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)
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)