I've got Mathematics for the Nonmathematician by Kline and that's kinda heading the right way, but what about whole courses of study? More books? It's more of an introduction than a thorough resource or course, and feels like it needs another four or five volumes and a lot more exercises to be really useful.
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 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...
But don't deny yourself an understanding of the meaning of limits. Almost all mathematics before calculus leaves you with a misimpression that neat formulas exist to solve problems. In reality, you've learned to draw straight lines with a ruler, and maybe a few curves with a compass. Before Calculus, you might actually believe that numbers that can be expressed as the ratio of two integers are typical, and that numbers like pi and the square root of two are "irrational" rarities (and until calculus, you probably don't know about Euler's constant unless it was introduced in precalc as another one of those odd and rare numbers).
Look out at nature, where are the triangles, rectangles, and circles? Maybe a wasp nest? Nah, not really. Try to draw a cloud, a tree, a tiger, or a human face. How useful is that straight line or compass? How useful is a line at all, other than to hint at something you can't actually draw, maybe by implying it exists as an ever vanishing limit from above and below? Math required calculus the instant humans decided to describe the world as it is, rather than by the limits of what we impose on it.
Also - in stats, how do you know what the area is under the probability density function?
What about meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory? In the post, the fieds under "electives" are by far the most interesting ones, IMHO.
And I fully agree, in-depth knowledge of probability theory as well as descriptive statistics and of course the application to systematic and sound decision making is absolute key, and ought to be taught to anyone from medic to policy makers (scary: Gigerenzer showed that medics tend to be confused about the difference between P(A|B) and P(B|A) - the very people whose job it is to diagnose whether you have cancer or not!).
If anything should be dropped from a highschool level its all of the insane memorization you have to do for some of the lower level math classes - I found that totally useless. You then learn some basic calculus and realize "I just wasted so much of my life" and never need to memorize those things again.
It's strongly self-selecting and as a result can afford to cover a truly insane amount of ground. The overwhelming majority of people who take it drop out (the usual dropout rate from people who take it in the first week is probably > 90%), but the people who stay almost all get As. And you will need to be almost entirely self-motivated because a lot (maybe most) of your waking hours will be thinking about math.
There's a very small minority of students for whom this is an optimal way of learning. For most students this is the quickest way to make them run screaming away from mathematics even faster than they already do.
Strong agree, but engineers probably need both. I'm currently watching a course on causal inference, and the tools are very much calculating gradients. And even if you just use someone else's MCMC, even in the models a differential equation or integral can randomly appear usefully.
In retrospect I should have taken a stats class in high school when I had that 1 hour gap for 1 semester, just to build a better intuition around the basic concepts.
I think math people may have a view of this question that is skewed in an interesting way.
Statistics is very useful and its common techniques are not difficult to apply.
But they seem to be very difficult to apply correctly. We have entire academic fields that are built mostly on the spurious application of statistical methods in contexts that make the whole project invalid.
And this sort of "techniques without theory" approach is what's being advocated for upthread and represents a failure mode that math people are unlikely to consider -- because they know that part of being able to apply a technique is being able to tell whether or not applying it is valid. Math people are unlikely to fall into the trap this approach sets. But the same people who want this approach are also likely to end up being hurt by it.
The only reason I took calculus is because it was the next math course after pre-calc and I still had a year left in high school. I didn’t even realize that there was a type of math that could be used to exactly calculate quantities related to continuously changing processes, but I was absolutely fascinated by how many real world problems calculus could solve - I realized that the problems given in algebra were contrived out of necessity to resolve in a neat manner. Now I had access to an entirely new vocabulary that allowed me to describe the world as it is, not as it needed to be to neatly fit in a 10th grade word problem.
Until that point I was interested in psychology (and had a vague notion that I’d drop out of school to make a career in music somehow), but I immediately dropped everything to take as many math and physics courses as I could. 14 years later and I work in a very math heavy engineering oriented field.
I know my story is probably atypical, and I have no clue what I’d be doing right now if I hadn’t taken calculus, but it’s one of those things that I look at in hindsight and think that I was that close to giving up on STEM, but for being forced to take Calculus. Instead, I earned my Ph.D. in applied math, and there’s not a day where I don’t use calculus of some kind.
And the basics of physics makes next to no sense without calculus, and even parts of intro chemistry make more sense after having taken calculus.
I remember in one economics class, which didn't have calc as a pre-req, the prof said "alright, for those of you who have taken calculus, this is an integral, you can now leave the classroom and come back tomorrow. Everyone else has to stay."
The ideas of calculus just seem so fundamental to me. It is sad that the American schooling system is so slow, and expectations so low, that it isn't taught to everyone. Meanwhile in other countries, everyone, artists to engineers, learns calculus.
It’s all integrals over pdf’s. A lot of integration by parts and other things in the core curriculum.
There's nothing a basic Stats 1 course needs calc grounding for (perhaps Riemann sums under a normal curve but that's more for deriving than the concept itself). As it stands now, AP Stats is used for kids that don't want to progress to an AP Calc, but want a math or need to hit a school or county requirement.
> 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.
A statistics course would be a much more brutal weed-out course.
Stats is not a generally useful skill (the concepts are, but can be taught in a data science course) but understanding how to work with data is.
Discrete math seemed to be the most applicable math class to my CS curriculum. I never took graphics so didn’t use linear algebra, and definitely never touched anything related to DiffEq.
I took calc 1 and stats 1 & 2. Much preferred the stats and it set me up for understanding all kinds of science lingo in articles and papers. I also indirectly use stats fairly often at work.
I'm all for giving tasters of as many different "beautiful ideas" as possible in school, but I think we should be elevating practical statistics into the top tier of subjects that we require kids to go through.
Because America has straight lines everywhere and no curves, unlike other places.
Calculus is absolutely required for most engineering tasks.
Maybe for CS students there could be more emphasis on statistics as well, without diluting the calculus.
Infact, where I studied, calculus was a prerequisite for certain statistics and probability classes. For good reason.
All fifty states or just continental?
I'm questioning your math pedagogy.
Calculus is the single most important math anyone, of any field, can learn as it's the first "practical math" you actually learn. Life behaves like calculus and in order to think about real life you need the concept of limits, derivatives, integrals, differentials, etc. It's patently absurd to say this should be replaced by statistics, which done to any rigor requires up to 2 years worth of calculus (through diff eq.) to even appreciate.
I'm shocked that you're a math major and didn't take away the biggest thing from learning analysis - the ability to think clearly through a problem and prove it correctly. While you may not be asked to vomit cantor's diagonalization onto paper for an interview the ability to think about problems you learned from doing these proofs translates to so many different fields, jobs, and life skills that I take the complete opposite view. If you want to understand anything you need to learn how to proof. I don't care if you're a nurse or an accountant. A rigorous proof based math course will change your life.
If by "learn statistics instead of calculus" you mean being able to mindlessly vomit today's new machine learning paradigm without understanding a thing then I think I can understand where you are coming from. Otherwise, I think this is some absurd parody of someone who studied math.
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.
Dumb question: what’s the best way to learn proof writing — let’s say for “fun”?
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...
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...
1) The problem with math in school is that there's not enough "real math".
2) Relatedly, insufficient exposure to "real math" in compulsory schooling is also (a major part of) why people think they don't like math.
My suspicion (again, without the actual background to make this claim with any authority) is that they are dead wrong on point 2—the "real math" parts probably contribute strongly to most folks' dislike of the subject, and the parts the mathematicians didn't like are probably relatively popular among people who don't go on to become mathematicians. This puts point 1 on some shaky ground (though it could still be true and well-justified, for other reasons).
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.
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
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.
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.
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.)
I find proofs and identities very hard to read. I assume it's a bit what having dyslexia feels like. I have to turn them, glyph by glyph, into something more algorithmic to make any sense of them. The only math-for-fun I've ever enjoyed are simple recreational math puzzles—proofs, reading or writing, are torture. I actually enjoyed the parts of math classes that mathematicians insist are bad and are the reason kids don't like math—the parts heavy on memorization and drilling the application of an algorithm—far more than anything that came later. Perhaps not coincidentally, those have also proven to be by far the best bang-for-the-buck of all my time spent in math classes over the years. I use that stuff every day.
It's called "engineering mathematics." (The books by Stroud would be an excellent choice here.)
I've forgotten 90% of the math stuff I learned
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.
I just tried clicking on some linear algebra topic, but it seems there are just videos
https://www.khanacademy.org/math/linear-algebra/alternate-ba...
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.
There are people in this very thread insisting that proofs are extremely useful in programming. I dunno if I just picked up the same skills elsewhere (Logic? Philosophy? Just... IDK, thinking and developing an absolute shitload of heuristics through years of experience?) or am entirely missing out and in fact don't have a clue how to program, but I don't see it (outside some rare niches where it probably is useful—coq exists, after all).
Sure, the word "exhaustive" can apply both to accounting for all (reasonably) possible problems in a block of code, and also to proofs, but the former doesn't feel at all like working on proofs, to me, to pick just one example (and some posts have seemed to imply that accounting for e.g. edge cases is exactly one case in which experience with proofs come in handy, but man, they feel like very different and barely-related activities to me).
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.
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.
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.
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.
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/
You'd have to know each octave doubles in frequency.
Side quest: When you play the bugle, the played frequency increases or decreases by MULTIPLES of the base frequency---NOT powers of 2. Suppose this base frequency is 250 Hz. There is an octave from 250 to 500, but there's a note between the octave from 500 to 1000 at 750 Hz, and a few notes between 1000 and 2000 Hz, which is the part of the musical scale something like Reveille is played. If Reveille jumped from octave to octave, it would just sound like the intro to Justin Hawkin's cover of This Town Ain't Big Enough.
So, if you know transformer hum is 50 or 60 Hz and Queen's frontman starts his singing at 100 Hz, then he can sing up to 1600 Hz, or four octaves. Mentally recalling what his falsetto sounds like, you can imagine a really high-pitched guitar solo an octave above this, and you can still imagine what an octave above that would sound like. (Maybe you're getting close to dog whistle territory in your imagination.)
This, then, is 6400 Hz you are imagining. The top of each sound wave to the top of the next is 6400 Hz. To record this, you'd need the top AND bottom of each sound wave, because the speaker cone moving from maximum to minimum displacement is how the sound is made. If you want to make sure you aren't accidentally recording the middle (zero crossing) of each wave, you can even take three or four or five samples per sound wave instead of two. It's a lot of thought, but you can reasonably decide that 25000 Hz is a good sampling rate for capturing much of the range of human hearing. Going too far beyond that, you're wasting storage space.
A CD holds a bit more than an hour of music, or 3600 seconds. If you've listened to Dire Straits, Eagles, Cyndi Lauper, Metallica, David Bowie, Led Zeppelin, ELP, or nearly any other band, you're probably aware the recordings have independent left and right channels.
Finally, each sample is going to be somewhere between "speaker fully retracted" and "speaker fully extended". With 5 bits, this gives 16 "stops" from the middle point to fully extended. But we know that music can get really quiet when it fades out, and a lot of volume knobs can go from zero to thirty and sometimes higher. When you have the volume at one, you can still tell the difference between loud parts and quiet parts, so you'd need an extra 5 bits just to get good dynamic range at loudest and quietest volume settings, or 10 bits. What happens when you double this? If you have 20 bits, you are probably close to wasting bits. You have a million places where the speaker coils can move to. For a speaker that moves a few millimeters, this means 20-bit resolution allows steps of a few nanometers. This is the scale of computer chips and color wavelengths. If you took the color blue and shifted its wavelength by a few nanometers, it would still be practically the same shade of blue! Without knowing about bit depth, you can reasonably assume 16 bits is good because it's a power of two and will give a lot of dynamic range. 8 would be too low. 32 is just wasteful.
With 32 bits, a speaker capable of moving 1 cm end-to-end will have 10 carbon atom diameters of linear resolution. The ears are impressive, but I don't know they can differentiate the air displacement of (speaker cone area) x (ten carbon atoms). Even having 0 to 100 on the volume knob, this leaves 25 bits of range at each volume setting. This is audiophile (and arguably, snake oil) territory.
So then, you can say 3600 seconds is pretty close to 3000 seconds, 2 channels is close to 3, 16 bits is close to 10, 25000 Hz is close to 30000 Hz... 3 x 3 x 3 x 10 x 1000 x 10000 ≈ 3,000,000,000. Since a byte has about 10 bits, divide by ten, and this yields a first approximation---based on logic reasoning of what we know---of 300 MB. It's wrong, but it's not "very" wrong. (It's off by a factor of two, not a factor of ten! Not bad for 4 rounded, intermediate conversion terms...)
(The idea is to round each term to a value starting with 1 or 3, because multiplying 3 and 3 is close to 10. The reason 2 is close to 3: 10^(1/2) = 3.16. This states that a good midpoint of 1 and 10 is 3.16, because if you square each term, you get: 1, 10, 100. Now, 10^(1/4) = 1.78. This means that any value less than 1.78 would be closer to 1 after squaring, and any value higher will be closer to 10.)
You can even take the analysis further and back-calculate things like how fast the CD might spin by guessing the track width and bit area, how long a track skip would be, whether the size limitation of the CD is due to optical or material properties, how far the laser would need to be to converge at one bit while being close enough that any deviation in the surface flatness doesn't send the return beam away from the sensor, etc. (This is all the info you'd probably use to begin the approximation if you weren't aware an audio CD holds an hour of music, like if you were asked in 1975 to "back of envelope" whether a compact, non-contact, vinyl-like, LP-length recording medium was possible.)
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.
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.
> 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.
So if you think about "real world usage", you can either use the math results and implicitly trust that they "just work", or you can dive a bit deeper, see if you agree with those results, or at least gain some insights from the proofs that have been presented.
And just to be clear, there is little to no "real world usage" math without academic math.
This might sound a bit dense, but the alternative is what 90% of programmers do every day.
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'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 do not, which is why I explicitly acknowledged that what I want is not "real math". I don't care about math for math's sake, even a little. Not my thing, never will be.
Basically I want to learn to apply useful results from hundreds-of-years-old "advanced" math the same way I learned math in grade school: memorization, pattern recognition, heuristics, intuition, and drilling, all with a focus on application. Keep the proofs far, far away unless there's some excellent reason I have to know them (and perhaps there's actually no way around that, but I suspect the current situation has more to do with the interests and world-view of people who design math curricula, i.e. mathematicians, than strict need, if you're mainly focused on application). Ideally, almost every single problem set would consist mostly of so-called word problems, drawn from realistic circumstances.
At some point, it would be good to get a a copy of Lyx and start to learn to write math in LaTeX - Then you can get feedback on your proofs online at math.stackexchange.com if you don’t know any math people locally.
Feel free to get in touch with me if you want to discuss further, happy to help!
I should also have mentioned this completely free book by richard hammack as an alternative: https://www.people.vcu.edu/~rhammack/BookOfProof/
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.
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.
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.
Has anyone else had an experience like that? (With math or other things?)
Don't leave us hanging, dude!
I really wish there was more opportunity for that. I'd love to take a few more classes, mostly in pure math, but there's simply nothing on offer for remote study past the 200ish level. (There are some remote masters programs in applied math, but nothing for pure).
I don't think I'd enjoy doing a PhD full-time. One or two classes per semester while working seems just about right. But the closest university is an hour away, so in-person isn't a realistic option.
See Brett Victor’s 2011 proposal: http://worrydream.com/KillMath/
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).
Your link doesn't even exactly talk about notation, but about pedagogy. Can you be more specific about which notation your consider "arcane"?
The assumption that there is a much better notation is one I tend to see only with the HN crowd. Outside of this group, even people who dislike the notation and/or struggle with math do not claim that a better/simpler one obviously exists.
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.
Some books I liked for self study because they have answers:
Introduction to Analysis, Mattock.
Elementary Differential Geometry, Pressley.
There is also recently Needham's Visual Differential Geometry and Forms, which is great.
Edit: I should also mention Topology without Tears (free, online, very good) https://www.topologywithouttears.net/
Too much emphasis on differential equations and not enough on things like topology, functional analysis and/or non-introductory parts of algebra like say representation theory.
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.
I just compared it to my undergrad's math curriculum and it matches up pretty well. Everything you mentioned is an elective.
- Calc I-IV
- Intro to proofs
- Linear algebra
- (Abstract) algebra I-II
- Real analysis
- Complex analysis
- Ordinary differential equations
- Partial differential equations
- (Others)
- 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 topologyFor real analysis it recommends as essential Abbott's "Understanding Analysis" and Rudin's "Principles of Mathematical Analysis". If you "haven't gotten your fill of real analysis" from those it recommends Spivak's "Calculus".
I'd consider promoting Spivak to essential, but using it for calculus rather than real analysis, replacing their recommendation of Stewart's "Calculus: Early Transcendentals".
By doing calculus with a more rigorous, proof-oriented introductory calculus book like Spivak, there is a good chance you won't need a separate introduction to proofs book so can drop the recommended Vellemen's "How to Prove It: A Structured Approach".
And the last(?) chapter where he uses induction to determine how to place an L-shaped figure on a grid...I never knew how to even approach that kinda' problem.
So yeah, I want actual practical applications ("exercises" != "applications") for math.
<climbs down from soapbox>
https://www.goodreads.com/book/show/8295305-a-book-of-abstra...
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.
I think it’s amazing how connected the fields are. It’s almost like “pick any two of analysis, algebra, geometry, number theory, topology, turn one into a adjective and you’ve got a new subject area”.
Topological algebra? Check (https://en.wikipedia.org/wiki/Topological_algebra)
Algebraic topology? Check (https://en.wikipedia.org/wiki/Algebraic_topology).
Geometric topology? Check (https://en.wikipedia.org/wiki/Geometric_topology).
Geometric algebra? Check (https://en.wikipedia.org/wiki/Geometric_algebra)
Algebraic geometry? Check (https://en.wikipedia.org/wiki/Algebraic_geometry)
Geometric number theory? Close (https://en.wikipedia.org/wiki/Geometry_of_numbers)
Mix algebra, number theory, and topology, and you may end up with arithmetic topology (https://en.wikipedia.org/wiki/Arithmetic_topology)
And don’t confuse that with arithmetic geometry (https://en.wikipedia.org/wiki/Arithmetic_geometry)
Maybe a more humble rewording of some of her statements e.g., "Anyone that follows and completes this curriculum will walk away with the knowledge equivalent to an undergraduate degree in mathematics." would be helpful.
Her suggested curriculum doesn't include anything from Number Theory, which is a foundational part of an advanced mathematics education. It is also one of, if not the most, beautiful topics one can study in mathematics.
I find it odd to call out "Introduction to Proofs" as a topic in and of itself. Proofs aren't really a topic in the way analysis or number theory is. At advanced levels, devising theorems and theirs proofs is what mathematics is.
I would suggest working through a proof-based linear algebra book in between to ease the transition. Axler's is a good one. Alternatives include Hoffman and Kunze and the more modern Friedberg, Insel, and Spence.
A lot of maths-related books, especially ones intended as textbooks will read like that in part because they aren't kidding about the 'abstract' in the title - they're trying to teach/re-summarize key concepts of mathematical abstraction. It's a good and true thing to notice.
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!
Also enjoyed Artin's Algebra.
> solving problems is the only way to understand mathematics. There's no way around it.
...without also understanding that doing problems is not a substitute for understanding.
(I'm still salty about that course. I've been doing linear algebra based puzzles nearly every day of my life and this professor somehow made the topic a boring chore.)
I complained about this to a friend who had also taken the course and he turned me on to Axler. I read through the first chapter, nodding along as I went. I got to the problem questions and couldn't believe what Axler was asking was even related to the material I had read through. I really struggled at first to understand. Axler was heavily juxtaposed to my previous experience. However, when I did understand, I didn't just understand, I grokked.
It was just such an awesome experience, and I credit that book in particular with breaking me out of a mathematics plateau and with liberating my mathematics education from a strict reliance on academia. The text is almost magical.
Kudos if the author is this talented.
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.
Why limit yourself to gender? Why not white vs other skin color? Why not the US vs another country? Why not democrat vs republicans? Why not western culture vs whatever other culture? Seriously, this kind of categorization is just ridiculous, especially when you speculate instead of showing evidence.
No, I won't be surprised if a STEM professor is reading topology. I will be surprised if a gender-study professor is reading topology. I will be also surprised if some stranger (i.e. I don't know the background of this person) who could only do pre-algebra in high school says Topology without Tears is the first book on Topology that they read and they immediately fall in love with topology. Possible, for sure. Surprising, of course. It's just a matter of probability.
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.
Something that might interest HN's demographic is Kevin Buzzard's Xena Project[2], centered around proof systems (in Lean). The natural numbers game [3] is particularly fun IMHO. I don't know if it counts as learning materials per se but it's certainly instructive.
[1] https://www.youtube.com/channel/UCIyDqfi_cbkp-RU20aBF-MQ/pla...
[2] https://xenaproject.wordpress.com/
[3] https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_g...
If you like getting into the nitty-gritty of problem solving then check out the books of Paul Nahin. They vary between "Level: Medium" and "Level: Difficult", with many of them reveling in the solution of equations, and integrals in particular. Although he recognizes the need for proofs, he makes a point of avoiding them in his books.
[1] https://www.johndcook.com/blog/2019/09/29/a-sort-of-mathemat...
https://www.gwern.net/docs/statistics/1957-feller-anintroduc...
this is linear algebra + combinatorics + probability + stats
If you understand the material in this one book it's reasonable to say that you are pretty good at math
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 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).
I learned all kinds of quantitative analysis and statistics in the CFA program ten+ years ago.
I had daily sheets that I would solve equations and answer all kinds of questions. I knew them forwards and backwards. I just looked at one now on fixed income - not sure if could answer any of the questions today to save my life.
And I work in finance daily!!
The alternative is something like "Oh, I should learn abstract algebra because it's a fundamental course in all curricula". You'll learn it, and then forget it.
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.
See https://press.princeton.edu/books/hardcover/9780691118802/th...
Not to be greedy, but do any of you know of other thorough curriculum guides like this? I know about https://teachyourselfcs.com already -- another amazing guide. Are there others? I would love to find one for statistics especially, but really any subject would be interesting.
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.
I don't want to study math. I want to know enough of it to solve some well-understood problems I've wanted to solve for decades. Simply learning how to diagonalize a matrix (and how to use such a thing) meant more than understanding a bunch of complicated matrix theory.
The discussion here has been much more interesting than the actual list, to me. If you wanted to master all of that material, I think a master's program is the way to go, not self-study.
I would be interested in hearing from people who _do_ successfully self-study. What makes it work?
The least interesting (from my point of view) is "already successful in a related field, applied my skills". That would include CS professionals studying maths, I think.
More interesting would be "unable to attend university because of X, did Y and really enjoyed it." Are you a person who completes MOOCs and get something out of them?
Step #0: make sure you know what math is
I can't stress it enough. I know this sounds funny, but when I went to study math a lot of people would drop out very quickly because they did not realise that what most people call math and what is taught at high school is completely different from actual theoretical math.
What most people think math is is a collection of formulas that you need to learn to "know math".
Math is actually a dynamic activity and is 100% about solving problems.
Just like programming is not about knowing programming languages. Programming is about solving problems (and knowing programming languages is necessary but not sufficient to be programming).
[1] website :- https://www.susanrigetti.com/philosophy
[2] discussion :- https://news.ycombinator.com/item?id=28367416
A shame I missed out on that discussion.
There are two main parts of calculus, and both can be well illustrated by driving a car. In the first part, we take the data on the odometer and from that construct the data on the speedometer. The speedometer values are called the (first) derivative of the odometer values. In the second part we take the speedometer values and construct the odometer values. The odometer values are the integral of the speedometer values. In notation, let t denote time measured in, say, seconds, and d(t) the distance, odometer value, at time t. Let s(t) be the speed at time t. Then in calculus
s(t) = d'(t) = d/dt d(t)
And d(t) is the integral of speed s(t) from time t = 0 to its present time.
Those are the basics.
Applications are all over physics, engineering, and the STEM fields.
Linear Algebra: The subject starts with a system of simultaneous linear equation. The property linearity is fundamental, a pillar of math and its applications. The STEM fields are awash in linearity. E.g., a concert hall performs a linear operation on the sound of the orchestra. E.g., in calculus, both differentiation and integration are linear. In the STEM fields, when a system is not linear, often our first step is to make an attack via a linear approximation. E.g., perpendicular projection onto a plane is a linear operator and the main idea in regression analysis curve fitting in statistics.
Most of math can be given simple intuitive explanations such as above.
That being said, I think you are missing out on an opportunity to reach a wider audience. It bugs me a bit that the requirements seem very American-centric. What I mean is the following bit:
> A high school education — which should include pre-algebra, algebra 1, geometry, algebra 2, and trigonometry — is sufficient.
And later the paragraph on "pre-calculus".
I know that many places don't have such names for courses in high school. In fact, often it's just called "Mathematics" and you either take it or you don't (obviously there is a spectrum here).
How is a prospective (non-American) student to know what is covered in Algebra 2 in an American high school?
I'm not asking you to change the article, I just hope I can nudge you into realizing that the text as it is now is more difficult than it needs to be for non-Americans.
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?
For other subjects, you can briefly substitute an intuition for the underlying structures with sufficient finesse in the presentation of the material (see the theory of knots and links, for an example), but calculus is not, in my experience, such a subject, and the early emphasis on it is harmful for the study of mathematics, which is supposedly what your list is for.
For some reason this is heresy, but I have honestly no idea how you are supposed to appreciate calculus from a mathematical perspective without being able to define the large stack of terms that constitute it. The situation is potentially different for a physicist, but if you want to study mathematics, the physical world is not the object of study, rather it is precisely the definitions that we have chosen.
Reading a math textbook is time consuming endeavor, regardless of underlying ability. The author herself mentions this in the introduction. I can think of a few factors that might make it possible for a busy person to go through all these books in a few years:
- They include books that were read partially while taking course.
- Consistency: allocating 1 - 2 hours per day for a few years.
- Doing exercises selectively: If you only do a handful of exercises per chapter this would dramatically increase the rate at which you go through a book. This would come at the cost of deeper understanding.
I have a large backlog of math books I'd like to read, but time is a constraining factor. If people have found strategies for reading these types of books, I'd like to hear about them.
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.
For many, many years I thought I did. I'd have a brief surge of interest for a few weeks, and then get completely bored of it. I'm not someone who finds it inherently easy, so boredom + difficulty = failure.
When I was foolish enough to do this in university, it meant doing great in the first few assignments, and then abysmally in the exam.
So my policy now is to never study maths for its own sake. Only when there's equations in a computer science paper I don't understand.
I'm partial to Jupyter notebooks lately - I run it locally from a docker container, and have a folder of notebooks. Mostly markdown cells, alternating between my narrative thinking and LaTeX math output.
For the other 99.99% of us, it would take many lifetimes worth of free time to make a substantial dent in these materials. To me, guides like these are too intimidating to even begin. Maybe what's needed is a meta-guide on structuring one's time and developing the necessary focus to be able to do this within one human lifetime.
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.
> [Mathematics] is the purest and most beautiful of all the intellectual disciplines. It is the universal language, both of human beings and of the universe itself. [...] That doesn’t mean it’s easy — no, mathematics is an incredibly challenging discipline, and there is nothing easy or straightforward about it
I am always, always going to condemn this unnecessary mystification and idealization of mathematics. It's exclusive and misleading.
"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."
:)
NO, NO, NO.
There is no real way to go up to the real deal without having understood elementary Functional Analysis, which the article doesn't even mention. FA is roughly what Linear Algebra would look like if instead of finite dimensional vector spaces we considered infinite dimensional vector spaces. It opens the rigorous path to non-linear optimization, analysis of pdes, numerical analysis, control theory, an so on. What this article mentions is a way to work around things, but nowhere near an undergraduate degree in mathematics.
I'm astonished that the PDE section has such books, they look like the calculus aspect of partial differential equations. A more appropriate book would be L. C. Evans' Partial Differential Equations. Same with ODEs, no mention of Barreira's or Coddington & Levinson's books.
There's a very good reason for this: FA is not all that useful if you're more on the algebra side of things.
Introduction to Higher Mathematics (Bill Shillito) - https://youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6KzmkUQ...
Some of my thoughts (mostly drawn from personal experience, feel free to disagree):
1. IMO "learning math" is really about learning how to recognize patterns and how to generalize those patterns into useful abstractions (sometimes an infinite tower of such abstractions!). So it really doesn't matter if one does abstract algebra or linear algebra or combinatorics or number theory or 2D geometry or whatnot at the beginning. Any foundational course in any branch of mathematics, or any book on proofs, will fulfill this need. People learn in different ways and have affinities for different topics, so some subjects will be easier and/or more interesting for them, so aspiring mathematicians should start with a topic they're at least initially entertained by. If you don't know where to start, one fun (for me) topic is the game of Nim; other combinatorics topics are also elementary and entertaining to think about. I'm fairly sure that if I had to take this suggested curriculum as an undergraduate, I would have picked a different major entirely, I personally find analysis quite difficult :(
2. One's first foray into a topic should be a one-semester course, not a textbook. Lecture notes for many courses are freely available online also, so you don't have to pirate the books you want if you aren't willing to pay $100 :P The reason is this: courses are curated by a mathematician to teach students the basics of a topic in one semester, so they will better highlight what you need to know, like important theorems, and have a more careful selection of problems. If you're confused, you can read the relevant textbook chapters. On the other hand textbooks are more like comprehensive references - reading a textbook through and doing all the problems will make you an expert at the material, but it's not as time-efficient (or interesting) as a course.
3. There are benefits to diving very deeply into a topic, but IMO one's mathematical experience is much richer if there's more consideration for breadth, especially when you're starting out. A student learning basic real analysis would benefit from understanding some point-set topology (not just the metric topology that usually begins these courses) and seeing how (some of the) pathologies of topological spaces disappear when you impose a metric and then you get things like being Hausdorff or having many different definitions of compactness coincide. After learning real and complex, of course one could move onto differential equations, but there are so many other ways to branch out, like exploring differential topology or learning about measures & other forms of integration, which also meshes very nicely with statistics. Exploring different branches emphasizes that there are so many directions you can go with math, even when you're just starting out, and gives you a better feel about how "math" is done, as opposed to just the techniques for a specific topic.
This is my first comment on HN, so please let me know how I can improve this comment!
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.
As opposed to... the ebook?
Only reading does