Question

Who invented Order of Operations?

Answer 1

Algebra is not invented. It can only be discovered. So there is no 'inventor'. This means, nobody can invent(!) an other way for order of operations. Mathematics is like the nature. You look at it, and you try to understand it. You develop new 'tools' (limit, derivation etc.) to understand it better.

Answer 2

Nobody. It's a bit like asking who invented punctuation? or who invented spelling?

The order of operations in a set of conventions that developed as people tried to use fewer technical terms and replaced them with grouping symbols and the tried to use fewer grouping symbols.

Instead of always writing either `4 xx (5 + 6)` or `(4 xx 5) + 6`
a convention was adopted to read `4 xx 5 + 6` as the second grouping.

It was not adopted at a big meeting or by voting or anything, it just happened that some people did this and it caught on.

Here's a Wikipedia link: (not my source, but it's ok)
http://en.wikipedia.org/wiki/Order_of_operations#Mnemonics