In reality, the fact that parallel lines do not intersect does follow from the definition of the word “parallel”. Therefore, the error results in several of the paragraphs in the original post being meaningless or untrue.
The trouble is that in 2-D Euclidean space, there are many equivalent definitions of “parallel”. It just so happens that straight lines that don’t intersect also have the same slope,will intersect any transverse line at congruent angles, and are always the same distance apart (and vice versa). However, these properties need not be equivalent in non-Euclidean geometry.
The OP’s issue seems to be that defining parallel lines as those which do not intersect is artificial. It’s a workaround Euclid developed to smooth over his presentation. He could not use local properties of lines and angles to prove parallel lines didn’t intersect. So, he defined them as lines that don’t intersect, introduced the parallel postulate, and then used those to prove the other properties of parallel lines. Later mathematicians found this to be rather inelegant and tried to prove parallel lines didn’t intersect using only properties of lines and angles.
Sure, it’s an error if you use Euclid’s definition of parallel, but I wouldn’t call the discussion meaningless. It touches on a very important issue of how to define things and what properties we want to retain when we generalize a notion.
The discussion would be helped if people consulted what Euclid wrote.
The trouble is that in 2-D Euclidean space, there are many equivalent definitions of “parallel”
It would be better to put that as there being many concepts which, in the presence of the 5 postulates, are all equivalent to the one that Euclid calls “parallel”. When one is not considering the foundations of geometry, it does not matter which of these properties one calls “parallel”, as one understands that when any of these properties is satisfied, all are. When one is considering the foundations, it does matter, and only confusion can result from using any but Euclid’s.
But the issue of the 5th postulate is not about definitions. Euclid’s 5th postulate does not mention parallelism at all. There are many other 5th postulates one can substitute for Euclid’s and get the same geometry, but the problem (so I gather from a few minutes wiki-ing) was that all of them seemed rather more complicated than the other four, leading many mathematicians to search for a proof that would render them all unnecessary.
Bolyai and Lobachevsky (and Gauss before them, but unpublished) settled the matter by working out what looked like a consistent theory of hyperbolic geometry. I say “looked like”, because mathematical logic was yet to be invented, and even Hilbert’s axioms were still in the future. Models of hyperbolic geometry within Euclidean geometry were found, still in the 19th century, definitively settling the matter.
When it comes to neutral geometry, nobody’s ever defined “parallel lines” in any way other than “lines that don’t intersect”. You can talk about slopes in the context of the Cartesian model, but the assumptions you’re making to get there are far too strong.
As a consequence, no mathematicians ever tried to “prove that parallel lines don’t intersect”. Instead, mathematicians tried to prove the parallel postulate in one of its equivalent forms, of which some of the more compelling or simple are:
The sum of the angles in a triangle is 180 degrees. (Defined to equal two right angles.)
There exists a quadrilateral with four right angles.
If two lines are parallel to the same line, they are parallel to each other.
It’s also somewhat misleading to say that mathematicians were mainly motivated by the inelegance of the parallel postulate. Though this was true for some mathematicians, it’s hard to say that the third form of the parallel postulate which I gave is any less elegant, as an axiom, than “If two line segments are congruent to the same line segment, then they are congruent to each other”. Some form of the latter was assumed both by Euclid (his first Common Notion) and by all of his successors.
A stronger motivation for avoiding the parallel postulate is that so much can be done without it that one begins to suspect it might be unnecessary.
When it comes to neutral geometry, nobody’s ever defined “parallel lines” in any way other than “lines that don’t intersect”. You can talk about slopes in the context of the Cartesian model, but the assumptions you’re making to get there are far too strong.
Well, Euclid was the standard textbook in geometry for a long time. There was a movement in the 1800s to replace the Elements with a more modern textbook and a number of authors used different definitions, which just ended up requiring them to introduce other axioms to get the result. Lewis Carroll ended up satirizing the affair.
It’s also somewhat misleading to say that mathematicians were mainly motivated by the inelegance of the parallel postulate.
If it were elegant, mathematicians wouldn’t have spent 2,000 years trying to prove it from the other four postulates. I very much doubt Euclid himself liked it. Intuition suggests that the result should follow from more elementary notions.
It was a workaround to let Euclid get on with his book and later mathematicians looked for a more elegant formulation.
Though this was true for some mathematicians, it’s hard to say that the third form of the parallel postulate which I gave is any less elegant, as an axiom, than “If two line segments are congruent to the same line segment, then they are congruent to each other”.
Is it obvious from the definition of parallel l lines that this ought to be true? That equality should be transitive seems like so obvious an idea that it’s barely worth writing down.
EDIT: It’s worth noting that classical mathematicians had very different ideas about what axioms should be. To them, axioms should be self-evident. Modern mathematics has no such requirements for its axioms. These are two very different attitudes about what axioms ought to be.
There was a movement in the 1800s to replace the Elements with a more modern textbook and a number of authors used different definitions.
What other definitions of “parallel line” do you have in mind?
Is it obvious from the definition of parallel l lines that this ought to be true? That equality should be transitive seems like so obvious an idea that it’s barely worth writing down.
Congruence and equality are not the same thing. One of these axioms says that being parallel is transitive; the other says that being congruent is transitive. I agree that both notions become much less useful if transitivity does not hold, but a non-transitive congruence relation is not nonsensical.
The trouble is that in 2-D Euclidean space, there are many equivalent definitions of “parallel”. It just so happens that straight lines that don’t intersect also have the same slope,will intersect any transverse line at congruent angles, and are always the same distance apart (and vice versa). However, these properties need not be equivalent in non-Euclidean geometry.
The OP’s issue seems to be that defining parallel lines as those which do not intersect is artificial. It’s a workaround Euclid developed to smooth over his presentation. He could not use local properties of lines and angles to prove parallel lines didn’t intersect. So, he defined them as lines that don’t intersect, introduced the parallel postulate, and then used those to prove the other properties of parallel lines. Later mathematicians found this to be rather inelegant and tried to prove parallel lines didn’t intersect using only properties of lines and angles.
Sure, it’s an error if you use Euclid’s definition of parallel, but I wouldn’t call the discussion meaningless. It touches on a very important issue of how to define things and what properties we want to retain when we generalize a notion.
The discussion would be helped if people consulted what Euclid wrote.
It would be better to put that as there being many concepts which, in the presence of the 5 postulates, are all equivalent to the one that Euclid calls “parallel”. When one is not considering the foundations of geometry, it does not matter which of these properties one calls “parallel”, as one understands that when any of these properties is satisfied, all are. When one is considering the foundations, it does matter, and only confusion can result from using any but Euclid’s.
But the issue of the 5th postulate is not about definitions. Euclid’s 5th postulate does not mention parallelism at all. There are many other 5th postulates one can substitute for Euclid’s and get the same geometry, but the problem (so I gather from a few minutes wiki-ing) was that all of them seemed rather more complicated than the other four, leading many mathematicians to search for a proof that would render them all unnecessary.
Bolyai and Lobachevsky (and Gauss before them, but unpublished) settled the matter by working out what looked like a consistent theory of hyperbolic geometry. I say “looked like”, because mathematical logic was yet to be invented, and even Hilbert’s axioms were still in the future. Models of hyperbolic geometry within Euclidean geometry were found, still in the 19th century, definitively settling the matter.
When it comes to neutral geometry, nobody’s ever defined “parallel lines” in any way other than “lines that don’t intersect”. You can talk about slopes in the context of the Cartesian model, but the assumptions you’re making to get there are far too strong.
As a consequence, no mathematicians ever tried to “prove that parallel lines don’t intersect”. Instead, mathematicians tried to prove the parallel postulate in one of its equivalent forms, of which some of the more compelling or simple are:
The sum of the angles in a triangle is 180 degrees. (Defined to equal two right angles.)
There exists a quadrilateral with four right angles.
If two lines are parallel to the same line, they are parallel to each other.
It’s also somewhat misleading to say that mathematicians were mainly motivated by the inelegance of the parallel postulate. Though this was true for some mathematicians, it’s hard to say that the third form of the parallel postulate which I gave is any less elegant, as an axiom, than “If two line segments are congruent to the same line segment, then they are congruent to each other”. Some form of the latter was assumed both by Euclid (his first Common Notion) and by all of his successors.
A stronger motivation for avoiding the parallel postulate is that so much can be done without it that one begins to suspect it might be unnecessary.
Well, Euclid was the standard textbook in geometry for a long time. There was a movement in the 1800s to replace the Elements with a more modern textbook and a number of authors used different definitions, which just ended up requiring them to introduce other axioms to get the result. Lewis Carroll ended up satirizing the affair.
If it were elegant, mathematicians wouldn’t have spent 2,000 years trying to prove it from the other four postulates. I very much doubt Euclid himself liked it. Intuition suggests that the result should follow from more elementary notions.
It was a workaround to let Euclid get on with his book and later mathematicians looked for a more elegant formulation.
Is it obvious from the definition of parallel l lines that this ought to be true? That equality should be transitive seems like so obvious an idea that it’s barely worth writing down.
EDIT: It’s worth noting that classical mathematicians had very different ideas about what axioms should be. To them, axioms should be self-evident. Modern mathematics has no such requirements for its axioms. These are two very different attitudes about what axioms ought to be.
What other definitions of “parallel line” do you have in mind?
Congruence and equality are not the same thing. One of these axioms says that being parallel is transitive; the other says that being congruent is transitive. I agree that both notions become much less useful if transitivity does not hold, but a non-transitive congruence relation is not nonsensical.