alt Hacker NewsThis isn't all that significant to anyone who has done Calculus 2 and knows about Taylor's Series.
All this really says is that the Taylor's expansions of e^x and ln x are sufficient to express to express trig functions, which is trivially true from Euler's formula as long as you're in the complex domain.
Arithmetic operations follow from the fact that e^x and ln x are inverses, in particular that e^ln(x) = x.
Taylor's series seem a bit like magic when you first see them but then you get to Real Analysis and find out there are whole classes of functions that they can't express.
This paper is interesting but it's not revolutionary.
There is a huge number of people who understand Taylor series, know how to compute them, and the things you can do with Taylor (and other kinds of) expansions. Yet none of us identified that this binary operation spans that lot of them, but I'm willing to read references to predating observations of the same kind. The author does mention alternative systems (incomplete in some specific sense) in the paper.
I did however keep thinking there was a lot of attention to trying to include special constants even though we don't know that much about these constants yet, while comparatively little attention went to say elliptic integrals etc.
When aiming for a wide catch, you'd include some of those esoteric functions, or erf() etc...
I also wished they had attempted to find a representation for derivative and integrals.