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 What's the *real* issue with Euclid's parallel postulate? 
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Joined: 13 May 2009, 13:10
Posts: 4
Post What's the *real* issue with Euclid's parallel postulate?
I'll begin by saying that I'm not a complete geometry noob and I understand in general terms what the issue is. I'm also aware that Euclid's postulates have since been refined and reformulated, and that they are no longer considered as precisely defined as they were back in his days (Hilbert for instance made an updated list of postulates in the early 20th century). So, this is more a question I guess concerning how the parallell postulate was viewed historically than today though in principle the issue is the same today as it was then.

Namely, the issue that seemed to bother many great minds in the past was the fact that parallell lines continue forever and that therefore there is no way to directly observe and verify something of infinite length. The difference with non-parallell lines being that they intersect at a finite point in space and therefore some direct observations can be made.

So, the idea was that since parallell lines go on forever and because you can't possibly verify an infinite length, then you can't be sure that two parallell lines never meet. This dilemma in turn opened up a can of worms which contributed to the subsequent discovery of other, non-Euclidean, geometries. At least this is the "official" version that you get repeated at every site dealing with this issue.

However, I must say that I find it very strange that infinity was seen as such a big issue in the parallell postulate. Because, as far as I can comprehend Euclid relied on the concept of infinity in his other postulates as well.

For instance, in postulate 2 he says that line segments can be extended indefinitely, i.e. to infinity. As I see it, his basic definition of a line is the root of the parallell postulate issue, because this is where Euclid introduces the concept of an infinite line. So, if postulate 2 isn't problematic then why is postulate 5, i.e. the parallell postulate problematic when it only reaffirms the point about infinite lines made in postulate 2? Because if extending something finite to infinity is acceptable, then proving something to be parallell over any finite length and then extending it to infinity shouldn't be an issue either.

Also, in his "common notions" that accompany the five postulates, Euclid by stating his notions about equality, sums and wholes assumes that space treats objects identically wherever they are located. This is yet another assumption that to me seems to rely on the concept of infinity but which apparently didn't bother anyone. If you accept that space treats objects equally everywhere, then doesn't this assumption automatically also imply that parallell lines always will remain parallell no matter where in space they are located or extended?

So, just to sum up my post, I would appreciate some clarification on why specifically this parallell postulate was seen as being so controversial while everything else Euclid stated wasn't? To me given my above reasoning this doesn't make much sense, yet all the great minds of the Middle Ages and beyond seemed to question only the parallell postulate. Given this, I must be missing something rather big in my above reasoning? Yet I can't seem to find it...


05 Oct 2009, 02:10

Joined: 13 Aug 2009, 00:08
Posts: 13
Post Re: What's the *real* issue with Euclid's parallel postulate?
I can only give you my take on this. I don't think the problem was ever anything to do with infinity and non-verifiability, and I don't recall ever seeing the fifth postulate ever couched in those terms in the many text books I have read over the years.

The issue seems to be that while postulates 1-4 are simple, concise, and "obvious", the fifth is long, complex, and takes a couple of readings to understand; and for this reason (and probably others), mathematicians thought that this was not a postulate but was, instead, a consequence of the other four. Most of the history of the fifth postulate down the centuries is about people trying to prove that -- and failing in their attempts.

So I think the real issue is that people thought it was not as "obvious" as the other four -- and, in that, they were correct. It is an independent postulate that can be replaced by other postulates that lead to other (non-Euclidean) geometries. So, again, I don't think the issue was to do with non-verifiability of infinity.

Just my take -- what do others think?


11 Oct 2009, 20:58
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Joined: 07 Jun 2008, 08:21
Posts: 235
Post Re: What's the *real* issue with Euclid's parallel postulate?
I was intrigued by your post bgd and did a little research. At first what I found was very similar to what flashbang found. But then I looked at my copy of Visual Complex Analysis looking for guidance with a completely different problem and found by chance that there was a chapter dealing with Non-Euclidean geometries.
The attached document is a summary of what I found out.
Attachment:
EuclidianGeometry.pdf [40.82 KiB]
Downloaded 253 times


14 Oct 2009, 16:24

Joined: 11 Jul 2009, 20:45
Posts: 25
Post Re: What's the *real* issue with Euclid's parallel postulate?
Parallel is a concept, the concept does not have an issue. To the degree that you can imagine,,,those are your wings!
IT'S A FRIGGIN AXIOM!!
BTW; It's based on Left/Right vs Depth-- Human Vision. Human Vision is being tuned out by other senses input, the Twittering!, The absolute, 24/7 Cable Noise, the unchanging wish to be happy in Schadenfreude. Religion, any Religion,,will only be done away with when the Priests are dead!


07 Nov 2009, 04:53

Joined: 29 Jan 2010, 19:21
Posts: 11
Post Re: What's the *real* issue with Euclid's parallel postulate?
Hello all. My first post here. I am no expert. I´m an engineer not mathematician. So forgive my petulancy :) . But this is something I thought about. My view is a little different. I think, as others here, that it is is mainly a question of choice. I would say that the choices are to permit or not a line to be curved. If you permit a line to be curved you permit a plane to be 'not' flat. And then you go with hiperbolic/no-euclidian geometry.
In my way of thinking, a paralel is only meaningful in if you are in a plane. If you are not in a plane you can create some paralel like concept, but it is a different thing.
And if you think about the number pi, it has a definite value (irrational and transcendental) only in the plan.
Euclides didn´t know everything we know about pi. If you 'tie' the 5th postulate to the pi definition you have interesting consequences (like pi can be rational for instance in some 'places').

Rodrigo


25 Feb 2010, 15:59
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Post Re: What's the *real* issue with Euclid's parallel postulate?
Quote:
If you 'tie' the 5th postulate to the pi definition you have interesting consequences (like pi can be rational for instance in some 'places').

That sounds very interesting rods! Do you know of any internet links about it? I'd like to read more.


26 Feb 2010, 11:15

Joined: 29 Jan 2010, 19:21
Posts: 11
Post Re: What's the *real* issue with Euclid's parallel postulate?
vasco wrote:
Quote:
If you 'tie' the 5th postulate to the pi definition you have interesting consequences (like pi can be rational for instance in some 'places').

That sounds very interesting rods! Do you know of any internet links about it? I'd like to read more.

No. Never heard anyone saying that.
I think this is a very powerful idea. If you look at RR 2.4 (Hyperbolic Geometry) Penrose talks about Lambert's formula $\pi - (\alpha + \beta + \gamma) = C\Delta$. But he does not comment that (or if) $\pi$ should (shoud it ?) be defined on a plane.
I see no problem on having $\pi$ with different values from the usual. But as far as I understand Lambert's formula uses the 'flat' $\pi$. And so does the non-euclidian geometry. So they somehow depend on the flat 'plane' definition.
A different idea but having some commom points.
Look at the uncertainty principle as you look to a irrational number. Thing about \pi=\displaystyle\frac{p}{d} being a irrational number (but it is a reason!) and notice that you can't have both $p$ and $d$ rational at the same time !
rodrigo


26 Feb 2010, 12:32

Joined: 29 Sep 2010, 08:23
Posts: 1
Post Re: What's the *real* issue with Euclid's parallel postulate
On occasion, I re-read various Chapters of R2R

So with regard to the parallel postulate it struck during the latest re-read that we tend to project rules into different realms of values based purely on an assumption that we are entitled to do so but with no proof that we truly can.

Consider two absolute real numbers:
zero (0)
positive infinity (+ oo)

Note that it is not a given that a "continuous" line exists between these values.

Now consider the spectrum of ranges between these two value points as being broken up according to the following ranges:

range #1: from zero (0) to 1 over infinity (1/oo)
range #2: from 1 over infinity (1/oo) to infinity minus (1/oo)
range #3: from infinity minus (1/oo) to infinity (oo)
range #4: from infinity (oo) to infinity plus (1/oo)

The physical world that we observe resides in Range #2
We can't see the ultra small (of size below one over infinity (1/oo))
We can't see the ultra large (of size above infinity minus (1/oo)) at least because the limit on speed of light (c) prevents us from doing so.

The Euclidean postulates appear to hold true in Range #2.
But what gives us the right to extend them into the other ranges (#1, #3 and as Buzz Lightyear might say, beyond infinity into Range #4)?

That is just another way of paraphrasing Penrose's questions about "ripples" in the fabric of the physical world. Stated otherwise, what makes us so sure it is a continuum and not a mesh? And if it is a mesh, what makes us assume it is a homogenous mesh?


29 Sep 2010, 08:52

Joined: 11 Jul 2009, 20:45
Posts: 25
Post Re: What's the *real* issue with Euclid's parallel postulate
flashbang
So, again, I don't think the issue was to do with non-verifiability of infinity.
_________________________________________

So, you can see to INFINITY?
Of course two "LINES" that keep the same "DISTANCE" will never meet, if the lines go on forever with the same "DISTANCE" between them!!!
FUGGID ABOUT ANGLES!!!

It is a EUCLID CONCEPT!!


17 Oct 2011, 17:33
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