This Will Make You Better at Mathematics FOREVER

[@whitb62]

I came to mathematics from a background in philosophy and math never really “clicked” for me until I hit proof writing. It felt very similar, proving tight logical arguments in the apriori world of abstraction. Proof writing really should be taught more in K12. Once you fall in love with the logical flow, it’s a game changer.

[@TheMathSorcerer]

Totally agree, proof writing is super cool.

[@LBT-sy4dp]

Man, i'm a former lawyer and i love doing self study math and financie, it help me a Lot as a profesional!!!

[@aravartomian1]

In most geometry classes in high school you do proofs. At least I had to do proofs in the class. I don't know how the curriculum has changed since I was in high school.

[@whitb62]

@@aravartomian1 Yea, why I said "more." Geometry proofs are really the only types you see and usually it's not that many or rushed. And maybe some mathematical induction stuff in a precalculus class but usually not. I think a logic course with formal proofs would be invaluable in k12. The only downside is it does feel pretty abstract and getting kids interested would be a huge challenge. Although the applications are enormous nowadays with computation and coding. But just the impact it could have in careful/clear/precise thinking/reasoning would be worth it alone.

[@whitb62]

@@LBT-sy4dp That's awesome. I had a calculus professor who told us, while working through the few proofs we covered, that if we could follow them, we should consider becoming either a math major or a lawyer. That comment always stuck with me and fascinated me. I've always loved logic the most as a field of study because it truly reveals the interconnectedness of all the disciplines of human understanding.

[@duncanbleak3819]

The rigor, attention to detail, patience and humility that is required to handle theorems and proofs can, potentially, make one better at "everything" difficult 'forever'. I tell myself this, as I recall struggling through Linear Algebra proofs decades ago in college.

[@whitb62]

I’m not originally a math person, but have gotten heavily into it recently mostly due to watching things like this channel. I’m getting deeper into proof writing and I truly feel my understanding of math has advanced significantly in a relatively short time. It really might be the key to understanding what mathematics really is, what makes it so unique as a discipline. Thanks for all the work you put into these videos, they’re invaluable.

[@basiliolopez9223]

My favorite math class was titled Foundations of Higher Mathematics and the professor used "The Book of Proof" as the course textbook. It's a great book that I will always remember because it helped me get through all of the proofs I had to do as an undergrad in Computer Science. Thank you for sharing these books and the rest of your videos!

[@TheMathSorcerer]

Oh wow that's awesome that they used that book!

[@Chalisque]

How to Prove It was the textbook for my first year undergrad course. Love this book. The chapter on infinite sets had me captivated.

[@TheMathSorcerer]

That's so cool that you used it for an actual course in college.

[@rafayalahmed6012]

​@@TheMathSorcerer I had this book as the main text in my discrete math class, taught by Professor Velleman himself. He's maybe an even better teacher in person, he didn't really follow the book too closely, he taught in his own way independent from the book, and after 15 years I remember and use almost all of what I learned regularly.

[@CaryInVictoria]

I'm an octogenarian who has kept one book from my university days: Principles of Mathematical Analysis (2nd. Ed.) by Walter Rudin, which among mathematicians is referred to as "Rudin". I understand that is still being used (3rd. Ed., around 1976) in some undergraduate analysis courses. It made me think more deeply than I thought possible, instilling the elegance and beauty of mathematical proofs.

[@snood101]

Velleman's book is a gem; the book is straight forward and has a lot of great examples

[@bayazid314]

They don't even talk about these books where I study higher mathematics 😅 Thank You for reviewing these books.

[@MaxPicAxe]

This is great because i started a couple of years a pure maths course where every homework question and exam question is rigorous proofs, but we never got any explanation of anything to do with logic or quantifiers or proofs, and so it came as such a shock. I'm sure if we had a dedicated module covering all of this, the rest of our course would have been so much easier.

[@hanskywalker1246]

What a coincedence! Just started reading the How to proof it book and im exactly at the page you opened

[@UnrealEntity]

"Mathematics is a behavioral science, I have never seen math happening by itself, there is always a human around" Richard Bandler

[@xray788]

3:00 that is so me, I always smell nostalgic books, takes me back in time.

[@arctic215]

me too!

[@LUCTIANITO]

the first times I loved math proofs was with geometrical demonstration of pitagora´s theorem and with the demonstration of sin^2(a) + cos^2(a) = 1

[@martinhawrylkiewicz2025]

Beautiful book review! Last year I studied Book of Proof which I found to be pretty good actually based on your recommendation. This year I am devoting my time and energy to studying How to Prove it by Velleman (3rd edition). I really like how Velleman shows you his motivation and thought process in the scratch work before every proof. Really like his in-depth discussions and challenging problems.

[@TheMathSorcerer]

Yes exactly! That is what I love about Velleman too!

[@argonwheatbelly637]

I have the two books on either end, but I'm missing the middle book. Math is a language, and Proof Writing is English for Math Students. I'm tutoring a few students in Geometry, who thought that it would only be about shapes, and finding area and volume. The difficult part is teaching them proofs. They're first learning how to think logically, and also learning about the basics of rigor. It's extremely challenging, because they're still learning how to write well in English and support their topics in essays. It's worth it.

[@Nidhsa]

I'm going to read the book of proof for my first proofs based class starting in January! I am planning on reading part 1 over the break

[@t0xic_g4s]

I bought some time ago the third edition of How to Prove it and I was amazed at how helpful it would have been if I had a book like that when I was first learning proof writing. In my EE&CS degree we didn't even have a class to learn proof writing, it was just glossed over in a discrete math course (which was optional!). If there wasn't the common course for grads/undergrads that I took back in the day, I wouldn't even have the chance to follow my dream. Thank you math sorcerer for covering such interesting books, some of us would not be able to navigate in the convoluted mess of academia without you 😂.

[@claws61821]

That's by design. The majority of academic administrators and coursewriters either don't want you to succeed at whatever your motivation to take the course is or at the course itself or truly don't care whether you succeed. That's also true of a minority of educators at all levels. The reasons and intents behind that malice and apathy are greatly varied, including but not limited to avarice and true malice and benevolent malice all in the general sense, but it is overwhelmingly the state of the field.

[@MaxPicAxe]

The great thing about it is that everyone can learn this stuff. Picking up some quantifiers and proof strategies is arguable 10 times easier than a lot of the mathematics covered in school, and yet, with this understanding, you realize you have never truly studied mathematics until now.

[@bradleybeauclair8282]

THIS IS ALL TRUE. I took Discrete math 12 years ago and it was so much fun. It was all that math that was easy, but I had always wondered about. It should of been my introduction to proofs but I did this Geometry w/ proofs book first and I did it line by line. I read 100% of it. I don't know the name of the book and I don't have it anymore. Cliffs notes word problems was super helpful and I recommend before pre-calc and trig.

[@ddgyt50]

I very recently did my first proof, or at least I think I did. during the process of learning how to do it I ran across the Book of Proofs along with a few other documents for download. Book of Proofs is a little daunting but I'm going to try to wade through it over time in my old age

[@OmarThomas-cq3jn]

The mats sorcerer sounds like Jason from maths an science....

[@gmcenroe]

I like Daniel Velleman's book. I have seen several other proof books reviewed here but I find Velleman's book to be very readable and easy to follow.

[@dilbyjones]

Been a while since I’ve heard “publish or perish” good stuff.

[@NilodeRoock]

I like the book " Proof Patterns " which is an actual toolkit to do proofs.

[@jennifertate4397]

Book of Proof is excellent!!! And has usually detailed answers for ALL odd numbered problems!!! I'm self-studying.

[@julienarpin5745]

I was watching your reviews of How to Prove It last night wondering if you still cared about this book. Thank you! I'm finally at the end of it.

[@TheMathSorcerer]

It’s a great book!

[@jimbobago]

I haven’t found the quote that’s mentioned at 3:52 in Solow but it’s on page two of the sixth edition of “A First Course in Abstract Algebra” by John B. Fraleigh.

[@TheMathSorcerer]

That's were I saw it!! YES!! How funny that you also remember lol.

[@jimbobago]

@ It just so happens that Fraleigh was my the text when my son took it and I found it in our basement. I decided to look through it shortly after seeing your video.

[@jimbobago]

@@TheMathSorcerer What is the probability (and as someone with a B.A. in math, I don't use the phrase lightly) that (A) My son would take Abstract Algebra as a non-math major, (B) He kept the book, (C) I watched your video, (D) I found the book, (E) I looked in the book, (F) I found the statement on page 2?

[@jimbobago]

I've only started "Book of Proof" online so I can't say much about it. Honestly, I don't get the adoration for Velleman. There's a lot of math I've studied that I can't even see how you could apply anything from the book to help you do the proofs. It's been a while since I went through it so maybe another look will change my thinking. Solow, for me, is THE proof book. I like how it explains all the basic types of proof techniques and does it in plain English without lots of symbols. And the techniques could be used all the way down to high school geometry. Also, every book in this video has updated editions.

[@khaled_ibnali]

you always delivered with books recommendations

[@bakhyou0012]

Thank you, Sir

[@UnrealEntity]

"this one is free, I bought it on amazon" 🤣🤣🤣 just having a bit of fun, despite joking a bit I love your videos

[@mcpuffinmuffin]

Have you had a look through the Beast Academy books or the follow-up series Art of Problem Solving? If so, what are your thoughts?

[@robfielding8566]

every mathematician being great at a computer theorem prover will be a good thing.

[@joshuagonzalez3880]

I read solow a lot of times and even though I got some good ideas from it, it was until I read velleman that I truly undersstood logic and proof structure.

[@lonelycubicle]

Hey, in your video “learn math start to finish” you recommended 2 other books for proofs (proof books is second set of recommendations.) Why the difference? If no difference, please clarify which proof book is best to learn from?

[@Ronenlahat]

What about Proofs by Jay Cummings? Is it good?

[@TheMathSorcerer]

Yes I have that book, it's awesome!

[@arctic215]

i always do the little whiff before reading a book and it's awesome that you do it as well!

[@japethspeaketh7034]

I wished I had known about these books back in my 1st year undergrad.

[@avosdelhevia-y7f]

While doing integration, we use LIATE or ILATE. But i dont understand why it makes sense, like what is the logic behind this rule? Hope u make a video on this.

[@jeycee1617]

Hi, The Math Sorcerer! I have this textbook called " Alan Agresti, Statistical Methods for the Social Sciences (5th edition). New Jersey, Prentice-Hall. " and it is for my Sociology Statistics Course. I was wondering if you by any chance have the book and would be willing to talk about it in a video, as I am kind of understanding the material and I am passing my class. However, I will be using the book again for more advanced statistics in the next semesters, so it has been a struggle to get all the necessary info for exams. Thank you!

[@terry2346]

Not one of these books in my Library!! Found the Book of Proof PDF!

[@richardcasey4439]

I have the book How to Prove It. Excellent text

[@sitrakaforler8696]

Timestamps (Powered by Merlin AI) 00:05 - Learning proof writing revolutionizes your understanding of mathematics. 02:01 - Learning proof writing is essential for mastering mathematics. 03:38 - Understanding mathematical definitions aids effective communication in proofs. 05:24 - An excellent free math book by Richard Hammock enhances learning in proofs. 07:13 - A comprehensive book on mathematics essential for undergraduates. 09:07 - A highly praised math book enhances understanding through excellent explanations. 11:02 - Understanding vacuous truths enhances mathematical reasoning. 12:51 - Understanding vacuously true statements enhances mathematical reasoning.

[@kevinbuenoartbywave]

I've been constructing a family Library for the last 8 years its full of subjects from Music, to Law, religion, and Maths and Physics; I'm currently enjoying Music, have a rough idea for my library and am content with its overall shape, I have 6 months learning guitar and I already know that it's not going to last forever eventually I will be content with my overall skills at that moment I know I will want to try my hand at Maths; I'm 28 years old if I play guitar for 10 years I know I will be 38, I'll travel for 2 years, at 40 I'm going to sit down and revisit Euclid and probably start with Set theory and discrete mathematics before I start algebra 1 and 2. My question is am I stupid to consider a math journey at 40? Is 40 too old to be a beginner?

[@danilo666]

Another great one book is "Mathematical Proofs A Transition to Advanced Mathematics"

[@EricWei-bi1np]

Hello Math Sorcerer! I just finished calc 3 and am hungry for more math! I really want to start self-studying differential equations. Do you have any recommendations for a diff eq's book? This is what I want for Christmas!

[@MBSall]

Hi math sorcerer I love watching your videos Keep on reviewing books Do you have any idea about a book on solving PDEs with numerical methods especially in fluid dynamics?

[@Foxtrot77777]

I purchased Velleman’s “How To Prove It” after you recommended it in a previous video. I expect it to be delivered later today. I look forward to learning from it! ❤

[@dimitrijestankovic6199]

I just found third edition Book of Proof from Richard Hammack online for free.Thank you for informations.

[@user-pu5qd1xz7e]

Would these books help with me doing A-Levels Maths and Further Maths?

[@kogenruns]

I never learned how to write proofs in high school and I've been struggling in university math classes because of it (many kids at my college went to private schools where they took math past calculus, but I went to public school). Thank you for these suggestions! I knew that I needed to get a better mathematical foundation if I am to do well in these hard classes, but I had no idea where to start

[@franciscovictor8187]

Hey math sorcerer. Can you please review Calculus on Manifolds by Spivak?

[@franciscovictor8187]

I hear a lot of people saying Micheal Spivak was a very good writer, mentioning his Calculus book. I wanted to know if Calculus on Manifolds is also good.

[@TheMathSorcerer]

It's excellent!

[@arijitmoitra1018]

There is a Calculus book available for free on MIT open courseware called Calculus by Gilbert Strang which I have been following for a long time. It begins with basic ideas and ends with vector calculus. How does it compare to other calculus text books ?

[@khaled_ibnali]

what do you think about The Story of Proof: Logic and the History of Mathematics by John Stillwell

[@Don_Gorgan]

Can you recommend math books for adult beginners learners.

[@waltercoady-jz2rf]

I remember years ago I took an abstract algebra course in university. The majority of the class struggled big time because nobody knew how to write proofs. We never had a proof writing course at our university. Is this a common thing?

[@paperart3224]

In order to fully understand a topic in mathematics. Is it enough to make our own examples and problems on that topic and then work on it. Instead of doing the exercises in the book.?

[@mariamtamer6567]

I want to know when to start learning proof writing, i am in grade 10 Egyptian system and i am relatively good at maths, i am starting my journey in self studying math and i got a discreet maths book, so now i wanna know should i start with proofs before i even start my journey or when exactly?

[@TheMathSorcerer]

Start now:)

[@АлексейСтафеев-р5б]

Are there one of these books on Amazon, but in Russian?

[@cdververfv3-d3g]

Hello, me super dumb in math, should i learn this first, me know basic arithemetic, should i learn the proof thingy?

[@anotherway917]

I am just starting this book (How to prove it) and even when I am just going through the introduction I feel like an idiot, although I do think I kind of understand it eventually. I have not really touched maths for a while and my level is around algebra. I wasn’t that bad at maths before when I took the public exam 10 years ago for university but made other career choices, but ultimately see the need of maths and feel like I need to catch up. Is it possible for someone like me to understand this book?? It is almost 400 pages and I only got through 15 pages in 4-5 hours (and only did the exercises with answers). I am scared!!

[@DANKUMGYE]

Had to watch this🙌

[@TheMathSorcerer]

Thank you!!

[@dennisalbert6115]

any book you can suggest on computer science compiler construction task-based theorem and can it be used to create proofs?

[@surrealistidealist]

Can proof-based math be taught to high school students or even younger students? It seems like at least some level of it should be taught to kids as early as possible.

[@argonwheatbelly637]

Geometry -- It's not only English for math students, but when taught properly, it introduces them to basic proofs.

[@surrealistidealist]

@ I was thinking about that, too.

[@crackersnip6505]

This video us so good, these books look awesome and I cant wait to buy how to prove it soon and learn propfs in highschool Btw I would really love to purchase your signed copy of stewarts calculus if you ever wish to sell it😊

[@kabmeisterkabs1220]

I want to get started with maths. Where do I start, any books or recommendation will be great thanks.

[@TheMathSorcerer]

Start here, with one of these. Try book of proof, it's free:)

[@ziggedr4y]

there have been a couple other videos on this channel titled "how to learn mathematics from start to finish" and in fact these books are recommended in those videos to read very early with the caveat that you don't need to 100% understand them by the time you are done. They are a foundational subject that you will revisit again and again and will help you read and understand other more intimidating texts earlier. but It's like learning music theory before picking up an instrument. some of the advanced stuff will not come easy until you have more practical experience in other areas.

[@saratsahoo4198]

How to get these books for personal use.

[@QaqawuliX87]

Math Soscer, every time you review that book by professor Dan Velleman it gives me goosebumps because he was my professor in math at university. But I never encountered that book. Hopefully, I will get my hands on it in due course.

[@QaqawuliX87]

Math Socerer, professor Velleman, as a teacher in the classroom he was awesome. We, students were in awe of him and we considered him a god.

[@AsikMondal-l2k]

I love you sir❤❤❤❤❤❤❤❤❤❤

[@AbdullahAlhamdda]

Can I read these books while in my second year of college?

[@TheMathSorcerer]

Yup

[@billylee5624]

Proof methods are structured in a way that if the conclusion is true than the propositions must be true. Sometimes this is called a vacuous truth. Sometimes though, the antecedent can be false, but the consequent must always be true for it to be a true statement. That is part of forming arguments in Real Analysis. Jk.

[@nosferatualucard2613]

"People often times study math and they get to these proof writing classes, and they can't do it, and so they change their majors" I laughed so hard at this. Clear sign of not actually loving your major there.

[@onenhere6458]

Proper scan or didn't happen. Educational material is either free for studying or datawalled for fearing.

[@Polymathmusician]

Hey, would you mind if I use this footage in my cinema?

[@Kriojenic]

I was gifted how to prove it by a temporary sub of mine for a physics class in HS. I think about him oftem

[@muhamedshehu7535]

I am taking real analysis in my first semester as an undergraduate, without a background in calculus and proofs, so its killing me🙃

[@b07_ashishsuradkar49]

Wow

[@TheMathSorcerer]

:)

[@jluiselizondo4483]

I apparently already know how to make basic proofs 💀 I study computer science, and I have had to make logical proofs in a couple subjects.

[@evans383]

How to Prove It is my favorite proof book

[@TheMathSorcerer]

Yeah it's really good. I even love the physical size of the book!

[@youssefn.3437]

سأترك هذا التعليق صدقة جارية بعد مماتي فمن يراه يستغفر الله ويصلي علي النبي

[@ulysses_grant]

Are those books aimed at laymen?

[@TheMathSorcerer]

Yes!! Most colleges will put in like a pre-req of calc 2 or something similar. But honestly, it's just words. If you can read carefully you can get through it and learn!!

[@ulysses_grant]

​@@TheMathSorcerer Wow, thank you for taking the time to reply! And thank you for clarifying that. For a math John Doe like me, your recommendations are pure gold.

[@SidibeAboubakar-ir1ud]

❤❤❤

[@rationalsceptic7634]

It is not free

[@erickgomez7775]

It is free if you pirate it

[@RONIIAL]

I love books and i hate the lack of time to speedrun books within 24 hours.

[@billylee5624]

Math cannot be without it's foundations of axioms and the various algebraic structures that form from them. Jk.

[@rynyue]

0123

[@periklisspanos7185]

Prove I will win top prize in lottery , logic and reason is out of the window

[@adambernache3210]

The fact that you recorded this video is enough math proof for me, as is the fact that you are even existing and talking about a whole lot of nothing.

[@Soltaiyou]

I disagree that “All Unicorns are purple” is vacuously true. Unicorns are fictional constructs and, as such, are multi-colored. Therefore, either unicorns don’t exist, in which case they have no color because there is no color associated with non-existence, or they exist in a fictional sense, in which case they have whatever color the artist chooses to give them.

[@temperedwell6295]

The fact that there are books and courses on proof writing shows how dumb the world has become, or at least, how low education has sunk.

[@CalBruin]

STOP recommending either _How to Prove It_ by Vellman and _How to Read and Do Proofs_ by Solow. Both are bad books and there exist far better books, including one you reviewed. Including that book, another I suggest you recommend instead is my favourite of the bunch, _Bridge to Abstract Mathematics_ by Ronald Morash

[@TheSTEMSphere7]

❤❤❤❤❤