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I decided to ask this question after being attracted by how much one can do with the knowledge in computer science, including iOS application development, game(or mods) development, website creating, etc., but having trouble with matching what I learned with what I really understood.

Realizing the lack of knowledge in programming, I once decided to learn programming in a specific language, like Objective-C and Swift. After I tried every single textbook on OC programming, I found myself wasting time in the "explanation" part and getting lost with the real "theoretical" part. It seems to me that those textbooks prefer a long and loose explanation in describing a concept without telling me what exactly it is(Like the notion of "Object" and "Pointer") and skip the real important "thinking" before the implementation of the code. Although after many times of reading I could create a very simple application(like a simple calculator), I was still very lost in how it really works.

That reminds me how I began to learn mathematics. In middle school "function" is just "$y=f(x)$", where $f(x)$ can be $x^2$, $\sin(x)$, and other operations about $x$. However, now for me $f$ is a set of tuples satisfying the unique mapping property, or an arrow with domain and codomain. More generally, I can give a affirmative response to every question like "what is (the definition of) this?(in a math sentence.)" So I asked myself, "what is most basic among all subjects in computer science"?

Based on the belief that computer system and programming can be completely theoretical, I decided to learn from the very basic. That's when I began to have trouble because of the lack of knowledge in the computer science world. Where is the code for my programming environment?(For me, Xcode) and how can I understand it? What is a computer made of (up to the transistors), and how do them work together? But then I found out that there is no book with the name similar to "how a computer works" that reach my satisfaction, which I thought is my vocabulary problem.

I hope that in the book I'm looking for, every keyword is rigorously defined, and without any waste of time in explaining what a notion "means" or how it is developed stage by stage historically. I expect that it gives only the goals(like the adding machine) and how to build it from the given definitions(a graph with the nodes of value "0" or "1"). Every textbooks in mathematics(like the GTM series) reach this goal, and for me, that is MUCH easier to read and understand. I think most of the math student like me will prefer this style of learning.

What is a good (book) route for a math student to self study computer science from the basic? I wish there are some theoretical textbooks that build the whole operation system, compiler, or some of this kind, that is readable to math students. Any answer would be much appreciated. Thank you.

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    $\begingroup$ You should be able to program perfectly well without really understanding how transistors really work. Just like you can play tennis without knowing how to make a tennis racket and tennis balls. $\endgroup$ – Peter Shor Jan 2 at 0:44
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    $\begingroup$ Besides seconding Peter's comment, just a little note: you may not have meant it that way, but the sentence "That's when I began to have trouble because of the lack of knowledge in the computer science world." may not be the best phrasing... It implies the lack of knowledge is in the CS world itself, not in your understanding of it (which I assume is what you meant). $\endgroup$ – Clement C. Jan 3 at 4:58

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