"It develops a complete theory of planar Euclidean geometry over a general field without any reliance on 'axioms'."
This statement alone is mock-worthy. Back in my college days, I poked through Russell and Whitehead's Principia Mathematica. (I was bored. The QA section of the college library was practically my second dorm-room.) Even though that massive thing took half a book to prove the existence of '1', and then went on to work on '2', never mind '+', it still started from some kind of first principles.
And I'd *love* to see how he gets around that awkward little postulate[1] about parallels.
[1]Merriam-Webster defines the two words as equivalent, so I'll hold Wildberger to it.
If I'm drawing the correct conclusions from the summary, it looks like he's talking about avoiding pure geometric axioms, and instead deriving geometry from arithmetic. That's a totally reasonable thing to do.
In fact, it's the complement of what Euclid did, and is arguably a lot more useful, since everybody doing formal geometry can be presumed to know arithmetic, so you can let your axioms be implicit without worrying too much that you're deluding yourself. Yes, getting arithmetic formally right requires axioms; but because arithmetic is used in everyday life and works, you don't have to mess with that unless you want tol.
Sorry, but if that's what he means he would be better off phrasing it as "without geometric postulates" instead of "without 'axioms'" I distinctly remember the thrill of analytical geometry when suddenly algebra and geometry were fused. I still wouldn't say that it was done without axioms.
(no subject)
This statement alone is mock-worthy. Back in my college days, I poked through Russell and Whitehead's Principia Mathematica. (I was bored. The QA section of the college library was practically my second dorm-room.) Even though that massive thing took half a book to prove the existence of '1', and then went on to work on '2', never mind '+', it still started from some kind of first principles.
And I'd *love* to see how he gets around that awkward little postulate[1] about parallels.
[1]Merriam-Webster defines the two words as equivalent, so I'll hold Wildberger to it.
(no subject)
In fact, it's the complement of what Euclid did, and is arguably a lot more useful, since everybody doing formal geometry can be presumed to know arithmetic, so you can let your axioms be implicit without worrying too much that you're deluding yourself. Yes, getting arithmetic formally right requires axioms; but because arithmetic is used in everyday life and works, you don't have to mess with that unless you want tol.
(no subject)