First off, he does have some interesting mathematical work. He has thought up some alternate formulas for computing Riemann integrals and performing basic calculus but ultimately it's nothing that amazing. But what makes him so special? John HATES irrational numbers.

He's a pretty frequent writer of Google knols (units of knowledge) and though his knols on calculus are pretty lucid and well-written, if you get him started on irrational numbers he goes off the deep end... fast.

For example, he thinks that the real numbers are countable

http://knol.google.com/k/are-real-numbers-uncountable

Well, not really, because he really thinks that real numbers (irrational numbers) are not "well-defined"

http://knol.google.com/k/are-the-real-n ... l-defined#

What makes him so lovable:

- His arguments are riddled with contradictions and circular logic (can you spot his assumption that all numbers are rational?)
- His primary reference is himself and he does so in third person. He'll quote himself in his own articles and then put "-John Gabriel" at the end. Or He'll say "...using John Gabriel's integration formula."
- He frequently interrupts his own arguments to start ranting about how everyone else is a blind fool
- He deletes all comments on his knols and disables comments on his blog. Just like anyone interested in academic discussion
- He thinks real analysis is complete bullshit

What sort of disturbs me:

- Almost all of his knols are really well-rated. How?

Basically he doesn't seem to understand how axiomatic mathematics work, he's incredibly close-minded when it comes to alternate definitions of concepts, and he seems to have a fundamental misunderstanding of what limits are. He'll often talk about the limit of a number... which is meaningless.

His knol page: http://knol.google.com/k/john-gabriel/-/nz742dpkhqbi/0#

His blog: http://mathphile.blogspot.com/