263 lines
8.5 KiB
Markdown
263 lines
8.5 KiB
Markdown
-----
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isHidden: false
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menupriority: 1
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kind: article
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created_at: 2010-08-01T10:04:31+02:00
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title: Undecidabilities
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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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- mathematics
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- science
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- philosophy
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- indecidability
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-----
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begindiv(intro)
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<%= tldr %> I prepend to be the creator of a simple world.
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And I use this world to give examples of different meanings behind the word _undecidability_:
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- Undecidability due to measure errors,
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- Big errors resulting from small initial measure error,
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- Fractal undecidability ;
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- Logic Undecidability.
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After that, I try to find what is the best we can do to discover truths.
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enddiv
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newcorps
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# The Undecidabilities
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begindiv(intro)
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If a demiurge made our world, he certainly had a great sense of humor.
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After this read, you should be convinced.
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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 one of the curse on this world: the *undecidability*.
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The inability to know if we had find the *truth*.
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The inability to predict many things that should be natural.
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Here begin the story.
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enddiv
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<%= leftblogimage("genesis.png") %>
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In 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 of the act I will accomplish.
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A last tense moment and... I _create_ the _Universe_.
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An incredible _Universe_ which will exists only the time of this read.
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I'm the _demiurge_ of this universe and you are its observer.
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I construct this world using only 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 probably _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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Lets the work begin.
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I create an _Earth_.
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I populate it with intelligent people, the _Ys_.
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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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I am a God who likes jokes.
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The first joke I make to Ys is to make their sense imperfect.
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Furthermore it is not possible to make perfect precise measure in my world.
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I let Ys ameliorate their technology but there is a theoretical limit to the best precision they can reach.
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I'd like to precise that 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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Now, let the game begins.
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Lets start easily, _errors can cause undecidability_.
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## Undecidability due to measure errors
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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 π radiants (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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<%= leftblogimage('triangle_3_angles.png') %>
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Some began to formalize the problem.
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They end by writing a mathematical proof.
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Marvelous!
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The proof seems correct, but, a problem remains.
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The proof is based on rules and axioms.
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How to be certain these rules and axioms are right in their world?
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They will try to measure again and again the sum of the angles of triangles.
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The measure will never fail.
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But they'll never be certain the rules and axioms are right.
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Because then only way to verify all axioms depends of observation.
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And as a facetious god, I forbid perfect measure in observation.
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Of course, they prey, they call me to help.
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And as any respectful god, I don't answer.
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Ah ah ah! I've always loved to make these kind of thing.
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Let's act as if I don't exists.
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What a good joke!
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They feel sad. But they have some hope:
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_Hope_
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> If we make small measure error, we will make small predictive error.
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## Growing errors Undecidability
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<%= leftblogimage('3_corps.png') %>
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Unfortunately, the three bodies problem will crush this _hope_.
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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.
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Until there all seems OK.
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But now, add another body.
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All errors will grow.
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Errors will grow at a point that any prediction will be unusable.
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Even with this bad news there is the _hope_ to _control_ the error.
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> May we should know the maximal measure error we can handle to predict something.
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> And we should at least determine what we can predict and what we cannot.
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Once again, this should not terminate has they hope.
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## Fractal Undecidability
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Consider the following question:
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<%= leftblogimage('mandelbrot.png') %>
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Consider some GPS coordinates on a point around the cost of the "Bretagne" in France.
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The coordinates are 3 feet precise.
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Is the point in the water or on Earth?
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For some coordinates it is not possible to know.
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Even if we are authorize to move a bit to dodge the borders.
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Because there are some zone in which all point could be a "border" for any size of the zone.
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We can even imagine some mathematical structure where _all_ points are at the border[^2].
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[^2]: The set $$\mathbb{R}\setminus\mathbb{Q}$$ has this property.
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## Logical Undecidability
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<%= leftblogimage("stackOverflow.png") %>
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Until there all problem were undecidable because of measure _errors_.
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May be in a controlled world without any _error_ we should be able to predict anything.
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I'm sorry to say no.
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Even in a self-contained mathematical world it can be possible to create object with an unpredictable behaviour.
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It is the _halting problem_.
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Theorem: It is undecidable given a description of a program, whether the program finishes running or will run forever.
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The idea of the proof is simple enough to be part of this article.
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And this is with pleasure I give you one here.
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> Suppose a program able to decide if any program halt exists.
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> More precisely:
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>
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> Hypothesis: there exists a program `P` such that:
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>
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> - `P(x,y)` return "stop" in a finite amount of time if `x(y)`[^1] will stop running.
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> - `P(x,y)` return "loop" in a finite amount of time if `x(y)` will never stop running.
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>
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> Remark: Any program can be represented as a string. Therefore, a program can be used as the input of another program.
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> It is authorized to write `P(x,x)`.
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>
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> Let `Q` be the following program using the return value of `P`.
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> <pre class="twilight">
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> Q(x) :
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> if P(x,x)="stop" then I enter in an infinite loop
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> if P(x,x)="loop" then I stop
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> </pre>
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>
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> Now, what is the value of `P(Q,Q)`?
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>
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> - if `P(Q,Q)` returns "stop" that imply by construction of `Q` that `P(Q,Q)` returns "loop".
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> - if `P(Q,Q)` returns "loop" that means by construction of `Q` that `P(Q,Q)` return "stop".
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>
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> Therefore there is a contradiction the only way to handle is by the non existence of the program `P`.
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[^1]: Meaning `x` taking `y` as input.
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I am the demiurge of this imaginary world.
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And I cannot know the future of this world.
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Therefore, creative power isn't equivalent to omnipotence.
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After all this, it becomes difficult to know what we can believe.
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But it would be another error to throw away all our knowledge.
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The next section discuss about this.
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newcorps
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# What could we do then?
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## Boat Serendipity
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<%= leftblogimage("triangle_on_sphere.png") %>
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The Ys have boats. And they navigate, the send two boats.
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The first to the North and the second to the West.
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The first boat stop after one week.
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The second make a $$\frac{3}{4}π$$ turn to the right.
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After what should have been $$\sqrt{2}$$ weeks, the second boat should have encounter the first boat.
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But the north boat seems to have disappeared.
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And after sometimes, they discovered the boat is very far to the north north east.
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The proof is here.
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There was a triangle were sum of its angle is not π radiant (180°).
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Yes! Good joke!
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As God I'd like all these jokes.
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Their world isn't _flat_.
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That means, two parallel line have two common point.
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They live on a spherical planet.
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Huge enough for they didn't notice it for a long time.
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Ah Ah Ah! How funny I am!
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But I don't know why some don't like my jokes.
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The firsts who discovered this little surprise are happy to tell the world and some are annoyed.
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Annoyed to the point to burn some of them.
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They apparently are bad losers within these people.
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And yes, you can draw triangle were the sum of angle is 270° instead of the 180° on a flat world.
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For triangle small than many kilometers the error is minimal.
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## Rational fractions
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<%= leftblogimage("controled_error.png") %>
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