Mathematics, for me at least, is a symbolic language such that its words are abstract objects and its sentences are abstract concepts and relationships. Things like arithmetic, geometry, calculus and algebra are topics we discus in this language because its abstractness allows as to get more far and deep in these topics through generalization or abstraction as a main tool, and after this abstraction one sees that a lot of things are more connected in a deeper level.
Informally, I think the language of mathematics is the language of (structured-)sets and maps between them, and a lot of the fundamental ideas of math is about what maps tell us about these structures, like if two structures have an isomorphism between them then they are similar in a way, the easiest example is that two sets have the same number of elements if there is a bijective map between them, and this was probably discovered more than one time in history by those who discovered numbers, and humans subconsciously mostly compare quantities of things in this way by seeing if there is a 1-to-1 correspondence, then a number came as an equivalence class of things with the same quantity, this is an example of a kind of abstraction that all humans can think of, without the mathematical terminology of course.
Another easy generalization is the generalization of integers with addition, automorphisms of a set with composition to the more abstract group, one basically notes that both integers and automorphisms are sets with an operation defined for it's elements that satisfies some properties, so we make this more general structure a thing and call it a group.
So sets and maps or more generally these objects and arrows categorical way of thinking is a natural thing in humans and mathematics just formalizes it along with the method of abstraction or generalization in a symbolic language.
Based on this I think there can be books that introduce this view of math for common people or anybody through some elementary category theory, a lot of people think that mathematics is about computations and such a book if existed will show that math is a language that formalizes and emphasizes the categorical way of thinking that probably all humans use in some things like understanding numbers.
Best Answer
The book that comes nearest to what you seem to have in mind is probably
From the blurb: “Conceptual Mathematics is the first book to serve both as a skeleton key to mathematics for the general reader or beginning student and as an introduction to categories for computer scientists, logicians, physicists, linguists, etc ... The fundamental ideas are illuminated in an engaging way.” And that's a reasonably fair description.