her.esy.fun/src/Scratch/en/blog/2010-07-09-Indecidabilities/index.html

202 lines
14 KiB
HTML
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<title>YBlog - Undecidabilities (part 1)</title>
<meta name="keywords" content="mathematics, science, philosophy, indecidability" />
<link rel="shortcut icon" type="image/x-icon" href="../../../../Scratch/img/favicon.ico" />
<link rel="stylesheet" type="text/css" href="/css/y.css" />
<link rel="stylesheet" type="text/css" href="/css/legacy.css" />
<link rel="alternate" type="application/rss+xml" title="RSS" href="/rss.xml" />
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="apple-touch-icon" href="../../../../Scratch/img/about/FlatAvatar@2x.png" />
<!--[if lt IE 9]>
<script src="http://ie7-js.googlecode.com/svn/version/2.1(beta4)/IE9.js"></script>
<![endif]-->
<!-- IndieAuth -->
<link href="https://twitter.com/yogsototh" rel="me">
<link href="https://github.com/yogsototh" rel="me">
<link href="mailto:yann.esposito@gmail.com" rel="me">
<link rel="pgpkey" href="../../../../pubkey.txt">
</head>
<body lang="en" class="article">
<div id="content">
<div id="header">
<div id="choix">
<span id="choixlang">
<a href="../../../../Scratch/fr/blog/2010-07-09-Indecidabilities/">French</a>
</span>
<span class="tomenu"><a href="#navigation">↓ Menu ↓</a></span>
<span class="flush"></span>
</div>
</div>
<div id="titre">
<h1>Undecidabilities (part 1)</h1>
</div>
<div class="flush"></div>
<div id="afterheader" class="article">
<div class="corps">
<p>&lt;% # toremove_ %&gt;</p>
<div class="intro">
<p><span class="sc"><abbr title="Too long; didn't read">tl;dr</abbr>: </span> I pretend to create a world to give examples of different meanings behind the word <em>undecidability</em>:</p>
<ul>
<li>Undecidability due to measure errors,</li>
<li>Big errors resulting from small initial measure error,</li>
<li>Fractal undecidability ;</li>
<li>Logic Undecidability.</li>
</ul>
</div>
<hr />
<h1 id="the-undecidabilities">The Undecidabilities</h1>
<div class="intro">
<p>If a demiurge made our world, he certainly had a great sense of humor. After this read, you should be convinced. Ill pretend to be him. Ill create a simplified world. A world that obey to simple mathematical rules. And Ill tell you about one of the curse on this world: the <em>undecidability</em>. The inability to know if we had find the <em>truth</em>. The inability to predict many things that should be natural. Here begin the story.</p>
</div>
<div>
<img src="../../../../Scratch/img/blog/2010-07-09-Indecidabilities/genesis.png" alt class="left" />
</div>
<p>In the beginning there was only void. Then a blog post beginning to be written. I breath profoundly to feel the weight of the act I will accomplish. A last tense moment and… I <em>create</em> the <em>Universe</em>. An incredible <em>Universe</em> which will exists only the time of this read. Im the <em>demiurge</em> of this universe and you are its observer.</p>
<p>I construct this world using only simples rules. I decide that <em>real</em> rules of this world will be the one we believe are <em>true</em> for our world. Note the difference. For their world, everything we <em>believe</em> today is <em>true</em> for them. Their world is then probably <em>simpler</em> than our. Particularly, we can describe this world with axioms and mathematic rules. It is not so sure for our Universe. But well talk about that later.</p>
<p>Lets the work begin. I create an <em>Earth</em>. I populate it with intelligent people, the <em>Ys</em>. Of course they are curious. In particular they try to understand their world. They believe that if they know the rules of their world they will be able to predict the consequences of most of their acts. They are so naive. If only they knew. But Im here to help them.</p>
<p>I am a God who likes jokes. The first joke I make to Ys is to make their sense imperfect. Furthermore it is not possible to make perfect precise measure in my world. I let Ys ameliorate their technology but there is a theoretical limit to the best precision they can reach.</p>
<p>Id like to precise that these people believe their world is flat. Some believe it is possible to find the rules of their Universe. Now, let the game begins.</p>
<p>Lets start easily, <em>errors can cause undecidability</em>.</p>
<h2 id="undecidability-due-to-measure-errors">Undecidability due to measure errors</h2>
<p>Here is what one of them think:</p>
<blockquote>
<p>All triangle I observe seems to share the same property. Each time I sum up their angles I obtain π radiants (180°). It is certainly a rule of my Universe. But how to be certain all triangle in my Universe share this property?</p>
</blockquote>
<div>
<img src="../../../../Scratch/img/blog/2010-07-09-Indecidabilities/triangle_3_angles.png" alt="three triangles" class="left" />
</div>
<p>Some began to formalize the problem. They end by writing a mathematical proof. Marvelous! The proof seems correct, but, a problem remains. The proof is based on rules and axioms. How to be certain these rules and axioms are right in their world? They will try to measure again and again the sum of the angles of triangles. The measure will never fail. But theyll never be certain the rules and axioms are right. Because then only way to verify all axioms depends of observation. And as a facetious god, I forbid perfect measure in observation.</p>
<p>Of course, they prey, they call me to help. And as any respectful god, I dont answer. Ah ah ah! Ive always loved to make these kind of thing. Lets act as if I dont exists. What a good joke!</p>
<p>They feel sad. But they have some hope:</p>
<p><em>Hope</em></p>
<blockquote>
<p>If we make small measure error, we will make small predictive error.</p>
</blockquote>
<h2 id="growing-errors-undecidability">Growing errors Undecidability</h2>
<div>
<img src="../../../../Scratch/img/blog/2010-07-09-Indecidabilities/3_corps.png" alt="Three bodies" class="left" />
</div>
<p>Unfortunately, the three bodies problem will crush this <em>hope</em>. Using Newtons Universal Law of gravitation with two bodies, we can predict with precision what will be their position and speed in the future. Until there all seems OK. But now, add another body. All errors will grow. Errors will grow at a point that any prediction will be unusable.</p>
<p>Even with this bad news there is the <em>hope</em> to <em>control</em> the error.</p>
<blockquote>
<p>May we should know the maximal measure error we can handle to predict something. And we should at least determine what we can predict and what we cannot.</p>
</blockquote>
<p>Once again, this should not terminate has they hope.</p>
<h2 id="fractal-undecidability">Fractal Undecidability</h2>
<p>Consider the following question:</p>
<div>
<img src="../../../../Scratch/img/blog/2010-07-09-Indecidabilities/mandelbrot.png" alt="Mandelbrot set" class="left" />
</div>
<p>Consider some GPS coordinates on a point around the cost of the “Bretagne” in France. The coordinates are 3 feet precise. Is the point in the water or on Earth?</p>
<p>For some coordinates it is not possible to know. Even if we are authorize to move a bit to dodge the borders. Because there are some zone in which all point could be a “border” for any size of the zone.</p>
<p>We can even imagine some mathematical structure where <em>all</em> points are at the border<a href="#fn1" class="footnote-ref" id="fnref1"><sup>1</sup></a>.</p>
<h2 id="logical-undecidability">Logical Undecidability</h2>
<div>
<img src="../../../../Scratch/img/blog/2010-07-09-Indecidabilities/stackOverflow.png" alt="recursive stack overflow" class="left" />
</div>
<p>Until there all problem were undecidable because of measure <em>errors</em>. May be in a controlled world without any <em>error</em> we should be able to predict anything.<br />
Im sorry to say no. Even in a self-contained mathematical world it can be possible to create object with an unpredictable behaviour.</p>
<p>It is the <em>halting problem</em>.</p>
<p>Theorem: It is undecidable given a description of a program, whether the program finishes running or will run forever. The idea of the proof is simple enough to be part of this article. And this is with pleasure I give you one here.</p>
<blockquote>
<p>Suppose a program able to decide if any program halt exists. More precisely:</p>
<p>Hypothesis: there exists a program <code>P</code> such that:</p>
<ul>
<li><code>P(x,y)</code> return “stop” in a finite amount of time if <code>x(y)</code><a href="#fn2" class="footnote-ref" id="fnref2"><sup>2</sup></a> will stop running.</li>
<li><code>P(x,y)</code> return “loop” in a finite amount of time if <code>x(y)</code> will never stop running.</li>
</ul>
<p>Remark: Any program can be represented as a string. Therefore, a program can be used as the input of another program. It is authorized to write <code>P(x,x)</code>.</p>
<p>Let <code>Q</code> be the following program using the return value of <code>P</code>.</p>
<pre class="twilight">
Q(x) :
if P(x,x)="stop" then I enter in an infinite loop
if P(x,x)="loop" then I stop
</pre>
<p>Now, what is the value of <code>P(Q,Q)</code>?</p>
<ul>
<li>if <code>P(Q,Q)</code> returns “stop” that imply by construction of <code>Q</code> that <code>P(Q,Q)</code> returns “loop”.</li>
<li>if <code>P(Q,Q)</code> returns “loop” that means by construction of <code>Q</code> that <code>P(Q,Q)</code> return “stop”.</li>
</ul>
<p>Therefore there is a contradiction the only way to handle is by the non existence of the program <code>P</code>.</p>
</blockquote>
<p>I am the demiurge of this imaginary world. And I cannot know the future of this world. Therefore, creative power isnt equivalent to omnipotence.</p>
<hr />
<p>After all this, it becomes difficult to know what we can believe. But it would be another error to throw away all our knowledge. In a future next part, Ill explain what we can hope and what attitude we should have once weve realized most of truth are unaccessible.</p>
<section class="footnotes">
<hr />
<ol>
<li id="fn1"><p>The set Rhas this property.<a href="#fnref1" class="footnote-back"></a></p></li>
<li id="fn2"><p>Meaning <code>x</code> taking <code>y</code> as input.<a href="#fnref2" class="footnote-back"></a></p></li>
</ol>
</section>
</div>
<div id="afterarticle">
<div id="social">
<a href="/rss.xml" target="_blank" rel="noopener noreferrer nofollow" class="social">RSS</a>
·
<a href="https://twitter.com/home?status=http%3A%2F%2Fyannesposito.com/Scratch/en/blog/2010-07-09-Indecidabilities/%20via%20@yogsototh" target="_blank" rel="noopener noreferrer nofollow" class="social">Tweet</a>
·
<a href="http://www.facebook.com/sharer/sharer.php?u=http%3A%2F%2Fyannesposito.com/Scratch/en/blog/2010-07-09-Indecidabilities/" target="_blank" rel="noopener noreferrer nofollow" class="social">FB</a>
<br />
<a class="message" href="../../../../Scratch/en/blog/Social-link-the-right-way/">These social sharing links preserve your privacy</a>
</div>
<div id="navigation">
<a href="../../../../">Home</a>
<span class="sep">¦</span>
<a href="../../../../Scratch/en/blog">Blog</a>
<span class="sep">¦</span>
<a href="../../../../Scratch/en/softwares">Softwares</a>
<span class="sep">¦</span>
<a href="../../../../Scratch/en/about">About</a>
</div>
<div id="totop"><a href="#header">↑ Top ↑</a></div>
<div id="bottom">
<div>
Published on 2010-08-11
</div>
<div>
<a href="https://twitter.com/yogsototh">Follow @yogsototh</a>
</div>
<div>
<a rel="license" href="http://creativecommons.org/licenses/by/3.0/deed.en_US">Yann Esposito©</a>
</div>
<div>
Done with
<a href="http://www.vim.org" target="_blank" rel="noopener noreferrer nofollow"><strike>Vim</strike></a>
<a href="http://spacemacs.org" target="_blank" rel="noopener noreferrer nofollow">spacemacs</a>
<span class="pala">&amp;</span>
<a href="http://nanoc.ws" target="_blank" rel="noopener noreferrer nofollow"><strike>nanoc</strike></a>
<a href="http://jaspervdj.be/hakyll" target="_blank" rel="noopener noreferrer nofollow">Hakyll</a>
</div>
<hr />
<div style="max-width: 100%">
<a href="https://cardanohub.org">
<img src="../../../../Scratch/img/ada-logo.png" class="simple" style="height: 16px;
border-radius: 50%;
vertical-align:middle;
display:inline-block;" />
ADA:
</a>
<code style="display:inline-block;
word-wrap:break-word;
text-align: left;
vertical-align: top;
max-width: 85%;">
DdzFFzCqrhtAvdkmATx5Fm8NPJViDy85ZBw13p4XcNzVzvQg8e3vWLXq23JQWFxPEXK6Kvhaxxe7oJt4VMYHxpA2vtCFiP8fziohN6Yp
</code>
</div>
</div>
</div>
</div>
</div>
</body>
</html>