I just got through reading the first chapter of principles of mathematical analysis by Walter Rudin, the first chapter goes on and on about Dedekind cuts and then starts defining properties of them, talking about how there closed under certain mathematical operations and such, I just don't understand how defining the notion of a Dedkind cut is at all useful or interesting, and when I say useful I don't mean practical. I mean in the sense that it could contribute to other areas of mathematics, not including itself, I kind of had the same experience when I started studying parts of linear algebra, I know that notation and rigour is important but some of the parts seem like over kill in terms of rigor and the results don't seem very meaningful to me. I have only read the first chapter and don't know if I should continue, I would appreciate any advice.
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If you don't like Dedekind cuts, which of course you are entitled to, what do you mean when you write $\sqrt2$?
Regarding the book, you don't say why you are reading it, so it is hard to give you advice on what to do. A very natural question for anyone interested in math is whether the objects exist, and how. Going back to my question above, you could say "$\sqrt2$ is a number that when squared gives 2". Very nice, but how do you know such an object exists? Defining an object by one of its properties doesn't make it to exist. For instance, I can say "let $m$ be the real number that is bigger than all other real numbers", or "let $s$ be a number such that $1/s=0$".
The point of that chapter 1 is to show that the real numbers can actually be constructed out of the rationals in such a way that we get a field with the properties that we usually expect from the reals. After all, if the reals didn't exist, then Calculus would be just hot air.