<div dir="ltr"><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><span style="font-family:arial,sans-serif">On Wed, Jul 12, 2017 at 4:15 AM, Stuart LaForge </span><span dir="ltr" style="font-family:arial,sans-serif"><<a href="mailto:avant@sollegro.com" target="_blank">avant@sollegro.com</a>></span><span style="font-family:arial,sans-serif"> wrote:</span><br></div><div class="gmail_extra"><div class="gmail_quote"><br><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 class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">> </div>if space-time is discrete, it opens up a whole can of worms even at<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>the planck level once you enable more than a single dimension. For<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>example, lets say that for the sake of argument that the Planck length<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>(Lp) is the fundamental and thus indivisible unit of length. A pixle of<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>the universe if you will.<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>Then what is minimal unit of 2-dimensional area? If you say that it is<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>Lp^2, then that is wrong because the Pythagorean theorem says that<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> [...]</div></blockquote><div><br></div><div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"><font size="4">If</font></div><font size="4"> space-time is discrete<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> then the </div>Pythagorean theorem<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> is only a approximation that works pretty well as long as things don't get too small. But I don't see how the existence of the continuum is of any relevance to the question of a conscious AI. Neither a computer nor the human brain can count the number of points on a line, but both can use calculus to fine the exact area under a parabola; assuming of course that lines and parabolas actually exist and are not just useful fictions. </div></font></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"></blockquote><br><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"><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">
<font size="4">>I know this is a bit heretical but perhaps irrational numbers really do<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>have a last digit. If the computational resources of the entire universe<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>is insufficient to calculate the<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>10^100^100^100 digit of PI, and given that<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>there are only about 10^81 atoms in the observable universe that seems<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>like a reasonable assumption,<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>could the 10^100^100^100<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>digit of PI<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>even be said to exist?</font></blockquote></blockquote><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"> </blockquote><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 class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">> </div>There are two problems with this argument. First, the observable universe<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>is just a cosmological horizon, and we don't have any reason to believe<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>the universe ends at the horizon.</blockquote><div><br></div><div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><font size="4">There are only 3 possibilities: </font></div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><font size="4"><br></font></div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><font size="4">1) Nothing exists outside the cosmological horizon.</font></div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><font size="4">2) A Finite amount of stuff exists outside the cosmological horizon.</font></div></div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><font size="4">3) A infinite amount of stuff exists outside the cosmological horizon.</font></div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><br></div><font size="4">All 3 violate a cherished scientific principle and yet one of them must be true. If you assume #1 is true then the Earth occupies a special position, it is the center of a finite flat spacetime universe. If you assume #2 or #3 is true then it's OK for a scienctific theory to conjure up things that are in neither your past nor your future causal lightcone. And with #3 you must also conjure up physical infinity even though there is no evidence there are a infinite number of any physical object.</font><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">
</blockquote><br><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 class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">> </div>Secondly, given the set of all 10^81 atoms that you mention, there are<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>2^10^81 possible subsets of those atoms.</blockquote><div><br></div><div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><font size="4">Are there? I would argue (as a devil's advocate) that if finding all <span style="font-family:arial,sans-serif">2^10^81 subsets is beyond the computational capacity of the observable universe (and whatever the unobservable universe can do is of no help whatsoever) then saying all those subsets exist has no meaning. And besides, </span><span style="font-family:arial,sans-serif">2^10^81 is no closer to being infinite than the number two is.</span></font></div></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"><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 class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">> </div><font size="4">If perfect circles don't exist is there anything about them to understand?</font></blockquote>
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<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">> </div>Yes. How did something that does not exist become so fundamental in<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>describing so much of what we can see and observe?</blockquote><div><br></div><div><div class="gmail_default"><font face="arial, helvetica, sans-serif"></font><font size="4">The human mind is not infinitely powerful so in dealing with the staggering complexity of the world approximations are needed. The idea that the planets moved in perfect circles around the sun worked pretty well but Kepler showed that a more complex mathematical curve, the ellipse, worked better. And then Einstein showed that even a ellipse wasn't quite right, but to understand how and why Einstein said the planets move high school geometry is not enough, you need 4 dimensional Tensor calculus and hyperbolic spacetime. And even Einstein wasn't quite right because he didn't take quantum mechanics into account. So when a child asks you how planets move it's best to just say "in a circle".<br> </font></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 class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">> </div>Without perfect<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>circles, you can't have complex numbers. And without complex numbers you<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>can't have probability amplitudes</blockquote><div><br></div><div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"><font size="4">And without a brain made of atoms that obey the laws of physics "you" can't have </font></div><font size="4">probability amplitudes<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">, in fact you can't even have you.</div></font></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">
<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline">> </div>Math is like the soul of the universe and an infinite number of angles can<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>dance on the head of a pin.<div class="gmail_default" style="font-family:arial,helvetica,sans-serif;display:inline"> </div>Sorry, I couldn't resist the pun. :-)</blockquote><div> </div><div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><font size="4">Yes but are the number of angles on that pun, sorry I mean pin, countably infinite or can they be put into a one to one correspondence with the number of points on a line? </font></div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><font size="4"><br></font></div><div class="gmail_default" style="font-family:arial,helvetica,sans-serif"><font size="4"> John K Clark</font></div><br></div><div><br></div><div><br></div></div></div></div>