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Kindly comments on Mental Context for Model Theory - Less Wrong
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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.