ITT: Proofs

Proofbros, how did you master the art of the proof?

Can we get a thread where anons trained in the art of upper-echelon maths discuss their proof stories. How you learned them, easy or hard, importance of proofs in various disciplines, etc.

Other urls found in this thread:

people.vcu.edu/~rhammack/BookOfProof/)
twitter.com/SFWRedditImages

grad student here

just do a lot of them, that's how you get good at anything

>Read a book on proofs
>Read a book on set theory
>Read a book on math logic

/thread

The actual proof writing is pure common sense. You just explain why the result is true, using only what you know to be true and the rules of logic (with a constant care as a beginner to always honestly ask yourself "Am I spouting bullshit right now ? Am I trying to scam the reader ?").
Now finding the actual ideas and intermediate steps is the tricky part and there's no real thing to it. That being said, knowing your course material (and especially the proofs of the results) is invaluable. That's the first place you should look for inspiration.

Proof that: adding 1 to the negative false power of zero equals the root of true

>always use pictures to help people visualize

>The actual proof writing is pure common sense.
This.
If you're solving the exercises at the end of the chapter, the proofs are almost straightforward if you took the time to carefully study it and you know all the prerequisites.

care to recommend any?

You should start right away from:
Set Theory (Jech).

alright will look into it thanks m8

Book of Proof by Hammack (people.vcu.edu/~rhammack/BookOfProof/)
or
A Transition to Advanced Mathematics by Smith, Eggen, and St. Andre

Elements of Set Theory by Enderton
The Joy of Sets: Fundamentals of Contemporary Set Theory by Devlin

Introduction to Logic: and to the Methodology of Deductive Sciences (Dover Books) by Alfred Tarski
Introduction to Metamathematics by Kleene

Question...

For a STEM major - especially one majoring in Mathematics - when/ where does the transition from uh... problem-solving math (like algebra, trig, or calculus) to proof-based math occur? In your opinions, which one is more important? Useful? Which one is needed more in grad-school STEM? If I excelled at geometry, algebra, trig, and calculus, am I guaranteed to excel at proof-based math with the same amount of hardwork? Is there any guarantee? Please upper-tier math anons, give us some of your stories and your advice.

I just sum zero or multiply by one most of the time.

Not a math major but I'll give it a shot
>For a STEM major - especially one majoring in Mathematics - when/ where does the transition from uh... problem-solving math (like algebra, trig, or calculus) to proof-based math occur?
Right after calculus, usually during calulus even
>In your opinions, which one is more important?
For math, proofs by a fucking long shot. For engineering, science, etc. I'd say a mixture.
>Useful?
Absolutely inperative
>Which one is needed more in grad-school STEM?
Basically all of your undergrad math will revolve around proofs to some extent
>If I excelled at geometry, algebra, trig, and calculus, am I guaranteed to excel at proof-based math with the same amount of hardwork?
Not even close. I excelled at all of those subjects too and even had looked into some basic proof-based math before college and it still hit me like a truck.

As an addendum, proof-based math is much more interesting than problem solving, so it's a trade off

These books aren't very useful. I don't see how learning anything but the most elementary set theory will improve your ability to write proofs. The Kleene and Tarski books are both great in their own right, but only incidentally helpful for writing proofs.

You can learn to write proofs by just trying to write them, again and again, and asking for help when you feel stuck. Importantly, you should avoid looking up proofs done by others until you've really exhausted yourself trying to do it on your own.

You'll probably feel like a retard for the first fifty to a hundred or so proofs you write, but hey, we all went through it at some point, it's just plain required.

Learning how to do logic or set theory proofs is only marginally useful for teaching you basic deductive methods. I think such methods are better learned working with material you're already familiar with, trying to understand it from a new angle. I grade proofs, and a lot of them focus on proving all sorts of pedantic unnecessary shit, like the commutativity of integer addition, or whatever. Treating your proofs in subjects like analysis or geometry anything like proofs in logic or set theory will almost always make them frustrating for your grader/professor to read. Try working through a book on analysis or algebra, any introductory one will do. I recommend Fraleigh for algebra.

I am doing a joint honours of mathematics and physics, in the uk it sort of starts towards the end of the first year.

>These books aren't very useful. I don't see how learning anything but the most elementary set theory will improve your ability to write proofs. The Kleene and Tarski books are both great in their own right, but only incidentally helpful for writing proofs.

They help by giving you practice and mathematical maturity.

They're not introductory books, and the student of math would do better to learn how to rigorously derive the results with which she's already familiar, e.g. in analysis or algebra.

suppose you are to prove a certain theorem: first try to come up with a lot of examples to see that it really holds in certain cases. based on those observations, convince yourself that the theorem is true - this is the most important. then imagine you have to convince some other person, on an informal level: by drawing a picture or by expressing some common feature shared by the examples etc. usually the actual proof will be just a formalization of this one idea.

I treat proofs as the final step of the mathematical process.

It's cheating, but I never ever "prove" anything formally before I write a proof. I am haphazard, draw lots of pictures, try to wrap my head around the problem and look at the situation from all possible angles, especially the unnecessary and inefficient ones until after enough squeezing, puffing, strangling, and posturing... something nice falls in my lap. Once that nice thing falls in my lap, I then figure out how to derive it formally. What that really means is, "what are the standard terms again?" and "what is the cleverest way to show that this nice thing is implied by them?" After figuring that out, I write up a very neat, nice looking proof, concealing all the dirty work and making me look like a smarty pants to anyone who is unaware of how unruly the actual process was.

In all fairness, once in a while the formal result is obvious and I get a nice little buzz, but most of the time it's more like a wrestling match with myself.

proofs are like trickshot montages. they make the person look like a pro but what you don't see is the 400 other missed shots. that is to say, proofs are like anything else- not what they appear to be.

this too

>she's

What is this?

Math major here, I'll share my experience
>When does the transition occur?
For me, in my linear algebra course. Calc I-III was entirely computational.
>Which is more important?
In math, cs, or physics, I think the proof is generally more important. An understanding of the proof generally nets a much more deeper understanding of the subject than merely being able to use the proof's conclusion to solve problems.
>Useful?
For "deep understanding," see above. Otherwise, entirely subjective based on need.
>Which one is needed more in grad-school STEM?
STEM is vast and infinite. I gravitate towards saying proofs, though.
>Will I excel?
I was a god at computational mathematics since before I could remember, proofs still took a while to get used to. At first, I thought that writing a proof was entirely different than doing computational mathematics, but in time my old knowledge adapted itself (god, I fucking love being a human being) into a proficiency at writing proofs.

Autism

>why do you ask?

kek

There's no proofs, only persuasion.