Comment on Mental Context for Model Theory

Thu, 30 Apr 2015 20:46:08 GMT

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.

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