SO REFRESHING to see someone pitching the idea of a math major interested in more than rigid formalism and proofs, proofs, proofs. Recognizing that there is a time & a place for it, and that real world applications are an important motivator for many people. So often in my quest for good mathematics resources I stumble across people who sneer at any books that include applications, as if knowing how the math might be used in the real world is somehow a bad thing. What's wrong with starting with some historical perspective, learning about the problems people were and are trying to solve, and then--once a person's curiosity is piqued--dig deeper into the theory of the thing? So thank you! I appreciate you.
As a mid-age man who graduated from a decent Math department 20 years ago, don't remember much details but still can't help reading some old math books once in a while, I really enjoy listening to your genuine thoughts and experiences with these books. I'm sure younger generation will benefit even more from your sharing. Thank you, Professor Rubin, for making this great video!
Unless I'm completely mistaking your views from this, I think your (clear) opinions about Algebra and Grothendieck's vision of mathematics are just wrong. It was ultimately highly abstract algebraic techniques that lead to Wiles' proof of FLT--a very concrete, historical problem, indeed. While I agree with you that not everybody needs to walk that road (and I, too, think that mathematicians broadly disregard the virtue concreteness), I don't think that the solution is to dismiss abstraction and generality as useless; such an attitude is rightly seen as regressive. However, I will grant you that many mathematicians seem to have the opposite problem, and to a much greater extent. In that respect, I find your views somewhat refreshing and intriguing. Grothendieck was certainly right about one thing: you've got to be willing to stick to your guns and express your opinions! Sub earned. I look forward to seeing more of your content in the future!
my proof class helped me a ton personally, but it wasn't untill I took graph theory did I realize the importance of a well defined approach to describing mathematics and alongside direct applications. Now I apply that approach isomorphically from course to course.
Agreed, my intro proof class covered basic set theory and logic, plus taught us techniques like proof by contrapositive, induction, counterexample, etc, which my analysis course took for granted that you already knew.
Thank you for the list of great resources. It's surprising how university courses have stripped away most the history and context of math topics. After doing well in first year calculus I realized I'm only at the midpoint to truly understanding it. The rest of the subject falls on the student to complete on their own -- with the help of a few good book recommendations!
I think it's the overemphasis on "useful" maths driven by profit based institutions. It's taken for granted that the innovation behind all the profitable stuff requires immense creative energy and time yet that is where the value originates from, despite lack of funding.
The first and last time I used "differential forms" was in my highly abstract freshman year vector calculus class. I have a PhD! While I enjoyed the course I think I and certainly most other students would have been much better served with a concrete, historical approach. Thanks for your videos, I'm interested in reconnecting with the historical and technical basics of the subject and I appreciate your approach and recommendations. I already picked up the Fourier Analysis book which showed up in my algo. edit: Now that I've watched the whole video I want to expand on how valuable I think this advice is. I take it at face value that you believe this approach is valuable for research-track mathematicians, and I am inclined to agree without being able to add any force to that. But as someone who went from a PhD into industry I think this advice is even more relevant and valuable. There is such a value to being able to formulate and solve concrete problems using a mathematician's approach, and the skills I learned in those few years of rigorous mathematics have paid continual dividends over my career. But I would have been even better off with more concrete facility in the basics, simply writing down, manipulating, and solving equations, having all the machinery of ODE, linear algebra and statistics at my finger tips, and the background in numerical methods to transform those facilities directly into code. Looking back on my undergrad and early graduate years, I can see clearly a fetishization of abstraction and generality when the essence of mathematics is solving problems. I hope that kind of hierarchical thinking can be done away with and replaced with a relish for tackling any concrete problem with rigor and insight.
Maybe it is a US thing, or in places close to old important established Universities. In my own experience, in Europe, there is a lack of pure Maths modules in many departments. You get your undergraduate degree with all the analysis you may want up to lebesque and measure, through complex, functional, PDE, Hilbert, etc. you get your basics in linear and abstract algebra but not that far, basics for probability and some geometry up to differential geometry, and a basic topology. It seems very complete but everything is there to help you in Physics and engineering, not pure Maths research. A typical undergraduate degree feels roughly 75% Applied/25% Pure at most, even if the applied is more rigorous than you can see in any engineering or even in Physics. BTW, there are also mandatory modules and optional in QM, GR, Cosmology, Astrophysics, etc. but nothing in number theory of any kind, and you can barely see graduate level courses with Lie, commutative algebras, representation, any advanced topic in algebra in fact, or topology or algebraic … anything. UK is different in that regard but you need to do graduate level Maths to find advanced topics in Analysis that you can find at undergraduate in many places in Europe. Edit: But I have to say, doing some good Physics alongside, Analysis was a lot more comprehensible, motivated and finally, easy to grasp in full. Even multivariate, after studying force fields, was like, but of course is reasonable. Disclaimer: I was doing a Physics degree not a Maths one, but the Analysis modules were the same for both undergraduate degrees.
The essence of mathematics IMHO is understanding the “logical structure” of the Universe. Solving problems is secondary and a natural consequence of that understanding.
super insightful and love the inclusion of the “genetic“ books. I think modern math classes sometimes focus too much on making people memorize a bunch of random theorems and formulas. Without actually trying to teach students the intuition or motivation behind certain rules or tricks in math. I’m the the type of person who can’t memorize any rule or formula unless I have seen it proven, in a clear way. You definitely helped me with all that extra work needed to find the right book definitely gonna check out all your suggestions. Thanks!
I have been a math teacher for over 20 years. I never understood the pedagogical importance of the historical approach. Your video made me want to start studying math again, reading books with more historical and applied perspectives. I know that students really like this approaches.
I listen to this front to back for an hour. I don't even understand some of the advanced math topics and courses you're talking about, but I enjoyed it. Feeling more excited about learning complex math in college!
The thing I like about this approach is that you are always grounded. Richard Bouchards also emphasizes looking at exmamples and seeing what is going on way before you develop or understand the general machinery. And I would say he's done pretty well for himself following this approach.
Great video! I would definitely have benefited from hearing this when I was finishing high school. Excessive formalism, lack of direction, and lack of motivation are definitely plaguing university-level education in mathematics. I was glad to see some of the great exceptions listed in your video (e.g. "Algebraic Number Theory and Fermat's Last Theorem", but really many other examples in your list; it's mathematics with a purpose!). I'd like to add a couple of suggestions to your list, that I honestly think would be a pity to not include. "Theory of Computation", by Michael Sipser, should fit in a modern undergraduate mathematics curriculum (e.g. proves a modern formulation of the Incompleteness Theorem); there's probably space for other courses in Logic. "Probability with Martingales", by David Williams, was also a great book; rigorous, yet practical; it's an analysis masterpiece, and also exposes basic measure theory in a way that's actually useful, leading all the way to a proof of the strong law of large numbers.
With Sipser and Rosen for Discrete Maths you can chew away any undergraduate CS major course in full… Kudos to you for the Sipser, a masterpiece in my opinion like the Spivak.
They could almost teach mathematics like a history course with motivating examples so people build confidence. Often books are written assuming a lecturer is illustrating and the book is there for exercises, which is a shame when "there was a tricky/interesting problem so someone formalised it and found patterns" is so much more fun
I am a 4th year undergraduate student in math and physics. I appreciate a lot the general point presented in this video. As a student with a background in physics in math courses, I feel kind of sad for fellow classmates who just don't have motivation to lean on when learning various math courses. It is mostly present at any analysis related courses, whether it's Calculus, Complex Analysis, ODEs and PDEs, Probability and so on. Even Metric and Topological Spaces which isn't necessaripy analytical, can be very baffling to many students. The reason for that is that as Daniel expressed, abstract theories with no motivations can only take you so far. I do believe that math is expansive enough on it's on and doesn't need other fields to justify it's existence, but I also believe that the real beauty and value of any subject truly shines only when considered in relation to other subjects from the same field and different fields. Many pure math students lack this kind of motivation and lense to view the subjects from, and I think it's such a shame. I encourage anyone who studies math to try and expand their point of view in some kind of way. It can be by reading books such as the ones Daniel suggested, I haven't read anyone of them but they seem promising enough. It can also be combining the math studies with other field, such as any science or engineering. I don't agree though that every single course should be tought that way. There is a huge merit to learbing things abstractly, as well as computationaly, as well as more intuitively motivated. A good mixture of different approaches to teaching math will give the best value to the students, in my opinion
Dear Mr. Rubin: very interesting video! I have discovered you lately and already enjoyed some of them a lot (your interview with dr. Hamkins or your apology of McKean and Moll's books (just bought a copy for myself). Here in Europe we don't really choose much within our Math major. I get the impression that you are of a very applied and analytic persuasion - closer to Morris Kline than G. Hardy. I think I have a softer spot for 'useless' math as, being from a Humanities background, I am accustomed to appreciating something because it is true and beatiful, and to not giving a damn about practicalities.
Thanks! I realize that students don't have complete say in courses or textbooks, but I hope my videos can help anyway. I greatly admire Hardy as a mathematician, but I disagree with his sentiments about the worth of math. I have a podcast episode about the idea of mathematical beauty you might find interesting: ruclips.net/video/s3GXKKyY-m8/видео.html
dude....i LOVED that !!..." Focus on SOLVING PROBLEMS. "...THAT. has been my own Mantra in my Mathematical Journey.....thank YOU SO much..." learn the TECHNIQUES. "....
Everybody has their own way. I'm older and studied calculus in my Freshman year at college. My high school geometry class used a translation of Euclid's Elements book one, so I knew how to do proofs. To my dismay at first, my university college taught all math including beginning calculus, rigorously. However, after a few months, I started really liking the rigorous approach, even though I was much better suited mentally to the practical applied approach. I struggled a little, but when I took my exams in graduate school (a different school than my undergraduate) it was easier than I thought it would be (e.g. one of my exams was based on "big Rudin" ).
Damn, I'm not a math major (Wish I did a double/triple major in math but kinda late now lol), but some of your points are indeed spot on. I took discrete math, theory of probability, and multivariable calculus this spring semester ('twas a tough semester haha but glad I got it through especially with covid lol). I really liked all of them besides calculus 3, but I wish professors have taught us "how did mathematicians came up with these theorems and definitions and what is the motivation behind them" so that we get a chance to think more like a mathematician. For example, why did mathematicians came up with vector calculus? What is it used for? What was the motivation behind the jacobian matrix? How did Gauss come up with Gaussian Curve? etc. I'm the type of guy if introduced to the motivation behind each theorem, definition, it really clicks and hence the "ah-ha moment". But with calc 3, the professor just taught us the definitions and how to plug numbers into the formula. It got so freaking dry that I end up not paying much attention and end up getting a bad grade. It is honestly my fault that I didn't do well but your way of approaching math will definitely help a lot of math and non-math majors.
Thanks so much! Glad to hear you're curious about the historical motivation of mathematics. Unfortunately, only a minority of math professors pay attention to it, so you're unlikely to see it in class. Trying to change that! 2nd Year Calculus by Bressoud should answer your questions about multivariable calculus if you're interested.
Great question! I don't know enough to have a definitive opinion. I took one course as an undergrad on Logic and I retained nothing because I've never used any of it. I think the overwhelming majority of practicing mathematicians can be successful without paying attention to set theory or logic. I'm also dubious of that work in these areas will have any philosophical implications for mathematics. But I'm aware of some issues I find interesting that do make contact with nontrivial parts of set theory and logic, like Ramsey theory or the Tarski-Seidenberg theorem in real algebraic geometry.
@@DanielRubin1 Hey Im sorry for keep asking you all kinds of questions here. I just recently changed my major this semester from statistics to math so I just need more guidance lol. Im taking real analysis this semester after taking spivak calculus. School is about to start in 3 days and the syallbus says that we will follow a so-called "Moore Method". I emailed the professor to learn more about this pedagogy that they are implementing this semester, turns out its something that is completely new to me. So basically, how it goes is that we will be given definitions only and using that definition, we have to prove given theorems. During class, one person will have to present his/her proof and the rest of us will have to critique the proof. In other words, the professor is just there to merely guide the discussion and lead us to the right direction. Do you personally think this is a good way to learn math? I think it will work well but Im a bit afraid that the topics that we will be learning will not be well motivated if that makes sense.
@@axisepsilon514 I've never been part of a class like that, but I've heard of the method. I'd guess that taking a class like that will force you to understand the definitions, theorems, and proofs really well, and give you experience presenting, which are both good things. But on net I don't think this method is a good idea. It's very easy to miss the point of what's going on if you focus on definitions and theorems and not motivating problems. You might have to look at other resources for some of the things I think are most important about real analysis. Working on some real analysis content for this channel, coming soon!
I’m an applied maths grad (4 years of maths). I’ve been working in the IT industry for the last 30 years, but recently started studying maths again. I can see you have a very practical bent, and I’m the same. In the vid you don’t recommend doing proof theory; just learn how to do proofs as you go. That was my attitude as well. But recently I’ve been learning mathematical logic, and I’m really enjoying it. What’s the use of logic? Well I guess it tells us something about the systems that we use to do mathematics and their weaknesses. So far I’ve done the Godel completeness theorem (which is not that straight-forwards) and will do the incompleteness theorems soon. There are plenty of techniques to be learned as well which I suspect will be valuable in other areas.
This video points out the weakness of math majors. Pure math majors will need an additional degree in order to find jobs outside academia. Something that this video says and repeat many times is the importance of solving problems over study abstract theories. In that sense the abstract algebra example in the video was excellent. The video points out several times the importance of numerical analysis and statistic, which are extremely important in order to have a complete landscape of mathematics and finding jobs, internships or summer jobs. Honestly, the best recommendation is a double major of math with computer science or business.
@@DanielRubin1 your observation is correct. My comment is valid for other majors too. The double major with math is the best choice. In fact, I read that math as part of a double major is the most popular choice in some universities in the US, I think that was at some university in Dakota.
Some comments: 0. If you think there is any chance that you will go into industry, I recommend classes in numerical analysis at the level of Rolston and Rabinowitz, computational statistics by Gentle, numerical optimisation by Nocedal and Wright, and numerical DE's like Arieh Isreles. 1. Another rigourous calculus text is Courant and John. 2. For Linear Algebra and vector Calculus, You might like Geometric Algebra, Linear Algebra, and Geometric Calculus with Alan MacDonald's books. For those of you who like Linear Algebra in the "Down with Determinants" style but hitting thinks like the Principal Axis Theorem, try Larry Smith's book. 3. In Advance Calculus and Classical Real Analysis, I found books written by applied mathematicians are better motivated in this vein. 4. A standard undergraduate Algebra textbook which gets to the insolvability of the quintic is Pinter's, which you can get from Dover. Saracino's looks good too. 5. Good introductions to probability and statistics include the introductory books by Ross and by Wackerly. 6. I think you should take a course in combinatorics and another in Graph Theory. I like Tucker for Combinatorics and Chartrand for Graph theory. 7. There are good introductory applied algebraic topology textbooks, like Ghrist. 8. I like Willmore's differential geometry textbook. DoCarmo's books are good too. 9. If you have a feel for what modern physics is like, I recommend books like Nakahara's for introductory topology and geometry. (There are others.) 10. If you are going to a mid-tier or higher grad school, you will need to know abstract point-set topology before you start. The level is the first 4 chapters of Munkres's _A First Course in Topology_.
Thank you for this video! I was fortunate enough to have professors over a decade ago who had similar teaching philosophies. But it is a rare view. I'm glad you shared both your focus on "building up from the concretes" philosophy and specific texts! It'll be great for some self study.
Wow, this really was a great video for someone going into math (myself); I have had many anxious thoughts about proof classes and number theory cause I am simply not interested in these areas as I stand currently, but I do love differential equations and numerical methods. Really happy to get this gem in my recommendations. Greetings from Sweden!
Glad you liked it! I think pursuing just the math you're interested in is a very good way to do it. If you ever need something else, then you'll have a good reason to learn it so it'll be more interesting.
I am a math student in Germany and unlike you i love theories and all abstract stuff. Actually i am going towards algebra (specialization) and i can remember that we had algebra groups rings and so on even in first term and i am really gald about it. I think you need to know all these stuff to just show and prove more stuff. All the computaional stuff are possible because people made it with these theories possible and thought through it. You need for example Rings and ideals so you can solve equations not numerically but symbolic with beatiful math and algebra you need Galios theory. You need algebraic topology in theoretical Physics. Despite the fact that I am doing Algebra i love measure and integration theory and i was wondering about your comment on measure and lebesgue integral. I am asking myself what would you say about Category Theory or invariants theory?
I like your "follow your curiosity" mentality I think that pushing that idea more in classes would lead to undergraduates learning more mathematics on their own and practicing math more often and thus be a great aid in their gaining of mathematical maturity. I would, however, be careful with the more utilitarian aspects of your philosophy. As someone who studies a lot of category theory, I may be biased, but I gain a great deal of motivation from seeing vast generalizations and seeing how far they may reach. Of course, the heavier machinery that you were talking about is outside the scope of undergraduate education but I fear that pushing students to favor more concrete points of view in terms of mathematics may bar them from viewing the beauty of math and discovering that they may enjoy math for math's sake (I also HEAVILY disagree with your recommendation of not taking any topology course, but I could just be biased because it was my favorite course as an undergrad and am now really into algebraic topology). But I also understand that I am often in the minority in regards to these things and approaching math like this and like an art leads to burnout and confusion for a lot of people, especially less mathematically mature undergrads.
You certainly picked up on some of my controversial attitudes about math. I do favor an approach to math that's about solving concrete problems, and I worry that doing math for it's own sake or seeking vast generalization will not lead to good results, and it's very dangerous for young researchers who need to find jobs. I don't think I mentioned category theory in this video; I think it's nice as a philosophy or organizing principle in math, but it's absolutely terrible as actual mathematics. I wonder: does category theory or any move to a much more general framework ever help solve problems in which we were already interested in a specific setting? I hope to address these issues and make my case in future videos. I have a podcast episode ruclips.net/video/s3GXKKyY-m8/видео.html that addresses the notion of beauty in mathematics. My Tricky Parts of Calculus series is a set of math lectures that emphasizes the hard problems and how things were done in history. I'm trying to show that progress in math was always made by solving concrete problems, not by developing at the start the appropriate general framework, even though this is how most math is taught.
@@DanielRubin1 what makes you say that category theory is a terrible way of doing mathematics? In many ways it is superior, or at least more natural, than more things such as set theory and in some areas of computer science is a far simpler and elegant point of view with regards to the theory of computation. I'm sure that computer science could offer some examples of problems you were talking about. But in addition just looking at mathematicians like Grothendieck and Serre allows one to see how these general points of view solved very difficult problems such as the Weil conjectures. In fact Grothendieck's whole approach to solving mathematical problems relied on him constructing theories around problems until the theory was powerful enough to, at least in his mind, render many already existing problems "almost trivial".
@Taylor You anticipate several topics I'm hoping to talk about in future videos! I don't like the study of category theory because I'm interested in problems, and it is almost always hopeless to try to solve a problem by abstracting away some properties and considering a more general situation; the more general theory must still capture the specific structure that the result depends on. Grothendieck really did believe that math was about finding the "correct" most general framework, out of which results would trivially pop out. I think that's a fantasy, not how math was ever really done. I have heard the claim that it was the Grothendieck approach that solved the Weil conjectures from many people, but that is very much up for debate. Hasse's proof of his inequality and Dwork's proof of the rationality of the zeta function came first and neither had anything to do with Grothendieck's work, nor did Deligne's proof use the theory Grothendieck thought was required. All these proofs required analysis; see also Stepanov's method for counting points on curves. Grothendieck was a great mathematician, but his philosophy has led mathematics astray.
To add another point to the debate here: Maths departments all around my country were half full at most between forever up to 2015-ish. 2007-2014 for example, Universities with 70 places struggling to get 35 students, and those were prestigious ones. Nowadays they have between 700 to 1000 applicants per year, for the same places. Grades to enter went to the roof and it is almost impossible to study. The reason? Statistics and Data Science. Post 2017 it is all what any student want to do. Also, in the long past, no banks used Maths graduates but graduates in Econ, Finance, etc. Today it is only Maths what they want, and a ton of graduate schools, PhD and Master’s in topics that lead you to the most competitive and rewarding jobs have Maths as a major or double degree as a prerequisite. Maths didn’t change. The problem now is the number of students that loves Maths is still 35 of those 700 applicants. And you may choose the wrong 35…
Great content and some really useful advice. I would like to disagree on the point that an introduction to proofs class (some programs call it a "transition to advanced mathematics" or something similar) is largely unnecessary. I would say that this class for me came right before Analysis I and II, and Abstract Algebra I and II, and was tremendously beneficial and a very practical precursor to these heavily proof-based classes. I would agree that, yes, it is more intuitively productive to actually apply methods of proofs to specific examples, but had I not received the conceptual and logical foundations of the language of proofs, methods used, and notational conventions, I would have certainly struggled more in the higher-level classes. I think that Analysis and Abstract Algebra are two of the most difficult courses for an undergraduate, and trying to learn the content of these subjects while simultaneously learning about proofs makes for a more challenging experience, in my opinion. I was able to focus much more on relevant concepts having already acquired a good solid foundation on things like set theory, the real numbers, completeness, cardinality, proof by contradiction, pigeonhole principle, induction, and even more specific methods like proving each set is a subset of the other to show equality, having to prove the converse for biconditionals, etc. These were things that immediately flipped a switch for me whenever I encountered them in later classes and I was able to more readily understand the logic behind such proofs. I would liken this to learning how to write programs that perform specific functions in, say, Java, for example. One could certainly learn the language by being introduced to particular concepts, control structures, classes, or functions while simultaneously learning their immediate application to specific uses. But I would argue that first having a logical foundation behind the concepts, how object-oriented programming works, how computers store information, what the computer actually _does_ when a certain function is called, etc., can make learning the examples much more effective and intuitive because the background is there. Again, this is just my opinion and you have clarified that your points are your own opinion as well, so there's no wrong answer here I don't think. I just thought I'd give a counterargument based on my own experience. This is wonderful content, though, and I would definitely like to see more of this kind of stuff on YT. I think you and The Math Sorcerer are really the two leading channels presenting information and advice on things _outside_ of the actual course material and more along the lines of career options, educational advice, learning techniques, and broader conceptual notions that really aren't touched on anywhere else. So thank you tremendously for that.
Would a course on “Discrete Mathematics” be the same as a proof writing class? At my university, I couldn’t find a specific class for proof writing, but the description for discrete math contains many of the same topics that you listed.
@@bendavis2234 I wouldn't say that Discrete Math is necessarily a "proof writing" class in that its focus is not on writing proofs but on studying discrete (finite/countable) objects. However, what it does do is introduce many of the logical tools you'll _use_ to write proofs, so it's kind of a precursor to proof writing. Most Discrete Math courses cover the basics of set theory including basic axioms, collections of discrete elements, integers, rational numbers, implications, tautologies/contradictions, conditional statements, and propositional logic, all of which will be used later on when writing formal proofs. However, there is a major difference between proofs in Discrete Math and proofs in higher level undergrad math such as Analysis, Algebra, and Topology. The difference being that those subjects deal with *_continuous_* sets of elements (such as the real numbers). As it turns out, this is a huge step up in complexity, and most of the major theorems in your undergrad courses deal with continuity (Calculus, for example, is literally *_about_* limits and continuity). So I think that most universities would consider your first *_real_* proof writing class to be Analysis, or possibly Linear Algebra. As I said, my university offered an introductory proofs class that came *_after_* my discrete math course, so it can depend. In fact, a lot of discrete math courses are more driven towards computer science (it's very important to programming). But still, you will learn the foundational tools and building blocks that will later be used for proofs, so it is still a very important course.
@@jinks908 Okay thanks that's good to know. Do typical courses in Real Analysis expect you to come into the class with knowledge of proof writing? Discrete math will at least cover the basics, but of course there are many key differences that you mentioned. Also, I'm not sure how much assistance you'd get in basic proof writing techniques in a class such as real analysis. It probably helps to have some previous experience when entering the class. It probably depends on the university, too.
@@bendavis2234 Most undergrad programs offer Analysis I and II. Analysis I (sometimes Intro to Analysis) will typically do a review of proof writing, techniques, and terminology during the first couple of chapters. For example, my university used Steven R. Lay's text which covers logic and proofs and set theory in Chapters 1 and 2. So in an introductory analysis course, you will usually review most of the material you'll use to write proofs. However, this is only a review, and so you'll need to at least be familiar with most of it. But you shouldn't be concerned too much as you'll need prerequisites for any Analysis course anyway. You won't be entering an analysis course without having taken some kind of proof/set theory class. Many universities have a Transition to Advanced Mathematics course which covers this. My Transition course actually covered most of the material that a Discrete Math course would, so I didn't even need to take Discrete. The other thing is that as you move into advanced math courses, (those beyond calculus/trig/algebra, etc.), your classes will shift their focus from computation and into proof-based study. So any university math curriculum will teach you proof writing first since all of your upper classes will use it. Your junior and senior years will be almost solely proof-based, so you'll have plenty of exposure to it. Not to mention all of your textbooks will have several proofs already in them. You will learn a TON of proofs just by example. Usually, you'll see a definition, followed by a theorem which is a consequence of that definition, and then a proof that shows why it's true. Then you'll read the next theorem and prove it yourself, and so on. In essence, proof writing is the bulk of your undergrad education. So it's not really a matter of coming into it with a knowledge of proofs. The main point of a math degree is to *_learn_* proofs. All of your courses will build on each other. Thus you'll certainly be given all the necessary ingredients, but it will still be up to you to make sure you understand it. So pay very close attention to your proofs/set theory courses as they will be the lifeblood of your math education.
@@jinks908 Thanks for the info. In my case, I’m doing an applied math major, so I’m not sure how much different it will be in terms of proof based coursework. For example, most of the classes are taught separately in the applied math department with courses like “Discrete Math for Applied Math”, “Advanced Engineering Calculus”, “Real Analysis with Applications”. I’m assuming that proof writing will still be important for these courses, but I’m not sure to what degree. It’s weird how little overlap is between the Pure Math and Applied Math departments at my school. I could go through my whole major without taking a course from the math department. I’m not sure if your school was similar in this way.
This is gonna be so useful for me, I'm in my last year of high school and after 2 years of college im starting a degree in pure math, thank you so much
You may have swayed me and a lot of other people from learning a lot of abstract, incomprehensible and boring (at least at the time of learning) mathematics so that we can get to the good stuff later. I, at least partially, have had the idea that I have to go through some amount of apathetic study so that one day everything will finally be clear and I'll be able to be creative and do what I wanted to do all along.
Indeed one of the main goals of this channel is to show that there's a better way to learn math--one that is guided by problems students are at each stage prepared to appreciate. I find this way better for comprehension, motivation, retention, and for training to solve novel problems. If you want the math you learn to be useful to you, you must be invested in it. It has to be concrete to you. It has to be a source of meaningful questions that you have some tools to investigate.
Looking from a completely abstract sense is the actual refreshment. Although I must say that I appreciate applications secondarily since it aids in accurate visualization. That's my personal view. Just do applied math if you absolutely hate abstract thinking (although you still need a lot of it, since it is mathematics, and it comes with the territory). Or... do engineering or physics lol. (Video is good, by the way, it is still important to get both sides in my opinion)
43:17 What? Functional Analysis isn't about the "fine properties of functions", a more apt description would be infinite dimensional linear algebra, because infinite dimensional vector spaces mostly arise as function spaces
The reason to study those spaces, all of which come with a norm or metric of some kind, is to gain control over some property of functions that is the key quantity that determines the behavior of solutions of a PDE or variational equation.
@@DanielRubin1 I disagree, you're looking at this from a very applied perspective that completely overlooks the heart of the subject! Ironically enough it's almost never about the functions themselves. Also not all spaces studied in FA are necessary metrizable, a good example being the space of distributions which is useful in PDE theory
I think your analysis bias is showing strongly in this video. There are plenty of students that will find abstract algebra intrinsically interesting. Of course, any algebra instructor worth their salt will begin by exploring examples, as the definitions are made to serve the examples that appear naturally, including symmetries of shapes, matrix groups, and actions on vector spaces. Symmetries of roots of equations can certainly be covered in an introductory group theory course - I think Galois theory, however, is far too much for a first algebra course. You also didn't devote much time in this video to the different branches of algebra. Groups, rings, modules, and fields / Galois theory all have a different "flavour", and not all of it is motivated by solving polynomial equations. Ring theory has its origin in number theory and geometry. Anyone who found linear algebra interesting will be intrigued by the generalization presented in module theory. And of course, going through these examples and seeing the terms "homomorphism" and "isomorphism theorems" pop up over and over will provide fair motivation for category theory. However, I do strongly agree that representation theory should make an earlier appearance. It could certainly take up a few weeks in a dedicated group theory course before students move on to a dedicated course. As an algebraic geometer, I completely agree with your thoughts on the subject and your recommendations. It's an extremely broad subject, and one really needs a strong foundation in the classical picture (i.e. without schemes) before moving onto more abstract sources. If you take an introductory alg geo course and you find it interesting, then I'd recommend taking commutative algebra and then start learning about schemes. Eisenbud and Vakil, respectively, are my preferred sources. Not everyone needs the full machinery of scheme theory, but most people could benefit from following at least the first few chapters of Vakil.
I certainly do have a bias. Essentially I want to encourage math students (and professors) to work towards addressing problems. I tried to recommend an approach to algebraic subjects based on problems students could appreciate at that point, as opposed to just learning theory. What would you say are the branches of algebra?
Two great books for topology are Topology by Munkres as an introduction to general/point set topology, and then Lee's Introduction to Topological Manifolds for Algebraic Topology. DON'T start with Hatcher, topology's actually really interesting on its own!
In addition, check out Introduction to Topology: Pure and Applied by Adams & Franzosa. It's full of applications of general topology and can be a stepping stone to recent books on applied algebraic topology topics such as topological signal processing and topological data analysis.
Great advice. Thank you a lot! I've subscribed instantly. By the way, I fully agree with almost everything you said. For example, multiple variable calculus should not be rushed and should take two semesters. I also could not agree more that numerical methods should not be ignored. But more importantly excessive formalism and generalizations are not good, especially for those who don't major in math. And I agree that even for a mathematician getting mired in them and tuning out everything else is wrong. A huge like for the video. I also think geometry should not be ignored in colleges and should not be de-emphasized as we have it nowadays (almost no geometry at all). As to books, I made a separate comment on them.
You make several comments along the lines of “these abstractions are not useful; learn the practical applications”. Perhaps this argument has some merit if one is planning to work in applied math or physics or engineering. That said, had I followed this advice as an undergrad 50 years ago I would never have made it through the 1st year of math grad school at Berkeley.
What I'm trying to say is that I'm very skeptical of learning abstractions for their own sake, and that progress in math has been driven by the need to solve concrete problems. I certainly find it easier and more enlightening to learn math by seeing how problems came up and how they were solved. Often the important math problems came from practical concerns, and I really do advise pure math students to be aware of real-world problems. It very well could be that in the artificial environment of some math departments appreciation of particular abstractions is prized much too highly. In the end the goal is to make worthy contributions to science.
@@DanielRubin1 I would say that in the end what matters in pure Mathematics is trying to understand the logical structure of the Universe. Mathematical applications should at least in principle follow naturally from pure Mathematics even if historically that process is often, perhaps usually, reversed.
Thank you for a great video! I am retired and self-studying 'all' the math I missed in my younger years. Currently finishing Calculus 2 and starting Linear Algebra. One of the subjects I'm contemplating for the future is 'Discrete Mathematics'. I didn't hear you mention it. Is it know by another name? Do you have any suggestions as to what subject(s) (textbooks) I should consider after Linear Algebra? I'd welcome your thoughts.
Thanks! But I'm no expert in applied math, and I wouldn't claim to know how to construct a curriculum for that field. With the limited exposure I have to the concerns of applied math, I'm trying to create a curriculum for students who want to do "pure" math that brings in some of the important and useful material from the applied world they might otherwise never learn about.
Our linear algebra was 2 semesters. First, we did applications, engineering, computer science approach. All calculations basically. Then we took the proof-based vector space semester. I don't know how anyone could have done the 2nd without the 1st to motivate it.
The genetic approach for LA is very different to the way it's done now. The whole subject was born out of complex analysis and quaternions, until Oliver Heaviside generalised - if that's even the right word - a quaternion into a vector, and split up the quaternion product into two different parts - the scalar part and the vector part, what is today a dot and cross product respectively. It would be a lot of effort to do LA like this at least for a first year, so while fascinating, it's a huge practical hurdle.
Dr. Rubin, Thank you for this very informative and useful video! I am sure you know the books by Richard Courant (and all) on Calculus / Analysis. They are very good / classic books. On this important and basic subject, in particular, the study and good understanding of multivariable calculus is essential for the study of PDE ( as you mentioned and I second it). Are you aware of the 2011 book “ Functions of several real variables” by M. Moskowitz and F. Paliogiannis? Perhaps worth mentioning these books. Thanks again.
I've said the same thing for years -- I'd rather have learned electrodynamics than calc 3 because the former was unmotivated and thus abstract in the wrong way.
awesome dude i LOve your basic, down to earth approach.. non pretentious and REAL talk about a very high intellectual subject...really enjoyable and already books you recommended..than YOU SO much !!
Thank you very much for this! Your preferred approach to Mathematics seems to coincide with mine. I love the abstraction, and I learn math for it, but it's got to be motivated first, preceded by computation of at least many concrete examples.
Gilbert Strang the author of the linear algebra book was/is a professor of math at MIT. So the book is top notch. Linear algebra is the most important math subject now. What if you have a thousand variables in the project you are working on ? You end up with a matrix of 1000 x 1000 = 1000000 members. Okay if you have a differential equations with 1000 variables you can solve the system of one thousand equations using the Runge-Kutta method on the computer.
Great advice on books. By the way there are a couple of superb books on statistics: _Mathematical Statistics With Applications_ by Wackerly et al, _Mathematical Statistics With Applications_ by Ramachandran, Tsokos. There's also another great book on differential geometry besides Manfredo: _Elementary Differential Geometry_ by Pressley. The author gives full solutions to all the problems in his book. It's not written in overly abstract way, nor does it focus on generalizations. There are books on algebra in the same vein too: _Contemporary Abstract Algebra_ by Gallian _A First Course in Abstract Algebra_ by Fraleigh The best books I know on linear algebra are _Linear Algebra With Applications_ by Kolman (Strang's text is just as good but I like Kolmans' better). _Linear Algebra, Theory, and Applications_ by Kuttler. This is a superb text. It's simply the best. BTW, it's more advanced than Kolman's. As to complex variables, I think the following book is great: _Complex Variables and Applications_ by Brown and Churchill.
What are the courses in this list that you have taught? How was your experience and *student feedback* when you taught courses using the following books? a) Second Year Calculus, Bressoud b) Analysis using A radical Approach to Real Analysis, Bressoud c) Number theory using Disquisitiones. Have you taught a first course in Algebra using Dickson's book? It does not introduce groups, rings and fields. Is that a good idea for a 1st course in Algebra? Also, did you really teach a 1st course on Algebra using Edward's book? While the book say that it needs only minimal prerequisites, it implicitly assumes the reader has already done some course in algebra (that is the probably reason why the book is a GTM).
Since aspects of linear algebra are used in multivariable calculus and ordinary differential equations, if I had it to do over again, I would have taken along with calculus II, saving time. --semiretired college mathematics and physics tutor, a former systems software developer.
There are differences of opinion on the introduction to proofs type of class. I'm in the camp that struggles with this. Because the focus is on techniques that can be used to prove things, I felt like I was working on proving things that I didn't really understand. That was frustrating for me and made doing the proofs harder--applying some technique I might understand to a concept I don't. I think studying some very basic mathematical logic and then paying attention to proofs whenever they arise (including in non-assigned exercises) in the lower-level math classes is a good way to go. Don't bother with a very theory laden mathematical logic class. Just study the deductive logic chapters in Schaum's Outline of Logic over summer or a break. It's very light (for math majors) and introduces a step-by-step logic system that can hide behind your paragraph proofs in math class. You can pick up other proof techniques as you go along. Induction is big in some of the more basic math classes that I have taken (I'm not a math major).
Advice from a guy in his mid 50s.....get rid of the word "should" from your vocabulary. It is used as a Guilt word, especially in religion, and parents use it so often. Guilt or repressed guilt is the cause for so many illnesses, literally and myself being an ex-maths student who is now a Relationship Trainer and Therapist who is Certified in Hypnotherapy and other modalities, knows what i am talking about. Use "supposed to" or "meant to' or "better off to" rather than guilt tripping people. Same with words such as "i'll get XYZ to call you back......as an example and not in this video. It is like, excuse me, how are you going to get someone to call me? What manipulative technique are you going to use? Rather you want to say or hear.....I'll ask XYZ to call you back. The #1 cause of cancer is someone with emotional conflicts and so they literally start eating away at themselves. Should is one way to start having ppl have those emotional issues. So stop it. 🙂 The stuff ppl including academics dont know and need to know.
I love this advice. As a CS major, I am irritated by the practicality gaps when you need to reach for some math. We want progress to accumulate without losing the progress with the loss of people. Put as much as possible into code; so that you can use the math to apply it to problems at huge scale. Deep Learning is mostly billion-variable-calculus ... implemented with nilsquares.
The Strauss text is standard for PDE courses, but it is dense and not easy to read. The Arfken text is horrific, yet Physics professors rely on it. The Mary Boas text for Mathematical Physics is another standard text, and is passably good.
If I had a Groundhog day, I'd love to spend a good 20 or 30 years on mathematics. Music and languages are currently taking too much of my time. Maybe it's being on the spectrum or whatever but I've never been able to get much into "real world" functionality as a person. I just want to study and try to understand everything about everything simply for its own sake.
I hear that. I.M. Gelfand said, "The most important thing a student can get from the study of mathematics is the attainment of a higher intellectual level.” The pursuit of exquisite beauty and ever-deepening insight is what gets many (most?) mathematicians revved up in the morning. Solving problems is part of that, just as creating music or sculpting involves plenty of problem-solving, but it's not necessarily the point of it, or what drives us. I do pretty much agree with Daniel Rubin that learning comes most naturally and easily if it proceeds on a need-to-know basis. But some people scratch that need-to-know itch primarily through the act of theory-building. I heard a fantastic self-quiz question the other day: do you learn theory in order to become a better problem-solver, or do you solve problems in order to understand theory better? 🙂
@@toddtrimble2555 To your question at the last, my answer would be We start by solving problems to understand the theory better till we understand the theory enough to solve the bigger and complex problems.
I think an intro to proofs or discrete math class is pretty essential before diving into analysis and abstract algebra. I dont think writing proofs come naturally to a lot of people and its essential to have a good foundation so you can feel comfortable when you take higher math classes. You can self study too, there are so many great books on proof writing that helped me jump into undergrad analysis and algebra. Personally i don't like math because of its applications, i like math because of its philosophical depth. And i believe that a lot of your book recommendations would be better suited for an applied math major.
My opinion ... it would be interesting and perhaps more beneficial, modern, and impactful to see a university develop a program where all math is taught as applied and computational aka applied numerical methods in a lab setting. I was lucky enough to have numerical analysis integrated into most of my ugrad courses thru special sections. But that course structure was cancelled after I completed the degree. I think people would grasp the concepts better. Especially for the long term.
Anyone here studied Spivak as a first calculus book? I was considering doing that (well, I did some calc at school, 20 years ago, which I've now mostly forgotten). Most other folks recommend Stewart or similar first. Which is actually how I ended up here - I was looking for recommendations for books that might get me up to speed quickly. Sounds like Prof Rubin thinks it's possible to jump straight in to Spivak? I do prefer that idea, just because the idea of re-learning something I already studied at school seems less appealing that learning something new. (Or rather, learning the same thing but in much more depth.) The exercises in Spivak are notoriously difficult.
I don’t feel that there is too much of a barrier to entering Spivak you can definitely jump straight in it’s just the exercises will be more difficult whether you’ve read another book or not (which I like haha). Also it’s more rigorous which contrary to this video I personally love.
@@nitroh7745 Thankyou! I needed to hear this. Since writing the comment above I did decide to go the easier route - I bought a copy of Stewart and started working through it, but I had already started to regret that decision - I was starting to realise I'd probably rather spend my time struggling with Spivak than breezing through something easier.
@@JimFarrand I would say it depends very much on how confident you are with proof writing, contrary to this video i feel like proof class (an introductory class not necessarily too deep into logic or set theory), is the most fundamentel class, sets are used in every single math subject so you ought to know how to manipulate them properly imo.
@@MonkeyDLuffy-gd6se i agree, a proof writing course covering propositional logic, sets, binary relations, natural numbers, induction etc. is a must in my opinion. otherwise i dont see how a reader should understand proofs by contradiction or proving the contrapositive of a theorem
contrary to the video, i strongly recommend grabbing a book on proof writing before a proof based book. trying to write proofs without knowing the underlying propositional logic behind proofs is like trying to learn how to read without knowing the alphabet. a classic book recommendation for this is "How to Prove It" by Velleman which has an abundance of exercises. A more modern book which is great for self learners is "Proofs: A Long-Form Mathematics Textbook" by Jay Cummings. His approach differs from traditional books bc he includes "scratch work" before giving the full proof so that the reader learns how proofs are actually formed
As a physicist, I have a very different view about abstract algebra. I love the subject, but I find the historical roots in the theory of polynomial equations boring. Much prefer to emphasize more modern applications.
I more or less agree with your recommendations. Strang is a classic, Rudin (also called Baby Rudin, to differentiate from his Real and Complex Analysis) is slick, but ultimately uninsightful. I had the Alfors complex analysis book when I took the course from Nahari. Nahari's Conformal Mappings is still in print and a classic. PDE's is a huge subject, I'm not sure what book I would recommend. I would that companion material to PDEs would include Integral Transforms (I love the book by Brian Davis: Integral Transforms and their Applications) , Calculus of Variations (with many examples from Goldstein's Classical Mechanics), and, of course, Finite Elements.
What do you recommend as someone who was a CS grad but loved math to continue interest if Grad school isn't an option? Tbh my math professors got me to love math more than I ever did in life. I hear the scary stories of self teaching because math is difficult to be a self taught approach since you can't have that direct one on one learning with a teacher of some kind. I dont think it's impossible but I feel like it can be done as a hobby. Any recommendations on approaching it? Just going to say great recommendations on books! I really want to get into coding theory and cryptography.
Yes, it's difficult to learn without a guide to let you know what's important and without the structure of program. It's a lot easier to learn if you focus on a particular problem you're interested in, even if it's just that you want to know how a specific piece of technology works. My advice may not be the best since I can't point you in a direction that is good for a career. But in terms of a general mathematical approach to coding and cryptography, it certainly helps to learn a lot of number theory. The books I mention in this video are good, and it might be a good idea to focus particularly on computational number theory (there's a great book by Shoup). Also complex analysis. I believe I mention in this video a book on coding theory by Stepanov that is interesting to me, but I have no idea how important the material in that book is in practice. You may find Conway and Sloane's Sphere Packings, Lattices and Groups interesting. I don't have a good general recommendation for cryptography at the moment.
Hi ! What the best calculus practical and problem solving critical thinking mathematics textbooks Calculus concepts 3 edition by Stewart Calculus 3 or 5 edition, Stewart Calculus 9 edition, Larson Calculus 12 edition, Thomas Thank you
Great to see some interest in the PROMYS program! At PROMYS, students are given daily problem sets whose material is ahead of the lectures. With the problems as a guide, students are forced to develop the theory on their own without a text. It's a tough but incredibly gratifying and valuable experience.
Wow, that's amazing. I wonder if it's possible to scale that approach for students who don't have access to promys.. like giving students online problems a little higher than their ability and helping them develop the theory independently
I like the idea of emphasizing applications as an undergrad math major. For me personally, working the proofs gives me confidence in my memory of the techniques, but maybe unmoored theory is best left mainly for the small few who go on for graduate degrees in math.
Could you please do a detailed review on differential geometry books like the 2 books by Barrett O' Neill and a few more complex analysis books like the one by Gamelin.
Wow, strong disagree about topology too. So much of modern math is figuring out what results you get from what assumptions. What comes from a set being a metric space, a vector space, a group, etc. And a topology is the minimum set of structure you can get on any continuous space.
@Daniel Rubin, I will be getting to uni next year and I overall value being conscious of the historical development of a subject, as I found this fruitful will supplementing a physics course I had at high school. So what resources would you recommend for doing this in math (as it seems like you are a math historian)?
I think Susanna Epp, Discrete Mathematics with Applications, is a reasonable text to use for a standard "proofs" course. Most proofs courses actually do use some real mathematics as a training ground for writing proofs, such as elementary number theory or elementary graph theory.
Good question. The texts I'm most familiar with treat the subjects separately, but I know some programs combine ODE and Lin Alg into one course, especially if the emphasis is on numerical methods. I think each subject deserves its own course, but since I like Gil Strang's other books, you might check out his Differential Equations and Linear Algebra.
It isn't exactly what you have asked for, but you may find useful "An Introduction to Linear Analysis" by Kreider, Kuller, Ostberg, and Perkins. I believe that it is out of print these days though. It does a decent job of developing enough linear algebra to support its main topics (DEs, Fourier analysis, orthogonal series, etc)
SO REFRESHING to see someone pitching the idea of a math major interested in more than rigid formalism and proofs, proofs, proofs. Recognizing that there is a time & a place for it, and that real world applications are an important motivator for many people. So often in my quest for good mathematics resources I stumble across people who sneer at any books that include applications, as if knowing how the math might be used in the real world is somehow a bad thing. What's wrong with starting with some historical perspective, learning about the problems people were and are trying to solve, and then--once a person's curiosity is piqued--dig deeper into the theory of the thing? So thank you! I appreciate you.
This is a gem of a video, like finding a hidden treasure on RUclips.
As a mid-age man who graduated from a decent Math department 20 years ago, don't remember much details but still can't help reading some old math books once in a while, I really enjoy listening to your genuine thoughts and experiences with these books. I'm sure younger generation will benefit even more from your sharing. Thank you, Professor Rubin, for making this great video!
Unless I'm completely mistaking your views from this, I think your (clear) opinions about Algebra and Grothendieck's vision of mathematics are just wrong. It was ultimately highly abstract algebraic techniques that lead to Wiles' proof of FLT--a very concrete, historical problem, indeed. While I agree with you that not everybody needs to walk that road (and I, too, think that mathematicians broadly disregard the virtue concreteness), I don't think that the solution is to dismiss abstraction and generality as useless; such an attitude is rightly seen as regressive. However, I will grant you that many mathematicians seem to have the opposite problem, and to a much greater extent. In that respect, I find your views somewhat refreshing and intriguing. Grothendieck was certainly right about one thing: you've got to be willing to stick to your guns and express your opinions! Sub earned. I look forward to seeing more of your content in the future!
my proof class helped me a ton personally, but it wasn't untill I took graph theory did I realize the importance of a well defined approach to describing mathematics and alongside direct applications. Now I apply that approach isomorphically from course to course.
Agreed, my intro proof class covered basic set theory and logic, plus taught us techniques like proof by contrapositive, induction, counterexample, etc, which my analysis course took for granted that you already knew.
Thank you for the list of great resources. It's surprising how university courses have stripped away most the history and context of math topics. After doing well in first year calculus I realized I'm only at the midpoint to truly understanding it. The rest of the subject falls on the student to complete on their own -- with the help of a few good book recommendations!
I think it's the overemphasis on "useful" maths driven by profit based institutions. It's taken for granted that the innovation behind all the profitable stuff requires immense creative energy and time yet that is where the value originates from, despite lack of funding.
The first and last time I used "differential forms" was in my highly abstract freshman year vector calculus class. I have a PhD! While I enjoyed the course I think I and certainly most other students would have been much better served with a concrete, historical approach. Thanks for your videos, I'm interested in reconnecting with the historical and technical basics of the subject and I appreciate your approach and recommendations. I already picked up the Fourier Analysis book which showed up in my algo.
edit: Now that I've watched the whole video I want to expand on how valuable I think this advice is. I take it at face value that you believe this approach is valuable for research-track mathematicians, and I am inclined to agree without being able to add any force to that. But as someone who went from a PhD into industry I think this advice is even more relevant and valuable. There is such a value to being able to formulate and solve concrete problems using a mathematician's approach, and the skills I learned in those few years of rigorous mathematics have paid continual dividends over my career. But I would have been even better off with more concrete facility in the basics, simply writing down, manipulating, and solving equations, having all the machinery of ODE, linear algebra and statistics at my finger tips, and the background in numerical methods to transform those facilities directly into code.
Looking back on my undergrad and early graduate years, I can see clearly a fetishization of abstraction and generality when the essence of mathematics is solving problems. I hope that kind of hierarchical thinking can be done away with and replaced with a relish for tackling any concrete problem with rigor and insight.
Maybe it is a US thing, or in places close to old important established Universities. In my own experience, in Europe, there is a lack of pure Maths modules in many departments. You get your undergraduate degree with all the analysis you may want up to lebesque and measure, through complex, functional, PDE, Hilbert, etc. you get your basics in linear and abstract algebra but not that far, basics for probability and some geometry up to differential geometry, and a basic topology. It seems very complete but everything is there to help you in Physics and engineering, not pure Maths research. A typical undergraduate degree feels roughly 75% Applied/25% Pure at most, even if the applied is more rigorous than you can see in any engineering or even in Physics. BTW, there are also mandatory modules and optional in QM, GR, Cosmology, Astrophysics, etc. but nothing in number theory of any kind, and you can barely see graduate level courses with Lie, commutative algebras, representation, any advanced topic in algebra in fact, or topology or algebraic … anything. UK is different in that regard but you need to do graduate level Maths to find advanced topics in Analysis that you can find at undergraduate in many places in Europe.
Edit: But I have to say, doing some good Physics alongside, Analysis was a lot more comprehensible, motivated and finally, easy to grasp in full. Even multivariate, after studying force fields, was like, but of course is reasonable. Disclaimer: I was doing a Physics degree not a Maths one, but the Analysis modules were the same for both undergraduate degrees.
The essence of mathematics IMHO is understanding the “logical structure” of the Universe. Solving problems is secondary and a natural consequence of that understanding.
super insightful and love the inclusion of the “genetic“ books. I think modern math classes sometimes focus too much on making people memorize a bunch of random theorems and formulas. Without actually trying to teach students the intuition or motivation behind certain rules or tricks in math. I’m the the type of person who can’t memorize any rule or formula unless I have seen it proven, in a clear way.
You definitely helped me with all that extra work needed to find the right book definitely gonna check out all your suggestions. Thanks!
I have been a math teacher for over 20 years. I never understood the pedagogical importance of the historical approach. Your video made me want to start studying math again, reading books with more historical and applied perspectives. I know that students really like this approaches.
Great to hear from a teacher!
I listen to this front to back for an hour. I don't even understand some of the advanced math topics and courses you're talking about, but I enjoyed it. Feeling more excited about learning complex math in college!
Thanks a lot! Glad you’re excited to learn math!
Brilliant!! Also, thank you for pointing out the additional texts!!
Great. Thanks for the recommendations for an eight-year undergraduate math degree.
_Motivation_ and _genesis_ provides an interesting approach to mathematics. Thanks!
Glad you think so!
The thing I like about this approach is that you are always grounded. Richard Bouchards also emphasizes looking at exmamples and seeing what is going on way before you develop or understand the general machinery. And I would say he's done pretty well for himself following this approach.
Listen to the 51:48 Summary and general advice at the end. Listen enough times to break the spell and until it truly sinks in.
Great video! I would definitely have benefited from hearing this when I was finishing high school. Excessive formalism, lack of direction, and lack of motivation are definitely plaguing university-level education in mathematics. I was glad to see some of the great exceptions listed in your video (e.g. "Algebraic Number Theory and Fermat's Last Theorem", but really many other examples in your list; it's mathematics with a purpose!).
I'd like to add a couple of suggestions to your list, that I honestly think would be a pity to not include. "Theory of Computation", by Michael Sipser, should fit in a modern undergraduate mathematics curriculum (e.g. proves a modern formulation of the Incompleteness Theorem); there's probably space for other courses in Logic. "Probability with Martingales", by David Williams, was also a great book; rigorous, yet practical; it's an analysis masterpiece, and also exposes basic measure theory in a way that's actually useful, leading all the way to a proof of the strong law of large numbers.
Thanks! I'll check those books out.
With Sipser and Rosen for Discrete Maths you can chew away any undergraduate CS major course in full…
Kudos to you for the Sipser, a masterpiece in my opinion like the Spivak.
They could almost teach mathematics like a history course with motivating examples so people build confidence. Often books are written assuming a lecturer is illustrating and the book is there for exercises, which is a shame when "there was a tricky/interesting problem so someone formalised it and found patterns" is so much more fun
Great video. I am studying mathematics by my own and this is very helpful
Complex analysis is easier than real analysis. Highly recommend Visual Complex Analysis as a companion text.
I am a 4th year undergraduate student in math and physics.
I appreciate a lot the general point presented in this video.
As a student with a background in physics in math courses, I feel kind of sad for fellow classmates who just don't have motivation to lean on when learning various math courses. It is mostly present at any analysis related courses, whether it's Calculus, Complex Analysis, ODEs and PDEs, Probability and so on. Even Metric and Topological Spaces which isn't necessaripy analytical, can be very baffling to many students.
The reason for that is that as Daniel expressed, abstract theories with no motivations can only take you so far.
I do believe that math is expansive enough on it's on and doesn't need other fields to justify it's existence, but I also believe that the real beauty and value of any subject truly shines only when considered in relation to other subjects from the same field and different fields.
Many pure math students lack this kind of motivation and lense to view the subjects from, and I think it's such a shame.
I encourage anyone who studies math to try and expand their point of view in some kind of way.
It can be by reading books such as the ones Daniel suggested, I haven't read anyone of them but they seem promising enough.
It can also be combining the math studies with other field, such as any science or engineering.
I don't agree though that every single course should be tought that way. There is a huge merit to learbing things abstractly, as well as computationaly, as well as more intuitively motivated. A good mixture of different approaches to teaching math will give the best value to the students, in my opinion
I think that the techniques and 'reasons why' of the theory is the most important part to learn the subject.
How say you are a fan of applied math without saying it loud for around 1hr? : This should be the title of the video.
Joking aside, great video!
Dear Mr. Rubin: very interesting video! I have discovered you lately and already enjoyed some of them a lot (your interview with dr. Hamkins or your apology of McKean and Moll's books (just bought a copy for myself). Here in Europe we don't really choose much within our Math major. I get the impression that you are of a very applied and analytic persuasion - closer to Morris Kline than G. Hardy. I think I have a softer spot for 'useless' math as, being from a Humanities background, I am accustomed to appreciating something because it is true and beatiful, and to not giving a damn about practicalities.
Thanks! I realize that students don't have complete say in courses or textbooks, but I hope my videos can help anyway. I greatly admire Hardy as a mathematician, but I disagree with his sentiments about the worth of math. I have a podcast episode about the idea of mathematical beauty you might find interesting: ruclips.net/video/s3GXKKyY-m8/видео.html
I would say for LA students: understanding Hilbert spaces is a must. Opening the floodgates to functional analysis becomes so incredibly useful.
dude....i LOVED that !!..." Focus on SOLVING PROBLEMS. "...THAT. has been my own Mantra in my Mathematical Journey.....thank YOU SO much..." learn the TECHNIQUES. "....
Everybody has their own way. I'm older and studied calculus in my Freshman year at college. My high school geometry class used a translation of Euclid's Elements book one, so I knew how to do proofs. To my dismay at first, my university college taught all math including beginning calculus, rigorously. However, after a few months, I started really liking the rigorous approach, even though I was much better suited mentally to the practical applied approach. I struggled a little, but when I took my exams in graduate school (a different school than my undergraduate) it was easier than I thought it would be (e.g. one of my exams was based on "big Rudin" ).
Damn, I'm not a math major (Wish I did a double/triple major in math but kinda late now lol), but some of your points are indeed spot on. I took discrete math, theory of probability, and multivariable calculus this spring semester ('twas a tough semester haha but glad I got it through especially with covid lol). I really liked all of them besides calculus 3, but I wish professors have taught us "how did mathematicians came up with these theorems and definitions and what is the motivation behind them" so that we get a chance to think more like a mathematician. For example, why did mathematicians came up with vector calculus? What is it used for? What was the motivation behind the jacobian matrix? How did Gauss come up with Gaussian Curve? etc. I'm the type of guy if introduced to the motivation behind each theorem, definition, it really clicks and hence the "ah-ha moment". But with calc 3, the professor just taught us the definitions and how to plug numbers into the formula. It got so freaking dry that I end up not paying much attention and end up getting a bad grade. It is honestly my fault that I didn't do well but your way of approaching math will definitely help a lot of math and non-math majors.
Thanks so much! Glad to hear you're curious about the historical motivation of mathematics. Unfortunately, only a minority of math professors pay attention to it, so you're unlikely to see it in class. Trying to change that! 2nd Year Calculus by Bressoud should answer your questions about multivariable calculus if you're interested.
Hi again. Just curious, what's your opinion on classes like Mathematical Logic or Set theory? Do you think they are just "math for math sake"?
Great question! I don't know enough to have a definitive opinion. I took one course as an undergrad on Logic and I retained nothing because I've never used any of it. I think the overwhelming majority of practicing mathematicians can be successful without paying attention to set theory or logic. I'm also dubious of that work in these areas will have any philosophical implications for mathematics. But I'm aware of some issues I find interesting that do make contact with nontrivial parts of set theory and logic, like Ramsey theory or the Tarski-Seidenberg theorem in real algebraic geometry.
@@DanielRubin1 Hey Im sorry for keep asking you all kinds of questions here. I just recently changed my major this semester from statistics to math so I just need more guidance lol. Im taking real analysis this semester after taking spivak calculus. School is about to start in 3 days and the syallbus says that we will follow a so-called "Moore Method". I emailed the professor to learn more about this pedagogy that they are implementing this semester, turns out its something that is completely new to me. So basically, how it goes is that we will be given definitions only and using that definition, we have to prove given theorems. During class, one person will have to present his/her proof and the rest of us will have to critique the proof. In other words, the professor is just there to merely guide the discussion and lead us to the right direction. Do you personally think this is a good way to learn math? I think it will work well but Im a bit afraid that the topics that we will be learning will not be well motivated if that makes sense.
@@axisepsilon514 I've never been part of a class like that, but I've heard of the method. I'd guess that taking a class like that will force you to understand the definitions, theorems, and proofs really well, and give you experience presenting, which are both good things. But on net I don't think this method is a good idea. It's very easy to miss the point of what's going on if you focus on definitions and theorems and not motivating problems. You might have to look at other resources for some of the things I think are most important about real analysis. Working on some real analysis content for this channel, coming soon!
I’m an applied maths grad (4 years of maths). I’ve been working in the IT industry for the last 30 years, but recently started studying maths again. I can see you have a very practical bent, and I’m the same. In the vid you don’t recommend doing proof theory; just learn how to do proofs as you go. That was my attitude as well. But recently I’ve been learning mathematical logic, and I’m really enjoying it. What’s the use of logic? Well I guess it tells us something about the systems that we use to do mathematics and their weaknesses. So far I’ve done the Godel completeness theorem (which is not that straight-forwards) and will do the incompleteness theorems soon. There are plenty of techniques to be learned as well which I suspect will be valuable in other areas.
I also think doing some work in mathematical logic is helpful for proof writing.
This video points out the weakness of math majors. Pure math majors will need an additional degree in order to find jobs outside academia. Something that this video says and repeat many times is the importance of solving problems over study abstract theories. In that sense the abstract algebra example in the video was excellent. The video points out several times the importance of numerical analysis and statistic, which are extremely important in order to have a complete landscape of mathematics and finding jobs, internships or summer jobs. Honestly, the best recommendation is a double major of math with computer science or business.
From what I understand, your recommendation is probably a good idea for majors in the other sciences as well.
@@DanielRubin1 your observation is correct. My comment is valid for other majors too. The double major with math is the best choice. In fact, I read that math as part of a double major is the most popular choice in some universities in the US, I think that was at some university in Dakota.
Some comments:
0. If you think there is any chance that you will go into industry, I recommend classes in numerical analysis at the level of Rolston and Rabinowitz, computational statistics by Gentle, numerical optimisation by Nocedal and Wright, and numerical DE's like Arieh Isreles.
1. Another rigourous calculus text is Courant and John.
2. For Linear Algebra and vector Calculus, You might like Geometric Algebra, Linear Algebra, and Geometric Calculus with Alan MacDonald's books. For those of you who like Linear Algebra in the "Down with Determinants" style but hitting thinks like the Principal Axis Theorem, try Larry Smith's book.
3. In Advance Calculus and Classical Real Analysis, I found books written by applied mathematicians are better motivated in this vein.
4. A standard undergraduate Algebra textbook which gets to the insolvability of the quintic is Pinter's, which you can get from Dover. Saracino's looks good too.
5. Good introductions to probability and statistics include the introductory books by Ross and by Wackerly.
6. I think you should take a course in combinatorics and another in Graph Theory. I like Tucker for Combinatorics and Chartrand for Graph theory.
7. There are good introductory applied algebraic topology textbooks, like Ghrist.
8. I like Willmore's differential geometry textbook. DoCarmo's books are good too.
9. If you have a feel for what modern physics is like, I recommend books like Nakahara's for introductory topology and geometry. (There are others.)
10. If you are going to a mid-tier or higher grad school, you will need to know abstract point-set topology before you start. The level is the first 4 chapters of Munkres's _A First Course in Topology_.
Thanks for the book recommendations!
Thank you for this video! I was fortunate enough to have professors over a decade ago who had similar teaching philosophies. But it is a rare view. I'm glad you shared both your focus on "building up from the concretes" philosophy and specific texts! It'll be great for some self study.
Wow, this really was a great video for someone going into math (myself); I have had many anxious thoughts about proof classes and number theory cause I am simply not interested in these areas as I stand currently, but I do love differential equations and numerical methods. Really happy to get this gem in my recommendations. Greetings from Sweden!
Glad you liked it! I think pursuing just the math you're interested in is a very good way to do it. If you ever need something else, then you'll have a good reason to learn it so it'll be more interesting.
I am a math student in Germany and unlike you i love theories and all abstract stuff. Actually i am going towards algebra (specialization) and i can remember that we had algebra groups rings and so on even in first term and i am really gald about it. I think you need to know all these stuff to just show and prove more stuff. All the computaional stuff are possible because people made it with these theories possible and thought through it. You need for example Rings and ideals so you can solve equations not numerically but symbolic with beatiful math and algebra you need Galios theory. You need algebraic topology in theoretical Physics. Despite the fact that I am doing Algebra i love measure and integration theory and i was wondering about your comment on measure and lebesgue integral. I am asking myself what would you say about Category Theory or invariants theory?
I like your "follow your curiosity" mentality I think that pushing that idea more in classes would lead to undergraduates learning more mathematics on their own and practicing math more often and thus be a great aid in their gaining of mathematical maturity. I would, however, be careful with the more utilitarian aspects of your philosophy. As someone who studies a lot of category theory, I may be biased, but I gain a great deal of motivation from seeing vast generalizations and seeing how far they may reach. Of course, the heavier machinery that you were talking about is outside the scope of undergraduate education but I fear that pushing students to favor more concrete points of view in terms of mathematics may bar them from viewing the beauty of math and discovering that they may enjoy math for math's sake (I also HEAVILY disagree with your recommendation of not taking any topology course, but I could just be biased because it was my favorite course as an undergrad and am now really into algebraic topology).
But I also understand that I am often in the minority in regards to these things and approaching math like this and like an art leads to burnout and confusion for a lot of people, especially less mathematically mature undergrads.
You certainly picked up on some of my controversial attitudes about math. I do favor an approach to math that's about solving concrete problems, and I worry that doing math for it's own sake or seeking vast generalization will not lead to good results, and it's very dangerous for young researchers who need to find jobs. I don't think I mentioned category theory in this video; I think it's nice as a philosophy or organizing principle in math, but it's absolutely terrible as actual mathematics. I wonder: does category theory or any move to a much more general framework ever help solve problems in which we were already interested in a specific setting?
I hope to address these issues and make my case in future videos. I have a podcast episode ruclips.net/video/s3GXKKyY-m8/видео.html that addresses the notion of beauty in mathematics. My Tricky Parts of Calculus series is a set of math lectures that emphasizes the hard problems and how things were done in history. I'm trying to show that progress in math was always made by solving concrete problems, not by developing at the start the appropriate general framework, even though this is how most math is taught.
@@DanielRubin1 what makes you say that category theory is a terrible way of doing mathematics? In many ways it is superior, or at least more natural, than more things such as set theory and in some areas of computer science is a far simpler and elegant point of view with regards to the theory of computation. I'm sure that computer science could offer some examples of problems you were talking about. But in addition just looking at mathematicians like Grothendieck and Serre allows one to see how these general points of view solved very difficult problems such as the Weil conjectures. In fact Grothendieck's whole approach to solving mathematical problems relied on him constructing theories around problems until the theory was powerful enough to, at least in his mind, render many already existing problems "almost trivial".
@Taylor You anticipate several topics I'm hoping to talk about in future videos! I don't like the study of category theory because I'm interested in problems, and it is almost always hopeless to try to solve a problem by abstracting away some properties and considering a more general situation; the more general theory must still capture the specific structure that the result depends on. Grothendieck really did believe that math was about finding the "correct" most general framework, out of which results would trivially pop out. I think that's a fantasy, not how math was ever really done. I have heard the claim that it was the Grothendieck approach that solved the Weil conjectures from many people, but that is very much up for debate. Hasse's proof of his inequality and Dwork's proof of the rationality of the zeta function came first and neither had anything to do with Grothendieck's work, nor did Deligne's proof use the theory Grothendieck thought was required. All these proofs required analysis; see also Stepanov's method for counting points on curves. Grothendieck was a great mathematician, but his philosophy has led mathematics astray.
To add another point to the debate here: Maths departments all around my country were half full at most between forever up to 2015-ish. 2007-2014 for example, Universities with 70 places struggling to get 35 students, and those were prestigious ones. Nowadays they have between 700 to 1000 applicants per year, for the same places. Grades to enter went to the roof and it is almost impossible to study. The reason? Statistics and Data Science. Post 2017 it is all what any student want to do. Also, in the long past, no banks used Maths graduates but graduates in Econ, Finance, etc. Today it is only Maths what they want, and a ton of graduate schools, PhD and Master’s in topics that lead you to the most competitive and rewarding jobs have Maths as a major or double degree as a prerequisite.
Maths didn’t change. The problem now is the number of students that loves Maths is still 35 of those 700 applicants. And you may choose the wrong 35…
Great content and some really useful advice. I would like to disagree on the point that an introduction to proofs class (some programs call it a "transition to advanced mathematics" or something similar) is largely unnecessary. I would say that this class for me came right before Analysis I and II, and Abstract Algebra I and II, and was tremendously beneficial and a very practical precursor to these heavily proof-based classes. I would agree that, yes, it is more intuitively productive to actually apply methods of proofs to specific examples, but had I not received the conceptual and logical foundations of the language of proofs, methods used, and notational conventions, I would have certainly struggled more in the higher-level classes. I think that Analysis and Abstract Algebra are two of the most difficult courses for an undergraduate, and trying to learn the content of these subjects while simultaneously learning about proofs makes for a more challenging experience, in my opinion. I was able to focus much more on relevant concepts having already acquired a good solid foundation on things like set theory, the real numbers, completeness, cardinality, proof by contradiction, pigeonhole principle, induction, and even more specific methods like proving each set is a subset of the other to show equality, having to prove the converse for biconditionals, etc. These were things that immediately flipped a switch for me whenever I encountered them in later classes and I was able to more readily understand the logic behind such proofs. I would liken this to learning how to write programs that perform specific functions in, say, Java, for example. One could certainly learn the language by being introduced to particular concepts, control structures, classes, or functions while simultaneously learning their immediate application to specific uses. But I would argue that first having a logical foundation behind the concepts, how object-oriented programming works, how computers store information, what the computer actually _does_ when a certain function is called, etc., can make learning the examples much more effective and intuitive because the background is there. Again, this is just my opinion and you have clarified that your points are your own opinion as well, so there's no wrong answer here I don't think. I just thought I'd give a counterargument based on my own experience. This is wonderful content, though, and I would definitely like to see more of this kind of stuff on YT. I think you and The Math Sorcerer are really the two leading channels presenting information and advice on things _outside_ of the actual course material and more along the lines of career options, educational advice, learning techniques, and broader conceptual notions that really aren't touched on anywhere else. So thank you tremendously for that.
Would a course on “Discrete Mathematics” be the same as a proof writing class? At my university, I couldn’t find a specific class for proof writing, but the description for discrete math contains many of the same topics that you listed.
@@bendavis2234 I wouldn't say that Discrete Math is necessarily a "proof writing" class in that its focus is not on writing proofs but on studying discrete (finite/countable) objects. However, what it does do is introduce many of the logical tools you'll _use_ to write proofs, so it's kind of a precursor to proof writing.
Most Discrete Math courses cover the basics of set theory including basic axioms, collections of discrete elements, integers, rational numbers, implications, tautologies/contradictions, conditional statements, and propositional logic, all of which will be used later on when writing formal proofs.
However, there is a major difference between proofs in Discrete Math and proofs in higher level undergrad math such as Analysis, Algebra, and Topology. The difference being that those subjects deal with *_continuous_* sets of elements (such as the real numbers). As it turns out, this is a huge step up in complexity, and most of the major theorems in your undergrad courses deal with continuity (Calculus, for example, is literally *_about_* limits and continuity).
So I think that most universities would consider your first *_real_* proof writing class to be Analysis, or possibly Linear Algebra. As I said, my university offered an introductory proofs class that came *_after_* my discrete math course, so it can depend. In fact, a lot of discrete math courses are more driven towards computer science (it's very important to programming). But still, you will learn the foundational tools and building blocks that will later be used for proofs, so it is still a very important course.
@@jinks908 Okay thanks that's good to know. Do typical courses in Real Analysis expect you to come into the class with knowledge of proof writing? Discrete math will at least cover the basics, but of course there are many key differences that you mentioned. Also, I'm not sure how much assistance you'd get in basic proof writing techniques in a class such as real analysis. It probably helps to have some previous experience when entering the class. It probably depends on the university, too.
@@bendavis2234 Most undergrad programs offer Analysis I and II. Analysis I (sometimes Intro to Analysis) will typically do a review of proof writing, techniques, and terminology during the first couple of chapters. For example, my university used Steven R. Lay's text which covers logic and proofs and set theory in Chapters 1 and 2. So in an introductory analysis course, you will usually review most of the material you'll use to write proofs. However, this is only a review, and so you'll need to at least be familiar with most of it.
But you shouldn't be concerned too much as you'll need prerequisites for any Analysis course anyway. You won't be entering an analysis course without having taken some kind of proof/set theory class. Many universities have a Transition to Advanced Mathematics course which covers this. My Transition course actually covered most of the material that a Discrete Math course would, so I didn't even need to take Discrete.
The other thing is that as you move into advanced math courses, (those beyond calculus/trig/algebra, etc.), your classes will shift their focus from computation and into proof-based study. So any university math curriculum will teach you proof writing first since all of your upper classes will use it. Your junior and senior years will be almost solely proof-based, so you'll have plenty of exposure to it. Not to mention all of your textbooks will have several proofs already in them. You will learn a TON of proofs just by example. Usually, you'll see a definition, followed by a theorem which is a consequence of that definition, and then a proof that shows why it's true. Then you'll read the next theorem and prove it yourself, and so on.
In essence, proof writing is the bulk of your undergrad education. So it's not really a matter of coming into it with a knowledge of proofs. The main point of a math degree is to *_learn_* proofs. All of your courses will build on each other. Thus you'll certainly be given all the necessary ingredients, but it will still be up to you to make sure you understand it. So pay very close attention to your proofs/set theory courses as they will be the lifeblood of your math education.
@@jinks908 Thanks for the info. In my case, I’m doing an applied math major, so I’m not sure how much different it will be in terms of proof based coursework. For example, most of the classes are taught separately in the applied math department with courses like “Discrete Math for Applied Math”, “Advanced Engineering Calculus”, “Real Analysis with Applications”. I’m assuming that proof writing will still be important for these courses, but I’m not sure to what degree. It’s weird how little overlap is between the Pure Math and Applied Math departments at my school. I could go through my whole major without taking a course from the math department. I’m not sure if your school was similar in this way.
This is gonna be so useful for me, I'm in my last year of high school and after 2 years of college im starting a degree in pure math, thank you so much
You may have swayed me and a lot of other people from learning a lot of abstract, incomprehensible and boring (at least at the time of learning) mathematics so that we can get to the good stuff later. I, at least partially, have had the idea that I have to go through some amount of apathetic study so that one day everything will finally be clear and I'll be able to be creative and do what I wanted to do all along.
Indeed one of the main goals of this channel is to show that there's a better way to learn math--one that is guided by problems students are at each stage prepared to appreciate. I find this way better for comprehension, motivation, retention, and for training to solve novel problems. If you want the math you learn to be useful to you, you must be invested in it. It has to be concrete to you. It has to be a source of meaningful questions that you have some tools to investigate.
Looking from a completely abstract sense is the actual refreshment. Although I must say that I appreciate applications secondarily since it aids in accurate visualization.
That's my personal view.
Just do applied math if you absolutely hate abstract thinking (although you still need a lot of it, since it is mathematics, and it comes with the territory). Or... do engineering or physics lol.
(Video is good, by the way, it is still important to get both sides in my opinion)
43:17 What? Functional Analysis isn't about the "fine properties of functions", a more apt description would be infinite dimensional linear algebra, because infinite dimensional vector spaces mostly arise as function spaces
The reason to study those spaces, all of which come with a norm or metric of some kind, is to gain control over some property of functions that is the key quantity that determines the behavior of solutions of a PDE or variational equation.
@@DanielRubin1 I disagree, you're looking at this from a very applied perspective that completely overlooks the heart of the subject! Ironically enough it's almost never about the functions themselves.
Also not all spaces studied in FA are necessary metrizable, a good example being the space of distributions which is useful in PDE theory
I think your analysis bias is showing strongly in this video. There are plenty of students that will find abstract algebra intrinsically interesting. Of course, any algebra instructor worth their salt will begin by exploring examples, as the definitions are made to serve the examples that appear naturally, including symmetries of shapes, matrix groups, and actions on vector spaces. Symmetries of roots of equations can certainly be covered in an introductory group theory course - I think Galois theory, however, is far too much for a first algebra course.
You also didn't devote much time in this video to the different branches of algebra. Groups, rings, modules, and fields / Galois theory all have a different "flavour", and not all of it is motivated by solving polynomial equations. Ring theory has its origin in number theory and geometry. Anyone who found linear algebra interesting will be intrigued by the generalization presented in module theory. And of course, going through these examples and seeing the terms "homomorphism" and "isomorphism theorems" pop up over and over will provide fair motivation for category theory.
However, I do strongly agree that representation theory should make an earlier appearance. It could certainly take up a few weeks in a dedicated group theory course before students move on to a dedicated course.
As an algebraic geometer, I completely agree with your thoughts on the subject and your recommendations. It's an extremely broad subject, and one really needs a strong foundation in the classical picture (i.e. without schemes) before moving onto more abstract sources. If you take an introductory alg geo course and you find it interesting, then I'd recommend taking commutative algebra and then start learning about schemes. Eisenbud and Vakil, respectively, are my preferred sources. Not everyone needs the full machinery of scheme theory, but most people could benefit from following at least the first few chapters of Vakil.
I certainly do have a bias. Essentially I want to encourage math students (and professors) to work towards addressing problems. I tried to recommend an approach to algebraic subjects based on problems students could appreciate at that point, as opposed to just learning theory.
What would you say are the branches of algebra?
Great guide! I have always been bothered by how to schedule my math courses, so this is really helpful to me.
Glad it was helpful!
Two great books for topology are Topology by Munkres as an introduction to general/point set topology, and then Lee's Introduction to Topological Manifolds for Algebraic Topology. DON'T start with Hatcher, topology's actually really interesting on its own!
In addition, check out Introduction to Topology: Pure and Applied by Adams & Franzosa. It's full of applications of general topology and can be a stepping stone to recent books on applied algebraic topology topics such as topological signal processing and topological data analysis.
Great advice. Thank you a lot! I've subscribed instantly.
By the way, I fully agree with almost everything you said. For example, multiple variable calculus should not be rushed and should take two semesters. I also could not agree more that numerical methods should not be ignored. But more importantly excessive formalism and generalizations are not good, especially for those who don't major in math. And I agree that even for a mathematician getting mired in them and tuning out everything else is wrong. A huge like for the video. I also think geometry should not be ignored in colleges and should not be de-emphasized as we have it nowadays (almost no geometry at all).
As to books, I made a separate comment on them.
You make several comments along the lines of “these abstractions are not useful; learn the practical applications”. Perhaps this argument has some merit if one is planning to work in applied math or physics or engineering. That said, had I followed this advice as an undergrad 50 years ago I would never have made it through the 1st year of math grad school at Berkeley.
What I'm trying to say is that I'm very skeptical of learning abstractions for their own sake, and that progress in math has been driven by the need to solve concrete problems. I certainly find it easier and more enlightening to learn math by seeing how problems came up and how they were solved. Often the important math problems came from practical concerns, and I really do advise pure math students to be aware of real-world problems. It very well could be that in the artificial environment of some math departments appreciation of particular abstractions is prized much too highly. In the end the goal is to make worthy contributions to science.
@@DanielRubin1 I would say that in the end what matters in pure Mathematics is trying to understand the logical structure of the Universe. Mathematical applications should at least in principle follow naturally from pure Mathematics even if historically that process is often, perhaps usually, reversed.
WONDERFUL Video ... nicely dome and very helpful! Thank you!
Thank you for a great video! I am retired and self-studying 'all' the math I missed in my younger years. Currently finishing Calculus 2 and starting Linear Algebra. One of the subjects I'm contemplating for the future is 'Discrete Mathematics'. I didn't hear you mention it. Is it know by another name? Do you have any suggestions as to what subject(s) (textbooks) I should consider after Linear Algebra? I'd welcome your thoughts.
It seems like you're creating a de-facto applied math curriculum. Fantastic video!
Thanks! But I'm no expert in applied math, and I wouldn't claim to know how to construct a curriculum for that field. With the limited exposure I have to the concerns of applied math, I'm trying to create a curriculum for students who want to do "pure" math that brings in some of the important and useful material from the applied world they might otherwise never learn about.
Our linear algebra was 2 semesters. First, we did applications, engineering, computer science approach. All calculations basically. Then we took the proof-based vector space semester. I don't know how anyone could have done the 2nd without the 1st to motivate it.
Sounds like a good way to cover linear algebra. It's probably so important to enough people to justify 2 semesters.
The genetic approach for LA is very different to the way it's done now. The whole subject was born out of complex analysis and quaternions, until Oliver Heaviside generalised - if that's even the right word - a quaternion into a vector, and split up the quaternion product into two different parts - the scalar part and the vector part, what is today a dot and cross product respectively. It would be a lot of effort to do LA like this at least for a first year, so while fascinating, it's a huge practical hurdle.
Dr. Rubin,
Thank you for this very informative and useful video! I am sure you know the books by Richard Courant (and all) on Calculus / Analysis. They are very good / classic books. On this important and basic subject, in particular, the study and good understanding of multivariable calculus is essential for the study of PDE ( as you mentioned and I second it). Are you aware of the 2011 book “ Functions of several real variables” by M. Moskowitz and F. Paliogiannis? Perhaps worth mentioning these books. Thanks again.
Excellent video. Thank you very much!!
The book Elementary Number Theory by David M Burton is brilliant for a nice introduction to the subject.
I've said the same thing for years -- I'd rather have learned electrodynamics than calc 3 because the former was unmotivated and thus abstract in the wrong way.
awesome dude i LOve your basic, down to earth approach.. non pretentious and REAL talk about a very high intellectual subject...really enjoyable and already books you recommended..than YOU SO much !!
Really good advice. Thanks
Thank you very much for this! Your preferred approach to Mathematics seems to coincide with mine. I love the abstraction, and I learn math for it, but it's got to be motivated first, preceded by computation of at least many concrete examples.
Gilbert Strang the author of the linear algebra book was/is a professor of math at MIT. So the
book is top notch. Linear algebra is the most important math subject now. What if you have a thousand variables in the project you are working on ? You end up with a matrix of 1000 x 1000 = 1000000 members. Okay if you have a differential equations with 1000 variables you can solve the system of one thousand equations using the Runge-Kutta method on the computer.
awesome from Beginning to end...i stayed the ENTRE video. a 4 STAR job buddy...again thank YOU...SO much !!
Great advice on books. By the way there are a couple of superb books on statistics:
_Mathematical Statistics With Applications_ by Wackerly et al,
_Mathematical Statistics With Applications_ by Ramachandran, Tsokos.
There's also another great book on differential geometry besides Manfredo:
_Elementary Differential Geometry_ by Pressley. The author gives full solutions to all the problems in his book. It's not written in overly abstract way, nor does it focus on generalizations.
There are books on algebra in the same vein too:
_Contemporary Abstract Algebra_ by Gallian
_A First Course in Abstract Algebra_ by Fraleigh
The best books I know on linear algebra are
_Linear Algebra With Applications_ by Kolman (Strang's text is just as good but I like Kolmans' better).
_Linear Algebra, Theory, and Applications_ by Kuttler. This is a superb text. It's simply the best. BTW, it's more advanced than Kolman's.
As to complex variables, I think the following book is great:
_Complex Variables and Applications_ by Brown and Churchill.
What are the courses in this list that you have taught? How was your experience and *student feedback* when you taught courses using the following books? a) Second Year Calculus, Bressoud b) Analysis using A radical Approach to Real Analysis, Bressoud c) Number theory using Disquisitiones. Have you taught a first course in Algebra using Dickson's book? It does not introduce groups, rings and fields. Is that a good idea for a 1st course in Algebra? Also, did you really teach a 1st course on Algebra using Edward's book? While the book say that it needs only minimal prerequisites, it implicitly assumes the reader has already done some course in algebra (that is the probably reason why the book is a GTM).
Since aspects of linear algebra are used in multivariable calculus and ordinary differential equations, if I had it to do over again, I would have taken along with calculus II, saving time.
--semiretired college mathematics and physics tutor, a former systems software developer.
There are differences of opinion on the introduction to proofs type of class.
I'm in the camp that struggles with this. Because the focus is on techniques that can be used to prove things, I felt like I was working on proving things that I didn't really understand. That was frustrating for me and made doing the proofs harder--applying some technique I might understand to a concept I don't.
I think studying some very basic mathematical logic and then paying attention to proofs whenever they arise (including in non-assigned exercises) in the lower-level math classes is a good way to go.
Don't bother with a very theory laden mathematical logic class. Just study the deductive logic chapters in Schaum's Outline of Logic over summer or a break. It's very light (for math majors) and introduces a step-by-step logic system that can hide behind your paragraph proofs in math class. You can pick up other proof techniques as you go along. Induction is big in some of the more basic math classes that I have taken (I'm not a math major).
Advice from a guy in his mid 50s.....get rid of the word "should" from your vocabulary. It is used as a Guilt word, especially in religion, and parents use it so often. Guilt or repressed guilt is the cause for so many illnesses, literally and myself being an ex-maths student who is now a Relationship Trainer and Therapist who is Certified in Hypnotherapy and other modalities, knows what i am talking about. Use "supposed to" or "meant to' or "better off to" rather than guilt tripping people. Same with words such as "i'll get XYZ to call you back......as an example and not in this video. It is like, excuse me, how are you going to get someone to call me? What manipulative technique are you going to use? Rather you want to say or hear.....I'll ask XYZ to call you back.
The #1 cause of cancer is someone with emotional conflicts and so they literally start eating away at themselves. Should is one way to start having ppl have those emotional issues. So stop it. 🙂 The stuff ppl including academics dont know and need to know.
I love this advice. As a CS major, I am irritated by the practicality gaps when you need to reach for some math. We want progress to accumulate without losing the progress with the loss of people. Put as much as possible into code; so that you can use the math to apply it to problems at huge scale. Deep Learning is mostly billion-variable-calculus ... implemented with nilsquares.
The Strauss text is standard for PDE courses, but it is dense and not easy to read. The Arfken text is horrific, yet Physics professors rely on it. The Mary Boas text for Mathematical Physics is another standard text, and is passably good.
If I had a Groundhog day, I'd love to spend a good 20 or 30 years on mathematics. Music and languages are currently taking too much of my time.
Maybe it's being on the spectrum or whatever but I've never been able to get much into "real world" functionality as a person. I just want to study and try to understand everything about everything simply for its own sake.
I hear that. I.M. Gelfand said, "The most important thing a student can get from the study of mathematics is the attainment of a higher intellectual level.” The pursuit of exquisite beauty and ever-deepening insight is what gets many (most?) mathematicians revved up in the morning. Solving problems is part of that, just as creating music or sculpting involves plenty of problem-solving, but it's not necessarily the point of it, or what drives us.
I do pretty much agree with Daniel Rubin that learning comes most naturally and easily if it proceeds on a need-to-know basis. But some people scratch that need-to-know itch primarily through the act of theory-building.
I heard a fantastic self-quiz question the other day: do you learn theory in order to become a better problem-solver, or do you solve problems in order to understand theory better? 🙂
@@toddtrimble2555 To your question at the last, my answer would be
We start by solving problems to understand the theory better till we understand the theory enough to solve the bigger and complex problems.
I think an intro to proofs or discrete math class is pretty essential before diving into analysis and abstract algebra. I dont think writing proofs come naturally to a lot of people and its essential to have a good foundation so you can feel comfortable when you take higher math classes. You can self study too, there are so many great books on proof writing that helped me jump into undergrad analysis and algebra.
Personally i don't like math because of its applications, i like math because of its philosophical depth. And i believe that a lot of your book recommendations would be better suited for an applied math major.
My opinion ... it would be interesting and perhaps more beneficial, modern, and impactful to see a university develop a program where all math is taught as applied and computational aka applied numerical methods in a lab setting. I was lucky enough to have numerical analysis integrated into most of my ugrad courses thru special sections. But that course structure was cancelled after I completed the degree. I think people would grasp the concepts better. Especially for the long term.
You are talking here about this book ---> "Classic Problems of Probability". Does it contain the solutions ?
Bravo for recommending numerical methods!
thank you for you persistence and time
Anyone here studied Spivak as a first calculus book? I was considering doing that (well, I did some calc at school, 20 years ago, which I've now mostly forgotten). Most other folks recommend Stewart or similar first. Which is actually how I ended up here - I was looking for recommendations for books that might get me up to speed quickly.
Sounds like Prof Rubin thinks it's possible to jump straight in to Spivak? I do prefer that idea, just because the idea of re-learning something I already studied at school seems less appealing that learning something new. (Or rather, learning the same thing but in much more depth.)
The exercises in Spivak are notoriously difficult.
I don’t feel that there is too much of a barrier to entering Spivak you can definitely jump straight in it’s just the exercises will be more difficult whether you’ve read another book or not (which I like haha). Also it’s more rigorous which contrary to this video I personally love.
@@nitroh7745 Thankyou! I needed to hear this.
Since writing the comment above I did decide to go the easier route - I bought a copy of Stewart and started working through it, but I had already started to regret that decision - I was starting to realise I'd probably rather spend my time struggling with Spivak than breezing through something easier.
@@JimFarrand I would say it depends very much on how confident you are with proof writing, contrary to this video i feel like proof class (an introductory class not necessarily too deep into logic or set theory), is the most fundamentel class, sets are used in every single math subject so you ought to know how to manipulate them properly imo.
@@MonkeyDLuffy-gd6se i agree, a proof writing course covering propositional logic, sets, binary relations, natural numbers, induction etc. is a must in my opinion. otherwise i dont see how a reader should understand proofs by contradiction or proving the contrapositive of a theorem
contrary to the video, i strongly recommend grabbing a book on proof writing before a proof based book. trying to write proofs without knowing the underlying propositional logic behind proofs is like trying to learn how to read without knowing the alphabet. a classic book recommendation for this is "How to Prove It" by Velleman which has an abundance of exercises. A more modern book which is great for self learners is "Proofs: A Long-Form Mathematics Textbook" by Jay Cummings. His approach differs from traditional books bc he includes "scratch work" before giving the full proof so that the reader learns how proofs are actually formed
The following links are broken fyi:
Edwards, Galois Theory
Akhiezer, Elements of the Theory of Elliptic Functions
What book for Numerical method?
Numerical methods might be a computer science course.
Specifically, a scientific computing course, sometimes embedded in MS in Applied Maths or MS in Scientific Computing directly.
Nice video, sir!
I’m a math major, and I’m proud to say I could probably do 2 questions after 20 hours
As a physicist, I have a very different view about abstract algebra. I love the subject, but I find the historical roots in the theory of polynomial equations boring. Much prefer to emphasize more modern applications.
This video was like gold!
Great video. What is your opinion on numerical analysis as a course?
@s v thanks bro
I more or less agree with your recommendations. Strang is a classic, Rudin (also called Baby Rudin, to differentiate from his Real and Complex Analysis) is slick, but ultimately uninsightful. I had the Alfors complex analysis book when I took the course from Nahari. Nahari's Conformal Mappings is still in print and a classic. PDE's is a huge subject, I'm not sure what book I would recommend. I would that companion material to PDEs would include Integral Transforms (I love the book by Brian Davis: Integral Transforms and their Applications) , Calculus of Variations (with many examples from Goldstein's Classical Mechanics), and, of course, Finite Elements.
Daniel Rubin, where can I find those notes on differential geometry by Richard Hamilton. I could not find anything online.
What do you recommend as someone who was a CS grad but loved math to continue interest if Grad school isn't an option? Tbh my math professors got me to love math more than I ever did in life. I hear the scary stories of self teaching because math is difficult to be a self taught approach since you can't have that direct one on one learning with a teacher of some kind. I dont think it's impossible but I feel like it can be done as a hobby. Any recommendations on approaching it? Just going to say great recommendations on books!
I really want to get into coding theory and cryptography.
Yes, it's difficult to learn without a guide to let you know what's important and without the structure of program. It's a lot easier to learn if you focus on a particular problem you're interested in, even if it's just that you want to know how a specific piece of technology works. My advice may not be the best since I can't point you in a direction that is good for a career. But in terms of a general mathematical approach to coding and cryptography, it certainly helps to learn a lot of number theory. The books I mention in this video are good, and it might be a good idea to focus particularly on computational number theory (there's a great book by Shoup). Also complex analysis. I believe I mention in this video a book on coding theory by Stepanov that is interesting to me, but I have no idea how important the material in that book is in practice. You may find Conway and Sloane's Sphere Packings, Lattices and Groups interesting. I don't have a good general recommendation for cryptography at the moment.
Hi !
What the best calculus practical and problem solving critical thinking mathematics textbooks
Calculus concepts 3 edition by Stewart
Calculus 3 or 5 edition, Stewart
Calculus 9 edition, Larson
Calculus 12 edition, Thomas
Thank you
Thank you for so many helpful resources and advice! One question: at promys, you said they did problem sets without texts. What does that mean?
Great to see some interest in the PROMYS program! At PROMYS, students are given daily problem sets whose material is ahead of the lectures. With the problems as a guide, students are forced to develop the theory on their own without a text. It's a tough but incredibly gratifying and valuable experience.
Wow, that's amazing. I wonder if it's possible to scale that approach for students who don't have access to promys.. like giving students online problems a little higher than their ability and helping them develop the theory independently
I like the idea of emphasizing applications as an undergrad math major.
For me personally, working the proofs gives me confidence in my memory of the techniques, but maybe unmoored theory is best left mainly for the small few who go on for graduate degrees in math.
Could you please do a detailed review on differential geometry books like the 2 books by Barrett O' Neill and a few more complex analysis books like the one by Gamelin.
No mention of Structure Theory of Lie Algebras???????
Vector analysis is not included?
Wow, strong disagree about topology too. So much of modern math is figuring out what results you get from what assumptions. What comes from a set being a metric space, a vector space, a group, etc. And a topology is the minimum set of structure you can get on any continuous space.
Super likable mathematician.
Good information. Thanks.
@Daniel Rubin, I will be getting to uni next year and I overall value being conscious of the historical development of a subject, as I found this fruitful will supplementing a physics course I had at high school. So what resources would you recommend for doing this in math (as it seems like you are a math historian)?
I think Linear Alegebra should be coreq with Trigonometry :)
Great vid. Thanks
Brilliant! Cheers.
How much time did it take you to learn all these subjects?
Have you seen a good proof textbook. It teaches logic and does lots of examples of proof of early results. Would be very helpful.
I think Susanna Epp, Discrete Mathematics with Applications, is a reasonable text to use for a standard "proofs" course. Most proofs courses actually do use some real mathematics as a training ground for writing proofs, such as elementary number theory or elementary graph theory.
Any book on discrete math will suffice for proof writing and has some logic. Most standard texts in mathematical logic will be fine
Gonna have to do Linear Algebra for the 3rd time😭
Do you have any suggestions for books about linear algebra and differential equations that treat them in an integrating fashion?
Good question. The texts I'm most familiar with treat the subjects separately, but I know some programs combine ODE and Lin Alg into one course, especially if the emphasis is on numerical methods. I think each subject deserves its own course, but since I like Gil Strang's other books, you might check out his Differential Equations and Linear Algebra.
It isn't exactly what you have asked for, but you may find useful "An Introduction to Linear Analysis" by Kreider, Kuller, Ostberg, and Perkins. I believe that it is out of print these days though. It does a decent job of developing enough linear algebra to support its main topics (DEs, Fourier analysis, orthogonal series, etc)
Great video
Thanks!