scratch/content/html/en/blog/2010-07-09-Indecidabilities.md

-----
isHidden:       false
menupriority:   1
kind:           article
created_at:     2010-07-09T10:04:31+02:00
title: Undecidabilities
author_name: Yann Esposito
author_uri: yannesposito.com
tags:
  - mathematics
  - science
  - philosophy
  - indecidability
-----

begindiv(intro)

<%= tldr %> We can attribute different meaning to undecidability:

- Undecidability due to measure errors, 
- Undecidability with big error from small measure error,
- Fractal undecidability ;
- Undecidability without any measure error.

Is the situation so bad we cannot do anything?
We can find the truth, but we can never be sure we had reach it.
I don't think we can do better than that.


enddiv

newcorps

# The Undecidabilities

begindiv(intro)

If the world was made by a demiurge he certainly have a great sense of humor.
And this read will prove it to you.
I'll pretend to be him.
I'll create a simplified world.
A world that obey to simple mathematical rules.
And I'll tell you about the *undecidability*.
The inability to know if we had find the *truth* or only an approximation to it.
The inability to predict many things.

enddiv

<%= leftblogimage("genesis.png") %>

At the beginning there was only void.
Then a blog post beginning to be written.
I breath profoundly to feel the weight the act I will accomplish.
A last tense moment and... I create the _Universe_.
A _Universe_ which will exists only the time of this read.

I'm the _demiurge_ of this universe and you are its observer.

I like to control everything.
I construct this world using only some simples rules.
I decide that _real_ rules of this world will be the one we believe are _true_ for our world.
Note the difference.
For their world, everything we _believe_ today is _true_ for them.
Their world is then _simpler_ than our.
Particularly, we can describe this world with axioms and mathematic rules.
It is not so sure for our Universe. 
But we'll talk about that later.


Lets begin to work.
I create an _Earth_.
I populate it with intelligent people.
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 I'm here to help them.


As a God I like jokes.
The first joke I do it give them imperfect sense.
Furthermore I make my world such that it is absolutely not possible to make perfect precise measure.
I let them ameliorate their technology but there is a theoretical limits to the best precision they can access to.


These people believe their world is flat.
Some believe it is possible to find the rules of their Universe.
The game begins.

Let begin easily, _errors can cause undecidability_.

## Undecidability due to measure errors

Here is what one of them think:


> All triangle I observe seems to share the same property.
> Each time I sum up their angles I obtain 180°.
> It is certainly a rule of my Universe.
> But how to be certain all triangle in my Universe share this property?

<%= leftblogimage('triangle_3_angles.png') %>





## Growing errors Undecidability


<%= leftblogimage('3_corps.png') %>









<%= leftblogimage('mandelbrot.png') %>




<%= leftblogimage("Turing_undecidability.png") %>










## Des bateaux à la rescousse de la *vérité*.

<%= leftblogimage("3DTriangle.png") %>





## Fractions rationnelles

<%= leftblogimage("rational_fraction.png") %>