Click Here To Show Diagram Code

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Obviously the correct move is here... the rest of the solution is left as an exercise for the reader.

_________________
That which can be destroyed by the truth should be.
--
My (sadly neglected, but not forgotten) project: http://dailyjoseki.com

Maths troll

The obvious error in your post is that you didn't prove that PI = 4. You just show that the perimeter of a circle is four times the diameter, that is:
c = 4d instead of c = PI d.

By the way, with the same method, you can prove that the diagonal of the unit square has length 2

I leave the rest as exercise for the reader

Er, I don't understand this post. Showing that "the perimeter of a circle is four times the diameter" would indeed be equivalent to showing that pi equals 4. Whichever way you look at it, you get a conclusion that's clearly not true - the challenge is to figure out what's wrong with the reasoning. The argument "proving" that the diagonal of the unit square has length 2 works the same way.

Refuting the argument in a nice way doesn't seem to be easy at all. I think it's just a lesson that taking a limit as you do something infinitely many times can be trickier than you might think - especially in geometric settings.

That depends on the definition of PI. The most useful definition of PI is as limit of a particular series (*). Using that definition, the construction can not say anything about the value of PI, only about the perimeter of the unit circle. Anyway, that's just semantics and has nothing to do with the problem at hand


The content of that quote is wrong

(I can't tell you what's wrong without revealing the answer to the riddle, though.)


(*) More hints:

Robinz, the issue is that while a number may be introduced by one of its interesting properties, (i.e. "Pi is the ratio of a circle's circumference to its diameter"), the number is just fundamentally a real number. If it mysteriously turned out that the ratio wasn't 3.14..., it would just turn out that the ratio wasn't Pi.

In the same way, you may be told "Justin is the guy on the forums who does x y and z", but if you find out that it's really John who does those things, it doesn't mean Justin is actually John, just that you've been told something false about Justin.

The issue the distinction between a description ("the x that has some property") and a proper name ("Pi"). It's one that has greatly interested philosophers, and at one point getting clear on it was an important topic in the foundations of mathematics.

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Occupy Babel!

You're all getting into semantics, which has nothing to do with the original post.

I have a PhD in maths myself, so I'm well aware that there are definitions of pi which don't involve circles, or geometry of any kind. They're all equivalent (although it's not readily apparent that they are without some at least moderately heavy-duty maths).

So, if you use one of the other definitions (the limit of a certain series, say), then you are quite correct that the troll doesn't "prove" that pi=4*. But it does "prove" that the circumference of a circle (in the Euclidean plane) is 4 times its diameter - something which we all know to be false. (You don't even need school-level maths to know that - just find something circular and measure it.)

So, we have a superficially plausible argument, leading to a conclusion which is definitely false. Such things always challenge us to find where the apparently convincing argument is apparently wrong. In this case, it's clear that the flaw falls somewhere in the limiting part of the argument. Indeed, the very fact that it produces an absurd conclusion here can be taken as a proof that you can't just take limits in such an intuitive geometric way and expect it to preserve length.

*actually, it does, provided we accept that the circumference of a circle is indeed pi times its diameter. For then we have c=4d and c=pi*d both being true at once, and since d is clearly non-zero it necessarily follows that pi=4

About the circle

Um, I'm not a mathematician at all, but you don't have to be one to see what's wrong with the "proof":

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Patience, grasshopper.

@robinz Well, then sorry for the unnecessary explanation. Your response to flOvermind was easily mistaken for being confused, rather than telling him to avoid a pointless technicality (which it was, imho).

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Occupy Babel!

It wasn't just talking about "pointless technicalities" - I was simply pointing out that his statements:



were illogical. From c=4d it would *follow* that pi=4, since I assume we can all take it as given that c=pi d.

So, after claiming that the error was "obvious", he didn't actually address the issue at all. (But Redundant has since done that - he may be still an undergrad but he clearly knows a lot more about Banach spaces than I ever have or will - it's not something I ever studied at all )

@robinz Well, then the semantic discussion is necessary, because the question is whether 'pi' is semantically more like "the number that is the ratio of the circumference to the diameter of a circle" or whether 'pi' is more like a proper name.

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Occupy Babel!

The preceeding nine posts are evidence of some VERY succesfull trolling.

Funny, actually I didn't want to start a discussion on the semantics of pi. But it seems that to some people, it's not obvious at all what's wrong with the example. Perhaps my hints were a bit too cryptic .
I assert that the only thing wrong is the final step, going from the circumference 4 to the assertion that pi=4



Wrong. I say that the circumference of a circle with diameter 1 in the Euclidean plane is actually 4. Prove me wrong (Hint: You didn't specify which metric you use. I chose to use the Manhattan metric ).



It either follows that pi=4, *or* that c=pi d is wrong. For example, using Manhattan metric, c=pi d is actually wrong. (I'll leave the proof to the reader. It's actually a lot easier to calculate the circumference of a circle under Manhattan metric, since you don't have to integrate that stupid square root ).

That's the main point of my previous posts. Basically, if you omit the last step (pi=4), there is nothing wrong at all with the "troll proof" showing that the circumference of the circle is 4. Because it is (under the Manhattan metric). Same with the diagonal of the unit square being 2.


@Redundant: Your solution is of course correct, too, assuming we're operating under the usual pythagorean metric.

(PS: Trolling is fun )

OK, fair enough, but now you are just trolling (I wasn't aware that I was, but apologies if anyone took it that way - I was merely disagreeing with flOvermind's post, which I thought was perfectly fine provided I did it politely.) Since no-one specified a metric, it's natural to assume that we're talking about the usual Pythagorean one. So there *is* something wrong with the troll "proof", which is that, under the standard definition of length, you can't assume it is preserved under the limit used. It appears now that you understood this, in which case I apologise for assuming that you didn't - it's simply not how your first post read to me

PS I promise that this is my last post on the subject.

I don't understand this proof, how does it suddenly go from 4 to 4! = 24?

I thought that was the point of this thread?