For example, ordering a list efficiently takes algorithmic thinking, but not maths.
As a mathematician (and crappy programmer), I'd argue with this.
It doesn't take *arithmetic*, but it does take math. See, most of you will only know math as arithmetic but it's really a language and a philosophy. The language of mathematics is used to evaluate statements for logic - even in arithmetic.
Mathematics is so much more than just arithmetic and it's a darned shame how it's taught in schools today. They still teach it largely by rote without once ever mentioning the whole logic thing They don't even touch on the fact that it's a language.
1 + 1 = 2. We can evaluate that for logic and conclude that it's correct. (Well, mostly... Our friend Bertrand Russell might argue that it's just an arbitrary answer, having spent nearly 200 pages devoted to the subject.)
Algorithmic thinking is indeed mathematical thinking. In fact, we even have (in advanced math) symbology for it. You might recognize some, such as Boolean operands.
Just because you're not expressing it numerically doesn't mean it's not math. The following is math:
cats are to dogs as goldfish are to guppies (which can even be expressed with symbols like : and ||)
Just because there's no numbers involved, doesn't mean it's not math. It's a statement you can evaluate for logic. I encourage folks to skim the following two links:
en.wikipedia.org
https://en.wikipedia.org/wiki/List_of_mathematical_symbols_by_subject#Logic (see also group theory)
No, I'm not one of those weirdos that thinks math is everywhere and everything. The universe isn't math. Mathematics is just the language we use to make statements about the universe and then to evaluate those statements for logic (sometimes interpreted as truth). So, math isn't everything and everywhere - we can just use it to describe most everything and everywhere.