and 13 is a logical absurdity,<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>and that's true<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>but if the universe can't calculate<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>the<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>10^100^100^100 digit of PI<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>and the 10^100^100^100 digit of PI<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>can't effect the universe then is a claim of it's existence any less absurd?</font></div><div class="gmail_quote"><font size="4"><br></font></div><font size="4">We usually say a metal circle<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ made by a blacksmith​</div> is a<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​n​</div> approximation of a Euclidean circle, but maybe we've got <div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​it​</div> backwards, maybe a<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>Euclidean circle is a approximation of a<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ real​</div> physical circle made of matter. I guess it<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>all<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ </div>comes down to one<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ question: which​</div> is more fundamental, physics or mathematics?</font></div><div class="gmail_extra"><font size="4"><br></font></div><div class="gmail_extra"><font size="4">I don't know the answer to any of these questions<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ I'm just asking.​</div> </font></div><div class="gmail_extra"><font size="4"><br></font></div><div class="gmail_extra"><font size="4"><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">​ John K Clark​</div><br></font>  <br></div></div>