Sure, that speaks to the usefulness of mathematics and the way it's constructed: perhaps the API analogy is the way to go about explaining it to others.

However, I am mostly talking about people (mathematicians even) who do not understand that something as basic as 2+2=4 relies on our faith that: 1. there exists "one", and 2. there exists a successor to "one" (simplified, for interested readers, look for Peano axioms). The beauty of mathematics and the human mind are in that everything else flows from there!

Most of them would claim that this is universally true, whereas as a mathematician, one needs to understand the difference, and be open to, as you put it, playing with assumptions!

> However, I am mostly talking about people (mathematicians even) who do not understand that something as basic as 2+2=4 relies on our faith that: 1. there exists "one", and 2. there exists a successor to "one" (simplified, for interested readers, look for Peano axioms). The beauty of mathematics and the human mind are in that everything else flows from there!

I don't see any faith in here:

Math just says 'if something like 1 were to exist, and if successors were to exists (etc), then after a long chain of reasoning you could conclude that 2+2=4'.

The 'faith' perhaps comes in when you go from '2+2=4' to eg two hens in my coop and two more hens in my coop means that I have four hens in my coop.

(And that's not trivial! Not all things in the real world or even in math behave like that. For some not even as an approximation.)

You are getting caught up in what mathematics does say: and you are right about that.

I am contrasting that with what people think it says (i.e. 2+2 is 4), without understanding that there's an "if" in there. They take it for granted as indisputable truth, meaning that they have faith in those "ifs" being fulfilled.

You do need faith that the axioms you work with are consistent, unless you're working in one of the few provably consistent systems.

That's somewhat true. Though in practice, you can just accept it provisionally as an empirical observation.

(Math is just as applicable even if you have your doubts and are missing absolute certainty.)

Empirical observation is not considered much proof of abstract principles because of the measuring errors.

I.e. Newtonian mechanics is empiricaly observed to be true with a certain error margin during measurements. As you lower measurement errors, you start to notice things that are not consistent with it.

Newtonian physics was and still is very much applicable for many use cases, just like mathematics is.

You seem set on "defending" mathematics where nobody is attacking it.

My comment is not to discredit mathematics: it is the ultimate expression of the human mind, showing the limits of our comprehension. Limits that clearly show that we can start reasoning only if we accept a few things as a given (what I simplified to "faith").

It is beautiful and transcendent in its expression, and while people consider it the most complex of human expression, I consider it the simplest full expression, worked down to the smallest ambiguities our brain will let us have. It is a reflection of what our mind can understand and grasp, and we should accept it as such so we can both grow our understanding of the world, and test limits of our understanding.