247 lines
7.8 KiB
Markdown
247 lines
7.8 KiB
Markdown
-----
|
|
isHidden: false
|
|
menupriority: 1
|
|
kind: article
|
|
created_at: 2010-08-01T10:04:31+02:00
|
|
title: Undecidabilities
|
|
author_name: Yann Esposito
|
|
author_uri: yannesposito.com
|
|
tags:
|
|
- mathematics
|
|
- science
|
|
- philosophy
|
|
- indecidability
|
|
-----
|
|
|
|
begindiv(intro)
|
|
|
|
<%= tldr %> I create a simple mathematical world to talk about the different meaning of undecidability:
|
|
|
|
- Undecidability due to measure errors,
|
|
- Big errors resulting from small initial measure error,
|
|
- Fractal undecidability ;
|
|
- Logic Undecidability.
|
|
|
|
After that, I try to find what is the best we can do knowing many things are undecidable.
|
|
|
|
enddiv
|
|
|
|
newcorps
|
|
|
|
# The Undecidabilities
|
|
|
|
begindiv(intro)
|
|
|
|
If the world was made by a demiurge he certainly had 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 one of the curse on this world: the *undecidability*.
|
|
The inability to know if we had find the *truth*.
|
|
The inability to predict many things that should be natural.
|
|
Here begin the story.
|
|
|
|
enddiv
|
|
|
|
<%= leftblogimage("genesis.png") %>
|
|
|
|
In 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_.
|
|
An incredible _Universe_ which will exists only the time of this read.
|
|
I'm the _demiurge_ of this universe and you are its observer.
|
|
|
|
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 probably _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 the work begin.
|
|
I create an _Earth_.
|
|
I populate it with intelligent people, the _Ys_.
|
|
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.
|
|
|
|
|
|
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 limits to the best precision they can access to.
|
|
|
|
|
|
I'd 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.
|
|
|
|
Lets 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 π radiants (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') %>
|
|
|
|
|
|
Some began to formalise the problem.
|
|
They end by writing a mathematical proof.
|
|
Marvelous!
|
|
The proof seems correct, but, a problem remain.
|
|
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 triangle angles.
|
|
The measure will never fail.
|
|
But they'll 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.
|
|
|
|
|
|
Of course, they prey, they call me to help.
|
|
And as any respectful god, I don't answer.
|
|
Ah ah ah! I've always loved to make these kind of thing.
|
|
Next, I'll act as if I don't exists.
|
|
Another good joke!
|
|
|
|
They feel sad. But they have some hope:
|
|
|
|
|
|
_Hope_
|
|
|
|
> If we make small measure error, we will make small predictive error.
|
|
|
|
## Growing errors Undecidability
|
|
|
|
|
|
<%= leftblogimage('3_corps.png') %>
|
|
|
|
|
|
Unfortunately, the three bodies problem will crush this _hope_.
|
|
Using Newton's 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 error will grow.
|
|
Errors will grow at a point that any prediction will be unusable.
|
|
|
|
|
|
Even with this bad news there is the _hope_ to _control_ the error.
|
|
> 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.
|
|
|
|
Once again, this should not terminate has they hope.
|
|
|
|
## Fractal Undecidability
|
|
|
|
Consider the following question:
|
|
|
|
<%= leftblogimage('mandelbrot.png') %>
|
|
|
|
|
|
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?
|
|
|
|
|
|
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.
|
|
|
|
We can even imagine some mathematical structure where _all_ points are at the border[^2].
|
|
[^2]: The set R\Q has this property.
|
|
|
|
|
|
## Logical Undecidability
|
|
|
|
<%= leftblogimage("stackOverflow.png") %>
|
|
|
|
|
|
Until there all problem were undecidable because of measure _errors_.
|
|
May be in a controlled world without any _error_ we should be able to predict anything.
|
|
I'm sorry to say no.
|
|
Even in a self-contained mathematical world it can be possible to create object with an unpredictable behaviour.
|
|
|
|
|
|
It is the _halting problem_.
|
|
|
|
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 entry.
|
|
And this is with pleasure I give you one here.
|
|
|
|
|
|
> Suppose a program able to decide if any program halt exists.
|
|
> More precisely:
|
|
>
|
|
> Hypothesis: there exists a program `P`such that:
|
|
> - `P(x,y)` return "stop" in a finite amount of time if `x(y)`[^1] will stop running.
|
|
> - `P(x,y)` return "loop" in a finite amount of time if `x(y)` will never stop running.
|
|
>
|
|
> 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 `P(x,x)`.
|
|
>
|
|
> Let `Q` be the following program using the return value of `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>
|
|
>
|
|
> Now, what is the value of `P(Q,Q)`?
|
|
>
|
|
> - if `P(Q,Q)` returns "stop" that means by construction of `Q` that `P(Q,Q)` never stop running.
|
|
> - if `P(Q,Q)` returns "loop" that means by construction of `Q` that `P(Q,Q)` will halt.
|
|
>
|
|
> Therefore there is a contradiction the only way to handle is by the non existence of the program `P`.
|
|
|
|
[^1]: Meaning `x` taking `y` as input.
|
|
|
|
|
|
I am the demiurge of this imaginary world.
|
|
And I cannot know the future of this world.
|
|
Therefore, creative power isn't equivalent to omnipotence.
|
|
|
|
After all this, it becomes difficult to know what we can believe.
|
|
But it would be another error to throw away all our knowledge.
|
|
The next section discuss about this.
|
|
|
|
newcorps
|
|
|
|
# What could we do then?
|
|
|
|
## Boat Serendipity
|
|
|
|
<%= leftblogimage("triangle_on_sphere.png") %>
|
|
|
|
|
|
The Ys have boats. And they navigate, the send two boats.
|
|
The first to the North and the second to the West.
|
|
The first boat stop after one week.
|
|
The second make a $$\frac{3}{4}π$$ turn to the right.
|
|
After what should have been $$\sqrt{2}$$ weeks, the second boat should have encounter the first boat.
|
|
But the north boat seems to have disappeared.
|
|
And after sometimes, they discovered the boat is very far to the north north east.
|
|
The proof is here.
|
|
There was a triangle were sum of its angle is not π radiant (180°).
|
|
|
|
|
|
|
|
## Fractions rationnelles
|
|
|
|
<%= leftblogimage("controled_error.png") %>
|
|
|
|
|