Comment on Mental Context for Model Theory

Thu, 30 Apr 2015 16:46:44 GMT

I agree that this was not written in the respectful tone that I would like to see at Less Wrong. I wish Anatoly had phrased that differently.

I am however concerned that when true statements are downvoted, there is a real risk that readers misunderstand why the comment was downvoted and assume that the contents are untrue. For the benefit of those readers, I simply wanted to state for the record that the contents of his message were indeed true.

This is not just a matter of misstating the axiom. The original post reads:

One of these things is not like the other. The fifth axiom is the only one which requires some effort to understand. Intuitively, it states that parallel lines do not intersect. This statement irked Euclid for reasons apart from the ugliness of the axiom.

The fact that parallel lines do not intersect seems like it should follow from the definition of lines and angles. It doesn't seem like something we should have to specify in addition. That we must assume parallel lines do not intersect (rather than proving it) was long seen as a wart on geometry.

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.

Discuss