Advice For Writing Math Proofs

[@AngelMedina-eh1rb]

Thomas Garrity said that people struggle with proof writing. That doesn’t mean that it’s impossible. We need to keep trying. I totally love the advice on looking back and reflecting on why it is correct.

[@andreshombriamate745]

Every time you write "obviously" imagine you are the professor and an student in front of you says "I don´t find it obvious".

[@mohamedmounir6770]

What really makes Math genius people distinguish from normal math people is proof writing and the construction of the proof itself.

[@rafaelfermin8925]

You are fantastic, a great advisor/counselor on the subject of mathematics. I will take your calculus courses, but before I am refreshing up on the pre calculus material that is a little rusty. I guess that it is never too late to learn, I am more than 70 years old, had taken some calculus at university in the seventies, but now after being retired, I am self studying math and physics with the intention to write a math book at a pre college level! Thanks for your enthusiasm and positive encouragement to the math students.

[@mitrikoudsi8060]

As a way of appreciation, I never skip any ads. Thank you very much, MS.

[@rengsn4655]

you are truly a sorcerer! i was just struggling through a proof. follow up question: how does one improve their creative thinking needed for some proofs, those key steps in a proof that are like "how does anyone come up with that?"

[@TheMathSorcerer]

Practice. Plus the ideas are reoccurring. The more proofs you do the more techniques you learn and you basically become more creative.

[@rengsn4655]

@@TheMathSorcerer thanks!

[@alfonsobustamante6937]

Really good advice, even more pertinent for proofs on research context, where checking stablished or avant-garde proofs and re-visit our own proofs could make a huge difference between an acepted article or a rejected one. My self has been the game changer to step forward into math research. Thanks again Math sourcerer ⭐️

[@DrLiangMath]

As a math professional, I would say your explanation is wonderful!

[@aomoussynonymous8712]

I also like analysing a proof's steps backwards to check for any errors or missed steps.

[@srikanthtupurani6316]

Few days back i have see a talk by H Furstenberg. He worked in ergodic theory and dynamical systems. He talked about steiner lehmus theorem. He told that when he was a kid he tried to prove that theorem and he struggled for three weeks to solve the problem. If he told everyone that he had proved the theorem in one hour people will believe him as he is a top mathematician. People will believe in whatever he claims. But we should appreciate he spoke truth. This quality makes some people extraordinary.

[@TheMathSorcerer]

nice

[@TheAsteroth]

I love your vids so much! They gave me so much strength and inspiration. Thank you so much oh mighty sorcerer!

[@Marco-kd7jk]

Man, i was thinking about this yesterday. You read my mind.

[@cbunix23]

"How To Read and Do Proofs" by Solow is very helpful for learning several proof techniques.

[@chet3030]

I agree. This book is very under-rated.

[@fordfactor]

I was always tempted to answer proof questions with "this is left as an exercise for the examiner", just to get one back.

[@hamzamohamed7935]

Please, more vedios about proves actually we need a series.... Thank for the great content and respect from Sudan Africa

[@Penrose707]

Thanks for another great one Sorc, hope you're well today

[@pauljarski7590]

American math students are crippled by their lack of training in writing proofs. They memorize and apply formulas, but as soon as they encounter a problem that's phrased in an unfamiliar way or that doesn't lend itself to a straightforward application of the formula, they're stuck! Proof writing is necessary to gain a wider view so that ALL the problems within a particular topic become accessible, not only the ones contrived so that the whole class can get an A.

[@vaxer.7749]

not only american students your words can be apply to all students around the world.

[@pauljarski7590]

​@@vaxer.7749 Not only American students, but ESPECIALLY American students. High school students in many other countries are expected to memorize precise definitions and use them to write correct proofs. I believe The Math Sorcerer has observed the same phenomenon, i.e. that foreign math undergrads are far more prepared, see kzbin.info/www/bejne/aYa0mH2vmqyobqc

[@Maths_3.1415]

@@vaxer.7749 true In India no one focuses on proofs

[@jaydenclowers2616]

Thank you I need this going further into Mathematics

[@xiaonguyen6693]

just writing "qed" at the end will improve it by 200%!

[@TheMathSorcerer]

Hahahhaah

[@homamthewise6941]

Nice vid as always thanks man

[@tethyn]

I hope the video gets more views because it is important for any self learning. A community to encourage, teach and learn from each other is not just sufficient but necessary (see what I did there😉). A safe space to post proofs with real criticism out of the spirit or helping and improving is part of the failure that helps everyone doing proofs understand the extent of the material. Discord, stack exchange or Reddit can be helpful when the community is supportive. I had to self learn mathematics as a physics grad student and not paying attention to proofs, which many physicist do, has the consequence of a shallow understanding of the math. As a former grad student in theoretical physics you find the work you do is only as good as the foundation; and exploring new areas of physics is limited extremely if your depth and limitations of the mathematics is not well understood. Proofs are not just part of the learning… it is what makes the learning possible… if not it would all be basic algebra. My proof writing is ehh… but hopeful people watching the videos can connect with others to form small pods of learning to allow the self learners to advance and seek the knowledge they are looking for. Great and very important video. I know you are a busy man but I how some can send their proofs to you to help them or redirect to someone who can (undergrad elective? Grad students… not that they do not have enough to do!). Good luck and take care to all.

[@martinhawrylkiewicz2025]

Thanks for making this great vid! To improve my proof writing I took your advice and got myself a free, wonderful PDF book titled The Book of Proof by Prof. Hammack. What a treasure!! It's full of great examples, so many different proof techniques, lots of math problems with worked out solutions. Highly recommended for anyone who wants to improve their proof game. Especially if you like to self-study! 🔥

[@peteaustin5018]

Hammack's book is legit awesome.

[@martinhawrylkiewicz2025]

@@peteaustin5018 I really like it how he explain things step by step with lots of visuals too. Another great book that Math Sorcerer recommended is called Transition to Advanced Mathematics by Chartrang and Zheng. Great proof book!

[@chet3030]

I'm working through this book myself. A professor from Northeastern University posted her lectures on this topic. She uses this book. Well worth checking out: kzbin.info/aero/PLpcwHaLYiaEUk-DVrPHjPpGmwL04WMbNQ

[@mustafa1037]

Please give some advice about some higher level mathematics that a high school student can learn by self study along with his high school maths... It is necessary.

[@itsyagirlmay]

Same

[@PhillipRhodes]

Any thoughts on using automated proof assistants / interactive theorem provers? I mean things like Coq, Isabelle, Agda, HOL, Lean, Idris, etc?

[@chet3030]

I was going to ask this same question. I'm curious about thoughts if something like Mizar, Isabelle, or Lean would be useful to a self-studier who doesn't have someone to check their logic.

[@LaithMMA]

"The proof of the Riemann hypothesis is trivial and left to the reader as an exercise" .

[@mirbooks]

Great advice... Thank you very much...

[@satyavivekanandbattula1091]

Thank you sir. Useful video

[@Maths_3.1415]

Hi bro

[@satyavivekanandbattula1091]

@@Maths_3.1415 hey.

[@vnever9078]

"You don't have to believe in God, but you should believe in The Book." ~ Paul Erdos Proof-writing is a skill which can be improved by: 1. writing lots of proofs and then verifying them-- reflecting on the steps; and, 2. going through good proofs -- Paul Erdos, a great mathematician of the previous century had this idea that God keeps the best proof for a theorem in a special document called "The Book". Most of the classic mathematics textbooks aren't so famous because of their contents/coverage, but because of the unique proofs they contain.

[@clearlycleric5987]

Hey,i am no smart when it comes to mathematics. My basics are horseshi* am the opposite of the fact that asians are smart. (M from India btw.) Since a kid i always wished to be like Bruce Wayne or Tomy stark. But with the time I had some mental health issues n addiction issues n i was behind everyone my age. From where should I start?

[@Maths_3.1415]

Strengthen your basics in mathematics and move to higher level

[@randoomly39484]

I wonder what is the difference between "knowing how a theorem is true" and "knowing the actual story behind this theorem" ?

[@MoguMogu818]

I can’t seem to get anything higher than a C- in any of my assignments, and it’s annoying me. I try my best only to end up not proving anything. I get feedback saying, you did good, but you didn’t prove anything. What the heck did I do then?

[@Iamblindanddeaf23]

Thank you !!

[@ghostek7792]

jokes on you, i hate discrete mathematics. in all seriousness although i really hated that class ( i had to take it for software engineering because it also deals with boolean algebra) hilariously enough after i started taking calc 3, i realized i was mystically able to understand what used to look like ancient tiki tiki text. sounds silly but just being able to read thing basic proofs or verify that i was actually understanding the concept, i realized that proofs were really just delevoped to have a stream lined system for communicating a mathematics principle or truth by mathematical means. it provides a structure because ultimately, just as all things man made, mathematics is purely arbitrary and we need a system to communicate these incredibly complex and abstract ideas through this system we've developed. what makes it difficult is there are two main parts of proof writing. 1) the concept itself, do you actually know what you're trying to describe, and understand the actual idea itself? 2) the writing, there is a developed 'alphabet' for communicating maths and you can understand the concept, but perhaps you need more practice on how to actually write it using the proofs system. think about it this way, if you're a spanish speaker, but don't speak english, you're capable of thinking in very complex thoughts as anyone else, but they're in spanish. if you want an english speaker to understand you, your idea is only going to be as good as how well you can communicate it, which might not be that good if you hardly know any english. in that sense treat proofs like another language.

[@Dr.Fatherland]

Thank you Isaac Newton.

[@meteor8076]

yes, but a proof can be written in multiple ways, only simple proofs have one straightforward way

[@eehsnapid]

Being a jee student I don't need to write proofs

[@Silentkidgaming]

Jee is about doing 30 questions in 1 hour. 😂

[@eehsnapid]

@@Silentkidgaming 25 questions in an hour

[@satyavivekanandbattula1091]

@@eehsnapid you're 12th grade class ? Jee 2023 or 2024 .

[@satyavivekanandbattula1091]

@@eehsnapid yeah . 20 MCQ and 10 integers.

[@satyavivekanandbattula1091]

@@eehsnapid Doing those 25 questions is not easy. Mixed concepts, questions are very tricky and little bit hard , some are very hard, some are okay .

[@ujwal90]

newton

[@ishaangoud3180]

Proof 🙏🏻

[@kapoioBCS]

Yea but what about the proofs you can not produce 😅😅

[@vnever9078]

First

[@Maths_3.1415]

Good

[@kafrikotroll8610]

You are the best 💖 I love all your videos 👍👍👍👍👍👍👍👍👍👍👍👍👍👍