This seems to be meant in a sense that would be used in Theoretical Computer Science. That is, even if your book is written very rigorously in a fashion so no professional mathematician has anything bad to say about it; there is an additional level of formalism if you want to actually implement it on a computer (like your quote says) or if you want to argue rigorously about the formulae in the book - i.e., kind of a meta-formalism.

As a simple example from Theoretical Computer Science, you can write the rules for the Lambda Calculus (or SEP) in a way that is relatively easy to pick up by students in a CS 101 class; they would then possibly be able to write expressions in that calculus, evaluate them correctly, and maybe even argue about some easy theorems and proofs in that language.

But then you would maybe want to write down a different version of the lambda calculus that is extremely formal and rigorous, compared to the CS 101 one. I.e. every single statement would be grounded in some previous statements you have given, and all of them down to some tiny set of small axioms. The "descriptive" text would itself look like just a complete bunch of mathematical statements with a lot of set theory, 1st order predicate logic etc.

With the latter version, you might overwhelm a new student, but somebody implementing it in a computer, or somebody trying to do formal correctness proofs (in the sense that they are not just words that can be understood by a human, but actual mathematical formulae in a meta-language that describes the language of the Lambda Calculus; and in the sense that these correctness proofs can be evaluated by a computer program at compile- or run-time).

TLDR: A "user" of the book might not care about every single detail being super "formal", and might be able to use the information in there just fine. Someone else who wants to maybe do deeper research into the topic, or wants to build a whole new formalism on top of whatever it is that is described in your example, might be glad to have an even more rigorous presentation.

This seems to be meant in a sense that would be used in Theoretical Computer Science (which itself can be closer to maths than computers, depending on who talks about it). That is, even if your book is written very rigorously in a fashion so no professional mathematician has anything bad to say about it; there is an additional level of formalism if you want to actually implement it on a computer (like your quote says) or if you want to argue rigorously about the formulae in the book - i.e., kind of a meta-formalism.

As a simple example from Theoretical Computer Science, you can write the rules for the Lambda Calculus (or SEP) in a way that is relatively easy to pick up by students in a CS 101 class; they would then possibly be able to write expressions in that calculus, evaluate them correctly, and maybe even argue about some easy theorems and proofs in that language.

But then you would maybe want to write down a different version of the lambda calculus that is extremely formal and rigorous, compared to the CS 101 one. I.e. every single statement would be grounded in some previous statements you have given, and all of them down to some tiny set of small axioms. The "descriptive" text would itself look like just a complete bunch of mathematical statements with a lot of set theory, 1st order predicate logic etc.

With the latter version, you might overwhelm a new student, but somebody implementing it in a computer, or somebody trying to do formal correctness proofs (in the sense that they are not just words that can be understood by a human, but actual mathematical formulae in a meta-language that describes the language of the Lambda Calculus; and in the sense that these correctness proofs can be evaluated by a computer program at compile- or run-time).

TLDR: A "user" of the book might not care about every single detail being super "formal", and might be able to use the information in there just fine. Someone else who wants to maybe do deeper research into the topic, or wants to build a whole new formalism on top of whatever it is that is described in your example, might be glad to have an even more rigorous presentation.