What really helped me is using python to learn abstract algebra. Being able to compute with groups/rings/fields and just trying lots of coding examples really helped. If you had a stupid idea for something, you could try it out in code and then explore it further. Coding with mathematics really encourages exploration and construction of mathematics and I found that so helpful. I’m by no means a pro, but confident in using python for abstract algebra problems.
@@DiegoGonzalez-zs9wz I used sagemath, it is a massive collection of python code and python interfaces to numerous mathematical engines written in C/C++/Fortran etc. It lets you do alot, and has a lot of documentation.
I was a math major, and I am now a professional software engineer. The "programmer brain" I now have has made it easier to think about math since I now can classify mathematical objects better. I always think about the "type" of the object I am dealing with now. Whereas Mathematicians tend to be loose and reuse symbols for different object types.
Math (and physics) alumni here. Thank you so much for sharing this! I discovered this gem thanks to your video and am halfway through. Truly thankful for this discovery - I was never quite keen of the dry approach from my undergrad to abstract algebra - form for the sake of form - without really understanding the big motivational and historical key points that brought the subject to life (although I was initially interested in groups for better understanding QFT's, gauge theories and particle physics at large). Thank you so much!
This host needs to start writing textbooks. And he can charge whatever he wants; I'm pretty sure I'd buy it. His concerns, passion, and priorities simply resonate so strongly with people
@@samueldeandrade8535 Yeah his channel is great and unique. Unfortunately, there is not a high enough interest in these topics at large and it is difficult to run profitable ads on a channel like this. So there is no real financial incentive for him to continue operating the channel except for personal interest.
You're tempting me. Could be dangerous if I start to let loose on Bourbaki and Grothendieck. I do want to make that video at some point, just need the right approach. But in the end, it's probably more convincing if I do like Edwards and provide good exposition of the math credited to that school of thought.
Thank you for the great recommendation. Textbooks should not only list the theorems and proofs, but should also give an insight into the thinking of the mathematicians who came up with these ideas. Hope to see more reviews of similar spirited books.
@@DanielRubin1 Is there some branch of geometry that explains geometrically all math subjects you need to master by the end of high school? Basically I want to have geometric proofs for all algebra and trigonometry content and even calculus by the time I finish high school. Has this work been done so far by some branch of geometry?
This is very intriguing, I’m very attracted to this style of mathematics exposition. I bounced off of abstract algebra pretty hard in grad school and ended up going the applied (engineering mathematics and numerical analysis) route. I could repeat definitions of groups, rings, and modules at you, but I’d be hard pressed to tell you capital “w” Why we would be interested, even though the importance of the general theory 100s of years later is evident. I’d love to see the historical and problem-solving approach, thank you for the recommendation!
Your experience rings true to me. I think a lot of students who learn algebra the standard way get very good at reciting the definitions and statements of theorems, and they get good at doing exercises where they are told "Show A implies B." But they rarely hold on to the techniques, and they can't conceive of how they could have come up with such nice definitions or conjectures themselves, which is what they'll need as mathematicians. The problem-solving approach gives this to you, plus the bonus of motivation.
@@DanielRubin1 This is a problem in software development as well. When you look object oriented design patterns and certain code architecture or design patterns, there are people who try explaining these designs as if they were created from thin air as opposed to logically following or progressing from a simpler design or technique. In my opinion, this layout is a result of people wanting to sound smart and instead of wanting to be clear. I have experienced this when learning from internal wikis at top software companies as well so I know the background, motivation, and attitude of these people. I have always documented my designs as a progression similar to the genetic approach that you mention in your videos. In fact, I thought about creating math courses but written with the genetic approach that you mention. I didn't know it had a name until I saw your videos.
I just wanted to say that i have really enjoyed all of your videos. You have a very unique approach to discussing math. As a math grad school drop out, your videos have gotten me to revisit quite a few topics many years later. Hope you can make more in the future.
Thank you so much for sharing this brilliant resource, Daniel! I find it important as a student to learn concrete ideas before touching the more abstract ones. This book, as I see it now, aids the same approach.
OK as someone who took two quarters of algebra studying groups, rings, and fields out of Dummit & Foote but didn't take the third quarter course on Galois theory, and thus never learned it, your review has gotten me really interested in this book, so thank you for the recommendation. This seems like it could be a nice change of pace from the usual math books I have read.
I just stumbled across your channel and I love it so far! I love getting a sense of some advanced and more abstract mathematical concepts even though I'm still working through some lower level classes. It's extremely motivating.
Love these videos on accessible books. You could even use keywords like friendly /accessible/self teaching as I feel a lot of people like to learn these topics as hobbies, post doing a more profession oriented degree with math minors
Despised "parentheses shuffling" in my first algebra semester (following Artin). Abstract, non-geometrical, symmetry (THE MAIN FKN POINT) not emphasized enough. Really dug the Galois magic in my second semester. Looking forward to peeking into this book...!
Thank you very much for the content you reveal, coming from analysis background, i always wanted to know more about different subjects of math like Galois Theory, Differential Geometry,...(at least basics). It is not easy to try to break the "code" of abstract definitions and theorems, without going back to motivations of those definitions, i always wanted books of "Edwards" type, thank you again.
This is why I love this channel, I took my first algebra class this semester, can't wait to pick up this book and learn the subject from a different perspective. Thx for the recommendation, keep them coming :)
I'd like to push back just a little on this recommendation if I may. Understanding the history of mathematical development is undoubtedly very important, but so is learning modern mathematics. From what I can tell, this book doesn't cover anything on extensions over F_p, and this can be a big disservice to students as it gives the impression that we only care about polynomials over C and subfields of C. When in reality, any text on modern number theory/algebraic geometry/representation theory is going to assume familiarity with the nuisances of field extensions over F_p including separable extensions, purely inseparable extensions, perfect fields, etc. Moreover, I think a text focused purely on Galois theory should include a discussion of some of the most interesting modern questions relating to the subject. The famous "inverse Galois problem" is an obvious contender, and a discussion of how Dirichlet's Theorem proves the case for cyclic groups is a great addition. Other potential topics to discuss include: the relationship between Galois theory and low degree group cohomology, transcendental extensions and transcendence bases, and infinite galois theory. I suppose I'd like to end with a question: besides a historical perspective, what does this book bring to the table that, say, Emil Artin's classic text doesn't? I don't think Artins text is optimal, but it is a short text with minimal background required that covers the standard material on Galois theory (up through the fundamental theorem and solvability by radicals).
This is perfectly good push back, so let me try to address your points. First, what I meant to get across is that the main reason I like expositions covering the historical development of math is for the emphasis on how problems were solved and how key ideas were introduced. This makes math more comprehensible to me than a treatment which begins by setting down the nice definitions that people only came around to much later, and using the properties of those objects to make theorems pop out very neatly. Learning about the history is just a bonus. And you're absolutely right that the priority in teaching has to be giving students something useful to them, which means getting to the frontier of modern mathematics efficiently. This book is mostly about math from the first half of the 19th century. But I think that this book can be useful to students starting out in the subject to learn the key ideas and techniques very well, and from there they will be well-equipped to investigate more modern material. There is some material on extensions over fields of characteristic p in the 7th and 8th Exercise Sets of the book. But I think it's fine that the book is focused on the problem of solving polynomials over Q, not on building the most widely applicable theory. The additional theory you mention can all be covered after going through this book, preferably in a context where those questions are natural, like number theory or coding theory. It would be nice to discuss some modern problems, but the Inverse Galois Problem is way beyond the scope of this book. This book only discusses finite groups occurring naturally as subgroups of permutation groups, and I think it would be good for students to learn more about finite groups and other kinds of groups after going through this book. And Dirichlet's theorem is also fair game for after this book (note that it appears in several of the books I've recommended: Stein and Shakarchi's Fourier Analysis, Edwards' Fermat's Last Theorem, and Knapp's Elliptic Curves).
Little is known that E. Galois planned to write a memoire on techniques of the modern duel, on ways how to defeat your opponents with the gun, the rapier and by kicking him with the boots.
Thanks for getting deeper into this great book, I had already bought it based on your recommendation in a previous video. I have bought some of the other books you mentioned, too and made a reading list of books I own (see below). I have studied mathematics a long time ago and I want to spend parts of my free time (re-)learning it and having fun. My current list goes like this and I have just started, I am half-way through Spivak. I am planning to do most of the exercises in all of the books that have them, so it will take quite some time. Calculus: Spivak, Calculus Multivariable calculus: Edwards, Jr., Advanced Calculus of Several Variables Ordinary Differential Equations: Braun: Differential Equations and Their Applications Linear Algebra: Strang, Linear Algebra and Its Applications Real analysis: Bressoud, A Radical Approach to Real Analysis Partial differential equations: Logan: Applied Partial Differential Equations Fourier analysis: Stein, Shakarchi: Fourier Analysis: An Introduction Complex analysis: Marshall: Complex Analysis Algebra: Edwards, Galois Theory Fulton and Harris, Representation Theory: A First Course Number Theory: Cox: Primes of the Form x2+ny2 Edwards, Fermat's Last Theorem: A Genetic Approach to Algebraic Number Theory Probability: Kolmogorov: Foundations of the Theory of Probability Differential geometry: Tu: An Introduction to Manifolds (as a preparation for the next book) Tu: Differential Geometry Algebraic geometry: McKean and Moll, Elliptic Curves
That's a great list! Ambitious! If you've got no exams or degree requirements, just remember there's no need to learn ALL of math. You're free to follow your curiosity and find one problem that you can get into really deeply. I think that's the most rewarding.
@@DanielRubin1 Yes, it will take quite a while and maybe there will be changes to the list. I am 53 now and I am doing it just for fun and enlightenment, it is actually much more fun than during university times, I also have better learning habits than back then.
I don't work in dynamical systems, and I'm not even sure of what is the full scope of math that is covered by that name. But I know a decent amount about the differential equations that arise in physics. If what you're talking about is a theory that begins with something like Arnold's Mathematical Methods of Classical Mechanics, then that's a really good book. Arnold has another book Huygens and Barrow, Newton and Hooke, which is historical and covers the very early work in mechanics and differential equations, including Newton's derivation of elliptical orbits of planets. I also like A History of Mechanics by Rene Dugas.
This is an interesting starting point of studying abstract algebra. However, for a person like me who does not have a traditional background of a math major may struggle to pick up on that book. I am not sure if this can be a good starting point for most math majors, but this book is going to be lovely for those that love abstract algebra and Galois Theory. I didn't learn a lot about Galois Theory for most of my time studying abstract algebra, but I heard that this is one of the most important subject areas in abstract algebra. My professors enjoy talking about that stuff. I started my study of linear algebra from an undergrad course and I used Charles C. Pinter's A Book of Abstract algebra. This book is a bit hard for me, but I like the presentation of this book. It covers Galois theory later, but I didn't covered that much in my class. I learned more about Galois Theory in my second semester of abstract algebra, but only parts of it, not a thorough coverage of that topic. I have a chemistry background, mostly in computational/theoretical chemistry and density-functional theory during my grad school days, so learning abstract algebra was hard. Fortunately, I learned about representation theory and my abstract algebra professor was amazed that point groups and symmetry groups in inorganic chemistry are connected with the concepts in representation theory. Do you think that representation theory will work well if it is covered in undergraduate classes to introduce linear algebra? I took a graduate level algebra class with representation theory and this was difficult. I may check out this book when I see it. What do you think about teaching about ruler and compass for abstract algebra students at the undergraduate level?
Representation theory is incredibly important for particle physics and for certain aspects of chemistry, and most math majors should learn something about the subject as well. I see a course in representation theory making sense after some group theory and a course in linear algebra. I don't know how it would work if concurrent with linear algebra since lin alg is of much wider applicability and taken by students with no exposure to group theory or the concerns of representation theory. As ruler and compass constructions are no longer part of the standard high school curriculum, the issue is often presented to students for the first time in a course on abstract algebra and Galois theory, when the historical problem is mentioned and then immediately answered by 19th century theory. It's at least worth bringing up. The issue is covered briefly in Edwards' book and I believe there are exercises about it.
I've learned more algebra taking electrical engineering in two quarters than however many years through jr high and high school. I wish certain levels of algebra weren't so abstract. I'll watch a Khan video and be more confused. Thanks for sharing.
bro I recently came across your calculus series. Could u pls suggest the books u used to develop those lectures, I really want to learn from books that give a historical aspect of the problem in addition to the motivation needed to solve it.
I will take a look, as the Springer Undergraduate Mathematics Series "Galois Theory Though Exercises" is a bit dense read and feels like its a secondary book when learning.
Darn! Now I am torn... I am approaching an introduction to abstract algebra course this Fall and I was reviewing from the course text, Algebra: Notes from the underground (the book takes a rings first approach) but now I am tempted to use this book as an introduction instead. Do you think there's any merit to studying my course text and using this as a supplement? Let me know your thoughts please, and take care!
There's definitely merit to having both books because they're completely different. I realize most students who take abstract algebra will have a course of the kind that I put down in this video, so I do encourage students who are interested to take a look at Edwards' book in addition to more standard texts. If you do get into Edwards' book, you'll have a much greater appreciation of algebra, including a rich set of problems about solving polynomials that it's used for. And if you like the book, see what the professor thinks about it.
How do you keep from getting rusty and forgetting this or that mathematics on a years-long basis? I find that even if I do every exercise in a chapter, several months later it's all gone
There’s a few things you can do to retain it for longer: 1) Do the standard homework and studying you would normally do, 2) After an exam, try talking about what you think you might have done right or wrong, 3) Try explaining the topic to someone else, 4) Indulge in your fascination for the subject (even if the subject isn’t very fascinating). That’s basically what I do, and a lot of my friends ask me this question, and this is basically the answer I give.
It's the same with all knowledge, of you don't maintain it, it slowly fades. Repetition is the key for retaining knowledge for a long time. I suggest you watch a video about it, and maybe even read something on how memory works.
I think the key to combating this is connecting the dots. Every time you learn a topic there will be loads of details to remember. It is unrealistic to expect that you will remember these details for any extended period of time after you stop learning said topic. Luckily we don’t need every detail! All of the formalizations of any math topic only serve to express IDEAS in rigorous ways. When we remember the core ideas and how they connect, we can easily relearn (and maybe even derive) any details we may need later. Luckily there are usually far less impactful concepts and ideas to remember than small details. Additionally, for me, these big ideas are usually more interesting than the small details, causing memory to not even be a factor (you naturally remember things you find interesting). Learning according to the big ideas is easier said than done. From my experience, many math books are absorbed in the details, leaving it up to the reader to translate the dense abstractions into a comprehensible idea. The skill of translating math to ideas comes with time so stick with it! I do not go to school thus I have no idea if this style of learning is effective in such a setting.
Great question! The truth is that I have forgotten a lot of math I once knew, including the material in Edwards' Galois Theory until I went through it again before making this video. It's only easy to hold on to the stuff you use on a regular basis. After a while in math you can see recurring themes in the kinds of arguments, so instead of trying to hold onto every detail you can just remember the type of argument or a key step, and then if necessary you can fill in the rest on demand. You also can come up with a kind of story as to how results are related and in what order they go in. That's where the problem-solving/historical mindset helps me. This is what graduate students usually learn to do when they prepare for their oral exams and begin to teach.
There are a lot of important topics in algebra, and personally I think one could ask whether Galois theory deserves the massive emphasis it receives over other worthy contenders. So while I have no reason to doubt that this is a fine text -- Edwards is a well-known expositor -- it might be healthy to reconsider one's priorities, or consider how much groupthink has been handed down over the generations. My own guess is that group representation theory, to name one example, may be more important to more people over the course of their mathematical lives than Galois theory -- I'm pretty sure that would be true for people who want to get more involved with physics. And that's just one example.
I completely agree. If we include linear algebra as part of algebra, then that is definitely the most widely useful, and then I don't have a ranking of priorities, but I don't put Galois theory ahead of representation theory. However I do think it is a good approach to start the study of algebra with a book like this because the theory of polynomial equations is the context in which fields, rings, and finite groups were first introduced. I would recommend more study of finite groups, representation theory, and Lie theory in an optional course that comes after this.
@@DanielRubin1 It makes a lot of sense to put linear algebra at the top of the list, yes. But also let's not forget category theory, which is making itself felt more and more throughout mathematics. (Full disclosure: that's my area.) Old prejudices die hard, but more and more young people are learning category theory because it turns out to be a very pragmatic thing to learn, and even to learn well. Finite groups is (I think) a case where fewer and fewer young people are getting involved; for example, the heroes who trail-blazed the classification of finite simple groups are aging out (or have already died), and there are only a handful of people left who have a real understanding of what's involved there. Semigroups on the other hand may be more in the ascendant these days. Don't mean to talk your ear off. I understand your reasons for liking this book; you sound like someone who would also appreciate Stillwell, who also puts a lot of work into the historical approach. Keep up the good work!
Hello ! Very intrigued by your approach to math focusing on applications/problems, computation and history. Did you at some point mention a book about the history of navigation and math or is that just a dream of mine ?
Dude, your enthusiasm is very compelling, and I actually watched the whole video, but this is definitely NOT an introductory book. It's a great way to get a bunch of otherwise curious people to sadly quit math.
Well, I hope otherwise curious people won't quit math because of this book. I view this book as a better introduction to algebra and Galois theory even than very friendly introductory expositions on the structures of abstract algebra (groups, rings, and fields) because from my perspective the story starts with this material, and all that abstraction came later. (Also all the methods of this book are still useful methods.) The text requires from the reader only what is necessary to understand some concrete nontrivial arguments, and the exercises can be tough but there are very good answers in the back, which means you can treat the harder ones like theorems proved in the text.
you know the dummit and foote's algebra book right? that thicc book that has a lot of different stuff, very comprehensive, is there a similar book that's very comprehensive and has a LOT of material but it's graduate level algebra, basically like that dummit and foote book but containing graduate algebra
Yes, Dummit and Foote is a very nice, fairly comprehensive TOME of algebra. I used it to study for a qualifying exam many years ago. Edwards' book is completely different in aim, scope, and presentation, and I think it's better for students to follow something like Edwards' approach rather than study a multitude of structures of Abstract Algebra like in D&F. To your question, it's not clear to me that there is some coherent body of material that could be called "graduate algebra" that goes much beyond the material in D&F. Researchers in different fields make use of various algebraic structures and there's no need for anyone to try to learn about all of them. I do not recommend Lang's Algebra, which is an attempt at a comprehensive algebra reference that I find totally useless, and in any case mostly overlaps with the superior D&F. Some people like Aluffi's book Algebra Chapter 0, which is another comprehensive treatment from the perspective of category theory, which holds little appeal to me.
@@DanielRubin1 thanks, I agree no one needs to know a lot about every algebraic structure, but my curiosity just wont let that happen lol anyway thanks i'll check out those books and see which one's good for me
Just discovered your video on the invention of natural logarithms. I have also been looking for the invention of statistics, but I have had a lot of difficulty. Any words of wisdom to aid a fellow knowledge seeker? Also, feel free to take this and run with it if you think it's worth sharing.
The historical development of statistics is a great subject and eventually I'll get around to making some videos on it. There are a bunch of good references. I believe I cited Hald's A History of Probability and Statistics and Their Applications before 1750 in the description of the Invention of ln(x) video. You can also look up Classical Topics on the History of Modern Mathematical Statistics by Gorroochurn, or for something short, look up Stahl's article "The evolution of the normal distribution."
Hey Daniel. I like how you know of a lot of good textbooks that include the historical motivation. Do you know of any such text for Geometry? Much appreciated.
The Line and the Circle by Carroll and Rykken (covers the beginning of Elements) and Heavenly Mathematics -- the Forgotten Art of Spherical Trigonometry by Glen Van Brummelen
I found Tom Leinster's notes on Galois theory on his website. His course assumes a prior course in algebra on the theory of groups, rings, and fields, whereas Edwards' book, despite belonging to the GTM series, does not.
I may review that book at some point. I heard good things, so I started looking through it. My preliminary thoughts are that it is one of these comprehensive algebra texts with the theory of all the most used algebraic structures, and the exposition is nice and inviting. The book aims to weave everything into the perspective of category theory, which may initiate students into a language used by some cohort of mathematicians, but whose use in solving novel problems I find doubtful. I can see why some people like this book and think it is useful, but it takes an approach which is philosophically the polar opposite of a book like Edwards, which puts problem-solving first. I would never teach a course out of Aluffi's book.
I can't help thinking that if Galois had gotten a good night's sleep before the duel, instead of staying up all night writing, he might have won the duel.
I wanted to learn why quintic equations couldn't have a general solution in radicals, but it seems like SO much work for just that little payoff. Life often feels dull without recreational mathematics, but when it gets at this high level it just wears me out. I'm not sure if I'd rather be bored or worn out. Neither is very fun.
I'd say the journey of learning all the math to get there is interesting in itself. The techniques you learn are very useful even beyond proving the insolvability of the quintic. Group and field theory (and algebra as a whole) are used a whole lot in other branches of math and even physics.
@@HilbertXVI Yeah, I guess so. When I wanted to learn about relativity I took a detour and learned more about surfaces instead. I mean, I didn't learn MUCH about them, but at least I learned something. I'm glad I did.
I think Serge Lang's "Algebra" is also great, IMO it has actually changed the way graduate algebra is taught in universities. Lang was a prolific writer on the subject of mathematics.
My advisor had Lang as a professor. He said Lang believed he should _always_ prove results in their most general form, and you can see this in Lang’s definition of say a group representation. I don’t think this is always beneficial for learning
Sounds nice! Price is a bit hefty, though, which would be much less of an issue if it didn't come along with Springer's really shoddy printing and binding quality - a result of its turn to a 'print on demand' model.
@@DanielRubin1 I am very nitpicky with books, I must confess. Having sewn bindings is a must for me (book opens easily and stays open, and will not lose its pages after heavy use in one's lifetime).
Historical perspectives always seem to me to be very motivating, except possibly linear algebra which I still think has no clean, natural introduction, but is just like, alright, here we go... 😅
Glad you like seeing the historical motivation! I've done some preliminary digging into the history of linear algebra, and I'd like to cover some of the key point in future math videos. Some highlights: people have solved simple linear systems since ancient times. The determinant and its relation to the solvability of a system is due to Cramer of Cramer's rule. Cramer, Bezout, Sylvester, Cayley, and others associated with finally putting down the theory of linear algebra were interested in problems in algebraic geometry and incidence geometry (how many points are in the intersection of a collection of curves, or what collection of curves pass through certain points). The multiplicativity of the determinant seems to have been discovered by Gauss in the course of his work on classifying binary quadratic forms. And Gauss also developed the method of least squares and the form of the normal distribution in the course of working out a theory of errors in measurement when trying to predict where Ceres would reappear.
great video thanks , what do you think about Elements of Algebra: Geometry, Numbers, Equations by john stillwell ?? . i think it is also a very nice book to start learning abstract algebra with historical motivation
Many mathematicians prefer to have the shiny latest version of a theory and focus on the ability to play with it right away. For some reason I value understanding what kinds of problems mathematicians wanted to overcome when devising a new theory. Maybe that's why I'm not a professional mathematician lol
I'm with you, and I actually suspect that most mathematicians also find it very difficult to work with a theory whose definitions involve conditions that seem obscure to them. If I don't understand the motivating problem and the examples that cause us to make those distinctions, then there's very little chance I'll be able to make use of that theory.
Haha, H.M. Edwards is my favorite math author too. Great book, I have it, and also have his other books. They're good too. His approach - example based with historical background - is the ONLY one I can study with. I just can't learn with the concise style of: definition, lemma, theorem. My rule: if it's from Springer and uses the default TeX font, skip it :-) EDIT: Other math books i love: Elliptic Curves by McKean and Moll (A gem packed with info, published by my favorite math publisher Cambridge) Primes of the form x^2+n y^2 by Cox
I mention "Abstract" Algebra to make it clear I am not talking about high school algebra. But the point of recommending this book is that it's not necessary to have a course on algebraic structures in the abstract. Edwards' book is about solving polynomial equations, and all of the structure revealed along the way is completely concrete.
hi Daniel, I came across another book that gives the historical background on this subject, titled "Equation that couldn't be solved: How mathematical genius discovered the language of symmetry" by Mario Livio, Please check it out and give your opinion.
Great review. I just got the book! I'm curious what you think of 'Topology: A Categorical Approach' by Bradley, Bryson and Terilla if you've checked it out.
Thanks! Hope you enjoy the book! I was not aware of the book you mention, though I'm familiar with Tai-Danae Bradley, who was the second host of PBS's UA-cam show Infinite Series. I checked it out. Everything I say comes with the caveat that I'm not a topologist and I've never had a use for category theory, but you can probably imagine that I would view that book with a lot of skepticism. The book's stated aim to "re-introduce basic, point-set topology from a modern categorical perspective" has nothing to do with solving problems. Maybe it could help someone who needs to absorb the categorical perspective in order to converse with experts in the field. I would look for a very different approach to topology.
@@DanielRubin1 Thanks! Yeah, you pretty much nailed the reason I would read it. I just watched your 'Math Major Guide' so now I have a better idea of where you're coming from. Just one more: how about the book 'Classical Topology and Combinatorial Group Theory' by Stillwell?
@@DanielRubin1 You should read the preface to the Stillwell book if you can find it. Sort of sounds like something you'd write :) "In recent years, many students have been introduced to topology in high school mathematics. Having met the Mobius band, the seven bridges of Konigsberg, Euler's polyhedron formula, and knots, the student is led to expect that these picturesque ideas will come to full flower in university topology courses. What a disappointment "undergraduate topology" proves to be! In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams. Pictures are kept to a minimum, and at the end the student still does not understand the simplest topological facts, such as the reason why knots exist [...]"
Using this book in an undergraduate course at an American University , is Academic Suicide for any professor dumb enough to do it.As attractive as this book might be , it has no place in the American System .UC Berkeley used to be number 1 ranked in the Country , in both Under grad , and Graduate and the Book "A first Course in Abstract Algebra " by Fraleigh which is an extremely gentle intro to Algebra was used for the under grad course ...I myself have taught Linear Algebra to undergrads and the typical student in such a class has trouble constructing the most basic proofs ...Let's not get carried away too much here....
You're probably right that most American departments will punish a professor who switches to a curriculum based on this book. You appear to suggest that the reason is that students will not be able to handle the material or do the exercises in this book. That's a shame. I would like to see someone do the experiment of teaching a course out of this book and report how it goes. I certainly think it would be helpful for students to have the prerequisite of linear algebra, which, though it is invoked very minimally in Edwards' book, would probably help students gain the experience with computation and proofs to tackle the exercises. But in any case, the material in Edwards' book is as concrete as could be, and is the real stuff of the beginning of algebra. How could a course on the abstract theory of groups, rings, and fields as covered by Fraleigh actually be easier while being as valuable?
@@DanielRubin1 In an ideal world , an undergraduate would open a terse book like Analysis by Rudin and rapidly complete one exercise set after another .....Unfortunately we do not live in such a world ....In fact most University administrators say that the real smart students study engineering not Math....Thus the typical math major does not have a very high level analytical ability ....Encouraging undergrads to look at graduate level texts is not in tune with reality in general ...I believe that you were also recommending a terse Graduate level Complex Analysis book for a first look at the subject ....Such things look nice on paper but rarely work out ...I used to be a Professor of Mathematics at a mid sized University and a Good Day was when someone understood an epsilon delta argument as to why x (squared ) is continuous or how to find the inverse of a 3x3 matrix....
I hate math and all things related to it. I never completed undergrad because I couldn't pass Algebra - a prerequisite. Completed 150 hours in everything else, but no math. It's a waste of time.
don't pay a hundred bucks for this book! if you dig for it you can get a PDF for free. Else some publishers let you access texts for free via your institution.
Bilge. No one *starts* a subject with a yellow peril Graduate Text. Also already at 0:25 we learn that the speaker does not know English. Nothing is _very unique_. It is unique or not unique. Not really unique or very unique or somewhat unique or uniquely unique or amazingly unique. Just unique, or not unique. In the set {17}, 17 is unique. In the set {17, -1/5}, it is not unique.
What really helped me is using python to learn abstract algebra. Being able to compute with groups/rings/fields and just trying lots of coding examples really helped. If you had a stupid idea for something, you could try it out in code and then explore it further. Coding with mathematics really encourages exploration and construction of mathematics and I found that so helpful. I’m by no means a pro, but confident in using python for abstract algebra problems.
Which modules did you use for abstract algebra if you have any references that can help learn that module? Thanks
@@DiegoGonzalez-zs9wz I used sagemath, it is a massive collection of python code and python interfaces to numerous mathematical engines written in C/C++/Fortran etc. It lets you do alot, and has a lot of documentation.
What do you mean that you compute exactly? Also, is there somewhere where you can find exercises like that?
@@navjotsingh2251wow, thanks for the recommendation.
I was a math major, and I am now a professional software engineer. The "programmer brain" I now have has made it easier to think about math since I now can classify mathematical objects better. I always think about the "type" of the object I am dealing with now. Whereas Mathematicians tend to be loose and reuse symbols for different object types.
Math (and physics) alumni here. Thank you so much for sharing this! I discovered this gem thanks to your video and am halfway through. Truly thankful for this discovery - I was never quite keen of the dry approach from my undergrad to abstract algebra - form for the sake of form - without really understanding the big motivational and historical key points that brought the subject to life (although I was initially interested in groups for better understanding QFT's, gauge theories and particle physics at large). Thank you so much!
This host needs to start writing textbooks. And he can charge whatever he wants; I'm pretty sure I'd buy it. His concerns, passion, and priorities simply resonate so strongly with people
Please make more videos. This is a real gem of a channel.
I agree.
True
It is a sad thing, isn't? He had a good channel.
@@samueldeandrade8535 Yeah his channel is great and unique. Unfortunately, there is not a high enough interest in these topics at large and it is difficult to run profitable ads on a channel like this. So there is no real financial incentive for him to continue operating the channel except for personal interest.
I would love to watch a full Bourbaki critique by you. I think you raise many interesting and valid points.
You're tempting me. Could be dangerous if I start to let loose on Bourbaki and Grothendieck. I do want to make that video at some point, just need the right approach. But in the end, it's probably more convincing if I do like Edwards and provide good exposition of the math credited to that school of thought.
Thank you for the great recommendation. Textbooks should not only list the theorems and proofs, but should also give an insight into the thinking of the mathematicians who came up with these ideas. Hope to see more reviews of similar spirited books.
Certainly I find that kind of insight really useful. More reviews coming!
@@DanielRubin1 Is there some branch of geometry that explains geometrically all math subjects you need to master by the end of high school?
Basically I want to have geometric proofs for all algebra and trigonometry content and even calculus by the time I finish high school. Has this work been done so far by some branch of geometry?
This is very intriguing, I’m very attracted to this style of mathematics exposition. I bounced off of abstract algebra pretty hard in grad school and ended up going the applied (engineering mathematics and numerical analysis) route. I could repeat definitions of groups, rings, and modules at you, but I’d be hard pressed to tell you capital “w” Why we would be interested, even though the importance of the general theory 100s of years later is evident. I’d love to see the historical and problem-solving approach, thank you for the recommendation!
Your experience rings true to me. I think a lot of students who learn algebra the standard way get very good at reciting the definitions and statements of theorems, and they get good at doing exercises where they are told "Show A implies B." But they rarely hold on to the techniques, and they can't conceive of how they could have come up with such nice definitions or conjectures themselves, which is what they'll need as mathematicians. The problem-solving approach gives this to you, plus the bonus of motivation.
LO mísmo dijo yo, barely I understand Fiels, Rings. Etc.
@@DanielRubin1 This is a problem in software development as well. When you look object oriented design patterns and certain code architecture or design patterns, there are people who try explaining these designs as if they were created from thin air as opposed to logically following or progressing from a simpler design or technique. In my opinion, this layout is a result of people wanting to sound smart and instead of wanting to be clear. I have experienced this when learning from internal wikis at top software companies as well so I know the background, motivation, and attitude of these people. I have always documented my designs as a progression similar to the genetic approach that you mention in your videos. In fact, I thought about creating math courses but written with the genetic approach that you mention. I didn't know it had a name until I saw your videos.
I love that Galois' memoir is actually included in this book!
I just wanted to say that i have really enjoyed all of your videos. You have a very unique approach to discussing math.
As a math grad school drop out, your videos have gotten me to revisit quite a few topics many years later. Hope you can make more in the future.
Thank you so much for sharing this brilliant resource, Daniel! I find it important as a student to learn concrete ideas before touching the more abstract ones. This book, as I see it now, aids the same approach.
Found your channel too late, but as many have said, you've got great content and insight. Hopefully someday you'll come back to making videos 🙂
OK as someone who took two quarters of algebra studying groups, rings, and fields out of Dummit & Foote but didn't take the third quarter course on Galois theory, and thus never learned it, your review has gotten me really interested in this book, so thank you for the recommendation. This seems like it could be a nice change of pace from the usual math books I have read.
I just stumbled across your channel and I love it so far! I love getting a sense of some advanced and more abstract mathematical concepts even though I'm still working through some lower level classes. It's extremely motivating.
Love these videos on accessible books. You could even use keywords like friendly /accessible/self teaching as I feel a lot of people like to learn these topics as hobbies, post doing a more profession oriented degree with math minors
Despised "parentheses shuffling" in my first algebra semester (following Artin). Abstract, non-geometrical, symmetry (THE MAIN FKN POINT) not emphasized enough. Really dug the Galois magic in my second semester. Looking forward to peeking into this book...!
Thank you very much for the content you reveal, coming from analysis background, i always wanted to know more about different subjects of math like Galois Theory, Differential Geometry,...(at least basics). It is not easy to try to break the "code" of abstract definitions and theorems, without going back to motivations of those definitions, i always wanted books of "Edwards" type, thank you again.
Absolutely! If you like this style, also check out the work of David Bressoud.
This is exactly what I want!!!! Thank you for sharing this!!!
I'm so glad!
This is why I love this channel, I took my first algebra class this semester, can't wait to pick up this book and learn the subject from a different perspective. Thx for the recommendation, keep them coming :)
Thanks, Rafik! Really glad to know that people enjoy these videos. Will try to put out some more!
Your first "abstract" algebra class?
@@Barnekkid yeah lol
I'd like to push back just a little on this recommendation if I may. Understanding the history of mathematical development is undoubtedly very important, but so is learning modern mathematics. From what I can tell, this book doesn't cover anything on extensions over F_p, and this can be a big disservice to students as it gives the impression that we only care about polynomials over C and subfields of C. When in reality, any text on modern number theory/algebraic geometry/representation theory is going to assume familiarity with the nuisances of field extensions over F_p including separable extensions, purely inseparable extensions, perfect fields, etc.
Moreover, I think a text focused purely on Galois theory should include a discussion of some of the most interesting modern questions relating to the subject. The famous "inverse Galois problem" is an obvious contender, and a discussion of how Dirichlet's Theorem proves the case for cyclic groups is a great addition. Other potential topics to discuss include: the relationship between Galois theory and low degree group cohomology, transcendental extensions and transcendence bases, and infinite galois theory.
I suppose I'd like to end with a question: besides a historical perspective, what does this book bring to the table that, say, Emil Artin's classic text doesn't? I don't think Artins text is optimal, but it is a short text with minimal background required that covers the standard material on Galois theory (up through the fundamental theorem and solvability by radicals).
This is perfectly good push back, so let me try to address your points. First, what I meant to get across is that the main reason I like expositions covering the historical development of math is for the emphasis on how problems were solved and how key ideas were introduced. This makes math more comprehensible to me than a treatment which begins by setting down the nice definitions that people only came around to much later, and using the properties of those objects to make theorems pop out very neatly. Learning about the history is just a bonus. And you're absolutely right that the priority in teaching has to be giving students something useful to them, which means getting to the frontier of modern mathematics efficiently. This book is mostly about math from the first half of the 19th century. But I think that this book can be useful to students starting out in the subject to learn the key ideas and techniques very well, and from there they will be well-equipped to investigate more modern material.
There is some material on extensions over fields of characteristic p in the 7th and 8th Exercise Sets of the book. But I think it's fine that the book is focused on the problem of solving polynomials over Q, not on building the most widely applicable theory. The additional theory you mention can all be covered after going through this book, preferably in a context where those questions are natural, like number theory or coding theory.
It would be nice to discuss some modern problems, but the Inverse Galois Problem is way beyond the scope of this book. This book only discusses finite groups occurring naturally as subgroups of permutation groups, and I think it would be good for students to learn more about finite groups and other kinds of groups after going through this book. And Dirichlet's theorem is also fair game for after this book (note that it appears in several of the books I've recommended: Stein and Shakarchi's Fourier Analysis, Edwards' Fermat's Last Theorem, and Knapp's Elliptic Curves).
Such book reviews are very valuable. Thanks for that! Keep those coming.
Thanks, will do!
I am uninformed in mathematics yet my grandfather taught me trig on a three sided slide rule once upon a time... thanks.
Little is known that E. Galois planned to write a memoire on techniques of the modern duel, on ways how to defeat your opponents with the gun, the rapier and by kicking him with the boots.
Thanks for getting deeper into this great book, I had already bought it based on your recommendation in a previous video. I have bought some of the other books you mentioned, too and made a reading list of books I own (see below). I have studied mathematics a long time ago and I want to spend parts of my free time (re-)learning it and having fun. My current list goes like this and I have just started, I am half-way through Spivak. I am planning to do most of the exercises in all of the books that have them, so it will take quite some time.
Calculus:
Spivak, Calculus
Multivariable calculus:
Edwards, Jr., Advanced Calculus of Several Variables
Ordinary Differential Equations:
Braun: Differential Equations and Their Applications
Linear Algebra:
Strang, Linear Algebra and Its Applications
Real analysis:
Bressoud, A Radical Approach to Real Analysis
Partial differential equations:
Logan: Applied Partial Differential Equations
Fourier analysis:
Stein, Shakarchi: Fourier Analysis: An Introduction
Complex analysis:
Marshall: Complex Analysis
Algebra:
Edwards, Galois Theory
Fulton and Harris, Representation Theory: A First Course
Number Theory:
Cox: Primes of the Form x2+ny2
Edwards, Fermat's Last Theorem: A Genetic Approach to Algebraic Number Theory
Probability:
Kolmogorov: Foundations of the Theory of Probability
Differential geometry:
Tu: An Introduction to Manifolds (as a preparation for the next book)
Tu: Differential Geometry
Algebraic geometry:
McKean and Moll, Elliptic Curves
That's a great list! Ambitious! If you've got no exams or degree requirements, just remember there's no need to learn ALL of math. You're free to follow your curiosity and find one problem that you can get into really deeply. I think that's the most rewarding.
@@DanielRubin1 Yes, it will take quite a while and maybe there will be changes to the list. I am 53 now and I am doing it just for fun and enlightenment, it is actually much more fun than during university times, I also have better learning habits than back then.
Can you recommend a text giving a historical perspective on dynamical systems?
I don't work in dynamical systems, and I'm not even sure of what is the full scope of math that is covered by that name. But I know a decent amount about the differential equations that arise in physics. If what you're talking about is a theory that begins with something like Arnold's Mathematical Methods of Classical Mechanics, then that's a really good book. Arnold has another book Huygens and Barrow, Newton and Hooke, which is historical and covers the very early work in mechanics and differential equations, including Newton's derivation of elliptical orbits of planets. I also like A History of Mechanics by Rene Dugas.
This is an interesting starting point of studying abstract algebra. However, for a person like me who does not have a traditional background of a math major may struggle to pick up on that book. I am not sure if this can be a good starting point for most math majors, but this book is going to be lovely for those that love abstract algebra and Galois Theory. I didn't learn a lot about Galois Theory for most of my time studying abstract algebra, but I heard that this is one of the most important subject areas in abstract algebra. My professors enjoy talking about that stuff. I started my study of linear algebra from an undergrad course and I used Charles C. Pinter's A Book of Abstract algebra. This book is a bit hard for me, but I like the presentation of this book. It covers Galois theory later, but I didn't covered that much in my class. I learned more about Galois Theory in my second semester of abstract algebra, but only parts of it, not a thorough coverage of that topic. I have a chemistry background, mostly in computational/theoretical chemistry and density-functional theory during my grad school days, so learning abstract algebra was hard. Fortunately, I learned about representation theory and my abstract algebra professor was amazed that point groups and symmetry groups in inorganic chemistry are connected with the concepts in representation theory. Do you think that representation theory will work well if it is covered in undergraduate classes to introduce linear algebra? I took a graduate level algebra class with representation theory and this was difficult. I may check out this book when I see it. What do you think about teaching about ruler and compass for abstract algebra students at the undergraduate level?
Representation theory is incredibly important for particle physics and for certain aspects of chemistry, and most math majors should learn something about the subject as well. I see a course in representation theory making sense after some group theory and a course in linear algebra. I don't know how it would work if concurrent with linear algebra since lin alg is of much wider applicability and taken by students with no exposure to group theory or the concerns of representation theory.
As ruler and compass constructions are no longer part of the standard high school curriculum, the issue is often presented to students for the first time in a course on abstract algebra and Galois theory, when the historical problem is mentioned and then immediately answered by 19th century theory. It's at least worth bringing up. The issue is covered briefly in Edwards' book and I believe there are exercises about it.
Excellent review! Will look it up
Thanks! Hope you enjoy it!
I've learned more algebra taking electrical engineering in two quarters than however many years through jr high and high school. I wish certain levels of algebra weren't so abstract. I'll watch a Khan video and be more confused. Thanks for sharing.
bro I recently came across your calculus series. Could u pls suggest the books u used to develop those lectures, I really want to learn from books that give a historical aspect of the problem in addition to the motivation needed to solve it.
Where can one get a thorough treatment of theory of equations over modulo n ring?
I will take a look, as the Springer Undergraduate Mathematics Series "Galois Theory Though Exercises" is a bit dense read and feels like its a secondary book when learning.
One may also start with Hartshorne's Algebraic geometry 😅
Thank you for recommending a fantastic book!
Darn! Now I am torn... I am approaching an introduction to abstract algebra course this Fall and I was reviewing from the course text, Algebra: Notes from the underground (the book takes a rings first approach) but now I am tempted to use this book as an introduction instead. Do you think there's any merit to studying my course text and using this as a supplement? Let me know your thoughts please, and take care!
There's definitely merit to having both books because they're completely different. I realize most students who take abstract algebra will have a course of the kind that I put down in this video, so I do encourage students who are interested to take a look at Edwards' book in addition to more standard texts. If you do get into Edwards' book, you'll have a much greater appreciation of algebra, including a rich set of problems about solving polynomials that it's used for. And if you like the book, see what the professor thinks about it.
How do you keep from getting rusty and forgetting this or that mathematics on a years-long basis? I find that even if I do every exercise in a chapter, several months later it's all gone
There’s a few things you can do to retain it for longer:
1) Do the standard homework and studying you would normally do,
2) After an exam, try talking about what you think you might have done right or wrong,
3) Try explaining the topic to someone else,
4) Indulge in your fascination for the subject (even if the subject isn’t very fascinating).
That’s basically what I do, and a lot of my friends ask me this question, and this is basically the answer I give.
It's the same with all knowledge, of you don't maintain it, it slowly fades. Repetition is the key for retaining knowledge for a long time. I suggest you watch a video about it, and maybe even read something on how memory works.
I think the key to combating this is connecting the dots. Every time you learn a topic there will be loads of details to remember. It is unrealistic to expect that you will remember these details for any extended period of time after you stop learning said topic. Luckily we don’t need every detail!
All of the formalizations of any math topic only serve to express IDEAS in rigorous ways. When we remember the core ideas and how they connect, we can easily relearn (and maybe even derive) any details we may need later. Luckily there are usually far less impactful concepts and ideas to remember than small details. Additionally, for me, these big ideas are usually more interesting than the small details, causing memory to not even be a factor (you naturally remember things you find interesting).
Learning according to the big ideas is easier said than done. From my experience, many math books are absorbed in the details, leaving it up to the reader to translate the dense abstractions into a comprehensible idea. The skill of translating math to ideas comes with time so stick with it!
I do not go to school thus I have no idea if this style of learning is effective in such a setting.
@@JM-us3fr ñ
Great question! The truth is that I have forgotten a lot of math I once knew, including the material in Edwards' Galois Theory until I went through it again before making this video. It's only easy to hold on to the stuff you use on a regular basis. After a while in math you can see recurring themes in the kinds of arguments, so instead of trying to hold onto every detail you can just remember the type of argument or a key step, and then if necessary you can fill in the rest on demand. You also can come up with a kind of story as to how results are related and in what order they go in. That's where the problem-solving/historical mindset helps me. This is what graduate students usually learn to do when they prepare for their oral exams and begin to teach.
Love these videos. Thank you.
Glad you like them!
Thank you for the recommendation
Does anyone know a course that uses this book so I can find accompanying problem sets? I've looked everywhere.
There are a lot of important topics in algebra, and personally I think one could ask whether Galois theory deserves the massive emphasis it receives over other worthy contenders. So while I have no reason to doubt that this is a fine text -- Edwards is a well-known expositor -- it might be healthy to reconsider one's priorities, or consider how much groupthink has been handed down over the generations. My own guess is that group representation theory, to name one example, may be more important to more people over the course of their mathematical lives than Galois theory -- I'm pretty sure that would be true for people who want to get more involved with physics. And that's just one example.
I completely agree. If we include linear algebra as part of algebra, then that is definitely the most widely useful, and then I don't have a ranking of priorities, but I don't put Galois theory ahead of representation theory. However I do think it is a good approach to start the study of algebra with a book like this because the theory of polynomial equations is the context in which fields, rings, and finite groups were first introduced. I would recommend more study of finite groups, representation theory, and Lie theory in an optional course that comes after this.
@@DanielRubin1 It makes a lot of sense to put linear algebra at the top of the list, yes. But also let's not forget category theory, which is making itself felt more and more throughout mathematics. (Full disclosure: that's my area.) Old prejudices die hard, but more and more young people are learning category theory because it turns out to be a very pragmatic thing to learn, and even to learn well.
Finite groups is (I think) a case where fewer and fewer young people are getting involved; for example, the heroes who trail-blazed the classification of finite simple groups are aging out (or have already died), and there are only a handful of people left who have a real understanding of what's involved there. Semigroups on the other hand may be more in the ascendant these days.
Don't mean to talk your ear off. I understand your reasons for liking this book; you sound like someone who would also appreciate Stillwell, who also puts a lot of work into the historical approach. Keep up the good work!
Would you recommend Abel’s Theorem in Problems and Solutions?
The irony of Rubin finally dissing analysis and embracing algebra :)
That's not how I would describe what's happening here. Note that this is an analyst's approach to the subject.
Very interesting recommendation
I recommend aluffi chapter 0
Hello ! Very intrigued by your approach to math focusing on applications/problems, computation and history. Did you at some point mention a book about the history of navigation and math or is that just a dream of mine ?
Dude, your enthusiasm is very compelling, and I actually watched the whole video, but this is definitely NOT an introductory book.
It's a great way to get a bunch of otherwise curious people to sadly quit math.
Well, I hope otherwise curious people won't quit math because of this book. I view this book as a better introduction to algebra and Galois theory even than very friendly introductory expositions on the structures of abstract algebra (groups, rings, and fields) because from my perspective the story starts with this material, and all that abstraction came later. (Also all the methods of this book are still useful methods.) The text requires from the reader only what is necessary to understand some concrete nontrivial arguments, and the exercises can be tough but there are very good answers in the back, which means you can treat the harder ones like theorems proved in the text.
you know the dummit and foote's algebra book right? that thicc book that has a lot of different stuff, very comprehensive, is there a similar book that's very comprehensive and has a LOT of material but it's graduate level algebra, basically like that dummit and foote book but containing graduate algebra
Yes, Dummit and Foote is a very nice, fairly comprehensive TOME of algebra. I used it to study for a qualifying exam many years ago. Edwards' book is completely different in aim, scope, and presentation, and I think it's better for students to follow something like Edwards' approach rather than study a multitude of structures of Abstract Algebra like in D&F. To your question, it's not clear to me that there is some coherent body of material that could be called "graduate algebra" that goes much beyond the material in D&F. Researchers in different fields make use of various algebraic structures and there's no need for anyone to try to learn about all of them. I do not recommend Lang's Algebra, which is an attempt at a comprehensive algebra reference that I find totally useless, and in any case mostly overlaps with the superior D&F. Some people like Aluffi's book Algebra Chapter 0, which is another comprehensive treatment from the perspective of category theory, which holds little appeal to me.
@@DanielRubin1 thanks, I agree no one needs to know a lot about every algebraic structure, but my curiosity just wont let that happen lol anyway thanks i'll check out those books and see which one's good for me
can you make a lecture about this theory in a simplest way to understand it
That's a big job. It would have to be a whole series and I'd have to come up with the right take on it. Maybe at some point. Thanks for your interest!
@@DanielRubin1 can you guide me what's I need to know previously to understand Galios theory
Until he finds a way to present this in his style, try the "Exploring Abstract Algebra II" playlist from Matthew Salomone.
Just discovered your video on the invention of natural logarithms. I have also been looking for the invention of statistics, but I have had a lot of difficulty. Any words of wisdom to aid a fellow knowledge seeker? Also, feel free to take this and run with it if you think it's worth sharing.
The historical development of statistics is a great subject and eventually I'll get around to making some videos on it. There are a bunch of good references. I believe I cited Hald's A History of Probability and Statistics and Their Applications before 1750 in the description of the Invention of ln(x) video. You can also look up Classical Topics on the History of Modern Mathematical Statistics by Gorroochurn, or for something short, look up Stahl's article "The evolution of the normal distribution."
Hey Daniel. I like how you know of a lot of good textbooks that include the historical motivation. Do you know of any such text for Geometry? Much appreciated.
Euclid's Elements lol.
The Line and the Circle by Carroll and Rykken (covers the beginning of Elements) and Heavenly Mathematics -- the Forgotten Art of Spherical Trigonometry by Glen Van Brummelen
You have a PDF of it?
Excellent review.
Thanks!
really very interesting, please do more book reviews
Thanks, meteor! Will do!
I agree with your overall opinion. But yellow GTM at the start? nope. I recommend Tom Leinster's rendition.
I found Tom Leinster's notes on Galois theory on his website. His course assumes a prior course in algebra on the theory of groups, rings, and fields, whereas Edwards' book, despite belonging to the GTM series, does not.
Love your book recommendation series! Thank you!
Glad you like them!
Aluffi's Algebra chapter 0 is the way to go
I may review that book at some point. I heard good things, so I started looking through it. My preliminary thoughts are that it is one of these comprehensive algebra texts with the theory of all the most used algebraic structures, and the exposition is nice and inviting. The book aims to weave everything into the perspective of category theory, which may initiate students into a language used by some cohort of mathematicians, but whose use in solving novel problems I find doubtful. I can see why some people like this book and think it is useful, but it takes an approach which is philosophically the polar opposite of a book like Edwards, which puts problem-solving first. I would never teach a course out of Aluffi's book.
I can't help thinking that if Galois had gotten a good night's sleep before the duel, instead of staying up all night writing, he might have won the duel.
I wanted to learn why quintic equations couldn't have a general solution in radicals, but it seems like SO much work for just that little payoff. Life often feels dull without recreational mathematics, but when it gets at this high level it just wears me out. I'm not sure if I'd rather be bored or worn out. Neither is very fun.
I'd say the journey of learning all the math to get there is interesting in itself. The techniques you learn are very useful even beyond proving the insolvability of the quintic. Group and field theory (and algebra as a whole) are used a whole lot in other branches of math and even physics.
@@HilbertXVI Yeah, I guess so. When I wanted to learn about relativity I took a detour and learned more about surfaces instead. I mean, I didn't learn MUCH about them, but at least I learned something. I'm glad I did.
I love your videos!
Well, +1 thing to the list of thing "to do one day"
I think Serge Lang's "Algebra" is also great, IMO it has actually changed the way graduate algebra is taught in universities. Lang was a prolific writer on the subject of mathematics.
Good for you if you can get something out of Lang's Algebra. I found that book impossible to learn from.
My advisor had Lang as a professor. He said Lang believed he should _always_ prove results in their most general form, and you can see this in Lang’s definition of say a group representation. I don’t think this is always beneficial for learning
@@JM-us3fr Formalism for the sake of formalism.
Sounds nice! Price is a bit hefty, though, which would be much less of an issue if it didn't come along with Springer's really shoddy printing and binding quality - a result of its turn to a 'print on demand' model.
Interesting. I really like the printing and binding of the AMS books.
@@DanielRubin1 I am very nitpicky with books, I must confess. Having sewn bindings is a must for me (book opens easily and stays open, and will not lose its pages after heavy use in one's lifetime).
Great job, very interesting
Thank you! Cheers!
Historical perspectives always seem to me to be very motivating, except possibly linear algebra which I still think has no clean, natural introduction, but is just like, alright, here we go... 😅
Glad you like seeing the historical motivation! I've done some preliminary digging into the history of linear algebra, and I'd like to cover some of the key point in future math videos. Some highlights: people have solved simple linear systems since ancient times. The determinant and its relation to the solvability of a system is due to Cramer of Cramer's rule. Cramer, Bezout, Sylvester, Cayley, and others associated with finally putting down the theory of linear algebra were interested in problems in algebraic geometry and incidence geometry (how many points are in the intersection of a collection of curves, or what collection of curves pass through certain points). The multiplicativity of the determinant seems to have been discovered by Gauss in the course of his work on classifying binary quadratic forms. And Gauss also developed the method of least squares and the form of the normal distribution in the course of working out a theory of errors in measurement when trying to predict where Ceres would reappear.
great video thanks , what do you think about Elements of Algebra: Geometry, Numbers, Equations by john stillwell ?? . i think it is also a very nice book to start learning abstract algebra with historical motivation
Thanks! I'll check that book out.
Many mathematicians prefer to have the shiny latest version of a theory and focus on the ability to play with it right away. For some reason I value understanding what kinds of problems mathematicians wanted to overcome when devising a new theory. Maybe that's why I'm not a professional mathematician lol
I'm with you, and I actually suspect that most mathematicians also find it very difficult to work with a theory whose definitions involve conditions that seem obscure to them. If I don't understand the motivating problem and the examples that cause us to make those distinctions, then there's very little chance I'll be able to make use of that theory.
Haha, H.M. Edwards is my favorite math author too. Great book, I have it, and also have his other books. They're good too.
His approach - example based with historical background - is the ONLY one I can study with. I just can't learn with the concise style of: definition, lemma, theorem. My rule: if it's from Springer and uses the default TeX font, skip it :-)
EDIT: Other math books i love:
Elliptic Curves by McKean and Moll (A gem packed with info, published by my favorite math publisher Cambridge)
Primes of the form x^2+n y^2 by Cox
Great to find someone else who appreciates Edwards' books and the book by McKean and Moll. I'll have to check out that book by Cox.
When is the great Daniel Rubin making a return?
Abstract algebra? Is all algebra not abstract?
I mention "Abstract" Algebra to make it clear I am not talking about high school algebra. But the point of recommending this book is that it's not necessary to have a course on algebraic structures in the abstract. Edwards' book is about solving polynomial equations, and all of the structure revealed along the way is completely concrete.
ME GUSTA VER LIBROS of Math QUE QUIERO TENER, Gracias a Daniel Rubin que puedo conocer. DÉSDE México in English, of course.
Well just thanks
Galwa theory?
hi Daniel, I came across another book that gives the historical background on this subject, titled "Equation that couldn't be solved: How mathematical genius discovered the language of symmetry" by Mario Livio, Please check it out and give your opinion.
1:21 Galois course 101 :D
Great review. I just got the book!
I'm curious what you think of 'Topology: A Categorical Approach' by Bradley, Bryson and Terilla if you've checked it out.
Thanks! Hope you enjoy the book!
I was not aware of the book you mention, though I'm familiar with Tai-Danae Bradley, who was the second host of PBS's UA-cam show Infinite Series. I checked it out. Everything I say comes with the caveat that I'm not a topologist and I've never had a use for category theory, but you can probably imagine that I would view that book with a lot of skepticism. The book's stated aim to "re-introduce basic, point-set topology from a modern categorical perspective" has nothing to do with solving problems. Maybe it could help someone who needs to absorb the categorical perspective in order to converse with experts in the field. I would look for a very different approach to topology.
@@DanielRubin1 Thanks! Yeah, you pretty much nailed the reason I would read it. I just watched your 'Math Major Guide' so now I have a better idea of where you're coming from. Just one more: how about the book 'Classical Topology and Combinatorial Group Theory' by Stillwell?
@@DanielRubin1 You should read the preface to the Stillwell book if you can find it. Sort of sounds like something you'd write :)
"In recent years, many students have been introduced to topology in high school mathematics. Having met the Mobius band, the seven bridges of Konigsberg, Euler's polyhedron formula, and knots, the student is led to expect that these picturesque ideas will come to full flower in university topology courses. What a disappointment "undergraduate topology" proves to be! In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams. Pictures are kept to a minimum, and at the end the student still does not understand the simplest topological facts, such as the reason why knots exist [...]"
Probably nice to read on the side, but if one wants to go the PhD route and whatnot, this won’t prepare them.
Using this book in an undergraduate course at an American University , is Academic Suicide for any professor dumb enough to do it.As attractive as this book might be , it has no place in the American System .UC Berkeley used to be number 1 ranked in the Country , in both Under grad , and Graduate and the Book "A first Course in Abstract Algebra " by Fraleigh which is an extremely gentle intro to Algebra was used for the under grad course ...I myself have taught Linear Algebra to undergrads and the typical student in such a class has trouble constructing the most basic proofs ...Let's not get carried away too much here....
You're probably right that most American departments will punish a professor who switches to a curriculum based on this book. You appear to suggest that the reason is that students will not be able to handle the material or do the exercises in this book. That's a shame. I would like to see someone do the experiment of teaching a course out of this book and report how it goes. I certainly think it would be helpful for students to have the prerequisite of linear algebra, which, though it is invoked very minimally in Edwards' book, would probably help students gain the experience with computation and proofs to tackle the exercises. But in any case, the material in Edwards' book is as concrete as could be, and is the real stuff of the beginning of algebra. How could a course on the abstract theory of groups, rings, and fields as covered by Fraleigh actually be easier while being as valuable?
@@DanielRubin1
In an ideal world , an undergraduate would open a terse book like Analysis by Rudin and rapidly complete one exercise set after another .....Unfortunately we do not live in such a world ....In fact most University administrators say that the real smart students study engineering not Math....Thus the typical math major does not have a very high level analytical ability ....Encouraging undergrads to look at graduate level texts is not in tune with reality in general ...I believe that you were also recommending a terse Graduate level Complex Analysis book for a first look at the subject ....Such things look nice on paper but rarely work out ...I used to be a Professor of Mathematics at a mid sized University and a Good Day was when someone understood an epsilon delta argument as to why x (squared ) is continuous or how to find the inverse of a 3x3 matrix....
I hate math and all things related to it. I never completed undergrad because I couldn't pass Algebra - a prerequisite. Completed 150 hours in everything else, but no math. It's a waste of time.
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I wanted to hear your point, but three minutes was too long to wait.. bye
You've certainly piqued my interest
don't pay a hundred bucks for this book! if you dig for it you can get a PDF for free. Else some publishers let you access texts for free via your institution.
"Very unique" is a pleonasm. I'm not criticizing your speech. I've just been waiting for an excuse to use the word "pleonasm". :)
Bilge. No one *starts* a subject with a yellow peril Graduate Text. Also already at 0:25 we learn that the speaker does not know English. Nothing is _very unique_. It is unique or not unique. Not really unique or very unique or somewhat unique or uniquely unique or amazingly unique. Just unique, or not unique. In the set {17}, 17 is unique. In the set {17, -1/5}, it is not unique.
Shut up