Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrentev.
Personally read only the first chapter, but the book is praised by lots of people. I bet Mr. Nikolaevich has read it.
You can find it on Amazon https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163
Besides the Napkin Project I mentioned, which is a genuinely good resource? I got a coordinate-free treatment of linear algebra in my school's prelim. abstract algebra course. We used Dummit and Foote, which must be prescribed by law somewhere because I haven't yet seen a single department not use it. However, in reviewing abstract algebra I instead used Hungerford, which I definitely prefer for its brevity. But really, you can pick any graduate intro algebra text and it should teach this stuff.
The texts by Stewart and Larson are the two most common introductions to calculus, and both are fantastic. Calculus is such a powerful tool that you don't need to worry about tailoring it to one field. Just as addition and multiplication will be used by economists and physicists alike, so will integration and differentiation.
Damn, son. That's way bigger than my guesstimate.
The amazon prices I checked out pinned the collection closer to $400, which granted is still really, really impressive.
In case you're curious this was my textbook. It's come down by a lot in price over a couple years. Brand new it was $365 in the shrink wrap from my school's store!
Eh, either way I'm wrong, just by a different amount.
A Book of Abstract Algebra by Charles Pinter. Don't let it scare you, it is a really easy read and a great text to learn some "real" mathematics.
I took a class in college called "Scientific Revolutions" about the shifts in scientific paradigm throughout history. One of the textbooks in the course was The Structure of Scientific Revolutions by Thomas S. Kuhn. It was rather enjoyable and extremely informative. I wish I kept my copy. Amazon has a bunch of related suggestions as well. (https://www.amazon.com/Structure-Scientific-Revolutions-Thomas-Kuhn/dp/0226458083) Perhaps this will spark some ideas?
I would say start with learning how proofs work. They can sometimes be confusing at first and (in my experience at least) calc doesn't do a great job of explaining them.
I learned from a book called A Transition to Advanced Mathematics it's been around for years, I know you can find copies of it for ~$5 without much trouble. It's got a good introduction to basic proof structure and ideas as well as dipping it's toes into combinatorics, algebra, analysis, and topology.
Another book worth looking into might be Book of Proof I personally don't have much background with this book, but it's the one used by my old university for their introductory course to proofs.
Yeah, I've just never been shown a problem where this stuff gives deep insight, and until I see one and understand it these are just gonna be arbitrary definitions that slide right out of my brain when I'm done reading them. I'll definitely give the book a look - is it motivated with examples?
The only book I have on category theory is Conceptual Mathematics: A First Introduction to Categories, and I must say, I'm not a fan of it - too intuitive, not detailed enough, not well organized, not formal enough - should have gone for MacLane instead.
In learning higher level math, books are invaluable. However, to be frank, the range of material you want to cover (algebra through pre-calc), I'm fairly confident any intelligent person could learn within five or six months, with or without books to supplement lectures they watch. Here's a book with good reviews for algebra, anyways:
https://www.amazon.com/Quantum-Mechanics-Theoretical-Leonard-Susskind/dp/0465062903
book on quantum mechanics
I see, makes sense.
And apologies: I said Taylor, I meant Stewart.
So my recommendation would be to start with pretty much any edition of Stewart, like for instance https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552/ref=sr_1_4?crid=27TLBWSQDEBRF&keywords=stewart+calculus&qid=1666117266&qu=eyJxc2MiOiIzLjAyIiwicXNhIjoiMi45NyIsInFzcCI6IjIuOTYifQ%3D%3D&sprefix=stewar...
The later editions just exist to be more expensive but don't really offer any additional value.
Once you've mastered everything in that up to Taylor series, I would maybe switch to a book on Linear Algebra. There are many good ones, and you might just "try out" the books by Strang, Lay, Friedberg, ... if you're really ballsy you can attempt Axler's. Axler's is not recommended as a first book, but you might get something out of the attempt at least.
Anyway, it's hard to imagine you'll even get to the Linear Algebra book before class starts. But if somehow you manage to get through Calculus and Linear Algebra, a very profitable thing to do is look into logic and discrete math. (This would cover, for instance, combinatorics among other things.) A good book for that is by Rosen.
https://www.amazon.com/Calculus-Early-Transcendentals-James-Stewart/dp/1285741552/ref=mp_s_a_1_4?keywords=james+stewart+calculus+8th+edition&qid=1661608473&sr=8-4 this is book essentially, I’ve tutored this for years know and I’m actually surprised you need to know the techniques since the application is more in depth, good luck!
This is the textbook I used when taking AP calc AB and BC.
Stewart’s Calculus is also a pretty common book at the university level.
Or, why you should learn Abstract Algebra (eg.) to get deeper in FP :).
An amazing talk that I was very fortunate to watch live.
Yeah I find this comes up for me a lot. I don't have a science background, dropped physics in highschool, can't do math without a spreadsheet to save my damn life.
That being said, a couple of these authors here were a huge jumping off point for me to become excited and energized at the concept; they may not go into the nuts and bolts of things but in terms of illuminating concepts and translating nearly undefinable ideas to a brain like mine it's essential.
I don't know, I think people like to pass judgement, but I find with QM there's as much art to the explanations as science, at least when you're starting to learn; you can hear the same explanation ten times, and then the right author comes along and number 11 is the one that breaks the concept wide open for you.
For what it's worth, that Halpern book I think is pretty well regarded as a historical account, I think the Carroll one is also good. Both little books are meant to be summations.
If you're anything like me and want to go "next level" on this stuff, I started with the Theoretical Minimum by Susskind and Friedman. It seems to hold up to a lot of scrutiny and is a text that appears in first year classes a lot. I'd be lying if I told you I understood it and it didn't kick my ass, but it may be what you're looking for as a next foray.
Finally Rovelli is a damn treasure and his face should be on money. Fight me.
If you have sometime, take a look through this book (<— online pdf of A Book of Abstract Algebra by Charles Pinter, which is a great (and cheap) book to get a physical copy of if you want it).
I think basic abstract algebra, which that course seems to be — it doesn’t sound like it goes beyond intro. group theory (equivalent to Ch 16 in the book I linked above), is barely even a proof oriented course. Because the objects and rules are so clear cut. The proofs are, very literally, almost entirely algebraic.
That said, everyone has different challenges walking thr same course so I’m not one to say.
But off-hand I wouldn’t stress it too much. Especially if you’re a junior. I’d go with both. But, talk to the prof and some other students.
I have a physics bg, but become increasingly interested in mathematics as years passed by. So I got into the habit of self-studying, mostly from translated Soviet Dover books.
I found that the following provides a nice overview of the basic math sub-fields, and it should be helpful for you too:
https://www.amazon.es/Mathematics-Content-Methods-Meaning-Dover/dp/0486409163/
You might like my goodreads list too.
Keep up the good work, believe in yourself, look out for friendly math communities either online (eg. this one) and offline, and never again let others define what you like and what you don't like :).
I liked this book when I was younger: https://www.amazon.com/Transition-Advanced-Mathematics-Douglas-Smith/dp/0495562025. It will help him make the jump from what's typically available in high school to what he'll encounter later. People are recommending Spivak's calculus, which is awesome, but I'm glad I had the one I linked under my belt before reading Spivak.
I have the book Conceptual Mathematics on my shelf that is good for beginners but I found it was too advanced for me at the time. As someone with a math degree the book would probably be excellent for you.
I always suggest this as an overview of the most important math fields up to the 70s, written by famous Soviet mathematicians.
I consider it pretty accessible to anyone with an interest in higher math, with a strong background of high school math and physics and a deep desire to learn more.
Well you can understand/get-the-frelong-of some stuff/concepts by reading high level books such as this:
https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163
But imho, if you want to understand a mathematical concept deeply, you'll have to go through the hard road of solving actual problems.
Hey OP, I was in a really similar position. I found that this book, basic mathematics by serge lange was exactly what I was looking for. It starts from the very beginning, but has you prove everything along the way and did wonders in teaching me how to think about math. It's not easy, but it's also not a bunch of rote memorization and gets you good as fast as you're willing to work at it, up to a place where you're ready to hop into calculus and other things.
There's also pdf versions put there. If I had to learn from one place, that book would be it.
A Book of Abstract Algebra - Charles Pinter
Given time constraints you may want to get a tutor, but regardless I highly recommend the above book. It does a great job of motivating ideas as it introduces them. It also has a lot of exercises to help. And the chapters are very short and to the point. (Book is also crazy cheap — as it’s a Dover book.)
Visual Group Theory - Nathan Carter
This ^ book gets a more mixed review as a stand-alone book, but can be an excellent supplement. The writing is… subpar, and the pacing is haphazard. But if you have some sense of AA ideas from elsewhere then the book is genuinely very helpful at giving you another way to see abstract algebra patterns. A lot of proofs and ideas become trivial once you’re able to visualize groups as Cayley diagrams. (Also just pretty and fun to do.)
A lot of the exercises reference some free visualization software online that the author wrote — use it — it looks a little sketch at first, but it’s actually great stuff and really helps you get familiar with various simple (and not so simple) groups and different ways of seeing their relationships. It’s very helpful.
Anyway — both of those books can help a ton, I think. But the cost is that it takes time to read them. It’s basically additional study materials. Depending on where you are right now you might need a tutor or friend to help you. But I would highly recommend the above. (Top one’s dirt cheap, other is like $70 — but if you have no choice you can find it online.)
I hope this helps and I hope you have luck. Abstract algebra is really playful and fun once you grok it.
You'll definitely want to read a theoretical book on calculus (Spivak is a good choice). Basic abstract algebra is fun, accessible, and very conceptual, and is used in physics and topology all the time. If you want to get properly into higher maths, linear algebra and multivariable calculus are essential. Once you have calculus, linear algebra, and basic abstract algebra, most other topics are pretty open for you to start learning.
The book that was used in my Intro to Proofs course was A Transition to Advanced Mathematics by Smith, Eggen, and St. Andre. Maybe the difference in presentation will make things click, but what I think might help better is a course or lecture series - like this for example. Both Book of Proof and the textbook I used start from what I'd call the basics - sets and propositional logic. Most textbooks will.
You might also want to look into materials for Discrete math courses. These tend to be courses mainly going over logic and sets, and are what a lot of students take before their Intro to Proofs course. That extra focus on those topics may be what you're missing.
<u>The Structure of Scientific Revolutions</u> could fall out of the sky and land square on his head -- he would just call it an ad hominem.
This sub will be overwhelmingly supportive of any idea you have (which is typically a very good thing), but it sounds like you may benefit a lot from reading books meant for an audience of people without a background in physics before you jump into this. This one is great, and there are other books with fewer equations you might also like. Getting a physics degree will require a very strong math background, and on top of that it’ll be two years or so of slogging along before you get to the sorts of things you’re envisioning (quantum mechanics, particle physics). If you’re worried your interest may be gone by then, it may be good to step back and do more thinking first.
Susskind’s book (Quantum Mechanics: The Theoretical Minimum https://www.amazon.com/dp/0465062903/ref=cm_sw_r_cp_api_glc_fabc_n5f9FbT9VTEFY ) and the corresponding free lectures which cover the same material (https://theoreticalminimum.com/courses/quantum-mechanics/2012/winter).
I also took a course on the subject which used Griffith’s text, but I feel I got more out of The Theoretical Minimum, honestly.
Abstract algebra is worth getting. It fun and exciting.I highly recommend this little book for self-study: A Book of Abstract Algebra by Pinter. It’s really good at motivated instruction. And remarkably well written forma rigorous book. Bunch of exercises to do — part of the pedagogy — but if you like math you’ll probably enjoy them. Little logic puzzles.
If you have a semi-decent understanding of algebra then a book to consider is: Visual Group Theory The book is a mess. It was someone’s thesis. It’s aimed at non-math majors. It has writing problems amd pacing problems. Problems! But it has strong redeeming qualities too! I used it (with copious commentary and auxiliary instruction) to teach math phobes group theory, but while that’s it’s intention — what influences was the visuals really reinforced my own group theory intuitions and made some concepts buttery intuitive.
Again, the Pinter book is amazing for reading on your own. The other book is a mixed bag, but offers some great insights if you have a basic abstract algebra framework going in (and are okay skipping/skimming the early chapters for more mathohobic types.)