So you want to study mathematics

My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult.

The analytical type of thinking that proof-writing is certainly useful, but you can make much the same argument of many other curricula, and besides, it's not like most intro calc courses even do any proofs. The vast majority of them, I would assert, are simply pre-med weed-out courses.

I still remember freshman year, showing up to the standard intro calc course, and dropping it as quickly as I could in favor of my uni's equivalent of Math 55 (i.e. the hardest u/g intro math course) because of how asinine I found the content...

I did a degree in applied math. You’d think this would be “math you’ll use,” but the fact is that despite my program having a CS concentration, most of the stuff I did was not really applicable in practice.

However, one thing that has been VERY applicable is proofwriting. Although math proofs are far more rigorous than most real world stuff, the discipline I learned in writing proofs has carried over into pretty much everything from programming (will this algorithm work every time?) to executive decisions (why, specifically, should we believe X?). Obviously in the former case I wind up doing actual proofs, and in the latter I make strong arguments based on logical consequences of established or presumed facts, or find flaws or gaps in arguments that are being considered.

I really wish I’d spent a lot more time on proofwriting than say, vector calculus.

Of course you may want specific math to solve real problems, and that’s a real need too! Not to diminish your point at all, just advocating for proofs to be seen in a practical light.

I'm actually currently writing a book with exactly this emphasis!

The working title is "Practical Math for Programmers," and the idea is to build a collection of 60-75, 3-page long, _compelling_ demonstrations of mathematics used in production settings, biased toward stuff a generalist programmer might find useful. Not going into proofs or foundations, but providing lots of references to further reading.

I'm aspiring for it to be like a Hacker's Delight, or Programming Gems, but just for math that is genuinely useful.

Read more here: https://buttondown.email/j2kun/archive/a-week-of-book-writin...

Sign up to hear when it's released here: https://jeremykun.us11.list-manage.com/subscribe?u=99aa071e9...

I have some sympathy for this sentiment, and there is no doubt that the producers of mathematics could and probably should spend more time making life easier for the users and consumers of mathematics. This is true even /within/ the discipline, for very theoretical stuff. In the end this takes vastly more work than people expect. That's not an excuse, but it's true.

However, I also have a fundamental objection. I don't see how you can be an intelligent tool user without at least a little curiosity about how your tool functions. Maybe you can apply your tool, even be highly effective, in certain instances. But this is inherently brittle knowledge. When the parameters of your problem change and you don't understand your tool well enough to adapt, you're lost.

"Math for people who just want to use it" is very broad. What do you want to use it for? Physics, biology, chemistry, computer science? Sociology? Economics? There might be some shared stuff, but for all of these disciplines there is a vast space of mathematics that might be relevant.

I think Eliezer Yudkowsky's idea of a book (series) covering "The Simple Math of Everything" is fantastic. I would love to read that book.

https://www.lesswrong.com/posts/HnPEpu5eQWkbyAJCT/the-simple...

> Are there any "math for people who just want to use it" tracks in math pedagogy?

Use it for what? That is the question. If you pursue the what, you will inevitably be exposed to genuine ways that mathematics may be employed by it.

The academic standard for "learning math" is like "learning programming" by reading the C++ language/STL spec from front to back. No one productively learns programming that way, and even if someone did, they would hardly be well off when faced with a real-world production C++ codebase that follows $BIGCORP's inhouse programming style.

Swinging by to plug my personal favorite resource for refreshing my self on Bayesian stats: https://www.youtube.com/playlist?list=PLwJRxp3blEvZ8AKMXOy0f...

I think statistics is by and large the most proportionally underrated subject proportional to its utility. A good command of stats and probability expands your power to use data to reason about answering questions. The channel author, Ben Lambert, has an alternative playlist where he uses some of the techniques taught in this playlist to solve problems in econometrics. However, a lot of what is taught here builds a great foundation for other domains, on everything from bioethics to data journalism to computer vision.

Another great channel that focuses a bit more on the machine learning side of things is StatQuest with Josh Starmer: https://www.youtube.com/c/joshstarmer

Math education has become torturously miserable in the US by moving extraordinarily slowly. You have multiple years of working on only slightly more complex equations and concepts and naturally people get sick of it.

You have generations of teachers who barely know math and view it as a punishment, teaching kids and instilling the same views in them.

And then you have outsider still saying "can't we have condense it and simplify it further so we won't have to learn all these useless abstractions" and the curriculum bends further this way. But these actual situation of math is that not understanding what's happening is the thing makes it an empty and unpleasant activity.

Edit: Also, yeah, 90%-99% of math can be accomplished with some math software. It's just for the remaining small percentage of stuff you need some understanding and for a small percentage of that you need lots of understanding. So most of this seems useless but 99% correct is actually not enough in some significant number of technical situations, etc.

Well you can think of it like going to the gym. You don't exactly see people doing squats during their daily life, but you can see the results of people having a good physical core.

This needs to be explained further during education and motivated appropriately. We have a short-term utilitarian perspective, and we need to take a step back at times and recall that it takes time and lots of sculpting to transform a wood log to a art piece.

As you can jog everyday for fun and/or for the challenge you can also jog to improve your physical health. And not doing proofs is like declaring a guy can weight-lift by just watching videos on youtube and never lifting a weight. Or a guy can "code" without writing a line of code.

Sure, here's a fun one !

https://web.stanford.edu/~boyd/vmls/

(I'd even replace Strang's "linear algebra" recommendation with this book.) Imo, proofs are useful in so far as they are enlightening (e.g., the proof that a problem has a minimum is often useful in so far as it tells you how to solve it!) but in many cases they are less so.

Math is pretty fun, though, proofs and all, and I'd recommend trying your hand at it as a cool little side hobby! It can often help with "clarity of thought" :) (In many cases, proofs are just one or two lines that tell you something interesting, too, not page-long arguments that are mostly definitions chasing.)

> "math for people who just want to use it"

It's called "engineering mathematics." (The books by Stroud would be an excellent choice here.)

I just want a place I can go to weekly for 15min to practice math. It keeps track of what I know and gives me exercises of stuff so I can retain the skills.

I've forgotten 90% of the math stuff I learned

If you have a solid background in calculus, I'd recommend Zill's Advanced Engineering Mathematics, which is pretty much basic math for physicists and engineers (aka for people who need to "use it").

If you just want to learn math for purely practical reasons, Khan Academy (http://www.khanacademy.org) is great. It might have added more lessons, but when I went through it, it went up through 1st year college calculus.

The thing that makes it VASTLY better than most self study math programs or books is that there are hundreds of exercises that you can do, and see if you got the right answer. If you didn't, it will in most cases explain how to do the problem so you can try again with a completely different problem, so you're not just memorizing the answers.

Another thing that makes it great is you can do a little bit a day, start and stop, and come back to it and it will remember your progress and where you left off.

Khan is also a gifted teacher. Unlike a lot of math teachers, he has great pronunciation and handwriting and you can watch his lessons as many times as needed.

The list presented here, calculus, odes + pdes, linear algebra, is essentially that, mathematics for people who just want to use it. It's all undergrad level. There's several layers on top of this - set theory, rigorous probability theory, algebraic geomeetry, topology, that are less useful but interesting to mathematicians.

YES! This is exactly my complaint about math. I find the abstract proof-oriented math kinda interesting (although How to Prove It is an exception that I discuss below) but I _really_ want practical real-word applications of these maths.

Even though I've learned linear algebra decades ago, Andrew Ng's example of using a matrix to encode 5,000 images then doing linear algebra on it blew my mind. I've since used that perspective in many other fields. Not once have I applied a proof to solve a programming problem.

I've thought of publishing, i.e.blogging, examples that I've come across but that would just be a mish-mash of stuff I've read elsewhere with no overarching theme/framework. Besides, someone else must have done this, no?

EDITED: Used the correct book title.

I dunno, at that point, are you really doing math? Math is proof. Now, granted, nobody expects you to know exactly how everything is proved. But there is an expectation that you can prove most of the stuff. Otherwise, it's very easy to stray from the truth into the plausible-sounding, but incorrect.

That said, you're absolutely correct that more justification and motivation is important. So much of math can be taught with problems from physics, computer science, etc. Perhaps a good book for you would be Concrete Mathematics by Knuth? I haven't read it but people swear by it.

I am literally working on exactly that, as an project I'm calling Math: ELI5, all open source on GitHub: https://github.com/EternityForest/AnyoneCanDoIt/blob/master/...

The catch is that I myself don't actually understand this well. It's just "What I learned googling stuff while working as a programmer. I've had help reviewing it but there could be errors.

I'm trying to cover all areas of math that a nontechnical person, or typical non math focused programmer would need to know, but I treat actually doing any of it by hand on paper as an arcane thing for the really dedicated, so there's not really any excercises.

Instead of actually teaching a real understanding of math, which I can't do, because I don't know it well, I just explain what people who do understand it use it for and why you might want to go actually learn it.

I also have any historical math related stuff that I find interesting.

I can't tell you how to derive or prove Euler's equation , but you don't need to know math to understand the emotional impact from a humanities perspective, and be amazed that all those constants fit together like that, and that someone could discover it.

Ultimately, I think traditional math education has it totally right. My life would be so much better if I knew it, because there's jobs that seem to require exactly what they tech in math class.

It's not directly useful for non STEM types, but the idea seems to be give everyone a head start since so many do want STEM jobs.

I think you really do have to get to the being able to do proofs level to make use of it in the real innovative applications.

The common everyday applications math people like to cite can usually have a dedicated software package. It's not like we still need to add two numbers on paper. If you want to build something, we have RealThunder's FreeCAD.

Excel's Goal Seek and CAS systems do the stuff people say we will use algebra for.

But if your in tech eventually you run into the wall and need to do something like a Kalman filter or calculate stresses in a bridge and you're screwed.

It depends on which department you take math course in. Any course offered in a math department, even statistics and probability will be treated formally there because that's what mathematics is. What you probably need is to take math courses from an applied mathematics department like physics or engineering.

But math is a broad field so you're going to have to pick specific courses. For example, partial differential equations are quite common. But if you learn it in a math department it'll have proofs and full rigour just like a course in set theory. While some techniques for solving them will be covered, we mostly study the underlying mathematical structures, why certain techniques work, etc. If you take it in a physics department they'll teach you techniques and numerical methods to solve a certain class of problems like heat equations or fluid dynamics. But learning through direct applications will inevitably limit you to those techniques while learning things in full generality tends to make it easier to pick up specific techniques when needed.

> Are there any "math for people who just want to use it" tracks in math pedagogy?

If you're talking about college level, "math for people who just want to use it", is basically all it is (outside of math departments and perhaps outlier curriculums in some elite places) and that's a problem.

Learning just the applications without picking up the theorems and without a true understanding of the concepts makes more advanced work in whatever discipline one chooses more difficult. Why? Because you'll have to follow mathematical arguments and it's so much easier to do that when you got the background to fall back on.

I think students before college need more focus on mastering the basics. They're rushed so much, and tragically, math is very cumulative. Any one who has tutored before will notice that it's disturbing how many people don't really understand how to manipulate fractions as they enter a calculus course.

Street Fighting Mathematics.

It's still a bit much to absorb, but you can't really ever say the author didn't attempt to describe each lesson in real world terms. You memorize or learn to graphically/logically derive a few things, like how log(1+x)≈x for small x or how you can sorta guess for most quantities or plots. (One book example figures the data capacity of a CD if you know its music storage using informed guessing of each conversion factor from seconds of music to bits.)

As another example: powers of cos x from -pi/2 to pi/2... you can sketch cos x and then roughly sketch the second power, and then it's clear that the more times you do that, you get a bit more of a bell curve. One in the middle will always stay one in the middle, but you get less area with each iteration as the rest of the function approaches zero. If you wanted an integral, you eyeball where the plot is at 0.5, draw a full-height rectangle from the left 0.5 value and the right 0.5 value. Because the height is 1, the width is pretty close to the true area. You can decide at this point---if you want more precision---to visually guess if the tails or the main part of the function should have more area and adjust your answer.

There's an Open Access PDF at https://mitpress.mit.edu/books/street-fighting-mathematics

There were lecture videos on MIT's Open Learning Library (and somewhere else before that), but they're currently down.

The class site is https://ocw.mit.edu/courses/mathematics/18-098-street-fighti... as well as http://web.mit.edu/18.098/

I have 2 kids who are in college now. The beauty of math that inspired me to study math (as an undergrad math major) was treated as an aside in K-12 math, and has now been replaced with Grind. I learned to love math thanks to just the enthusiasm of my teachers, and stuff that I did outside of class. Both of my kids got great math scores, good enough to skip college math altogether, which one of them has done.

Today, kids need to be told: Math is nothing like what you learned in high school.

I think K-12 math should be divided into 4 quadrants, taught in a soft of spiral:

1. Arithmetic (symbol manipulation, through algebra and calculus) 2. Computation (numeric and symbolic) 3. Learning from data 4. Theory (sets, proofs, etc.)

Some of these things could be blended with the science curriculum.

You don’t need proofs, but you still need an intuition for why things work the way they do.

Otherwise you will be lost as soon as you leave the textbook territory.

Proofs are just one way to build intuition.

The best way to learn applied maths and get intuition is an “Introduction to Maths for Physicists” 101 course.

The proofs can help immensely in remembering theorems and knowing their needed assumptions and application area. Mainstream math education is also already light on difficult proofs in introductory (undergraduate) texts. You can always skip the proofs anyhow.

This is more of a university perspective, but......

> Are there any "math for people who just want to use it" tracks in math pedagogy

It's called Industrial Engineering /s (but only kind of)

It is tame as compared to a pure math or physics B.S. But, you pretty much cover the gamut of every math tool that is useful in the real world. From stats, supply chain, search, calculus to combinatorics and so on. Each concept is squarely grounded in a field where it is used.

I was surprised at how well my mechE (usually the closest undergrad degree to Industrial) degree prepared me for applied math and ML coursework for my CS masters. In comparison, a lot of CS undergrad peers struggled in those courses.

i second this. there should clear distinction between academic math and "real world usage" math.

I think, the problem is, proofs are what makes math useful.

This might sound a bit dense, but the alternative is what 90% of programmers do every day.

> I want a mathematics education designed for all those kids (likely a large majority?) who spent math from about junior high on wondering, aloud or to themselves, why the hell they were spending so much time learning all this. One that puts that question front and center and doesn't teach a single thing without answering it really well, first.

The problem is that the answer will depend heavily from person to person and from field to field and often the most sensible answers require a mathematical maturity that creates a chicken-and-egg kind of difficulty.

Do you care about engineering? Well you'll need some calculus for that. Do you care about prediction modeling? Well there's some stats you'll need for that. Do you care about finding patterns in the world? Well there's abstract algebra for that. Do you care about reasoning itself? Have fun with mathematical logic. But because humans have different motivations, there's no one-size-fit-all motivation-based approach.

This is especially painful for mathematics because I think most people who have learned some amount of pure mathematics will relate heavily to what helpfulclippy says in a sibling comment: "However, one thing that has been VERY applicable is proofwriting. Although math proofs are far more rigorous than most real world stuff, the discipline I learned in writing proofs has carried over into pretty much everything from programming (will this algorithm work every time?) to executive decisions (why, specifically, should we believe X?)."

The skills of rigorous and abstract thinking that pure mathematics provides is both nearly-universally helpful, but also simultaneously as a result very difficult to motivate. "This will help you think better across everything you do" is lofty-sounding, but generally not a convincing sell unless someone is already curious. But it's true that being able to wrap one's mind around pure abstraction (after rattling off a rigorous definition for an abstract question: Question: "But what is X really?" Answer: "X is just that. No more, no less.") has ramifications for all that one does.

And the most painful part of all of this is if you try to start by teaching the wonders of pure mathematics instead of all the messy, boring rote stuff, students' eyes are liable to glaze over even more because of the aforementioned chicken-and-egg issue with mathematical maturity.

In this way it's similar to trying to motivate someone to read and write. The key that unlocks that interest for everyone is going to be different and it's very hard to explain the near-universal benefits that reading and writing bring to one's way of thinking (but I can always just have a computer transcribe it or read it aloud to me!) without some inherent curiosity in them.

I didn't take any of these classes personally, but I do remember in college there was a whole host of Calculus for Business and Economics type courses that were much more focused on practical application than theory. Maybe picking up some of those textbooks would be the way to go.

Proofs certainly aren’t the only aspect of what one might call “understanding mathematics.” Aren’t proofs generally limited to very specific mathematical courses (at least until you get to the cutting edge where new results inherently require proofs)?

Theoretical physicists use probably more math than other subjects. Generally the litarature to use math is called 'mathematical methods for physicists'. The book by Arfken is considered standard by many. Did you mean something like that?

History of mathematics is a cool and relatable subject in so far as you can imagine the problems of ancient Egyptian worker paid in bread, and land survey/architecture. problems to Nazi code breaking, yeah?

There's even a package for that, it's called numpy.

Yes, all tracks of math involve people using math. Proof writing is a part of use for a lot of it. There are also very many tools within the use of proof writing where a lot of education is around learning a bunch of tools so that you can better "recognize which tool to apply, then apply tool", in a future real-world use where you want or need to prove something.

I'm a little snarky, but you have a broken idea of what math is. It's not even your fault. I don't even claim an unbroken idea for myself, I went through public education too, though I do think it's less broken. Somehow compulsory education has managed to get near universal basic literacy, but seems to have failed on whatever equivalent some sibling comments have hinted at exists for math or at least mathematical reasoning. A lot of algebra work taught for junior high can be understood as just a foundation to be able to understand later things (though you can of course use some of it directly as taught without having to learn more for every-day things like some boy scouting activities, or helping with putting together a garden or a fence, or programming -- and some of course is entirely useless). But instead of pushing algebra even earlier, states are instead moving to push it even later. (Let alone trying to spread awareness of even a hint of the subtle divide between more general algebra and analysis that a lot of STEM undergrads don't even really get a whiff of except maybe knowing it's often said to be a thing.)

To try and be more helpful, I'll suggest you don't actually want to learn math at all. So don't! At least, not directly. Instead, find something you want to learn more about in science, engineering, or technology/programming, and dig into it until you start hitting the math being used. For many things, especially at the introductory level, it's fundamentally no more complicated than being able to read a junior-high-school level equation. Occasionally you'll need to know about some functions like square root, or sine, or exponentiation, or some other new functions that will be explained (like a dot product) in terms of those things. When you don't understand something, you may need to find an outside reference (or a few) for it, if the book itself doesn't cover it enough or to your liking. Even then, you can often find outside presentations of that thing which are still motivated by the general field and are thus not proof-heavy.

However sometimes the best explanation may still be found in a "pure" book just about the thing, and if you can get over whatever problem you have with proofs you can learn to see how they can be used to build your understanding of the thing in smaller pieces, not just as tools to say whether this or that is true or false. In other words, proofs can serve the same function as repetitive problem-solving exercises, and are often given as exercises for that reason.

I'm a fan of the Schaum's Outlines series of books just for the sheer amount of exercises available in them, I just wish I had better self-discipline to actually do more exercises. Though they maybe aren't the best resources for a brand-new introduction to something.

To give a small example, maybe you're interested in game programming, and eventually want to dive into studying 2D collision detection more specifically so you can implement it yourself instead of using someone else's library, so you might stumble on a copy of "2D Game Collision Detection: An introduction to clashing geometry in games". Its explanation of the dot product comes early (its whole first chapter is on basic 2D vectors), consisting of 2 diagrams and two code examples (the first mostly defining dot_product(), the second using it as part of a new enclosed_angle() function) and some text all over 2.5 pages. It gives things in programming notation instead of mathematical notation, apart from some ² squared symbols occasionally. It gives a few equivalences like a vector's dot product with itself is its length squared, shown as dot_product(v, v) = v.x² + v.y² = length², without proving them, and points you to wikipedia of all places if you want to know more about how that or another detail are true. Why learn it? It's used immediately after in explaining projection, and then later in collision detection functions. Generally that book is structured as: learn the bare minimum of vectors, use them to implement collision detection for lines, circles, and rectangles.

I'm not saying this is a great book but it's representative of what you'll find that I think you're really after, which is motivated use of some bits of math. If you don't like that book's treatment of vectors, there are a billion other game programming books that cover the same thing as a sub-detail of their main topic, and maybe even better for you because it'd be grounded in e.g. a graphical application you've already got setup and running to see results rather than a standalone library. Or there's special dedicated math books like "Essential Mathematics for Games and Interactive Applications". Or you can go find dedicated "pure math" books on linear algebra if you want. Or maybe your junior high / high school math education was good enough you can more or less skip most of this and move on to something more interesting, like physically based rendering (https://www.pbr-book.org/) which also of course has vectors and dot products with brief explanations. Or maybe you don't care at all about game programming, and want to learn about chemical engineering, or economics, or the mechanics of strength and why things don't fall down, or...

I looked up the Chartrand Mathematical Proofs book and it's been a while since I had to buy a textbook, but $175 for hardcover and $75 for paperback or ebook? That's nuts. If I were a student today, I'd pirate that and feel absolutely no remorse for doing so.

Yes! Combinatorics, graph theory, linear algebra, and even some parts of theory of computation and grad-level abstract algebra and algebraic number theory can be learned without calculus (I think!).

N.b. This is not to say that these are all easier than calculus, and I wouldn’t even really recommend learning say Galois theory before calculus, I’m just saying it seems that one could.

How old were you when you started into maths past the highschool level, and how many hours a week were you working in the unrelated field? I’m 29 and have a sincere interest in math, but I’m relegated to self-learning because I earned on.y a 2.3 GPA in undergrad. I’ve been working threough Hammond’s proof book as well as some vector calculus, and while I seem to completely understand the material as I read about it, the exercises leave me pretty bruised up.

I don’t know anyone who knows or values math much past algebra. “I don’t need it for my career, so why would I spend tome on it?” I don’t plan to switch to engineering, so I often find myself distracted & wondering what value math will add to my life other than making my interests more obscure and distant from the everyday people I meet. All I have to go on is “I’m interested & I trust that I’ll find it helpful once I know it. Also some people who are good at math make pretty good money.” But when I get 60% on a set of exercises, it’s challenging to keep faith.

Well,

I feel like one can easily get a bunch of "Really you should start with X" statements concerning math. Really you should start with proofs, really you should start with problems, really you should start with these concepts. I started with concepts rather than proofs or problem and I too went to a MA and various study. I tackled both proofs and problems but I don't think I'd have done as well if I'd jumped on these immediately.

So, altogether for someone wanting to get into advanced math, I'd say to look at the variety of advice out there and follow the kind that seems to help your progress.

On the other hand I sometimes wonder why Calculus is made a big deal out of. It was probably the easiest among all the parts of maths. You just have to imagine an small unit and how to extrapolate for integration and imagine how to break it down to small parts for differentiation. When I originally learnt Calculus in high school it was couple of days of learning the concepts and the a week of deriving everything from that and move on!

I'm doing a MSc in mathematics. I tried to read that book once and I thought it was terrible. One of the few books I dropped before finishing it and also the reason why I have been reluctant to pick up other Murakami books. With respect to your phenomena of leveling up your mathematical maturity without doing mathematics, that has happened to me a few times and I would say yoga and psychadelics are the primary source of inspiration. Of course you need to do a lot of reading and studying (or if you have courses then you can sit back and have them feed it to you). One thing is making the information known and the other is to understand it, this is where the things that can spark random enlightenments come in.

I haven't read the book, so I'm curious to know what it was about the way they described maths that helped you so much?

C'mon! With a build-up like that and you don't mention the name of the book?!

Don't leave us hanging, dude!

Texas AM has a program that gets somewhat close though it definitely has a computational focus. Here's a list of their recently offered courses: https://www.math.tamu.edu/graduate/distance/openletter.html

Where did you take your classes?

My favorite accessibility-increasing tool is the computer. Doing math shouldn't involve so much circus math, i.e. doing things just for show, since a computer does so much immediately and accurately. We already use graphing calculators, and notation-wise graphs can be a useful tool in themselves in elaborating an idea (see also Feynman Diagrams, or electric circuits), but there's so much more calculators can do, let alone actual PCs, cell phones, and web apps. By chance in 9th grade "Intermediate Algebra/Algebra 2" I had a teacher not wholly opposed to modern technology and so he only had us do a small amount of those "solve this system of equations using a 3x4 matrix by hand, showing each matrix transformation to reach the row reduced form, taking up some pages of paper" problems before he brought in a classroom set of chonky TI-92 calculators and showed us the rref() function. That Christmas I asked my mom to upgrade me from my non-graphing scientific calculator that had served since elementary school to a TI-89 Titanium that served me even through college until I learned and got used to various PC programs. The lesson that there were powerful tools around stuck with me pretty fast though, and I wrote some simple programs on the calculator for that and other classes throughout HS (mainly just automating calculator input steps, no fundamentally new algorithms the calculator couldn't already do); in HS physics I also had learned more programming and did a little simulation with pygame and it was fun to enter numbers in the program, run it, see the mass trajectory animate and show some computed values, and then do the actual experiment and get the same results. I only wish I had been shown some PC programs earlier.

I met a friend many years later who sadly was still forced to do that rref()-by-hand for even larger systems of equations in university! That left no time to actually learn anything useful in linear algebra. Madness.

https://theodoregray.com/BrainRot/ has some nice ranting about this (though it does go a bit off the rails when it starts talking about video games).

Why is it that every time any subject about mathematics comes up there is always a complaint about notation?

Your link doesn't even exactly talk about notation, but about pedagogy. Can you be more specific about which notation your consider "arcane"?

I wrote an essay response to this proposal that you might enjoy: https://jeremykun.com/2020/05/17/musings-on-a-new-interface-...

Sure, an introductory notation could be created to bridge the gap, and that could be more human friendly.

That said, as someone who uses notation for math regularly, I want to keep using notation. It’s a helpful tool, and it is an efficient and precise language.

I think learning Real Analysis from baby Rudin is like learning Probability Theory from Wikipedia. It's so encyclopedic that if it's your first look at real analysis, it will be too dense to understand, but if it's your second or third look, you will find beauty in its brevity.

Very pretty book (Needham's), will check it out! I think over 20 years ago I actually attended a house party that Needham was giving in SF. It's a small world.

Agree that Baby Rudin is VERY difficult to study on its own. I recommend only studying it alongside the other two books I listed: Abbott's Understanding Analysis and Spivak's Calculus (which has a solutions manual). Abbott in particular is very straightforward (at least in comparison with baby Rudin haha)

Those are good, I also really like Visual Complex Analysis

Consider Analysis 1/2 by Terence Tao for introduction to analysis.

Surprised Arnol'd isn't mentioned for ODEs.

Yep, I came here to say this.

I think it is important that anyone who wants to study math understand that real math is not at all like what you learn in a physics or engineering department. In these departments you will always hear people say things like

>"proofs are not useful, all you have to do is memorize the 'trick' they use. Once you know which trick to use, it is easy"

or you will hear them say.

>"Math isn't about understanding, it is just about learning rules and symbols on paper".

This is not mathematics. These things do happen.... in a physics and engineering department. It is, in fact, a descriptions of a physics education, not a description of a mathematics education.

For this reason I would be careful taking mathematics advice from physicists too seriously as they may, unintentionally, lead you very far astray.

> This felt like it was written by a physicist or engineer.

I just compared it to my undergrad's math curriculum and it matches up pretty well. Everything you mentioned is an elective.

guilty as charged! :)

As someone studying teaching math, I found it interesting to compare her suggestions:

  - Calc I-IV
  - Intro to proofs
  - Linear algebra
  - (Abstract) algebra I-II
  - Real analysis
  - Complex analysis
  - Ordinary differential equations
  - Partial differential equations
  - (Others)

to my program plan:

  - A couple teaching courses (including one for roughly grades 5-8)
  - Calc I-IV
  - Statistics (one without calc, one with)
  - Linear algebra
  - Discrete
  - Geometry
  - Number theory
  - History of math (apparently not just a history class, I haven't taken it yet)
  - Abstract algebra and into to topology

I'll second this. "How to Prove It" gets recommended a lot, but I couldn't get through it. I found it terribly boring and unmotivated. Some people can power through dry material but I'm not one of them. I found it much easier to learn to write proofs when they were related to topics I was interested in.

Spivak is a better analysis book than Abbot.

Speaking of group theory, I can recommend "A book of abstract algebra". I think that it's a very approachable introduction to the topic. As a person with a CS degree doing ML, it changed my perspective on so many different topics, I can't recommend it enough.

https://www.goodreads.com/book/show/8295305-a-book-of-abstra...

It’s amazing how different the subjects of mathematics are. It’s like the difference between a drum and flute.

You listed some of my favorite stuff. Weirdly, when I was 11, my math tutor told me I’d probably really like finite mathematics. She turned out to be right.

Both Strang and D&F are extra-relevant for cryptography (I was struck by how much the earliest parts of D&F --- which I haven't gotten much further beyond --- read like the mathematics background chapter of a cryptography book), and I've been in study groups for both of them with non-mathematicians that went OK. But the D&F study group fell apart for logistical reasons, so maybe it would have hit a wall after a couple more months.

Gross's review of linear algebra from his MIT algebra course bridges the gap: http://wayback.archive-it.org/3671/20150528171650/https://ww.... A combination of that and then chapter 11 in D&F should cover whatever readers didn't get from Strang.

That being said, Axler is an excellent book. I don't know if I would replace Strang with it, but I should add it as a supplement to the next edition of this guide!

Second Axler. "Linear Algebra Done Right" is probably the pure mathematics textbook I've most enjoyed reading ever (but be warned you will learn very little about applied methods from it if that's what you care about).

Also enjoyed Artin's Algebra.

Strang's latest book DE&LA is disappointing, it is linear algebra and its applications with the abstraction taken out and mushed together with supplementary notes from ODE videolectures. Mattuck's ODE course is good.

For many years MIT students would go from Strang in year 1 to Artin in year 2. Artin != D&F of course though many would say it does less hand holding than D&F

I’d be hard pressed to recommend D&F to a self study audience. Perhaps it’s just me, but the endless enumeration of examples and counterexamples at the beginnings of most chapters seemed to destroy my intuition rather than reinforce it.

I have been using Morris in my class.

I agree that this does not on its surface seem possible, but I can think of a few explanations.

1. I recently spent a week on one section of one chapter of a math book. I was able to follow it within an hour on the level of "these are the rules and this is the sequence of their application," but I have stuck with it since then because I wanted to understand it well enough that the proof they chose to use would seem obvious to me. If you saw "understanding math" like the peak of a mountain, you'd get there a lot more quickly, but if you want to try out every permutation of every device and condition anything can take forever.

2. Algebra seems simple in retrospect, and my teenage self was kind of dumb. Maybe with my complete adult brain I'd be able to finish highschool starting from scratch in a few months. Evidence to that point is the pacing of college remedial math classes. Maybe, to a certain extent, people have an innate math setpoint that they will snap to very quickly when given the chance.

3. Intelligence is equally distributed between genders, but most professional physicists are men, which means that for every professor there is almost exactly one corresponding woman who has equal potential but isn't in the system. If you heard that the department chair at a university sat down and read a book about topology without a lot of trouble you wouldn't be surprised at all. In other words, it's not surprising that someone can do this, it's surprising that someone who can do this is not in the social bucket for people that do it, but if you think about the other things you've heard about that, you realize you already knew.

I am inclined towards #3 out of all these explanations but all may be true at once.

> I find it hard to believe that the author started to appreciate physics by reading The Feynman Lectures on Physics before any exposure to physics

I'm a physics major. My first exposure to physics was Feynman, and it made me want to become a physicist. I think that's a statement more about Feynman than me though, as I'm not particularly talented. It was a common story among other physics majors too.

More like one in 50-100 million brilliance, and such people do exist. It's a statistical certainty they exist. Terrance Tao for example.

At a first glance that book completely fails for an undergrad lin alg course, and looks weak in other areas too.

Examples: the words nullity and kernel don't appear, rank of a matrix is not in it, and, well, every topic I can think of for undergrad linear algebra is simply not in the book.

It's equally bad for stats: no mention of many common distributions a student would learn for example.

I hesitate to contradict someone who has gone through the whole Math PhD process, but I have to say that the best mathematicians that I know treat the problems they're working on as games, or riddles to be solved... and they've taught their kids (and others) this same method of thinking about these problems. There's a huge mental tool set, and often a lot of grinding to get to a solution, but it's just a game (and there are often more elegant solutions)!

I enjoy provoking interest in complex numbers and exponentials among precocious teens, but I've never been more humbled than having a Galois theory joyously explained to me by an 11 year old. (p.s. I'm an engineer so I follow your technique).

Was programming a 3-axis machine in early college back in 90s. After a few months I was mostly re-inventing trigonometry. The I actually took trig later on. That would have been super handy at time.

I think the question needs to be asked, "What am I hoping to get from this?"

If its tools to help you solve some problem, great. If its because you find it fun, great.

But if its because you feel you want to 'better yourself' or perhaps feel like it would somehow prove your intellectual worth, you probably won't get a lot out of it.

This is true. When I started my first job, I tried casually learning more from where I stopped after I finished school. My motivation slowly waned as I realized I vastly preferred playing video games than watching lectures and trying to do some problem sets.

Correct me if I’m wrong, but I assume you’re looking at this through the wrong lens. As a PhD you’ve learnt something hard at a considerable depth but this isn’t what I, or most people, hope to achieve by learning maths themselves. It’s normally to a far lower depth that’s far more achievable like Calculus I or some aspects of number theory or knowing what all the funky symbols such as Summation mean.

I experienced something like this recently. I struggled to grasp linear algebra when I took it in undergrad and as a result was always intimidated by the subject. Now, years later, I'm taking a course in graphics and naturally needing to relearn linear algebra and suddenly everything just clicks.

POTUS is a very low bar...

I really wonder how some people can be so productive and prolific. In comparison I feel like an uneducated slob.

Susskind’s Theoretical Minimum is fantastic for physics self study: https://theoreticalminimum.com/

I posted recently asking for exactly this... but for medicine.

I have a body, everyone I know has a body.

The operate pretty much the same everywhere in the world.

I would like to know how it works.

The solution to russel's paradox is that sets can not contain themselves. There is a carefully crafted set of rules describing what sorts of sets exist, and it is designed to avoid paradoxes such as that one. These rules are the ZFC axioms (https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html)

Do we have good universal descriptors for math levels? I'm a big fan of accessibility, and I think your idea about tweaking language to reach a wider audience could be a big win for increasing the article's impact.

To update the article to include your recommendations, the author would probably need some kind of "cross-walk" which would map the American perspective to a more universally understood framework. Would you happen to know what "pre-calculus's" opposite number would be in the universal framework?

That’s what the real analysis course is usually for, and why it comes after linear algebra and algebra.

For me, it’s all about consistency. It’s spending a few minutes to a few hours every day, forever, til the day I die.

I usually can squeeze in 30 minutes to an hour every day to study something (whether it’s math or something else — right now I’m studying cinematography). Sometimes that’s in 15-minute chunks if it’s a busy day. Usually it’s before bed or while I’m eating lunch or if I have extra time on the weekends while my kids are napping.

It’s all about just doing a little bit every day. That’s been successful for me.

Agree, but I have a feeling that statistics is more like (theoretical) physics, in the sense that it is "not math."

I find paper and pencil works pretty well.

I will tell you the secret: just jump in.

Grab a pen and paper, open up the first book in any of the guides, and start reading. Read for 15 minutes, or 20 minutes, or whatever time you have on your lunch break or before you go to bed or while you’re using the bathroom. Do it again the next day. And the next. And the next.

That’s how I did it. There’s no brilliance involved. It’s just jumping in. The more you do, the easier it gets.

Check out my physics guide: https://www.susanrigetti.com/physics. It has both the physics core curriculum AND the math essentials you need to know in order to understand the physics essentials. (And thank you!)

On the mysticism note, I want to add that math is perhaps the only subject that forces one to engage the upper "abstract" mind. The lower mind is concerned with modeling real world phenomenas, while the upper mind works with purely abstract things, aka the "true reality" in mysticism.

You cut out the middle of that paragraph, which says:

"Sadly, there is all sorts of baggage around learning it (at least in the US educational system) that is completely unnecessary and awful and prevents many people from experiencing the pure joy of mathematics. One of the lies I have heard so many people repeat is that everyone is either a “math person” or a "language person” — such a profoundly ignorant and damaging statement. Here is the truth: if you can understand the structure of literature, if you can understand the basic grammar of the English language or any other language, then you can understand the basics of the language of the universe."

:)

I'm a fan of functional analysis, but even in my (very competitive) undergraduate curriculum, it wasn't required for a bachelor's in mathematics. I think Susan's guide covers most of what the undergraduate programs I've seen require.

This is certainly not universally the case, even in very well-regarded departments. The University of Chicago, for example, does not require it: http://collegecatalog.uchicago.edu/thecollege/mathematics/.

My undergrad institution does not require FA to graduate. It doesn't even list as an elective.

There's a very good reason for this: FA is not all that useful if you're more on the algebra side of things.

You may be surprised to discover that many/most physics undergrad curricula do not mandate a course on probability.

Agree, your appoach is much benefit for the mass

All over the world many people try to learn French. Almost any would tell you that learning a second language is hard. Nobody will fight you on this.

Math is a language to express entirely new concepts thar have nothing to do with everyday thinking and often include things like recursion that makes them impossible to reason about without more than just a few bytes of mental RAM.

There is no step by step algorithmic process to do it. If there were, a computer would be able to do it and far fewer people would want to learn.

Even at the level where there is an algorithm, it's far too big, nobody hand executes the source code of XCas by hand.

Math seems to require not just a skill that can be learned, but an inate ability to deal with multiple connected pieces of information at once, and to see abstract patterns in things.

In programming, if you have to understand more than one tiny bit at a time, you might consider tossing it and starting over. In math it's just normal for lots of ideas to connect.

True! But sometikes getting better at something is all people really want, and that's ok. I see that most of my CS students just want to be able to not see math as an obstacle when learning new/interesting things.

Yeah, iirc the ebook formatting made the book difficult to read.