The mathematics is a kind of 'formal science'.
I have no idea about the meaning of the word, 'formal',in this expression.
From my perspectives,the word,'formal',should stands for some qualities like 'nothing to do with the essence','a series of connections linked by logics',etc.But I can't find such an explanation in my dictionary.
I would not speak of mathematics as "formal science". Strictly speaking I would not call it science at all, although much of science uses mathematics.
Mathematics can be formal or informal. formal mathematics strictly follows rules of reasoning, indicating each step clearly. Proofs, for example, are normally done in a formal style. In that context "formal" means following he defined rules and procedures exactly.
The word "formal" has different senses in different contexts. One can speak of an event or occasion as formal if social rules and patters are more strictly followed than at an informal or casual event. Formal language more carefully adheres to correct grammatical forms, and uses standardized phrases in many cases. There are other senses of "formal" as well.
The 'formal' here need to be understood mathematically. (So are many other terms in a math book.) It means the 'math science' is about axioms and deductions using axioms and certain deduction rule(s), that is, no real-life means of gaining knowledge is taken into consideration when doing a math problem. That said, the nearest definition given by merriam-webster is:
relating to or involving the outward form, structure, relationships, or arrangement of elements rather than content.
As you can see in Hilbert's Foundations of Geometry, which is one of the most famous books written in the spirit of formalisation among mathematicians, every theorem is gained through rigorous logic deduction. Here, the method of drawing a picture to see something intuitively is not considered a valid proof[1]. However, methods like drawing a picture are widely used in other science or technology areas to help you get some result or prove something, which is one of the traits that differentiate 'formal' from 'not formal'.
Having some knowledge of formalism, mathematical logic and formal system may help you better understand the connotation of 'formal' in math. By the way, I feel like this question is more about philosophy of mathematics than English.
footnote:
[1] You may see there are some pictures in that book, but they are intended to help understanding rather than serve as a proof. In a research level math book, all pictures of geometric shapes, which are rare if not nonexist, can be removed without any reduce of the validity of the proofs there.
It would be helpful to see more of the context within the paragraph. It sounds like it may be getting used in the sense of some kind of synonym. If you have access to a Thesaurus, that would probably be helpful to you.
For example, the adjective formal can have a sense of scholarly or academic depending upon contextual usage. It can also have meaning systematic or authoritative.
