[Python-ideas] Why operators are useful

[Steven D'Aprano]

On Mon, Mar 18, 2019 at 05:51:08AM -0700, Rémi Lapeyre wrote:

> Maths’ typing is explicit so you don’t need to spend brain cycles to
> determine them.

Surely that depends on how formal you are being?

Maths can vary hugely in formality, even at a professional level. It is 
a terrible overgeneralisation to state that maths is always explicitly 
typed, unless your definition of mathematics is defined so narrowly as 
to exclude the majority of maths done in the world.

In my own personal experience, there is a lot of mathematics done using 
implicit typing. I've never seen anyone explicitly declare that the i, j 
or k loop variables in a sum or product is an element of ℤ, they just 
use them:

∞
∑ expression
i=0

Likewise it is very common to assume that n is an integer, x and y are 
Reals, and z is a Complex. Perhaps not in formal papers, but in less 
formal contexts, it is very common to assume specific convections used 
in the field rather than spell them out fully. For example:

https://en.wikipedia.org/wiki/Volume_of_an_n-ball

You might not give Wikipedia much credence, but I trust you won't 
object to John Baez and Terry Tao as examples of actual practicing 
mathematicians:

https://johncarlosbaez.wordpress.com/2019/03/15/algebraic-geometry/

https://terrytao.wordpress.com/2019/02/19/on-the-universality-of-the-incompressible-euler-equation-on-compact-manifolds-ii-non-rigidity-of-euler-flows/


Similarly, I've never seen anyone explicit declare the type of a 
variable used for a change in variable. Even if we've explicitly stated 
that x is a Real, we might write something like:

    let u = x^2 + 3x

in order to apply the chain rule, without explicitly stating 
that u is also a Real. Why would you need to?

Its not like mathematics has a compiler which can flag type errors. We 
declare types only when needed. The rest of the time, we can use 
convention, domain-knowledge or inference to determine types.


-- 
Steven