ÿþ<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"><!-- InstanceBegin template="/Templates/Base.dwt" codeOutsideHTMLIsLocked="false" --> <head> <!-- InstanceBeginEditable name="head" --> <link href="../../SpryAssets/SpryCollapsiblePanel.css" rel="stylesheet" type="text/css" /> <link href="tabs.css" rel="stylesheet" type="text/css" /> <link href="cpstyl.css" rel="stylesheet" type="text/css" /> <script src="../../SpryAssets/SpryCollapsiblePanel.js" type="text/javascript"></script> <link href="base.css" rel="stylesheet" type="text/css" /> <link href="ovr.css" rel="stylesheet" type="text/css" /> <!-- InstanceEndEditable --> <!-- InstanceBeginEditable name="doctitle" --> <title>PLATO HODIE - " !/</title> <!-- InstanceEndEditable --> <style type="text/css"> <!-- /* Font Definitions */ @font-face {font-family:"Cambria Math"; panose-1:2 4 5 3 5 4 6 3 2 4;} @font-face {font-family:Tahoma; panose-1:2 11 6 4 3 5 4 4 2 4;} @font-face {font-family:"Arial Black"; panose-1:2 11 10 4 2 1 2 2 2 4;} /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {margin:0mm; margin-bottom:.0001pt; font-size:12.0pt; font-family:"Times New Roman","serif"; color:#1A1A1A;} a:link, span.MsoHyperlink {color:saddlebrown; text-decoration:underline;} a:visited, span.MsoHyperlinkFollowed {color:saddlebrown; text-decoration:underline;} table.MsoNormalTable td p a { text-decoration:underline !important; } table.MsoNormalTable td p a { ; } p.MsoAcetate, li.MsoAcetate, div.MsoAcetate {mso-style-link:"Balloon Text Char"; margin:0mm; margin-bottom:.0001pt; font-size:8.0pt; font-family:"Tahoma","sans-serif"; color:#1A1A1A;} span.BalloonTextChar {mso-style-name:"Balloon Text Char"; mso-style-link:"Balloon Text"; font-family:"Tahoma","sans-serif"; color:#1A1A1A;} p.msochpdefault, li.msochpdefault, div.msochpdefault {mso-style-name:msochpdefault; margin-right:0mm; margin-left:0mm; font-size:10.0pt; font-family:"Times New Roman","serif"; color:#1A1A1A;} .MsoChpDefault {font-size:10.0pt;} @page WordSection1 {size:595.3pt 841.9pt; margin:20.0mm 42.5pt 20.0mm 30.0mm;} div.WordSection1 {page:WordSection1;} p.LPar { font-family:Tahoma, Geneva, sans-serif; font-size:11pt; } div.MsoNormal1 {margin:0mm; margin-bottom:.0001pt; font-size:12.0pt; text-align:center; font-family:"Tahoma","sans-serif";} div.aff1 {mso-style-name:!5:F8O; margin-top:18.0pt; margin-right:0mm; margin-bottom:6.0pt; margin-left:14.2pt; text-align:center; page-break-after:avoid; font-size:14.0pt; font-family:"Tahoma","sans-serif"; font-variant:small-caps;} div.aff2 {mso-style-name:>45@0B>@K; margin-top:0mm; margin-right:0mm; margin-bottom:6.0pt; margin-left:14.2pt; text-align:center; page-break-after:avoid; font-size:11.0pt; font-family:"Tahoma","sans-serif"; font-variant:small-caps;} li.MsoNormal1 {margin:0mm; margin-bottom:.0001pt; font-size:12.0pt; font-family:"Tahoma","sans-serif";} li.aff1 {mso-style-name:!5:F8O; margin-top:18.0pt; margin-right:0mm; margin-bottom:6.0pt; margin-left:14.2pt; text-align:center; page-break-after:avoid; font-size:14.0pt; font-family:"Tahoma","sans-serif"; font-variant:small-caps;} li.aff2 {mso-style-name:>45@0B>@K; margin-top:0mm; margin-right:0mm; margin-bottom:6.0pt; margin-left:14.2pt; text-align:center; page-break-after:avoid; font-size:11.0pt; font-family:"Tahoma","sans-serif"; font-variant:small-caps;} p.MsoNormal1 {margin:0mm; margin-bottom:.0001pt; font-size:12.0pt; font-family:"Tahoma","sans-serif";} p.aff1 {mso-style-name:!5:F8O; margin-top:18.0pt; margin-right:0mm; margin-bottom:6.0pt; margin-left:14.2pt; text-align:center; page-break-after:avoid; font-size:14.0pt; font-family:"Tahoma","sans-serif"; font-variant:small-caps;} p.aff2 {mso-style-name:>45@0B>@K; margin-top:0mm; margin-right:0mm; margin-bottom:6.0pt; margin-left:14.2pt; text-align:center; page-break-after:avoid; font-size:11.0pt; font-family:"Tahoma","sans-serif"; font-variant:small-caps;} --> </style> <link href="../2014/tabs.css" rel="stylesheet" type="text/css" /> <link href="../../design/plato.css" rel="stylesheet" type="text/css" /> </head> <body> <!-- SPBU top panel loader --> <script type="text/javascript"> (function(){ var sc = document.createElement('script'); sc.type = 'text/javascript'; sc.async = true; sc.src = '//topbar.spbu.ru/loader.js'; var ls = document.getElementsByTagName('script')[0]; ls.parentNode.insertBefore(sc, ls); })(); </script> <!-- /loader --> <table class="Pwrapper"> <thead> <tr> <td colspan="2"> <a href="../../index.htm">;0B>=>2A:>5 D8;>A>DA:>5 >1I5AB2></a></td> </tr> </thead> <tbody> <tr> <td class="Pmenu"> <a href="../../index.htm"><img src="../../design/logo.gif" width="166" height="74" alt="Plato" /></a> <div><a href="../../ABOUTUS/index1.htm"> =0A</a></div> <div><a href="../../AKADEMIA/index2.htm">:045<88</a></div> <div><a href="../index3.htm">>=D5@5=F88</a></div> <div><a href="../../SUMMERSCHOOL/index4.htm">5B=85 H:>;K</a></div> <div><a href="../../PROJECTS/index5.htm">0CG=K5 ?@>5:BK</a></div> <div><a href="../../DISSERTATION/index6.htm">8AA5@B0F88</a></div> <div><a href="../../TEXTS/index7.htm">"5:ABK ?;0B>=8:>2</a></div> <div><a href="../../RESEARCH/index8.htm">AA;54>20=8O ?> ?;0B>=87<C</a></div> <div><a href="../../COMPANIONS/index11.htm">!?@02>G=K5 8740=8O</a></div> <div><a href="../../PARTNERS/index9.htm">0@B=5@K</a></div> <!-- <div><a href="../../INFO/index10.htm">=B5@=5B-@5AC@AK</a></div> --> <hr width="70%" /> <div><a href="../../pfsforms/index.html"> &laquo;;0B>=>2A:>5 D8;>A>DA:>5 >1I5AB2>&raquo;</a></div> </td> <td> <script type="text/javascript" language="javascript"> function SetBodyProps() { document.body.onresize="BodyResized()"; document.body.onload="BodyResized()"; } var tInit = SetBodyProps(); </script> <!-- InstanceBeginEditable name="MainContent" --> <blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'> <p align="right" style='text-align:right'><b><span style='font-size:10.0pt; font-family:&quot;Tahoma&quot;,&quot;sans-serif&quot;'><a href="../index3.htm">  !!# $ &</a></span></b></p> </blockquote> <div style="text-align: center;"> <center> <table> <tr> <td> <img src="IMAGES/PPhS.png" border="0" alt="" /> </td> <td style="text-align: center; white-space: nowrap; vertical-align: middle;"> <span style="text-align: center; font: Georgia; font-size: 40px; font-weight: bold; font-variant: small-caps;">The Universe of Platonic Thought<br /> #=825@AC< ?;0B>=>2A:>9 <KA;8</span><br /> <span style="text-align: center; font: Georgia; font-size: 20px; font-weight: normal; font-variant: small-caps;">26th International Conferecne&nbsp;&nbsp;&middot;&nbsp;&nbsp;XXVI 564C=0@>4=0O :>=D5@5=F8O</span><br /> <span style="text-align: center; font: Georgia; font-size: 14px; font-weight: normal">28&ndash;30 August 2018&nbsp;&nbsp;&nbsp;St Petersburg, Russia&nbsp;&nbsp;&middot;&nbsp;&nbsp;28&ndash;30 023CAB0&nbsp;2018&nbsp;&nbsp;&nbsp;!0=:B-5B5@1C@3, >AA8O</span> </td> <td> <img src="IMAGES/pla150.png" border="0" alt="" /> </td> </tr> </table> </center> </div> <hr /> <div class="d1"> <table class="ovrBtn"> <tr> <td class="c1" style="vertical-align:bottom !important;"> <img src="IMAGES/ovrbrd_11.png" alt="" /> </td> <td class="c1"> <img src="IMAGES/ovrbrd_12.png" alt="" /> </td> <td class="c7"> </td> <td class="c1"> <img src="IMAGES/ovrbrd_15.png" alt="" /> </td> <td class="c1" style="vertical-align:bottom !important;"> <img src="IMAGES/ovrbrd_16.png" alt="" /> </td> </tr> <tr> <td class="c1"> <img src="IMAGES/ovrbrd_18.png" alt="" /> </td> <td colspan="3" class="c8" > <a href="upt26en.htm">Back to the Conference Program</a> </td> <td class="c1"> <img src="IMAGES/ovrbrd_19.png" alt="" /> </td> </tr> <tr> <td class="c1"> <img src="IMAGES/ovrbrd_23.png" alt="" /> </td> <td class="c1"> <img src="IMAGES/ovrbrd_24.png" alt="" /> </td> <td class="c9"> </td> <td class="c1"> <img src="IMAGES/ovrbrd_26.png" alt="" /> </td> <td class="c1"> <img src="IMAGES/ovrbrd_27.png" alt="" /> </td> </tr> </table> </div> <table border="0" cellspacing="0" cellpadding="0" width="100%"> <tr> <td><img src="tabs/tii_48.png" border="0" /></td> <td style="background:url(tabs/tii_45a.png) repeat-x"></td> <td><img src="tabs/tii_56.png" border="0" /></td> </tr> <tr> <td style="background:url(tabs/tii_68.png) repeat-y"></td> <td width="100%" bgcolor="#F0FFF0"> <div style="text-align:right;font-family:Tahoma, Geneva, sans-serif;font-size:16px;"><a href="ther137.html" style="text-decoration:none"><img src="IMAGES/ru.png" border="0" alt=" CAA:0O 25@A8O" />&nbsp; #!!/&nbsp; !/</a></div> <div style="font-family:Tahoma, Geneva, sans-serif;font-size:14px;"><p class='pcAuthor'><span class='pcName'>Igor Grishkov</span><span class='pcAffil'>, St Petersburg Mining University</span><span class='pcPosition'>, student</span></p> <p class='pcTitleD'>Zeno, Plato, Aristotle and Heisenberg s uncertainty principle</p><p class='tcSumma'>The report proposes to compare the approaches of Parmenides, Zeno, Plato and Aristotle to a paradoxes role as a thought method, and to show, using Heisenberg s uncertainty principle, whether their approaches are relevant in terms of modern physical theory.</p><br /> <p class='tcKeywords'>Zeno, Parmenides, Plato, Aristotle, paradoxes, physics, universe, time, infinity, Heisenberg, uncertainty principle</p><br /> <p class='hcAbstract'>Zeno of Elea, the Parmenides disciple, explained his teacher s ideas about being, creating paradoxes, or aporias, which expose the absurdity of our understanding of physical movement, using some mathematical argumentation.&nbsp;</p> <p class='hcAbstract'>However, if Parmenides distinguished between the Path of Truth (the method of axiomatic cognition) and the Path of Opinion (an empirical method of recording any experience including contradictory ones) in the poem <I>On Nature</i>, Zeno attempted to overcome this division by applying the mathematical method for analyzing movement experience. He demonstrated the non-triviality (aporiasis) of such a problem.&nbsp;</p> <p class='hcAbstract'>Later, in the dialogue <I>Parmenides</i>, Plato returned the Parmenides opposition between experience and thought, but not in a logical, but in a dialectical form. He accepted paradoxes as cognitive method being not a dead end for thinking. Aristotle regenerated the Parmenides difference not dialectically, but in postulating some thought laws as the law of identity and the law of non-contradiction, excluding paradoxes as errors or sophistries. This is the Aristotle s approach for solving Zeno s paradoxes.&nbsp;</p> <p class='hcAbstract'>Aristotle tried to solve <I>The Achilles and the tortoise</i> paradox in terms of infinite concepts:  Hence Zeno s argument makes a false assumption in asserting that it is impossible for a thing to pass over or to come in contact with infinite things in a finite time& So while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility: for in this sense the time itself is also infinite [1]. However, this decision is not mathematically correct, as some contemporary mathematicians do insist. For example, David Bostock explains the error using the concepts of potential and actual infinity. Thus,  & ball s motion is certainly divided into infinitely many parts by the infinitely many points which mark the top of each bounce, and these must surely be admitted to have been actualized in the course of the bouncing. I conclude, then, that Aristotle does not after all have the right solution to Zeno s problem [3].&nbsp;</p> <p class='hcAbstract'>In the terms of modern physics, it is also possible to offer a solution for paradox in finite quantities. According to the theory of particle-wave dualism that includes Heisenberg s uncertainty principle, each body has properties not only of a particle, but also of a wave. Thus, the momentum of Achilles (as well any other macrocosm object) is huge in comparison with the Planck constant.Therefore, the uncertainty of the position in space is extremely small, but it is finite. In the mathematical model of the paradox, the problem is hidden in ever decreasing segments of space. So, that sufficiently small distances that can be calculated, using the momentum of Achilles (his mass multiplied by velocity), the uncertainty in space will become larger, than the subsequent parts of space before the turtle. Thus, the wave properties of Achilles  catch up with the turtle, covering the last segment, which cannot yet divide them. So, the sum of distances that Achilles must cross is finite (because the number of segments is finite until the moment, when the uncertainty in space becomes larger than the distance before the turtle).&nbsp;</p> <p class='hcAbstract'>Nevertheless, the fact that Zeno s paradoxes can be solved by modern physics, this decision just confirms what the pupil of Parmenides wanted to say. Any paradoxes solution ultimately leads to others, less obvious unsolvable postulates. In the case of Aristotle s solution the uncertainty of the theoretical concept of infinity and of the theoretical properties of infinite quantities appears on mathematical level. And the empirical uncertainty of the physical space between Achilles and the tortoise turns up in the case of a modern paradox  solution by Heisenberg s uncertainty principle.&nbsp;</p> <p class='hcAbstract'>&nbsp;</p> <p class='hcAbstract'>References&nbsp;</p> <p class='hcAbstract'>&nbsp;</p> <p class='hcAbstract'>1. <i>Bostock D.</i> Aristotle, Zeno and the potential infinite // Proceedings of the Aristotelian Society 73, 1972. .37 53. &nbsp;</p> <p class='hcAbstract'>2. <i>Grishkov I.V. </i> Heisenberg s Uncertainty Principle and Zeno s Paradoxes // -:>=><8G5A:89 25:B>@, 2018. !?5F80;L=K9 2K?CA:.!1>@=8: AB0B59 : III 564C=0@>4=>9 ABC45=G5A:>9 =0CG=>-?@0:B8G5A:>9 :>=D5@5=F88  -:>=><8:0 8 <5=546<5=B [Ekonomicheskij Vector. Special Volume. The 3d International Conference for Students  Economics and Management ], 2-3 April, 2018. P. 105  108. &nbsp;</p> <p class='hcAbstract'>3. <i>Hilbert D., Bernays P. </i> Foundations of Mathematics Vol. 1, 1934.</p> </div> </td> <td style="background:url(tabs/tii_69.png) repeat-y"></td> </tr> <tr> <td><img src="tabs/tii_72.png" alt="" /></td> <td style="background:url(tabs/tii_74.png) repeat-x"></td> <td><img src="tabs/tii_79.png" alt="" /></td> </tr> </table> <p class="MsoNormal" align="right" style='text-align:right'><span style='font-size:10.0pt;font-family:&quot;Tahoma&quot;,&quot;sans-serif&quot;'>© ;0B>=>2A:>5 >1I5AB2>, 2018 3. </span></p> <p align="right" style='margin-top:0mm;margin-right:3pt;margin-bottom: 0mm;margin-left:-36.0pt;margin-bottom:.0001pt;text-align:right'><b><span style='font-size:10.0pt;font-family:&quot;Tahoma&quot;,&quot;sans-serif&quot;'><a href="../index3.htm">  !!# $ &</a></span></b></p> <!-- InstanceEndEditable --> </td> </tr> </tbody> </body> <!-- InstanceEnd --></html>