Mathematical Proof Writing
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- Опубліковано 21 вер 2024
- My Courses: www.freemathvi... || I discuss math proof writing, the best ways to learn math proof writing, and book recommendations for learning to write mathematical proofs.
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Thank you:)
The cool thing about this man is that you can easily watch and enjoy his videos even if you’re really bad at math (like me); obviously if you are at a certain level you can enjoy his videos even more, but this channel is still the best math channel I’ve ever laid my eyes upon
Thanks Math Sorcerer.
I took a proofs class twice, flunking, then getting a B--.
I'm continuing to study proofs on my own, and I'm
much higher on the hurdle than before, so I just need to
continue, and I can see mathematics opening up for me
slowly but surely. It really is great!!!
I feel that the problem with proofs is that for many of us, we were thrown into proofs during linear algebra, rather than dedicating a year or a semester on the fundamentals of proofs of more familiar concepts to get the logic down.
I had a professor named James Lang at Valencia College who taught me how to write proofs in his Logic and Proof class. I still to this day structure my proofs the way he taught me. The book that we used in his class was Mathematical Reasoning - Writing and Proof.
Nothing like coming inside after a long day of work in 100+ degree weather and be meeted with a math sorcerer video. Great stuff man. I’m currently working through Jay Cummings book, going through a subsection a day and it has been helping me a lot. Really like the structure of his book too, makes things easy to follow and you get introduced to proofs immediately instead of having to learn sets and logic first. I’m actually on the sets section right now, and it is great.
AWESOME!!!!
This channel brought my love for maths again. I watch your videos daily with some maths proof writing in it🎉🎉🎉... Watching your content from South Africa.
Your abstract algebra and topology playlists have shaped my maths degree
Yes, it's recommended to learn set theory before taking your proof writing class. It makes your life 100 times easier.
Hammack is really good for self study because the answers are detailed, and sometimes exploratory, and every other problem is answered. I usually use the answers as "hints" by slowly revealing a part of the answer, then trying on my own to move forward, then looking at the next part of the answer if I get stuck, to see if I'm on the right track or to learn the next move. And sometimes I get on the right track, and can then finish the rest of the proof on my own without having looked at the entire answer until afterwards! That's why I think providing answers is important because those who really want to learn, won't overly depend on answers.
Off topic of math proofs I guess but: I got a Calculus book in the mail yesterday that is so easy. Its Calculus by Marvin L Bittinger. Its kind of like the Blitzer math books in the way it has easy to understand explanations and detailed explanations. I love that it has exercises in the margins next to the explanations. After an explanation, it will say go do exercise whatever and the exercises are right there in the margins next to the explanations. The back of the back has all the answers to the exercises in the margins and answers to the odd numbers of exercises at the end of each section. So there is a lot of exercises in it to do and you can check most of the solutions in the back. I think its probably either a textbook for a course in college level Calculus 1 or a high school course in calculus. Its not as challenging as the Stewart book. But its probably good for a beginner. Mine is the fourth edition. It was a book in college library because it has a old library cold holder in the back. National University Library San Deigo is tamped on the side and inside. So happy with this Caculus book. Its perfect for me.
I think a lot of people initially dislike proof writing because they had a bad experience in geometry in high school. Many of my friends hated geometry, and the teacher was awful. I wound up teaching myself, but i didn't just learn enough to pass a test. I enjoyed, "English for Math Students." So, I read more about proofs. This proved to be very helpful in college.
Learn to write. Learn to write proofs. You'll math a lot better!
(Yes, that was "math" used as a verb. 😊 )
Nah, I just hate it because my brain is stupid and slow and so the reward ratio is very low.
So thankful that I found your channel. I borrowed How to Prove It from my university's library and it's been a great help.
The transition to advanced math book is a gem! I found out about it recently and it's absolutely phenomenal, great to see you talk about it. Thanks for your fantastic videos.
I'm currently a first year graduate student in mathematics. Really struggling with extremely short course timing and professors who rush things.
I am usually very slow when I do things myself, so often times I have to look up solutions after I spend like hours on one problem. I've hated not being able to do many problems myself over time, but even few of those moments where I actually took a day or even more so to completely understand what actually went on with the problem, the logical structures, and the corresponding implications - and then wrote a very detailed proof with every step highlighted - those are moments of accomplishment to me, as I did exactly what I aspired as a child to do - to learn about something intricately.
Hence, the short timings and really oppressive deadlines has been a constant source of discouragement and self-deprecation. Thank you for your videos, as I feel inspired to keep moving forward. Good day, sir.
I love your videos.I am self-learning CS. Yours enthusiam for math is contagious. I have tranfered that enthudiasm to CS.
I can listen to you talk for hours
We experience years in Math classes doing computations. Doing proofs is completely different mindset from computations.
Thank you for this. This gave me back the confidence to re-learn again. I have my degree in Physics and am currently enrolled in grad school for my masters in math. I figured let me do a masters in math because the research I was doing as an undergrad in Physics was under the dept head of math at my school (my research is in fluid mechanics). The first class I took last semester was real analysis and we used Rudin's book as you mentioned and omg did I feel so out of place seeing none of my Physics knowledge or skills (even my math skills) could help and barely passed by the skin of my teeth. Now I am taking probability and its been 3 classes so far and was thinking my knowledge in statistical physics will help but it looks like its a no go again. Luckily there are resources like your videos and others and many books to help bring me up to speed and I feel more confident to tackle all of this again after feeling I am not good enough.
I've been loving your motivational shorts! You should create more!
Thank you! Will do!
I am a math major and a big fan of your content. Tomorrow I am having my first midterm of foundation of math ( proof writing) and I struggled to understand a the answer to a question. I thought of watching one of your vids about proofs and I considered reading a different book. 15 mins later I figured its the easiest question! Thanks for your input and keep up the good work 👏🏼 🙌🏼
Thank you so much for the book recommendations! I will complete Calculus III in December and as a transfer student to the University, my first required class is Intro to Proofs! I'm definitely going to purchase these books, because when you said you can be a rock star in Calculus, but the Proof level will make you want to give up (paraphrasing), that made me want to start studying now! My end goal is receiving my Doctorate in Mathematics Education, and I WILL do it; however, it's going to take a lot of hard work, dedication and studying, and I am up for the challenge. I've been subscribed to your channel for a while now and your videos are really helpful. Thank you again!
Thank you for this video. One of the best early reasons to learn math is to separate opinion from logic and deduction. We all could learn to spend more time thinking over the merits of our arguments (math or otherwise) before expressing ideas out loud.
Until you can assert your premises as true, you cannot claim to argue soundly. That epistemological tidbit can mean the difference twixt dialectic and debate.
Yayy! First one, here
In Australia the highest level of high school math covers basic proofs including proVe root 2 is irrational, inequality proofs often related to the AM GM inequality and “language of proofs” with the existential and universal quantifies as well as if, thEn statements. I hope this will serve me well when I start my math degree!
Which school are you going to?
I just finnished high school at Knox Grammar and will be attending UNSW to do Advanced Mathematics and Computer Science.@@mitchellhayman381
This question is a little peripheral to the discussion but could you make a video that talks about how mathematic research is accomplished? I can visualize how research is done in the physical sciences (hypotheses, lab experiments, data, etc.) but how do math grad students find a problem, develop research and write a dissertation on the subject?
im a 39 and i want to embark on a journey to get to the level you are talking in this video, i just want to thank you for the great public service you are doing with these videos.
Best thing I’ve found on Udemy were his courses. They go great with his book recommendations on YT
The Solow book has web lectures that go along with the book. Solow is a professor at Case western reserve university. If you go to his faculty page, there is a link in the middle of the page to the Wiley student companion site. There you can find lectures for all the chapters. There is also a solutions manual for SELECTED problems. It is NOT the instructors solution guide.
I studied Book of Proof by Hammack. Awesome for self study. I got it free as a PDF download and I printed it 2 chapters at a time at work , bounded the pages together with tacky glue, added a cover and that was my self study book till I finished it so I could print another 2 chapters ...
If you don’t mind, can I ask where did you get it? I really want to learn :
A statement in physics books that always used to set my teeth on edge: "We'll leave it for the reader to supply the details." It felt like being left on a cliff ledge, in the dark, with a storm in the offing. The day that I decided to bite back and work those pesky derivations through was the day I fell in love with physics. Math proofs, though.....hmmmmmm 😉
🩵🩵
Fermat did that, so yeah, I get that. 😊
Very nice stuff. Really appreciate your efforts. I am an electrical engineer pursuing PhD in control system. What i have figured out through my academic career is that a good mathematician is a boss in an electrical engg domain. Your videos are source of inspiration for me to learn mathematics. Make a video on self study book on optimization techniques. Regards
Love the vid! I'm using How to Prove it by Daniel Velleman (3rd edition) to teach myself proof writing based on your recommendation. I'm doing it every day for at least 1 hr and now I'm finishing chapter 4 on Relations. Wonderful book full of great, step by step detailed explanations. Some exercises are a bit tricky 💪
I'm now in my 5th year studying math and still it sometimes fells like hitting a screw with hammer because you couldn't find the right screwdriver. It works but you know that there must be a more elegent solution.
Yeah I know what you mean. It's crazy because you do get better. If you ever go to graduate school that's when you really start getting good. Good luck:)
Thank you, trying to get up to speed on doing some basic proofs, your recommended books have saved me! The How to Read and Do Proofs is a great place to start, it's a short book with simple advice and the author has a vid series on youtube for it too.
I do not want to sound arrogant but I have experienced jealousy from my former ''friends'' because I did a Bachelor's in Mathematics. Also their parents used to praise me.There is just something in mathematics which attracts energy from all people.
Lol. I was only in College Algebra, and a friend of my daughter's mom saw my math book and said, "Whose book is that!" Her husband pointed to me and said, "It's hers!" I'm like, WTF?
I think it's great that you got a Bachelors in mathematics.
@@sr2291 haha thats a fun story.
@@MathsScienceandHinduism People go into total meltdown when they find out you can do math. It's really fun to tell them and watch their heads explode.
@@sr2291I’d love to hear exactly which book was so massively impressive 😂
@@bimrebeats College Algebra. The class before PreCalculus.
I think the top of mathematics is being able to do proofs.
I read a research paper on strategy in proof writing when one of my students came in struggling with this. I realized I was still struggling with some of the strategy. I realized you have to be a little psychic on how it's going to turn out to make it a strong proof. So now I tell my students that; when solving your problems "think ahead and be a little psychic as to what's going to make the most elegant result."
I really feel that these types of 'talks' should be a necessary part of the first few classes in first year university, and maybe for ANY course that's taken at the college level. Although it's wasn't always the case, my experience was that, you'll spend most of the year trying to become adept at a particular subject, and then only near the end of the course, when you have all of the skills to do a problem (I'm talking about problems that are not directly spoken about in the course), do you get the reveal, a demonstration of how those tools are used. Prior to that moment, you might not have even grasped exactly what the tools are for and why they'll be useful.
Intuitively, it feels as if that part of the process, at least the explanation of it, could come earlier on. I don't know, it's just an intuitive feeling I have. A simple class named 'why you need this skill' might give a little bit of extra motivation. One of the classes I really did enjoy was Graph Theory because, early on we were given tasks that showed exactly what the tools are good for. I was like, "WOW... you can do that with this?...". I even found the answer to a line puzzle that I had struggled with as a child, and gave up, thinking it was impossible; turns out it WAS impossible, but I learned to prove it, from the theorems we were taught in that class. There were quite a few 'WOW' moments throughout that course.
In a University setting there should be room for the student own thinking and for him/her to be challenged. I know about the pressure with the grades. But this is not about the grades, it is about to learn how to think and how to be an independent thinker. To accomplish this, the student needs to feel the struggle and to learn how to overcome it. Part of this process is to think why you need to do the things you are doing. You need to reach some idea on your own, even a wrong one, that works temporary for you. Only then, you will appreciate the insight. In my case, after many issues, I learnt the hard way to appreciate this setting in University. It is what makes those places what they are, in fact.
I took an introduction to real analysis as my second class in uni, man, the struggle was worth every second. I love math!
Wow, you're way ahead then. Have you already done vector calculus? Linear algebra?
Thank you for recommending Book Of Proof. I'm considering perhaps ordering a hard copy.
Can you please make a t shirt or mug that says "...so I recommend you get some books"
haha great idea
I think a lot of us got our first proofs from a HS Geometry teacher who didn't really understand them to begin with. It took me to adulthood to really understand them.
Doing contruction proofs in Geometry is what finally turned-on my Math Lightbulb.
Apparently Einstein created his own proof for the Pythagorean theorem upon first learning of it at the age of 12. Some have natural talent!
I'm a CS student in paul sabatier University in toulouse France
We had discrete math as well , at first I liked the topic I also saw your videos about set theory and proofs that really helped me ! After that came more complicated topics such as countable / uncountable sets .. cantor diagonal , my professor was one who wrote REALLY quickly so many arguments and most of my friends in that class were at that level also !
It's like programming once getting the vibe it becomes sort of a language and no needs to think about what algorithm you used or anything.. the argument makes sense
I kind of had to think everything tell I took another class in proving logical statements with logical trees
Do you think AI will help in proofs and which AI do you think is the best at it right now?
Needed this for my linear algebra course
Thanks I was looking for a video on math proofs (I speak spanish but this will do)
I believe he has these videos in Spanish, too.
To be honest your videos rekindle my passion for mathematics
@4:19ish: teacher or professor=underrated. Hats off to all teacher and professor🎩👒🤠
There should be if there aren't already somewhere some class that goes over proof in exquisite detail. It's used in several classes, but the schools I went to didn't offer any proof classes. I had the most fun with it, while still knowing barely anything about it, in Discrete Math. I think PMI, which we find in Discrete Math, is used for quite a few proofs iirc.
Oh wow no proof classes crazy! Yeah PMI is key, it's good you at least had discrete math:)
@@TheMathSorcerer I had it twice since the UoH teacher didn't post my grade for some reason. I liked the Rosen book they used better than the one from APUS by Kholmer, Busby, & Ross. But at APUS they went over some proof writing in both Discrete Math & Real Analysis classes.
Sincerely love this man. What a great contribution he's making. Truly appreciate you.
Writing proofs is hard at first. I saw the first proofs in middle school (just before going to Uni) and that was just fine. But then in the first year at EPFL, in the LinAlg course, we had to come up with our own proofs of stuff. For the first few weeks it was just blank minds and blank stares, but then it was as if a switch had been flicked. Hence, in my view, at a beginner's level, the best way of learning how to prove stuff is trying to prove stuff. Once you've got that you can start working on improving the style and so on.
Thanks!
Dude I'm the exact same as you. I'm a CS major and I had a discrete math class and I have a proofs class right now!
Sorcerer I believe, if time and other constraints permit, you should make a UA-cam course on Proof writing.
Thank u so much for suggesting such books wallah i found what i really was looking for. Our analysis teacher is not that good so i really need to study proofs alone. u literally saved me teacher thank u so much again
Hi Math Sorcerer ! I'm a 3rd year econ major facing real analysis this semester. I suck at maths but I'm really determined to learn math. I've watched your other videos on proofs & real analysis. I don't understand mathematical induction & I feel lost. Coud you suggest some areas of math i should have good command over to start proof writing & then real analysis? Regards.
Professor Math Sorcerer, thank you for the video.
Bouncing off another guys comment , could you do a video on doing "pure" and/or "applied" mathematical research? I feel like there is a lot of resources on finding REUs , but not as many on finding a math background-appropriate original problem to work on (say as part of an undergraduate or masters thesis , or forming an independent REU type experience using funding from a research scholarship) .
How do people who serve as advisees for a math REU even go about finding topics that would be appropriate for a reasonable motivated undergrad to tackle, and how does that process differ from finding such projects for a master's student , or people in the initial years of their PHD.
Day 1 of asking for a MathSorcerer lifting video
LOL!!!
The Goldbach conjecture is true, every even number x is always the sum of 2 prime numbers because with every increase in value of x (always 2 integers more than the last) then all odd numbers below x/2 move one further away from x/2 and all above x/2 move one closer, so the odd numbers always pair with another odd number. So if one odd number a distance k below x/2 is a multiple of a Prime (Pn) then we can rule out it and the number a distance k above x/2 as being a prime pair. So by eliminating all multiples of P
This is driving me nuts: what is San Francisco built for?
There is no such thing as too many books!
There's loads of clumsy stuff out there. My favourite pet grievance is when - in a proof by the absurd (is it really called like this in english ? - the "absurd" assumption is made way too early.
Does the math wizard ever talk about Aristotelian/Classic Logic & Euclid’s Elements?
what's that joint button for? have you tried fresh Cilantro, Mint, Lime, and Jasmine rice fried in a pan over an electric stove?
I have a question, I want to teach in france because i will be a master student with an scholarship for a year sptember 2024 to sept 2025, i come from a Peru( spanish country) should I start publishing videos in youtube doing lessons of advanced, intermediate or basic math (obviously i'll do it in french) with which level of math should I start, or is better if I start just studying what I'll be studying in the master.
Please plane trigonometry and cordinate geometry by sl loney
Hey love your channel and may I ask a question:
If in set theory, I can create a relation which takes a set of elements which are propositions (like set a is a subset of set b) and map it to a set of elements containing “true” and “false”, then why is it said that set theory itself can’t make truth valuations?
I ask this because somebody told me recently that “set theory cannot make truth valuations” Is this because I cannot do what I say above? Or because truth valuations happen via deductive systems and deductive system is technically not part of say, the first order set theory ?
Can someone please post the link to the set theory and other playlists he talked about? There are 10s of playlists with just one or two videos and I couldn't find it myself. Thank you
Here is the set theory playlist:) ua-cam.com/play/PLO1y6V1SXjjOYeXXWeIXQrblyHOf1Fu3W.html
Here is a playlist with easy proofs for beginners: ua-cam.com/play/PLO1y6V1SXjjNQg75KAdGNW9cD6zUul4Gs.html
Thank you!! I'm a student who struggles with the abstraction of measure theory. Do you have any advice?
The language in The solow book i found to be very confusing, the vellerman book is much better with its self contained chapters on logic and clear exposition.
I have a question. I email you regarding to research guidance. Plz answer it.
I currently have a copy of the book Discrete mathematics by susanna epp and before I commit to it I was wanting to know if this is a sufficient book for learning Proof writing?
What’s the best time to start learning how to write proofs ? I’m a beginner and right now i styduing algebra, so what you think ? Should i delay proofs studying to calculus or something or should i start now ?
I’m doing proof now 😫
How rigorous is zhangs book on a transition to advanced mathematics
If I take your courses, will you give me feedback?
"I have a problem I have too many books"
Bibliotaph!
*I* don't have this problem. ... well, only a little.
Greetings professor, can you please tell me the best books to study C++ from? From a very basic level to advanced.
I would be grateful if you replied professor 😊
My favorite is Programming Principles and Practice by the creator.
Which is larger: the set of all possible proofs, or the set of all possible logics?🐒📚
Second here, but haven't watched the video yet.
You have to prove the answers so it doesnt matter if you have the answers.
How to proof neN ~ 0
I have so many books on proof writing and math in general, but i havent even read more than 3 pages from any of those 30+ books. I know how useful books are but when I open one its so hard for me to focus and I get so overwhelmed knowing that each book has 300+ pages of material I need to learn. Thats the main reason why I am always so behind in classes. Any tips on how to improve that?
Take it page by page. And set goals for yourself. 10 pages per day for example. At that rate you'll finish the book in one month. You have to start somewhere, 10 pages isn't so bad, right? ;)
@@devon9374 Ya 10 pages isnt bad but i am taking 5 univeristy classes so thats 50 pages a day. On top of that classes move fast so I have to do about 25 pages a day for each classes so its just gets very overwhelming
@@successorof68 I understand, I went to uni for engineering and understand completely. In your case, you have to do what you can for survivalal lol, when your down with uni you and go back and learn it in detail. But in uni, you'll capture the key points of the topics. However, I do know classmates that never went to class, just read the textbook and showed up for tests.
@@devon9374 ya i m just having a very hard to i guess managing my time. I only have 2 more semester left counting this one so its getting really tough.
@@successorof68 Hang in there, you'll finish no problem. I used to wake up at midnight to work on a hw assignment for one class so I could focus. It used to take me at least 6-8 hours
Hi
could you please challenge us with writing a proof. and anyone who wants you to grade them will send you money. ❤😊
Did you ever do a proof review or did you ever ask another expert to review your proof?
yes of course! multiple times:)
Peace be upon you Sir
I have a bachelor degree in Mathematics and I have a certificate in IELTS with score 6 OVERALL, in the beginning I wanted to study master degree in mathematics but I hate it now and I am really interested in aerospace engineering , I could study it in master or I need to study bachelor again ?
Here’s a proof by contradiction:
You’re not a master because there are proofs that have yet to be completed 😂
abraham lincoln and euclid over law school
five dah-lah
i think I'm being censored at the library
Only bla bla bla nothing about proof
I call you you professor to look into the proofs of Islam..
I promise, if you look into them without biased perspective, I mean with open mind and heart -->> you would realize how shining is the truthfulness in Islam as the only religion revealed by the Creator.
Love this channel