I recently started studying advanced mathematics on my own, after taking two classes in college (discrete math and advanced mathematical logic) that spurred my interest in the topic. However, I am struggling a little bit on things that I assume others find easy. I have started a real analysis book (Rudin), and find some of the topics less intuitive and more (seemingly to my brain) arbitrary when compared to the stuff I learned in logic and discrete math.
I specifically feel like I have issues retaining things that I have proved, even if I understand the steps of the proof, because they have no clear intuition in my mind. To give a simple example, consider the following theorem for a field:
Consider $x,y,z \in F$
If $x+y=x+z$, then $y=z$.
The proof for this was fairly easy to me, but I feel like I don't have any intuition on how to remember this. Obviously, this statement is true for the rational numbers, which is a field. But how do I intuit that this is true for all fields? I understand the proof, but don't intuitively understand why its the case by necessity for any field. To me, a field just seems like an arbitrary construct with no real intuition behind it. I feel like I ultimately just have to memorize the result, because without reproving it, I would have no idea whether it was necessarily true or not.
Any suggestions?
Thanks!
I think you've picked an example that does lend itself rather poorly to intuition, so it's not shocking that you're having this problem. It's okay to think of this proof as being purely formal.
The reason is that the field axioms were chosen to be a minimal set of laws from which it is possible to derive the main laws of arithmetic. It is quite possible to understand intuitively why $x + y = x + z$ implies $y = z$ in the familiar field of rational numbers (and even more easily for positive rational numbers), but it's a different matter to pick a voluntarily small number of axioms and start deducing logical consequences from them.
The intuition you should have of a field is that once all of these logical deductions have been done, the familiar rules of arithmetic mostly hold for the four operations $+, -, \times, \div$.
You learn by focusing on the details of the logical development, and by experience with counterexamples, that certain properties true in $Q$ are not true in all fields. For example, $1 + 1 + 1 \ne 0$ need not hold in a field.