Line is a Circle

Originally posted by Sev
Ok, I was referring to both.

Well - that would explain the difference then....the language is clear and unamboguous. The statements it can produce can contain ambiguities. A simple example would be that we can formulate valid statements than can be neither true nor false...which means that they are ambiguous.

For anyone not well up on maths, one of the simplest analagies in the english language is : "This statement is false"

A slightly mathsier example is the idea of the set of all sets which do not contain themselves. If this set is a member of itself, it shouldnt be. If it is not a member of itself, it should be. Therefore, its membership is ambiguous.

As I said, I had an exception to the very low level proof of axiomatic statements. Those, I feel, are ludicrous and unecessary and that only reassert themselves in a circular way.

But they dont, really.

The Incompleteness Theorem shows that my views are entirely justified. My line from the very beginning of this argument, was that some things must be taken on "faith", or assumed for the sake of mathematics, whether it was in gest or not.

OK - you are correct in asserting that some things must be "taken on faith". More formally, some things are nothing more than the unproveable base definitions of our formal system.

Godel proved that any finite base of such definitions (or articles of faith) will lead to an incomplete or inconsistent system. (The obvious assertion which follows is that an infinite base may not lead to incompleteness or inconsistencies.)

However - it is important not to confuse this with the right to decide what articles must be taken on faith, and which must not.

so going back to the example we had in hand, you can "take it on faith" that 1+1=2, but the fact remains that this can be proven from the base axioms of our mathematics.

I agree that the proof is incredibly tedious, and more than irrelevant to the likes of you and me - or indeed almost everyone - but because it can be proven, it is not axiomatic.

Thus, it would be like saying "I take the truth of Fermats Last Theorem on faith, because its evidently true". Before it was proven, modern computers had searched to ridiculous lengths to verify that this was evidently true, but it still required the proof to be fully acceptable in mathematics. Any additional proof which used the results of Fermats' Last as part of its logic had to be qualified as being true "assuming Fermat's Last Theorem is true". Now, it was a reasonably safe assumption, but it still develops an element of doubt...especially when you start having theories which are true if and only if N other theories are proven to be true, where N is a large number.

So - the point I was making is that you, I, or anyone taking something on faith (or not) does not matter to the proveable correctness (or otherwise) of something.

The point you were making is that some things have to be taken on faith.

So, as the "ambiguity" being discussed by both sides was different (ambiguous even), it is fair to say that the point you were making is as valid from the context you had as mine is from the context I had. They are both, in effect, correct

jc

So are you agreeing or disagreeing?

Interesting way to dismiss an argument. Can you not answer? or are you afraid to damage your own credibility?

I haven't read through the thread, and my head is geared for maths today of all days (had two maths exams today, shotgunned).

I propose the following as one way of looking at a line:
x,y axes.
elipse centred at origin.
negligible length along the y-axis (lim x->0 of y component)
length across the x-axis = |legnth| of the "line".

The first post is just wrong in my opinion, for if the line is a circle of infinite diameter (as you look onto the edge (of negligible thickness I might add)), then the "line" the observer sees must also be of infinite length, from the observer's perspective.

That has more then likely been said already.

I saw devore mention tangents and I almost started hyposthesising whether or not my navel is best described as an infinite loop or square consisting of 720*.......

It's been a long day, I'm off to get wasted....

BTW - theres an interesting read over at http://mtnmath.com/whatth/ which is at the very least tangentially related to the discussions here. The section on Formal Mathematics is a nice easy introduction to some of the stuff discussed here.

Whole thing is a pdf download too I think.

jc

Originally posted by SOL
My point is not actually any of the above, really I was just waffelling but also that maths is only as accurate as you are

Waffling after how many pints? Mathematical logic and systems as they are abstract (though can be representational of the real world) are independent of the theorist or the observer(for want of a better word.)

A line is a circle? Yeah whatever...

Originally posted by SOL
So it follows that the line between two points is ... not a line

Alarm bells might ring in one's head when one reads such a sentence that clearly defies logic. If they do, then you are clearly reading into his mere supposition far too literally.

Regardless of the fact that he may have accidentally and unconsciously stated illogically that a line is not a line for the sheer effect of emphasising the main concept of such an elementary thought experiment, he has still provided the context and framework for you to construct such an idea, (that as the subject of the thread clearly states) that a line is a circle.

Yes you might have proven that a line is indeed a line, and that it can never not be a line, and it was clearly wrong of him to say then that his argument had made a line no longer a line... but you have not disproven that although a line may be a line, it could also be a circle, an assumption that he has made, founded upon the statement (that you have accepted for the purpose of your rejection), that a circle of infintite radius is a line, regardless of said error. An assumption that you have not yet disproven, as you claim to have.

If anything you have simply corrected a minor technicality in the wording of his origional supposition. I hope you never took what he had said to literally as a formal proof, he's clearly just toying about with an abstract idea. I just cant see why you would want to even bother yourself to try to disprove it. But calling him an idiot? That's hardly called for. Either you have missed the humour in his initial post, or I have missed the humour in your reply.

It would seem, despite the irony of this final comment, that...
Humour Ï Maths.

So you can prove that two circles of infinite radius, placed less than infinity apart intersect? I would have thought meddling with infinity would plunge you into mathematically absurdity and ambiguity.

What I'm getting at is if you were to construct an inductive proof for circles of all values "to infinity" would that really qualify as a proof "at infinity" too? Or is such a concept too intangible?

Because if you could'nt do that, then I could give a rather cheeky reason why...

Originally posted by bonkey
You would need to show why a circle of infinite radius behaves differently to all other circles to invalidate the proof

Because it's a line.