{"id":120,"date":"2015-02-07T05:21:40","date_gmt":"2015-02-07T05:21:40","guid":{"rendered":"http:\/\/aristotle2digital.blogwyrm.com\/?p=120"},"modified":"2021-11-25T21:21:04","modified_gmt":"2021-11-26T02:21:04","slug":"the-dangers-of-being-equal","status":"publish","type":"post","link":"https:\/\/aristotle2digital.blogwyrm.com\/?p=120","title":{"rendered":"The Dangers of Being Equal"},"content":{"rendered":"<style>\ntable, th, td {<br \/>\n    border: 1px solid black;<br \/>\n    border-collapse: collapse;<br \/>\n}<br \/>\nth { text-align: center !important; }<br \/>\n<\/style>\n<p>One of the favorite themes of this blog is how language and reasoning affect each other, sometimes to the detriment of both.\u00a0 The overlap between logical reasoning and mathematical language is particularly ripe with possibilities of confusion because of the way certain concepts are used contextually.\u00a0 In an earlier post I discussed the seductive properties of the humble symbol $\\infty$.\u00a0 A far more deadly symbol is the highly overloaded glyph described by two parallel horizontal lines \u2013 the equal sign &#8216;$=$&#8217;.<\/p>\n<p>There are so many contextual uses of the equal sign that it is hard to know where to start.\u00a0 And each and every one of them is sinister to the untrained.\u00a0 Like some kind of bizarre initiation ritual, we subject students of all stripes to this ambiguous notation, and then we get frustrated when they don\u2019t grasp the subtle distinctions and shaded nuances of meaning that we take for granted.\u00a0 This situation closely parallels the experiences many of us have had learning how to swim, or ride a bike, or ice-skate, or drive.\u00a0 Those of us who know how to do something, often can\u2019t remember how hard it is to learn when you don\u2019t know.<\/p>\n<p>Of course, this situation is not unprecedented in language.\u00a0 A simple internet search using the string \u2018word with the most definitions\u2019 returns the <a href=\"http:\/\/dictionary.reference.com\/help\/faq\/language\/t47.html\">following statement from Dictionary.com<\/a><\/p>\n<div style=\"background-color: #ff6666; border: solid 1px black;\">&#8220;Set&#8221; has 464 definitions in the\u00a0<em>Oxford English Dictionary<\/em>. &#8220;Run&#8221; runs a distant second, with 396. Rounding out the top ten are &#8220;go&#8221; with 368, &#8220;take&#8221; with 343, &#8220;stand&#8221; with 334, &#8220;get&#8221; with 289, &#8220;turn&#8221; with 288, &#8220;put&#8221; with 268, &#8220;fall&#8221; with 264, and &#8220;strike&#8221; with 250.<\/div>\n<p>So, functionally overloading a word with numerous meanings, some of them very closely related and some of them quite distinct, is commonplace.<\/p>\n<p>What makes the equal sign so frustrating is that it is mostly applied in highly technical fields where shades of meaning in thought can have large implication in outcomes.\u00a0 Consider the differences in meaning in the following equations:<br \/>\n\\[ \\pi = \\frac{C}{D} = \\frac{C}{2 r} \\]<br \/>\nand<br \/>\n\\[ \\pi = \\ln\\left( i^{-2i} \\right) \\]<br \/>\nand<br \/>\n\\[ \\pi = \\sqrt{6 \\sum_{n=1}^{\\infty} \\frac{1}{n^2}} \\; . \\]<\/p>\n<p>Each of them tells us something about the irrational number $\\pi$, but in very different ways.\u00a0 In the first equation, we think of $\\pi$ as the assigned value for the correlation between the diameter of a circle, $D$, and its circumference, $C$.\u00a0 This concept is purely geometric, and can be explored with rulers and compasses and pieces of paper. In some sense, it can even be regarded as a causative relation, telling us that, if we make a circle of radius $r$, then we are making an object whose perimeter is a distance $C$.\u00a0 The second equation is an identity in the purest sense of that term.\u00a0 It boldly states that one of the many disguises of $\\pi$ is an algebraic expression involving the natural logarithm and the imaginary number .\u00a0 The final equation is neither an assignment nor an identity, but a set of instructions saying \u2018if you want to know how to compute $\\pi$ to some accuracy, then set up a computing process that takes the first \u00a0integers and combines them in this funny way.\u2019<\/p>\n<p>The science of computing has long recognized that the usual ambiguity of human language would be inadequate for machine instructions.\u00a0 All programming languages to which I&#8217;ve been exposed clearly distinguish between the concepts of assignment, equivalence, and function definition.\u00a0 Using the pi equations above, one might express them in the programming languages <em>Python<\/em> and <em><a href=\"http:\/\/maxima.sourceforge.net\/\">Maxima<\/a><\/em> as<\/p>\n<table width=\"619\">\n<tbody>\n<tr>\n<th width=\"125\">Pi equation<\/th>\n<th width=\"222\">Python<\/th>\n<th width=\"276\">Maxima<\/th>\n<\/tr>\n<tr>\n<td width=\"125\">\\[ \\small\u00a0 \\pi = \\frac{C}{2r} \\]<\/td>\n<td width=\"222\">\n<pre>pi = C\/(2*r)<\/pre>\n<\/td>\n<td width=\"276\">\n<pre>pi : C\/(2*r)<\/pre>\n<\/td>\n<\/tr>\n<tr>\n<td width=\"125\">\\[ \\small \\pi = \\ln\\left(i^{-2i}\\right) \\]<\/td>\n<td width=\"222\">\n<pre>pi == ln(i**(-2*i))<\/pre>\n<\/td>\n<td width=\"276\">\n<pre>pi = ln(i**(-2*i))<\/pre>\n<\/td>\n<\/tr>\n<tr>\n<td width=\"135\">\\[ \\small \\pi = \\sqrt{6 \\sum_{n=1}^{\\infty} \\frac{1}{n^2}} \\]<\/td>\n<td width=\"222\">\n<pre>def sum_sqr(n):\n    sum = 0\n    for i in range(1,n+1):\n        sum = sum + 1.0\/(i*i)\n    return temp\n\ndef approx_pi(n):\n    sum = sum_sqr(n)\n    return (6*sum)**(0.5)\n<\/pre>\n<\/td>\n<td width=\"276\">\n<pre>calc_pi(n) := \nblock([sum],\n sum : 0,\n for i: 1 thru n do\n sum : sum + 1\/(i*i),\n ans : sqrt(6*sum));\n<\/pre>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Note that, in each case, there is a clear syntactical difference between assignment (&#8216;$=$&#8217; or &#8216;$:$&#8217;), the conditional test for identity (&#8216;$==$&#8217; or &#8216;$=$&#8217;), and functional definition (\u2018def\u2019 or &#8216;$:=$&#8217;).\u00a0 For anyone who\u2019s been programming for some time, switching back and forth between these ideas of assignment, equivalence, and definition is relatively effortless, but for the beginner it is one of the hardest concepts he will have to learn.<\/p>\n<p>The situation is even more complex in the physical sciences, for two primary reasons.\u00a0 First, and foremost, because man has been investigating the physical world longer than he has been writing computer programs.\u00a0 As a result, there has been more time for man to layer different meanings and subtle distinctions.\u00a0 Second, computers are intrinsically stupid and require a high degree of precision and clarity to function.\u00a0 A nice discussion of this last point can be found in the prologue of the book <a href=\"http:\/\/www.amazon.com\/Functional-Differential-Geometry-Gerald-Sussman\/dp\/0262019345\/ref=sr_1_4?s=books&amp;ie=UTF8&amp;qid=1423104141&amp;sr=1-4&amp;keywords=Jack+Wisdom\">Functional Differential Geometry<\/a> by Sussman and Wisdom.<\/p>\n<p>As an example, let\u2019s look at perhaps the most famous physical statement \u2013 Newton\u2019s second law.\u00a0 Many people, even those lacking formal training in science, know that the expression of the law is \u2018force equals mass times acceleration\u2019 or, in mathematical terms,<\/p>\n<p>\\[ \\vec F = m \\vec a \\; . \\]<\/p>\n<p>But what does the equal sign here mean?\u00a0 The concept of a force tells us that it is a vector quantity that transforms like a position vector.\u00a0 That means that a force relationship is the same in all frames.\u00a0 For example, the balancing of the pulls from three ropes tied to an object such that the object doesn&#8217;t move is an equilibrium condition that is independent of the frame in which it is expressed.\u00a0 An accelerating observer will make the same conclusion as an inertial observer. So the force on the left-hand side of $f=ma$ is geometric in its meaning.<\/p>\n<p>On the other hand, we understand that the acceleration appearing on the right-hand side is kinematic.\u00a0 It describes an object&#8217;s motion and it&#8217;s the kind of thing measured with rulers and clocks.\u00a0 It is fundamentally frame dependent when described by an accelerating observer.\u00a0 Just imagine the visual perception of someone on a merry-go-round.\u00a0 The mass, which measures the object\u2019s unwillingness to move under influence of a force, simply scales the acceleration and can be regarded as constant.<\/p>\n<p>So how do we reconcile what the equal sign is meaning here?\u00a0\u00a0 On one side is a geometric quantity as immutable and placid as a mountain.\u00a0 The other side is as ephemeral as rising mist or running water, flowing to and fro.\u00a0 How can they actually be equal?<\/p>\n<p>Well, the answer is that the equal sign should be regarded as relating cause and effect.\u00a0 If we regard the force as known (<em>e.g.<\/em>, Newton\u2019s universal law of gravity), then the equal sign allows us to deduce the resulting motion once the force is applied.\u00a0 If we regard the acceleration as known (<em>e.g.<\/em>, we film the motion and do a frame analysis), we can infer (via abductive reasoning) the force that caused it.<\/p>\n<p>Clearly, the innocent-looking &#8216;$=$&#8217; packs a lot more meaning than at first it appears. It is interesting to ponder why it is that the shortest of strings, such as &#8216;$\\infty$&#8217;, or &#8216;set&#8217;, or &#8216;$=$&#8217;, have the longest and deepest of meanings. Maybe it reflects on the subtly of the human mind.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>One of the favorite themes of this blog is how language and reasoning affect each other, sometimes to the detriment of both.\u00a0 The overlap between logical reasoning and mathematical language&#8230; <a class=\"read-more-button\" href=\"https:\/\/aristotle2digital.blogwyrm.com\/?p=120\">Read more &gt;<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-120","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/120","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=120"}],"version-history":[{"count":0,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=\/wp\/v2\/posts\/120\/revisions"}],"wp:attachment":[{"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=120"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=120"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/aristotle2digital.blogwyrm.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=120"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}