My Math Forum Euclid's Fifth Postulate

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July 17th, 2017, 11:35 PM   #11
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 Originally Posted by v8archie I just think your criticism is over the top. Many mathematical texts are difficult to understand if you don't already know what they are trying to say. Especially the pre-notation ones. I think it was some of al-Khwarizmi's writing I saw recently that sounded like utter gobbledy-gook. Such was the rhetorical tradition. In contrast "that which has no parts" is a reasonably good definition of a point from 1800 years earlier and without the benefit of any existing definitions to work from. Sure, it might not tell you much, but if you know what he means, it makes sense.
It's not my criticism even, what I said has been said by many commenters.

Example, Hartshorne (authority on modern geometry):

Quote:
 On the other hand, some of Euclid's other definitions, such as the first, "a point is that which has no part," or the second, "a line is breadthless length," or the third, "a straight line is a line which lies evenly with the points on itself." give us no better understanding of these notions than we had before. It seems that Euclid, instead of giving a precise meaning to these terms, is appealing to our intuition, and alluding to some concept we may alreadjr have in our own minds of what a point or a line is. Rather than defining the term, he is appealing to our common understanding of the concept, without saying what that is. This may have been very well in a society where there was just one truth and one geometry and everyone agreed on that. But the modern consciousness sees this as a rather uncertain way to set up the foundations of a rigorous discipline. What if we say now, oh yes, we agree on what points and lines are, and then later it turns out we had something quite different in mind? So the modern approach is to say these notions are undefined, that is, they can be anything at all. provided that they satisfy whatever postulates or axioms may be imposed on them later.
See also Heath's translation of Euclid for pages long discussions of his definitions and common notions, which I guess you'll definitely find over the top!

July 18th, 2017, 12:26 AM   #12
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 Originally Posted by Micrm@ss No it doesn't. It's a meaningless statement. "A point is that which has no parts" is merely a heuristic. It is telling us how to think of the concept of a point. Which sure, is useful. But it is not helping in any way to give a rigorous definition of a point. Want proof? He never actually uses this definition anywhere in the elements! He couldn't use this definition, since it's unusable. Surely you mean common notion 1? Yes, the common notions are somewhat more useful. Although I highly doubt Euclid had thermodynamics in mind haha. Still, the common notions are very incomplete. He very often uses common notions that he didn't state.
Firstly let me thank you for the correction to my senior moment.
I did indeed mean to refer to common notion 1 in the second part of my post.

Secondly whilst I agree with you that Euclid's lists were not exhausitve, I was disappointed with the casual dismissal of my first comment from someone whose posts are normally highly perceptive and a pleasure to read.

Criticism, yes. But constructive criticism please.

I can only assume you have not understood my point, which is particularly ironic considering your username.

When I first read Heath's translation I thought the lists so much fluff and nonsense, although a valiant attempt to define his working materials.
You cannot develop theorems about points, lines and so on until you have at least a working definition.

It was later I realised just how far reaching these definitions are.

"That which hath no part" goes right to the heart of the application of the (Newton's) Calculus to Physics.

July 18th, 2017, 02:39 AM   #13
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 Originally Posted by studiot Firstly let me thank you for the correction to my senior moment. I did indeed mean to refer to common notion 1 in the second part of my post. Secondly whilst I agree with you that Euclid's lists were not exhausitve, I was disappointed with the casual dismissal of my first comment from someone whose posts are normally highly perceptive and a pleasure to read. Criticism, yes. But constructive criticism please. I can only assume you have not understood my point, which is particularly ironic considering your username. When I first read Heath's translation I thought the lists so much fluff and nonsense, although a valiant attempt to define his working materials. You cannot develop theorems about points, lines and so on until you have at least a working definition. It was later I realised just how far reaching these definitions are. "That which hath no part" goes right to the heart of the application of the (Newton's) Calculus to Physics. Please define density for me.
Dear Studiot, I am deeply sorry you took my reply as a causal dismissal. It was not at all intended this way. In fact I agreed with your first post that the postulates were not all there were and that a lot of Euclid relied also on the axioms and common notions. I just wanted to elucidate to the OP that most of his definitions should not be taken as in the modern sense. I apologize if I came off as antagonizing, or as dismissing your accurate post. I can asure you it was not meant this way! I'll be sure to reread my posts whenever I post next time, as I don't wish for these misunderstandings to happen!

Also, more on point. I am not saying (or should not have said) that definition I is useless. It is a fine heuristic and it puts a very good idea in the head of the reader of what a point is. But you have to agree it is not like a modern mathematics definition. Nowadays, we wouldn't call this a definition, but rather an explanation or something else. That is all I meant with my comment.

I don't really understand the density thing. Do you mean it like "rationals are dense in the reals" or in another way? Maybe I'm being dense (pun intended).

Also, I do agree that most of Heath is fluff and only interesting for historians and not mathematicians. However, it is I think the most authorative English translation of Euclid nevertheless.

July 18th, 2017, 03:20 AM   #14
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 Originally Posted by Micrm@ss Dear Studiot, I am deeply sorry you took my reply as a causal dismissal. It was not at all intended this way. In fact I agreed with your first post that the postulates were not all there were and that a lot of Euclid relied also on the axioms and common notions. I just wanted to elucidate to the OP that most of his definitions should not be taken as in the modern sense. I apologize if I came off as antagonizing, or as dismissing your accurate post. I can asure you it was not meant this way! I'll be sure to reread my posts whenever I post next time, as I don't wish for these misunderstandings to happen! Also, more on point. I am not saying (or should not have said) that definition I is useless. It is a fine heuristic and it puts a very good idea in the head of the reader of what a point is. But you have to agree it is not like a modern mathematics definition. Nowadays, we wouldn't call this a definition, but rather an explanation or something else. That is all I meant with my comment. I don't really understand the density thing. Do you mean it like "rationals are dense in the reals" or in another way? Maybe I'm being dense (pun intended). Also, I do agree that most of Heath is fluff and only interesting for historians and not mathematicians. However, it is I think the most authorative English translation of Euclid nevertheless.
Thank you for returning to open mind mode.

My library does not run to multiple direct translations of Euclid. Heath is all I have.

But what I originally thought as fluff, I later came to think that Euclid was talking about dimensions in the first few definitions.

So a line has no width or height but has length.

Now imagine passing a line through a point that has dimensions.
The questions arise

Which part of the point does the line pass through?

If two lines pass through the same point can they not actually touch (meet) by passing through differnt parts of that point?

"That which hath no part" ie zero dimension in all directions avoids this.

So such apparently banal statements are necessary to develop the theory.

On to my remark about density - nothing personal intended, glad you could take it that way.

So much of Physics is founded on continuum mechanics which is about distributed or extensive properties, such as mass, charge, electric fields, stress etc.

Today we gaily integrate density over some region to obtain the total mass, but what is the density at a point which has zero volume, since density is mass/volume?

We must look to the theory of limits to answer this question.

Today if I want fire I have all sorts of sophisticated means of obtaining it.
I don't have to resort to rubbing two boy scouts together to start one.

 July 21st, 2017, 07:30 AM #15 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,347 Thanks: 227 AFAIK , the 5th postulate was not called 'The Parallel Postulate' by Euclid , but reading commentaries by others about it one gets the impression he called it so.

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