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Why are mathematical proofs so hard?
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whyarehardmathematicalproofs
Problem
I am an electrical engineer and trying to make a transition into machine learning. I read in multiple articles that I have to learn data structures and algorithms, before this I have to learn about mathematical proofs. I started studying it on my own using the material available on MIT's OCW, while I did grasp the concepts of induction and well ordering etc..
I've been struggling with the exercises for a very long time and it's really frustrating. I can easily deal with any type of proofs that I saw before (e.g. once I saw the proof of a recurrence question I became pretty good at proving them). My problems start when I face an unusual question. I feel like I am memorizing the proofs rather than learn how to prove.
Is there any way (or any resources) that can improve my proving skills in a way that whenever I see an unusual question (like the checkers tiles and chess tiles type of questions) I don't have to stare at them for 2 hours before giving up?
I've been struggling with the exercises for a very long time and it's really frustrating. I can easily deal with any type of proofs that I saw before (e.g. once I saw the proof of a recurrence question I became pretty good at proving them). My problems start when I face an unusual question. I feel like I am memorizing the proofs rather than learn how to prove.
Is there any way (or any resources) that can improve my proving skills in a way that whenever I see an unusual question (like the checkers tiles and chess tiles type of questions) I don't have to stare at them for 2 hours before giving up?
Solution
I feel like i am memorizing the proofs rather than learn how to prove
You can't learn "how to prove". "Proving" is not a mechanical process, but rather a creative one where you have to invent a new technique to solve a given problem. A professional mathematician could spend their entire life attempting to prove a given statement and never succeed.
I can easily deal with any type of proofs that i saw before ( eg. once i saw the proof of a recurrence question i became pretty good at prooving them). My problems start when i face an unusual question.
That is normal. Any mathematics "proofs" course isn't designed to teach you how to take an arbitrary problem you've never seen before and be able to solve it (since nobody, not even the best mathematics professors can do that). Rather, your learning goals are
-
Learn how to "read" proofs and judge their correctness
-
Learn how to "write" down a proof in the right mathematical language
-
Learn about known proof "techniques" and how to apply them
If you are working on a new, unknown problem, it is normal that you might not be able to solve it. However, knowing and having memorized other proof techniques may help you. Often proofs involve combining a new idea with existing known proof techniques. The more, and the more varied the proofs you already know are, the better your chance of being able to solve the given problem.
You are on the right track. You should simply keep studying proof techniques. The exercises you are doing are good. Don't worry if you get stuck. As you get more experienced and your "toolbox" of techniques grows, you will be able to solve exercises that are less "alike" the previous ones you have seen.
You can't learn "how to prove". "Proving" is not a mechanical process, but rather a creative one where you have to invent a new technique to solve a given problem. A professional mathematician could spend their entire life attempting to prove a given statement and never succeed.
I can easily deal with any type of proofs that i saw before ( eg. once i saw the proof of a recurrence question i became pretty good at prooving them). My problems start when i face an unusual question.
That is normal. Any mathematics "proofs" course isn't designed to teach you how to take an arbitrary problem you've never seen before and be able to solve it (since nobody, not even the best mathematics professors can do that). Rather, your learning goals are
-
Learn how to "read" proofs and judge their correctness
-
Learn how to "write" down a proof in the right mathematical language
-
Learn about known proof "techniques" and how to apply them
If you are working on a new, unknown problem, it is normal that you might not be able to solve it. However, knowing and having memorized other proof techniques may help you. Often proofs involve combining a new idea with existing known proof techniques. The more, and the more varied the proofs you already know are, the better your chance of being able to solve the given problem.
You are on the right track. You should simply keep studying proof techniques. The exercises you are doing are good. Don't worry if you get stuck. As you get more experienced and your "toolbox" of techniques grows, you will be able to solve exercises that are less "alike" the previous ones you have seen.
Context
StackExchange Computer Science Q#128980, answer score: 42
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