I agree on the idea of "proof" courses being an elaborate exercise to do simple definition matching. Although I could see the class being useful depending on the teacher. Reminds me of a story from Surely You're Joking Mr Feynman, where many of the books considered for California elementary schools had the pretense of being scientific, but then asked the most trivial questions that didn't use the science in any meaningful way. (e.g. the temperature of the stars are X, Y, Z. Find the average temperature :D ) . Maybe these "how to prove things" course is just an outgrowth of that kind of thinking at a college level.
Hi Daniel. I wanted to say that reading the Book of Proof in highschool and learning to write proofs was the crux of my "mathematical awakening". I then learned linear algebra and found the most gratifying parts of the course to be the most abstract ones (i.e. vector spaces) which led me to learn abstract algebra over the following summer. I did wonder where it was all going at first, but when I accepted the definitions at face value, I began to love the results (I proved Euler-Fermat in the first month and couldn't stop thinking about it). Sure, the problems were relatively straightforward and the theory didn't seem as "powerful" as Calculus. Furthermore, my summer ended with semidirect products, Galois theory a long ways away. However, as soon as I found a non-trivial problem -- classifying groups of various orders -- I dove into it with enthusiasm. And, when I consider my highschool classmates, I was the only one who fell in love with mathematics.
That's a nice story, and I'm glad to hear about your enthusiasm for math. In this video and others, I've expressed skepticism about courses designed to teach students how to write mathematical proofs and about abstraction for its own sake, but it's fine that they can sometimes inspire students. Certainly some people find it exciting when they first encounter abstract definitions and methods of proof because they put issues in clear focus and provide more satisfying resolutions than the methods emphasized in high school. But I'll stress that the undergrad math curriculum overemphasizes facility with abstraction, and largely misses the problem-solving process that mathematicians actually engage in and that led to that abstraction in the first place.
1:51:14 agreed. The current obsession with unnecessary abstraction and insisting that there is a "right" way of viewing things is unwarranted (at least from the perspective of an analysis student).
Definitely. Especially if the "right" way is meant to be general enough to handle other situations that are not relevant in a particular course. Challenge: Is there a good reason to introduce the open-cover-admits-a-finite-subcover definition of compactness in a real analysis course, as opposed to just using sequential compactness, for instance? Glad to see students are picking up on this.
@@DanielRubin1 I believe the definition of compactness using covers was only discovered in the 1930s. Since basically all the math covered in an undergraduate real analysis course is pre-1900 I don't see the need to give the more general definition.
@@DanielRubin1 I mean if that is really a bone of contention, why not decline to introduce the word, and instead prove properties of closed and bounded sets?
@Shashwat Avasthi That's not what I want to do. For one thing, closed and bounded sets are sequentially compact in R^n, but not in most of the infinite-dimensional spaces of functions we care about. My point is that the notion of sequential compactness is strong enough to guarantee the existence of limits, which is what we want, while also being a straightforward notion that is not too difficult to establish for sets with that property, like closed and bounded sets in R^n.
@@DanielRubin1 Sure but are say spaces of functions a concern of a typical real analysis course? ie if one say follows your idea of do not do theory until you need it, would it not follow that do not talk about sequential compactness until you get to say the context of those particular spaces? Rudin talks just a bit about uniform convergence, which atleast as far as I see it does not need to use sequential compactness in his treatment. I mean I am all for proving that these 2 are equivalent on R^n as a theorem, what i think might be a bit unmotivated is defining the word compactness. I might be speaking complete hogwash so pardon me if that is the case ( not a math major)
Thanks for the video! I'm an undergrad at with a-lot of the same concern; ie which graduate class to take, how to pick the right advisor, what kind of math is concrete giving a problem solving mindset. This is all wonderful! I've also seen PDE in Stauss and taking one on Evan's
I previously had Dan as a professor for partial differential equations at Cornell. I have been industry for a while doing a variety of mathematical related things including partial differential equations, numerical analysis and machine learning. Love the channel.
Hi Daniel. I think I am in a similar situation like you were in years ago and just want to take as much graduate math courses as I can . I was considering doing algebraic topology and John Lee’s text on smooth manifold. But what would be better options than just taking more harder math course ?
My view on this today is that the better option is to focus on a concrete problem you're really interested in, and learn the techniques and theory you need in order to get some new results. It's very good to find a professor you trust and whose work you like to advise you on a senior thesis. I strongly caution against trying to absorb more and more theory just because subjects are well-regarded, without a research agenda.
Would you say that exercises in textbooks that require a student to show some example meets the definition of mathematical object or idea is, in general not an effective exercise? (On average, of course it always depends on specific topics, but if you had to provide a general assessment) I.e. i know “it depends” but in general what is your assessment
My worry is that all a student might get out of doing exercises like that is an appreciation of the definitions. Of course, reasoning with abstraction is an important skill in mathematics, but it's a basic one. I think it's much more important for students to acquire the techniques in a subject, which more typically happens when the exercises ask students to establish nontrivial results.
Nah. The correct way to do a math major is the way I am doing it: major in physics and graduate as a mediocre student who only studied before tests, then somehow get into a PhD engineering program, then realize halfway through that math is actually what you like, then decide to finish out the program while studying math on the side with a private tutor, then graduate with no desire for an engineering job and wishing that you had studied math as an undergraduate.
I also have my own video with a guide to doing a math major: th-cam.com/video/EE7KpcReYw4/w-d-xo.html
I’m an engineer, but I always wanted to learn higher-level mathematics. Now I’m doing it by myself and these discussions help a lot. Thanks!
I agree on the idea of "proof" courses being an elaborate exercise to do simple definition matching. Although I could see the class being useful depending on the teacher.
Reminds me of a story from Surely You're Joking Mr Feynman, where many of the books considered for California elementary schools had the pretense of being scientific, but then asked the most trivial questions that didn't use the science in any meaningful way. (e.g. the temperature of the stars are X, Y, Z. Find the average temperature :D ) . Maybe these "how to prove things" course is just an outgrowth of that kind of thinking at a college level.
I remember that example of "finding the average temperature" from Surely You're Joking, Mr. Feynman.
Hi Daniel. I wanted to say that reading the Book of Proof in highschool and learning to write proofs was the crux of my "mathematical awakening". I then learned linear algebra and found the most gratifying parts of the course to be the most abstract ones (i.e. vector spaces) which led me to learn abstract algebra over the following summer. I did wonder where it was all going at first, but when I accepted the definitions at face value, I began to love the results (I proved Euler-Fermat in the first month and couldn't stop thinking about it). Sure, the problems were relatively straightforward and the theory didn't seem as "powerful" as Calculus. Furthermore, my summer ended with semidirect products, Galois theory a long ways away. However, as soon as I found a non-trivial problem -- classifying groups of various orders -- I dove into it with enthusiasm. And, when I consider my highschool classmates, I was the only one who fell in love with mathematics.
That's a nice story, and I'm glad to hear about your enthusiasm for math. In this video and others, I've expressed skepticism about courses designed to teach students how to write mathematical proofs and about abstraction for its own sake, but it's fine that they can sometimes inspire students. Certainly some people find it exciting when they first encounter abstract definitions and methods of proof because they put issues in clear focus and provide more satisfying resolutions than the methods emphasized in high school. But I'll stress that the undergrad math curriculum overemphasizes facility with abstraction, and largely misses the problem-solving process that mathematicians actually engage in and that led to that abstraction in the first place.
Mooney never ceased to amaze me with his brilliant intuition for abstract things. Besides he is such a nice guy and is always willing to help
1:51:14 agreed. The current obsession with unnecessary abstraction and insisting that there is a "right" way of viewing things is unwarranted (at least from the perspective of an analysis student).
Definitely. Especially if the "right" way is meant to be general enough to handle other situations that are not relevant in a particular course. Challenge: Is there a good reason to introduce the open-cover-admits-a-finite-subcover definition of compactness in a real analysis course, as opposed to just using sequential compactness, for instance? Glad to see students are picking up on this.
@@DanielRubin1 I believe the definition of compactness using covers was only discovered in the 1930s. Since basically all the math covered in an undergraduate real analysis course is pre-1900 I don't see the need to give the more general definition.
@@DanielRubin1 I mean if that is really a bone of contention, why not decline to introduce the word, and instead prove properties of closed and bounded sets?
@Shashwat Avasthi That's not what I want to do. For one thing, closed and bounded sets are sequentially compact in R^n, but not in most of the infinite-dimensional spaces of functions we care about. My point is that the notion of sequential compactness is strong enough to guarantee the existence of limits, which is what we want, while also being a straightforward notion that is not too difficult to establish for sets with that property, like closed and bounded sets in R^n.
@@DanielRubin1 Sure but are say spaces of functions a concern of a typical real analysis course? ie if one say follows your idea of do not do theory until you need it, would it not follow that do not talk about sequential compactness until you get to say the context of those particular spaces? Rudin talks just a bit about uniform convergence, which atleast as far as I see it does not need to use sequential compactness in his treatment. I mean I am all for proving that these 2 are equivalent on R^n as a theorem, what i think might be a bit unmotivated is defining the word compactness. I might be speaking complete hogwash so pardon me if that is the case ( not a math major)
Thanks for the video! I'm an undergrad at with a-lot of the same concern; ie which graduate class to take, how to pick the right advisor, what kind of math is concrete giving a problem solving mindset. This is all wonderful! I've also seen PDE in Stauss and taking one on Evan's
Glad it was helpful! Strauss is a great intro to PDE, and Evans is great for a second course.
I'd love to see you review CU Boulder's Applied Math major
I remember Connor lecturing me on Black Holes and possibility of time travel in 4th grade.
I previously had Dan as a professor for partial differential equations at Cornell. I have been industry for a while doing a variety of mathematical related things including partial differential equations, numerical analysis and machine learning. Love the channel.
Hi Daniel. I think I am in a similar situation like you were in years ago and just want to take as much graduate math courses as I can . I was considering doing algebraic topology and John Lee’s text on smooth manifold. But what would be better options than just taking more harder math course ?
My view on this today is that the better option is to focus on a concrete problem you're really interested in, and learn the techniques and theory you need in order to get some new results. It's very good to find a professor you trust and whose work you like to advise you on a senior thesis. I strongly caution against trying to absorb more and more theory just because subjects are well-regarded, without a research agenda.
Would you say that exercises in textbooks that require a student to show some example meets the definition of mathematical object or idea is, in general not an effective exercise? (On average, of course it always depends on specific topics, but if you had to provide a general assessment)
I.e. i know “it depends” but in general what is your assessment
My worry is that all a student might get out of doing exercises like that is an appreciation of the definitions. Of course, reasoning with abstraction is an important skill in mathematics, but it's a basic one. I think it's much more important for students to acquire the techniques in a subject, which more typically happens when the exercises ask students to establish nontrivial results.
Nah. The correct way to do a math major is the way I am doing it: major in physics and graduate as a mediocre student who only studied before tests, then somehow get into a PhD engineering program, then realize halfway through that math is actually what you like, then decide to finish out the program while studying math on the side with a private tutor, then graduate with no desire for an engineering job and wishing that you had studied math as an undergraduate.
Note: I was here when the channel was only 10k subscribers big.
Hi, Daniel. It's your personal troll.