I love this channel and it has kinda become my comfort channel. The way you make your videos is relatable and inspiring. I mean this in the sense that I see you and am like “oh, these math wizards are people too. I can be like them if I try.” I’m not the best at writing so I hope this makes sense.

11:10 I had a professor who would end a proof with w⁵. He told us it stood for “which was what we wanted”

You seem like such a relatable person. When I listen to you, I’m able to find some comfort. Love from 🇮🇹🙃

There seem to be two competing styles of mathematics pedagogy. One of them is what I think you and I would both lean towards, which is patiently teaching young math students how math really works by starting with the simpler concepts, building up to more complex ideas, and being rigorous about the math along the way. The other is to teach some rather advanced math to students in such a way as to skip the proofs and explanations, but get them to memorize and have some intuition for the methods. I really began to love math when I was a student at a community college where two of the math professors really taught their courses in a rigorous, logically clear way. The first of these courses I took was a linear algebra course. Instead of a matrix-oriented, memorization-based class for engineering majors, which evidently is the norm at some colleges, this was a rigorous class about the math itself. The professor was really enthusiastic and loved math. This was an entirely new experience for me, and it was the first time I really enjoyed math.

I appreciate your openness and your thoughts on being an academic in general. Although I am not a PhD student in Mathematics but in Finance, I can absolutely relate. Thanks a lot!

I agree we should introduce proof writing at a younger age. Proof writing is The Skill you get from studying mathematics formally, and there is no reason children in middle school or high school couldn’t be introduced to it at an elementary level. I used Hammack for math proofs class. It’s very accessible, I think, but what the reader gets is proportional to what they give. I wish I had tried harder in that class. Thanks for sharing

Nice to hear you reminisce about some things! My introductory proof writing course at university was centered on metric spaces and I learned a lot from that. It definitely helped a lot with real analysis, abstract algebra and topology later. Some middle school students can handle elementary proofs for sure. I didn't see any until high school geometry though. Nice video !

Best new channel on KZfaq. I could watch pen+paper/textbook videos from this white desk forever.

This channel is very interesting, I have pretty limited knowledge of math (B.S. in astronomy + one failed group theory class + binge-watching math channels on youtube) but I enjoy your content anyway. It's cool to see how my life could have looked like if I decided not to drop from M.S. and actually go deeper into academia.

As a young student who wants to go into math research, i'm so glad i discovered this channel because i really had no idea what the atmosphere of math research was like, i just love the subject. it's funny, i also started loving math by studying geometry, at the same age as you

i really love your videos!

Love your content!

You seem like the 25 year version of the math sorcerer. Love the videos!

Loving these vids: please keep it up so i can continue loving the vids

Can you do a video of all your algebra texts? That would be awesome! Love the content!!!!!

Cant wait for series on measure theory

Great video as always. Thanks for putting the effort into it.

Most of the time I don't comment on videos, but I just need to say that your channel is great! One of your videos appeared on my home channel randomly, I believe it was the first video you made? I would love if you went into more detail about the problems! Edit: I'm from europe, and they make us use proofs in middle school, and they really do help!

math proof would have definitely got me into maths more cuz all I wanted to know was how things work, without any practicality... i ditched it before getting into limits because I was so bad at using the calculator... now that I have went through history & philosophy UG to understand better how things work and a librarianship PG for me to organise knowledge, I find myself having to learn physics and neuropsychology to further my understanding in how things *actually* works, both are so interesting but just say I hope I am more familiar with the mathematical language...

Btw if you don’t mind, you should really upload an asmr math video of you solving questions or working on problems whatever? You sound rly cool and it might motivate the few of us at least to maybe kick start our study session too. Great vid tho!

lol I literally have that Richard Hammack book on my desk right now how small the world is haha

This guys the goat man

I took a proofs class in my uni a few years ago and barely passed it. Took a few more undergrad maths and fortuantely passed those but my marks were mid. I only understood proofs and proofwriting when i took analysis

I have started a channel doing proofs. Because I’m seeing a subject, this semester that is called advanced calculus. We have been through all of these, so I feel like motivated to upload some proofs to the channel because I think that some people will find them useful and will help them in their writing.

I found the Hammack book for free for download as a pdf. Not sure if it was supposed to be free, however I am happy to have found it.

as an undergrad maths student i kinda struggle in proof writing aswell

Hey, I was wondering if you had any textbook you'd recommend for self studying analysis? Or if you could do a video on different textbooks you'd recommend for different subjects, I'd very much like to see that, and I think I've seen other commenters echo the sentiment. Love your videos, keep it up!

Quick note: Having a master's degree or a PhD. will NOT guarantee you a job opportunity at the level you're expecting. Unless you're planning on staying in academia, companies couldn't care less if you did your best during your graduate studies; they only care about EXPERIENCE. So the focus of your career should be getting at least 2 internships during your studies and maximizing your networking connections, making sure that you will be guaranteed a job right after graduation. Tldr: dont get too lost on proofs. Focus on opportunities to apply what you've learned to the real world, unless you wanna stay in academia

Thanks, I learned from this video.

I’m pursuing my applied mathematics bachelors! Just about finished my 2nd proof course. My 1st professor in proofs HARPED on how we wrote ours in terms of it being clear etc. that really helped me out seriously, but still proofs are a little hard

I love this channel.

I think starting with proofs asap makes any math topic and book way easier. I don't think learning proof writing and reading is the hardest. Not many symbols you got to learn and you can get started after learning some basic algebra as you showed. I think because it's held off till the very last point till you get to real analysis, it's feels like a gigantic mountain to climb. But I blame real analysis and the assumption of where you are mathematically at that point is more at fault.

im not taking a math major in college but i can safe to say for me ill prob. stop at bachelors and maybe masters but going for PhD is kinda a waste of money/time at least for me.

In retrospect, declining my acceptance into a PhD program was one of the best decisions I ever made.

I got the book! Hoping the self study it

I was not too much into art but I like to draw and paint, but also asked myself what is this reality and this planet, what is the point of life. How is it that I did not exist a few years back, but now I do.But I won't forever exist. Then I read somewhere this quote from Michelangelo "The true work of art is but a shadow of the divine perfection. Only God creates. The rest of us just copy." The obvious conclusion from this is that God is the greatest artist. I always liked math, but I never had a good background for competitions, but I really liked to go to the core of each topic learnt in class, obviously never got to that point because for example a simple set can be as complex for a grad student as for a kid, but in this manner I always liked to think of math results as some type of divine brush that God once put his effort on. This kept me on the subject and still is, it's the purest form of art. I'm still an undergraduate, don't think I'm able to do research but I discovered trivial things on my own before actually learning, and made conclusions most of the times incorrect during lecture without previously reading before it, it's an amazing feeling to even have conclusions, doesn't matter if it's right or wrong. I have an ok background on Algebra, but the analysis part I need to improve, also number theory, really obscure results such as reciprocity law, Artin and Grothendieck were truly remarkable mathematicians I look up to, because they worked on the areas I want to work on. I have a personal goal which is to get kids to study mathematics and do research, while my "career" goal is to learn about the langlands program, at least from the algebraic number theory point of view. Having said all of that I was never a book person, only went to books once I learnt the subjects, I find it more enjoyable this way. This is why I would really enjoy a mesure theory series. I can do math for a whole day but I really need the base knowledge either from a video or a person in order to touch a book. This has been my weakness since school, with this given information I would like to know your second opinion about it. Thanks for all your videos, it really motivates me without even you trying to do so.

more book please

Jay Cummings book is the best book on mathematical proofs I have ever read

I agree that we should start teaching proofs earlier. I think more people would be interested in math if they had a taste of the philosophical aspect of it that you only really study in math major courses

i like this

im taking ap calc bc, im absolutely in love and learning about the inner machinations of the world is facinating

Crazy college stories next please

This video is going to blow up

Proofs with pawprint! Thats how make the tiring process of proving more fun.

Love the paw print! Keep it.

To anyone still on this video, I recommend the book "Mathematical Proofs - A Transition To Advanced Mathematics", I worked through it last summer holidays and it was a very very well written book. And I need ehat I've learned there all the time in contest math And to your comment about middle school: I am in highschool, so yeah this topic is very accessible for anyone interested

Idk if it's a wall or because I was homeless but this trig stuff is so confusing I was homeless in geometry so precalc or advanced algebra is really difficult I find algebra itself very easy no test less than 90% but that and logs are little tricky

I did my BS in math. That's the same intro to proofs book I used!

I thought first you are Richard Hammack ;)

See "so long and thanks for the PhD" and "PhD Comics"

I know it's trivial but for the proofs you presented, how would you prove that the sum of two integers n and m is also itself an integer? Is that an axiom?

I think maths should become a global subject, especially mathematical logic; in the broad view, you are not just teaching math but the principles of human thinking.

Some universities don’t allow students to take as many math courses they want. They prescribe certain number of credits.

If you want to be a researcher, go to the grad school. If you want to earn money, apply for jobs rather than go to grad school. I wanted to study the elliptic curve with my professor when I was studying in America(he is a well-known professor in his field), but I didn’t have enough money to make a living. Thus, I started to search for available job positions. Now, my goal is to make money for the math grad school I like and study more there, even though I will be in my 40s or 50s. I don’t want to wait just to die. I want to study math more.

I’d love to go into research and get a phd but I'm struggling to get my bachelor's and my grades are mediocre at best but I can’t bring myself to quit bc I really love my subject even if my grades don’t reflect it

Math is beautiful as it is. No need to make it "pretty". Just lucid.

if you can and want then yes

8:22 It is.

Hi, I love your videos. I have a question, though. I’m currently discerning what major to switch to (I’m in engineering but I’m not really keen on it). I love math and I love physics especially when I get to prove something, so I think I’m interested in maybe doing a double major or a major in math and a minor in physics. My biggest qualm with engineering is that it takes up so much of my time and really forces me to sacrifice the things I want to do (like learn more about math, take art classes, exercise regularly). Basically my question is does math require you to make more sacrifices than necessary? I’m not against working really hard for something. Especially, if it allows my some freedom, but before I jump into math I want to make sure I can be both a person who majors in math and a person who has a life. Would you say that’s possible?

Insta clicked this video

phd = permanent head damage

At what university are you doing your PhD?

hi new subscriber! I'm a woman currently in my second year of my co-terminal program (bachelors and masters) in applied math and statistics. currently there just aren't very many other women in my major, how does graduate school compare? What is the diversity like? Thank you!

"I don't even know if you want middle schoolers doing this" You say that like it's firearms training or acid base chemistry lol.

the answer is no

1st

all this complex things yet you still know nothing about Afterlife Read Quran please

When I teach proofs, I place heavy emphasis on providing specific justification for each step in the proof. Having a hard requirement for justification helps to prevent students from making up their own nonsense when they are doing proofs on their own. Here is how I might present the proof to my students: Definition: n ∈ ℤ is even if ∃ k ∈ ℤ, n = 2k. Theorem: ∀ n m ∈ ℤ, n even ∧ m even ⇒ (n+m) even. Proof: Let n ∈ ℤ and m ∈ ℤ with n even and m even. By definition of even, ∃ k ∈ ℤ, n = 2k. By definition of even, ∃ l ∈ ℤ, m = 2l. Then n+m = 2k+2l. By the distributive law, 2k+2l = 2(k+l). By transitivity of equality, n+m = 2(k+l). Hence there exists an integer x, namely x=k+l, such that (n+m) = 2x. By definition of even, (n+m) is even. QED Different variations are possible. I might leave out transitivity of equality (in fact transitivity of equality is already low-key needed to get n+m = 2k+2l). But I think I would not leave out the distributive law under any circumstances, since that to me is the crux of this proof.

Why is It not 2(n + m + 2) ∈ Z at the end? Was the 2(n + m + 1) ∈ Z a mistake, or am I missing something?