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Philosophy, Metaphysics & the Afterlife
Infinity:The headstart paradox
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<blockquote data-quote="Harte" data-source="post: 24492" data-attributes="member: 443"><p><strong>Re: Infinity:The headstart paradox</strong></p><p></p><p><div class='quotetop'>QUOTE(\"Alpha and 0mega\")</div></p><p> </p><p>A&O,</p><p> </p><p>You have here a variation on an ancient paradox. The Greek Xeno was the first to voice this paradox, at least as far as we know, thus it is called "Xeno's paradox." Xeno's example (slightly less complicated than yours, therefore better for illustrative purposes) was a man walking toward a wall. First the man had to cover half the distance to the wall, then half of that, then half of that, etc. This proves that all motion is impossible. As Iggy would say, the answer is calculus.</p><p> </p><p>Let d = distance to wall at start, then at the halfway points, the succesive distances to the wall are:</p><p>d/2, d/4, d/8, d/16, etc.</p><p> </p><p>We can form the sum d/2+d/4+d/8+d/16+...d/2^n = d (where n=>1->infinity)</p><p> </p><p>Calculus was invented for problems like these. The concept of <em>limit</em> (lim) allows you to bridge that last gap, as it were.</p><p> </p><p>You can see that 2^n gets larger as n gets larger. You then should know that d/2^n gets smaller and smaller as n gets larger. In the limit, where n has reached infinity, d/2^n becomes zero, that is, the distance to the wall at the limit is zero, you have reached the wall.</p><p> </p><p>This is a mathematical construct. It is a way to make mathematics match up with what we observe in the real world. It by no means suggests that infinity is an actual place or that anything can become infinitesimally small. Notice that both your statements and Xeno's begin with mathematics, then go on to show how mathematics cannot describe the real world. Calculus was invented to <em>describe</em> what we observe in the real world. No branch of mathematics can <em>explain</em> what we observe in the real world.:huh: </p><p> </p><p>H</p></blockquote><p></p>
[QUOTE="Harte, post: 24492, member: 443"] [b]Re: Infinity:The headstart paradox[/b] <div class='quotetop'>QUOTE(\"Alpha and 0mega\")</div> A&O, You have here a variation on an ancient paradox. The Greek Xeno was the first to voice this paradox, at least as far as we know, thus it is called "Xeno's paradox." Xeno's example (slightly less complicated than yours, therefore better for illustrative purposes) was a man walking toward a wall. First the man had to cover half the distance to the wall, then half of that, then half of that, etc. This proves that all motion is impossible. As Iggy would say, the answer is calculus. Let d = distance to wall at start, then at the halfway points, the succesive distances to the wall are: d/2, d/4, d/8, d/16, etc. We can form the sum d/2+d/4+d/8+d/16+...d/2^n = d (where n=>1->infinity) Calculus was invented for problems like these. The concept of [i]limit[/i] (lim) allows you to bridge that last gap, as it were. You can see that 2^n gets larger as n gets larger. You then should know that d/2^n gets smaller and smaller as n gets larger. In the limit, where n has reached infinity, d/2^n becomes zero, that is, the distance to the wall at the limit is zero, you have reached the wall. This is a mathematical construct. It is a way to make mathematics match up with what we observe in the real world. It by no means suggests that infinity is an actual place or that anything can become infinitesimally small. Notice that both your statements and Xeno's begin with mathematics, then go on to show how mathematics cannot describe the real world. Calculus was invented to [i]describe[/i] what we observe in the real world. No branch of mathematics can [i]explain[/i] what we observe in the real world.:huh: H [/QUOTE]
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