(Continued from previous dialog .) F: "Where is then the analogy with squaring the circle?
G: When I raised in this post the analogy between the motion of the bishop and the squaring of the circle, imagine a single bishop on a chessboard, with no other pieces present. Under those conditions, if the bishop is moved according to the rules of chess, you can never move from white to a black box.
F: "And it shows that it is impossible to square the circle?
G: No, definitely not showing it. It's just an image that serves to illustrate the idea of \u200b\u200babsolute impossibility in Mathematics. It is as impossible to square the circle as the bishop is to move from a white to a black box (under the conditions described previously).
F: But it is possible that the bishop move from a white box to a quarter in a game of truth. Do not invalidate this analogy?
G: the contrary. Squaring the circle asks, given a circle, construct a square that has exactly the same area, using only a straightedge and compass . Under these conditions the construction is absolutely impossible. But it is possible to build if we admit other resources in the same way that the bishop himself can change color if we add further complexities to the situation.
F: How?
G: For example, since the circle, put a string around its edge so that both edge and thread, coincide. Then pull the thread so it looks like a segment. Trace the segment determined by the thread. If the diameter of a circle is the measured segment pi and from it is very easy to draw the square order.
F: So is it possible or impossible to square the circle?
G: is impossible if we just use the resources it requires the classic problem (straightedge and compass). But it is possible if we allow the use of other elements. So its strange examples of games in which the bishop will change color, far from refute the analogy, make it more complete.
F: But you say that the analogy does not prove that it is impossible to square.
G: No, of course not. The show is based on three facts:
1. From a segment unit can only be built (using straightedge and compass) segments whose length is an algebraic number. [Unable to obtain all algebraic numbers, but that's not important now.]
2. The square circle is possible if and only if it can build a segment of length pi.
3. Pi is not algebraic.
The combination of these facts unavoidable results ... Well, what we already know: squaring the circle is impossible.
F: "The demonstrations of these three facts are easy to understand?
G: The demonstration of the fact (1) is not difficult, just requires a little knowledge of geometry. demonstrates the fact (2) is a bit more difficult. The fact of (3) is rather more complicated.
F: What if there was an error in the proof of fact (3)? (I say that because is the most difficult.)
G: The show has been revised again and again by generations of mathematicians who have unanimously ratified their validity.
F: What if, despite everything, there was a mistake? A mistake that would have overlooked all the thousands of mathematicians who reviewed the show?
G: Have you ever been to Paris?
F: You tell me, and it was you who created me.
G: Well . You never was in Paris, me neither.
F: I see, going to ask how I know that the Eiffel Tower really is.
G: Exactly. How do you know? How do you know that there is no universal conspiracy to make us believe (who never were in Paris) that there is something called "Eiffel Tower"? How do you know actually in Paris there nothing? How do you know if what happens is that every visitor to the tower case was recruited to be part of this conspiracy and to propagate the lie? How do you know all the supposed pictures of the tower are fake? Etc., etc. ...
F: I can not know for sure.
G: Exactly. In fact I could not even know if you were standing in front of the tower itself, because their senses could be being deceived. But infinitely more reasonable assumption is that the Eiffel Tower does exist and that all photographs that show (well, let's say almost all) represent a real object.
F: suppose you're right.
G: Similarly, exactly the same way, infinitely more reasonable assumption is that squaring the circle really is impossible, because generations of mathematicians have so found. The most reasonable, the only sensible thing is to abandon any attempt to solve the problem (using the classic method).
F: So why are there still people who tried?
G: I have no idea.
F: you allow me one last question?
G: course.
F: Following his words ... the same way, exactly the same way, the most reasonable assumption to me is to accept that I am talking to you?
G: Sure.
F: Yet I do not exist. As I said before, you created me. Where does that leave us?
G: I wonder what kind of statement is "I do not exist." Who is the "I" that asserts that there?
The End?
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