How to get better at math proofs?
I've read several books about proofs, and I'm rather good at them, but I'm wondering whether I could become even better by committing some general rules and heuristics to memory.
I already know the different proof methods - I need to learn how to speed up the process and be more effective.
pic unrelated
>>7951975
Your pic related is shit fyi.
If you're proving that two things are equal, then rely on the antisymmetry of an available order relation to get you through.
For example, to show that o(a)=n for an element of a group, say a, first show that o(a) <= n and then show o(a)>=n. Also, a much more common example is to show that two sets are equal by showing that they are mutual subsets.
If there is a statement about every element of an inductive set (not necessarily just the natural numbers,) then immediately try induction. If that doesn't work, then just see if you can prove it for a general element of the set.
If you're proving a bi conditional statement, say P<=>Q, I find it easier to prove P=>Q and ~P=>~Q in many cases instead of proving P=>Q and Q=>P. You're secretly proving the contrapositive of the converse of P=>Q when you do this.
>>7952021
Thanks, this helped.
>>7952021
>youre secretly proving the contrapositive
>secretly
op just follow an appropriately leveled textbook on a subject youre interested in learning and work through exercises and proofs given
>>7951975
Reading books about proofs is a colossal waste of time, honestly
You'll pick it up as you go along, implicitly or explicitly, just by reading and more importantly doing proof-based math.
It's something like learning a language. You can either study a big fat book of grammar and then try to awkwardly apply the rules as you do stuff, or you can start talking and eventually you'll learn to speak properly.
I guess some people like the first method but I much prefer the second.
tl;dr just get a textbook that looks interesting and start practicing with it
>>7951975
*ahem*
Logic is the one thing you can't shorthand. What that means is that you can't pattern yourself to be better at logic because logic is a pattern manager in the first place. Logical reasoning is the one skill you can't train your brain to perform, because the brain doesn't hold the state complexity necessary to perform it at all. The mind, which is emergent from the state representations present in the brain, is the thing that's capable of logic. TL;DR: You can't use logic without thinking. It can't be habitual.
Now, this is not to say you can't get better and faster at it. The problem is, if I don't say the above paragraph, you will go on to develop Schizophrenia in the name of getting better at proofs. I know this, because, logic: I love it. I wanted more of it, but it's a finite resources. So I had to resort to magical thinking, because if fiction was real then logic could be rearranged arbitrarily. This led me to fairy logic, which is nice, but not what you want. What you want is to do something other than what I did, which was to repattern my brain to possess the skillset of a Schizophrenic. You don't want to short your brain out trying to pattern what isn't a pattern. You don't want to spend the next 30 years trying to figure out if logic is finite or infinite without any way of noticing that that's the question your brain was asking itself this whole time. You need to use my experience negatively, as a pre-criticism to the art you're about to develop, and then you need to come back and tell me how to escape fairy logic.
My recommendation would be to read up on the various philosophers that have become popular as of late. By seeing how they process logic, your mind can integrate the reasoning behind their experiences and hopefully learning several different types of heuristics that'll help you give proofs faster and in a deeper number of formal systems. Basically, your job now is to learn to rephrase yourself without resorting to Schizoidism.
>>7952021
Rearranging proofs doesn't help at all, trust me. It gains no rigor and won't lead to new conclusions. I probably shouldn't have said that proofs can be rephrased; you should only need one proof per formalism. Proving it twice doesn't help, because it basically amounts to circular reasoning.
>>7954674
what
