I read through some of John Gabriel's the new calculus last night, he does seem quite arrogant but some of the things he says seems to make sense. For example is his way of taking derivatives based of parallel secant lines right? To me it just seems like rolls theorem. Is this guy correct with his new Calculus or can it be ignored?
There are interesting alternatives to the standard formulation of calculus; here I am thinking of Abraham Robinson's nonstandard analysis (see http://mathworld.wolfram.com/NonstandardAnalysis.html) and Errett Bishop's constructive approach (see e.g. https://books.google.dk/books/about/Foundations_of_constructive_analysis.html?id=Z7I-AAAAIAAJ&redir_esc=y). Robinson and Bishop were accomplished mathematicians.
However, anyone who makes the claim that
The New Calculus is not worth just one Abel prize, it is worth ten Abel prizes. Of course you will never know unless you study my work.
of his/her own work (cited from http://thenewcalculus.weebly.com) and writes that
Neither Cauchy sequences nor Dedekind cuts are valid constructions of real numbers.
is speaking the language of crackpots.