2010-07-09 15:26:00 +00:00
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isHidden: false
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menupriority: 1
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kind: article
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created_at: 2010-07-09T10:04:31+02:00
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2010-07-12 09:04:33 +00:00
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title: Undecidabilities
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2010-07-09 15:26:00 +00:00
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author_name: Yann Esposito
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author_uri: yannesposito.com
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tags:
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2010-07-12 09:04:33 +00:00
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- mathematics
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2010-07-09 15:26:00 +00:00
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- science
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- philosophy
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- indecidability
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-----
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2010-07-12 09:04:33 +00:00
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2010-07-09 15:26:00 +00:00
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begindiv(intro)
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2010-07-12 09:04:33 +00:00
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2010-07-15 15:12:29 +00:00
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<%= tldr %> We can attribute different meaning to undecidability:
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2010-07-13 15:29:29 +00:00
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2010-07-15 15:12:29 +00:00
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- Undecidability due to measure errors,
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- Undecidability with big error from small measure error,
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- Fractal undecidability ;
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- Undecidability without any measure error.
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2010-07-12 09:04:33 +00:00
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2010-07-15 15:12:29 +00:00
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Is the situation so bad we cannot do anything?
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We can find the truth, but we can never be sure we had reach it.
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I don't think we can do better than that.
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2010-07-09 15:26:00 +00:00
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2010-07-13 15:29:29 +00:00
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2010-07-09 15:26:00 +00:00
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enddiv
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2010-07-13 15:29:29 +00:00
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newcorps
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2010-07-12 09:04:33 +00:00
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# The Undecidabilities
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2010-07-13 15:29:29 +00:00
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begindiv(intro)
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2010-07-15 15:12:29 +00:00
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If the world was made by a demiurge he certainly have a great sense of humor.
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And this read will prove it to you.
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I'll pretend to be him.
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I'll create a simplified world.
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A world that obey to simple mathematical rules.
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And I'll tell you about the *undecidability*.
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The inability to know if we had find the *truth* or only an approximation to it.
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The inability to predict many things.
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2010-07-13 15:29:29 +00:00
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enddiv
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<%= leftblogimage("genese.png") %>
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2010-07-15 15:12:29 +00:00
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At the beginning there was only void.
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Then a blog post beginning to be written.
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I breath profoundly to feel the weight the act I will accomplish.
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A last tense moment and... I create the _Universe_.
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A _Universe_ which will exists only the time of this read.
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2010-07-13 15:29:29 +00:00
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2010-07-15 15:12:29 +00:00
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I'm the _demiurge_ of this universe and you are its observer.
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2010-07-13 15:29:29 +00:00
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2010-07-15 15:12:29 +00:00
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I like to control everything.
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I construct this world using only some simples rules.
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I decide that _real_ rules of this world will be the one we believe are _true_ for our world.
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Note the difference.
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For their world, everything we _believe_ today is _true_ for them.
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Their world is then _simpler_ than our.
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Particularly, we can describe this world with axioms and mathematic rules.
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It is not so sure for our Universe.
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But we'll talk about that later.
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2010-07-13 15:29:29 +00:00
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2010-07-15 15:12:29 +00:00
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Lets begin to work.
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I create an _Earth_.
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I populate it with intelligent people.
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Of course they are curious.
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In particular they try to understand their world.
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They believe that if they know the rules of their world they will be able to predict the consequences of most of their acts.
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They are so naive.
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If only they knew.
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But I'm here to help them.
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2010-07-13 15:29:29 +00:00
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2010-07-15 15:12:29 +00:00
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As a God I like jokes.
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The first joke I do it give them imperfect sense.
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Furthermore I make my world such that it is absolutely not possible to make perfect precise measure.
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I let them ameliorate their technology but there is a theoretical limits to the best precision they can access to.
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These people believe their world is flat.
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Some believe it is possible to find the rules of their Universe.
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The game begins.
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Let begin easily, _errors can cause undecidability_.
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2010-07-13 15:29:29 +00:00
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## Undecidability due to measure errors
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2010-07-15 15:12:29 +00:00
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Here is what one of them think:
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> All triangle I observe seems to share the same property.
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> Each time I sum up their angles I obtain 180°.
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> It is certainly a rule of my Universe.
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> But how to be certain all triangle in my Universe share this property?
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2010-07-13 15:29:29 +00:00
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2010-07-15 22:13:31 +00:00
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<%= leftblogimage('triangle_3_angles.png') %>
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2010-07-13 15:29:29 +00:00
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## Growing errors Undecidability
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2010-07-15 22:13:31 +00:00
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<%= leftblogimage('3_corps.png') %>
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2010-07-13 15:29:29 +00:00
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2010-07-15 22:13:31 +00:00
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<%= leftblogimage('mandelbrot.png') %>
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2010-07-13 15:29:29 +00:00
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<%= leftblogimage("Turing_undecidability.png") %>
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2010-07-13 21:15:27 +00:00
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2010-07-13 15:29:29 +00:00
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## Des bateaux à la rescousse de la *vérité*.
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<%= leftblogimage("3DTriangle.png") %>
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2010-07-13 21:15:27 +00:00
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2010-07-13 15:29:29 +00:00
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## Fractions rationnelles
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<%= leftblogimage("rational_fraction.png") %>
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2010-07-12 09:04:33 +00:00
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