<div dir="ltr"><div dir="ltr"><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><span style="font-family:Arial,Helvetica,sans-serif">On Tue, Jul 23, 2019 at 6:55 PM Will Steinberg <<a href="mailto:steinberg.will@gmail.com">steinberg.will@gmail.com</a>> wrote:</span><br></div></div><div class="gmail_quote"><div dir="ltr" class="gmail_attr"><br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="auto"><div><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div class="gmail_quote"><div style="font-family:arial,helvetica,sans-serif"><font size="4"><span class="gmail_default" style="font-family:arial,helvetica,sans-serif">>> </span>The atoms in a brain can perform calculations because they are organized in the only way that can, the way Alan Turing described. The atoms in Earth are not. </font></div></div></div></blockquote></div></div><div dir="auto"><br></div><div dir="auto"><i><span class="gmail_default" style="font-family:arial,helvetica,sans-serif">></span>There are MANY atoms in the brain that do not perform calculations. </i></div></div></blockquote><div><br></div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><font size="4">No atom in the brain by itself can perform a calculation but groups of them can if they're organized in the right way, and the right way always comes down to Turing's way.</font></div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="auto"><div dir="auto"><i> <span class="gmail_default" style="font-family:arial,helvetica,sans-serif">> </span>Plus, what about the noncomputable? </i></div></div></blockquote><div><br></div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><font size="4">Yes some things can't be calculated. For  example all </font><span class="gmail-il" style="font-family:Arial,Helvetica,sans-serif;font-size:large">Busy</span><span style="font-family:Arial,Helvetica,sans-serif;font-size:large"> </span><span class="gmail-il" style="font-family:Arial,Helvetica,sans-serif;font-size:large">Beaver</span><span style="font-family:Arial,Helvetica,sans-serif;font-size:large"> Numbers are well defined and finite, but we only know the first 4. They are:</span></div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><br></div><div><font size="4">BB(1) =1<br>BB(2) =6<br>BB(3) =21<br>BB(4) =<span class="gmail-il">107</span></font><div class="gmail_quote"><span style="font-size:12.8px"><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"><br></div></span></div><font size="4">But those are the only values we've be<span class="gmail_default" style="font-family:arial,helvetica,sans-serif">en</span> able to calculate with certainty, the problem is the <span class="gmail-il">Busy</span> <span class="gmail-il">Beaver</span> function grows faster than any computable function. <span class="gmail_default" style="font-family:arial,helvetica,sans-serif">Some</span> suspect that BB(5) is 47,176,870 but are far from certain <span class="gmail_default" style="font-family:arial,helvetica,sans-serif">of that, </span>and BB(6) is at least 7.4*10^36534 and BB(7) is at least 10^10^10^10^10^7 but could be much larger. Big as they are all <span class="gmail-il">Busy</span> <span class="gmail-il">Beaver</span> numbers are finite but after a certain point they are not computable and nobody even knows exactly where that point is. It has been proven that BB(<span class="gmail_default" style="font-family:arial,helvetica,sans-serif">1919</span>) is not computable <span class="gmail_default" style=""><font face="arial, helvetica, sans-serif">even in theory, not even if you had infinite resources.  B</font></span>ut what is the smallest non<span class="gmail_default" style="font-family:arial,helvetica,sans-serif">-</span>computable<span class="gmail_default" style="font-family:arial,helvetica,sans-serif"> Busy Beaver integer</span>? Nobody knows<span class="gmail_default" style="font-family:arial,helvetica,sans-serif">,</span> but I wouldn't be surprised if it were BB(5)<span class="gmail_default" style="font-family:arial,helvetica,sans-serif">. So there </span>are some things that a Turing Machine can not calculate<span class="gmail_default" style="font-family:arial,helvetica,sans-serif">,</span> </font><span class="gmail_default" style="font-size:large;font-family:arial,helvetica,sans-serif">but</span><font size="4"> nothing else can either,</font><span class="gmail_default" style=""><font face="arial, helvetica, sans-serif" size="4"> in point of fact nothing but a </font></span><span style="font-size:large">Turing Machine<span class="gmail_default" style="font-family:arial,helvetica,sans-serif"> can calculate anything at all.</span></span></div><div><span style="font-size:large"><span class="gmail_default" style="font-family:arial,helvetica,sans-serif"><br></span></span></div><div><span style="font-size:large"><span class="gmail_default" style="font-family:arial,helvetica,sans-serif"> John K Clark</span></span></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><br>
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