As opposed to... the ebook?

Only reading does

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.

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.

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.

> [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.

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.

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!

This is in the site guidelines: https://news.ycombinator.com/newsguidelines.html.

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/

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.

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...

They don't seem to have a problem intuiting what a plotted PDF is showing. I think that's because in some sense it can be read analogously to a histogram. Of course, they don't have the tools to generate or manipulate one. But that's honestly not something that applied social science researchers have to do often when using traditional methods.

https://www.khanacademy.org/math/linear-algebra/vectors-and-...

Someone without any background in the subject would probably find Dafny interesting.

For example, I might not remember how to code bisection search but I can figure out the loop invariant and from there it's easy enough to code a working binary search. And even if you have right bisection search down to muscle memory, you can modify the loop invariant to create left bisection search if you need it, while if I tried to keep two binary search algorithms in my head I feel like it would be more error-prone.

Those are indeed very useful for life/safety critical systems such as aerospace, air traffic control, medical devices, etc.

1. Students in math courses and their parents grow to prefer(through the overall institutional constraints) to have a simple exercise that guarantees them credit - while actually doing math is a matter of crossing the Rubicon into tough puzzle-solving, and it needs some guidance for unexceptionable students to start enjoying.

2. Math teachers, particularly in the lower grades where qualifications are lower, have a harder time teaching concepts than they do exercises. And they are also incentivized to hand out a grade, preferably one that satisfies the parents.

So no matter how the high level is set up, everything converges into giving the kids a worksheet to "plug 'n chug." Which is just a confusing, badly paced grind, and therefore an easy reason to hate math. Either you get it completely and are just sitting there chugging through the problem set, or you have no idea what's going on and it's due tomorrow so your grade rests on something you feel defeated by.

I actually think that for the parts that are currently treated as rote memorization work, the curriculum should lean into it and treat it like learning the alphabet, with worksheets where you literally fill in the dotted lines repetitively; hand them out to everyone as a portion of the homework. And then the logic and critical thinking aspects need to proceed like a philosophy course, with interaction through a step by step process, not "get the answer in the back of the book". This element is something I've long thought could be automated in some degree with computer systems that let you play with the concepts, and therefore correct your thinking.

So you're saying you know nothing yet are sure "experts" are wrong, based on no evidence. OK.

1. How the Body Works from DK.

2. The Machinery of Life by Goodsell.

Check them out.

DK books are great to get introduced to something new- get a lay of the land, learn introductory jargons, etc.

I think they’re both true. To the musician, the drum and the flute have a sameness. “You make notes in time with them”, the unconscious mastery. To the beginner (me) it’s worthwhile to see their separateness. I shouldn’t think “I don’t like playing instruments.” because of my experience with recorders in the third grade. Don’t let the trauma of high school trig keep you out of graph theory. They certainly feel different.

This is not hollow advice. Yesterday I bought a flute!

Algebraic combinatorics (imo) encompasses related structures in all three of combinatorics, algebra, and geometry, though.

Well, it is not only about people. You would also want to "convince" Nature (so that the aircraft you designed does not crash right after a takeoff), your own or, often, someone else's wallet (so that money is not spent on nothing).

I think this is a common first experience when first hitting pure mathematics. Mathematics often feels like very rote applications of rules drilled into one's mind, and then you hit a pure mathematics textbook and the questions become a step change in difficulty where you're expected to derive novel insights on your own that the text doesn't hold your hand in showing. A single problem can easily occupy days of your time before the "aha!" moment, but as you say, once you get the "aha!" you realize your understanding is quite profound as opposed to a shallower understanding of just how to apply a given set of rules.

For instance, let's take a very common "application" of statistic: given a test that has false positive probability p, you can apply the test n times, and the false positive rate of the composite test is p^n. (For e.g. this is used to analyze bloom filters, or the Miller–Rabin primality test, or hash tables). However this is only true if the tests are independent. If you have to apply this result to analyzing (for e.g.) some new data structure you wrote, you'd first have to prove that the tests are independent. And maybe it's just me, but I find that if I use heuristics to check if some events are independent, I really often get it wrong.

Another example: the uniform limit theorem (https://en.wikipedia.org/wiki/Uniform_limit_theorem) is quite useful, but to properly apply it, you have to understand the difference between uniform convergence and pointwise convergence, and maybe it's just me, but I found the difference very unintuitive when I first encountered them (in a standard proofs-based analysis class). Even now, if you gave me some random series of functions, I can't really imagine using heuristics to check it uniformly converges to its pointwise limit, I'd want to try to write down a proof to be sure. So this useful tool is gated behind understanding some (to me) subtle proofs.

You might want something less applied, but I highly recommend Burden & Faires "Numerical Analysis" and Trefethen "Numerical Linear Algebra". The various interpretations provided by Trefethen for understanding a matrix, eigenvectors, singular value decompositions, etc. are amongst the most insightful descriptions I've come across.

Drilling proofs is a valid kind of drilling and can be an effective way to learn something. Not necessarily the only way, sure. But there's nothing fundamentally different or "more real" about drilling proofs vs drilling grade school multiplication problems. You'll memorize things, you'll see patterns, develop heuristics, gain intuition.

Application can sometimes be tricky; did those grade school multiplication drills have application? Are they granted more application by phrasing things in terms of word problems around counting apples or whatever rather than the compressed a times b = blank expression? Well, sometimes the application of proofs will be more direct, sometimes less, and can be phrased better or worse, more realistically (and necessarily more complexly) or less so, like any other exercise, whether it wants a proof or not. CS proofs about big-O complexity are applicable to analysis of algorithms, which is pretty important if that's your focus. Though most problems you could find to drill specifically on big-O (as opposed to other parts of algorithm analysis, like recurrence relations) would likely take the form "find the complexity of this" or "given the complexity is such, estimate..." rather than "prove that...". There are many things no one knows how to prove that are still an area of study, clearly proofs aren't the be-all-end-all. Anyway, the mental processes involved between something like "find x, the hypotenuse of the triangle" and "prove the Pythagorean theorem" often aren't that different. There are multiple ways to prove it, you could drill on them.

And technically, computer programs themselves can be thought of as proofs (Curry-Howard correspondence) so if you've ever written a program that terminates you've written a proof... Proofs don't necessarily have the form or flow "by axiom 1, axiom 2, theorem 34, modus ponens on this, proof by contraposition on that which we'll name lemma 8, and by induction over the integers here, we have proved blah, QED".

And if you grant simple algebraic symbol manipulation as something you would do to solve a word problem, well, that itself is a style of proof. (There's a whole automated proof engine written entirely on the basis of substitution, the same process you use in a simple algebra problem of substituting x + 3 = 10 with x + 3 - 3 = 10 - 3 and reducing to x = 7.)

But fine, no proofs, not even in disguise! What is it that science, engineering, and technology focused subject books that use math only as needed without bothering to prove things when unnecessary (some having exercises and drills of word problems from realistic circumstances) don't do to solve your craving?

>The researchers say that as last author is usually associated with seniority, based on this data, their model predicts that it will be 258 years before the gender ratio of senior physicists comes within 5% of parity.

https://physicsworld.com/a/gender-gap-in-physics-amongst-hig...

Not at all, you are only prohibited from taking "freshman courses"[1]—that's what 55 is supposed to cover (definitely not all of the undergrad math curriculum), though some professors go beyond that. Many students go on to take at least some undergrad courses, with those in the 140's range being mathematical logic gems with no real counterpart in the graduate department.

[1]Students from Math 55 will have covered in 55 the material of Math 122 and Math 113. If you have taken 55, you should look first at Math 114, Math 123 and the Math 131-132 sequence. https://legacy-www.math.harvard.edu/pamphlets/courses.html

I wish for a service just like them but with many more difficulty levels and less (non-spaced) repetition.

Sigh. I miss having a printer.

I also read PDFs full-screen on my high-res, high-dpi laptop.

Try these two.

I have read tens of thousands of pages in PDF. (Yes, I checked)

i.e. |u X v| = (1 + (dx/dz)^2 + (dy/dz)^2)^(1/2) = (dz^2 + dx^2 + dy^2)^(1/2).

This was extremely frustrating to me for a while until I accepted that this was how Leibniz did it, so if it’s good enough for him it’s good enough for me.

Yup. You need to know what a proof is.

The rank of a matrix is fundamental to understanding linear transformations, since it given you knowledge about the dimensions of the "output space". It becomes more fundamental if the person goes on to study deeper math. It tells you how to compute the size of a basis for the target space. The uses go on and on.

>And distributions aren't everything; I can look up any distribution I want on Wikipedia

Yes, you can look up anything on wikipedia, but not in this book, which is why this book will not teach you the things the OP claimed it would.

>a proper foundation in what statistics means is much harder to come by

It's very hard to get that proper foundation from a book that does not cover those foundations. One is better served by using a proper book with a consistent and well laid out presentation of the needed concepts. Saying one can look fundamental stuff up elsewhere means the book is lacking.

Otherwise it just gets absorbed and forgotten like another 'useless information' class. Ask how many CS majors post college use or remember calculus vs understanding stats for the seven millionth badly designed A/B test they ran at work today.

I was a math major and this is not exclusive to calculus or analysis. If anything, algebra classes in things like group theory were much more instructive on this point. Everything from my third and fourth years (when I specialized into combinatorial optimization) was built on a cornerstone of proofs and theorems.

I do think logic is important, and teaching that instead of focusing on math proofs might be a good alternative.

Maybe it would help them not get scammed by keto diets and unsafe drugs and such though.

What sort of memorizations do you have in mind that you no longer need to memorize once you know calculus?

Apart from the applied stuff you mention, the real core of a mathematics education involves, I think, 4 main areas with significant overlap

Group A: number theory, graph theory, combinatorics

which shares concepts with

Group B: Algebra, Topology, complex analysis, differential geometry, metric spaces...etc

which shares concepts with

Group C: Functional analysis, measure theory

which shares concepts with

Group D: probability and statistics.

As for the applied math that you mention, you should really need to add vector calculus and I'd highly encourage anyone to take a course on fluid mechanics (from a mathematics department instead of an engineering department) to get a real feel for vector calculus in action.

though I am a little surprised that they have 1 course of differential equations in there instead of complex analysis as a required topic, as I think the latter is a better pure math topic. But it's MIT, so be it. Whether directly or indirectly, many of us learned to view math the MIT way by patiently working through foundational books like Artin and Munkres.

That said, my mention of non-introductory algebra topics probably is more of a personal idiosyncracy/interest.

What happens is you learn numerical approximation methods and then recreate the tables with human 'computers', like we used to do before electrical computers came into play. Or create mechanical analog computers like they did in the 1800s or greek times.

How would the typical person go about calculating the normal distribution by hand? Are we going to call everyone blind because they haven’t memorized e to arbitrary precision?

IDK, what does R do?

Honestly all of those feel more niche than calculus. I agree with you and joatmon-snoo on the usefulness of statistics and would probably support bumping calculus in favor of statistics, but meta-mathematics, topology, logic (which bleeds into meta-mathematics), combinatorics (which is kind of covered by stats), number theory, discrete mathematics, and graph theory are all much less useful even in adjacent STEM fields (discrete mathematics and graph theory matter more in CS, but far less for day-to-day programming). History of mathematics is effectively an entirely separate discipline and philosophy of mathematics has meta-mathematics/mathematical logic as a prerequisite.

Calculus unlocks much of physics and engineering (and lots of stats!). Large cardinal theory does not unlock any other field to the best of my understanding.

Linear algebra became a lot more interesting once you had cheap computers and Matlab.

I have some appreciation for calculus now but I really did not enjoy it much in high school or even in college. It turned me away from learning more math for some time which is unfortunate - linear algebra isn't my favorite either but I liked that much more off the bat, so I wish I had some exposure to it in high school. Then again, maybe the high school teaching style is what made me dislike calc to begin with.

And yes probability and statistics are fundamental. I was shocked a bit when I learned it was not taught in highschools world wide (i.e. not in the U.S.A.). But then again I had gotten numb with the current average level in the taught topics people arrive at undergrad at.

Note there is a lot of interconnectivity. To understand a new concept you might need concepts in another. E.g. number theory and probability.

In the first year (non US) you learn linear algebra, some real analysis and how to write proofs as well as important basics. In your second year you can choose all these electives, which then don't have to spend time introducing natural numbers, induction etc.

You're right; sorry for the typo. I meant the span of the coefficients (which is a space).

I'm not saying the concept of “rank” is useless, or anything. I'm saying a lot of linear algebra is based on high-level techniques and algorithms, but I don't think they're not truly fundamental concepts that you need to learn to understand linear algebra, if you learn / intuit a different high-level formulation instead.

> And a student trying to learn where and why stats is useful

That's not what I mean by “fundamentals”: I mean what stats is and how it works. A solid grounding in applied statistics is just as much about human psychology as it is about mathematics, and you never need know where your distributions come from so long as you memorise the rules.

> Sure, you can provide only a Gaussian,

Is that what the book does? If so, then I completely agree. But I don't see why you can't teach statistics using a Gaussian, a Poisson, a beta (three arbitrary distributions) and a general “some distribution”. I'm more of a pure mathematician, though, so if the answer's just “that doesn't prepare statisticians well for the kind of distributions that come up in the real world” then I can well believe it.

> Find a textbook for beginning students that does not cover a multitude of distributions.

… Aren't we discussing one?

My claim is that, given the finite amount of time allotted to mathematics in a secondary school curriculum, it's better to spend that time learning about medians/means, regressions, statistical tests, and so forth, instead of memorizing power laws and derivatives of trigonometric functions.

If you want people to understand that numbers are concepts / descriptions of something potentially infinite, and that it's ok to totally work with non-finite describable numbers without ever really writing out their values, just fucking start with that. I honestly think the start of most undergrad math curriculums should be a logic & foundations class with proofs, sets, number theory and the whole "numbers are more logical concepts, not really specific values" vs. calculus. Then teach calculus, with a big dose of 'what is infinity, really?'.

I think it's actually a great disservice and the hand wave that most calculus math classes do about those exact concepts really fucks over a lot of people. It makes calculus a weed out class because the fundamentals are not explained properly so a good amount of people just go into it as yet another thing they have to ape without real understanding. And for the people who don't work well with things they half understand, they really struggle, like I did. It really made my computer science degree a lot worse, because of the instance of starting all undergrad math with hand wavy calculus, for 3 or 4 classes.

I even wrote blog articles about it, it was probably the worse part of my computer science degree, and if math education was done differently where they didn't handwave, I probably would've had a much better time.

https://www.thoughtfunction.com/2019/11/doing-my-compsci-deg...

https://www.thoughtfunction.com/2019/11/what-i-wished-i-knew...

Most people don’t need mathematical statistics, but far more will find meaning/interest in the immediate applications of lighter statistics than the another-math-class-full-of-equations that calculus feels like to many.

Anecdotally, most universities are scrambling to add lighter data science courses to their humanities majors. These all teach basic stats, and many ignore the low-level calculus required for those methods (again, speaking of the humanities versions here).

I was a math major and didn't understand calc all that well until I took advanced calc, and I didn't understand statistics all that well until I took the upper-level mathematical stats courses.

On the other hand, less-than-popular culture still is the foundation for our popular culture today - the amount of times "Romeo and Juliet" or "Macbeth" has been adapted, referenced and deconstructed in (popular) media is astonishing, and without knowing the original, you wouldn't fully get the modern references.

You don't need to love Shakespeare (I don't exactly, though I like to see the "Scottish play" every few years) to reap these benefits.

School is not about what people like. If you go by that measure, there's no real need in math, because you will find a lot less people doing calculus for fun, as compared to people who play fantasy football. Of course, you can't really understand the ideas behind fantasy football if you don't know maths, but who cares, it's not popular...

Please note that I did not claim physical labour has no value and one should not know about it. I said that it's relatively easy to pick up, and that it may be more worthwhile to get a professional when the job demands it (a professional, I would like to add, who learned his craft after school, ideally in some kind of apprenticeship). In my country, to get a driver's license, you need to know basic first-aid procedures (and yes, there's a one-day course you need to attend). That doesn't make you a neurosurgeon, and if you have persistent headaches, it's probably better to ask a professional than to depend on me with my first-aid course - or some guy who "learned about surgery techniques" in high school.

Calling an m-tuple a coefficient and then calling the column span the "coefficient span" does not show understanding of the material.

>I consider it a more fundamental, more useful concept than your "rank".

The dimension of the span is the rank. It's exactly why rank is important. The nullity is the dimension of the kernel, and rank + nullity = n.

Saying rank is not important is like saying dimension is not important. The dimension of a vector space is the first and most important invariant that describes the space. Rank is that dimension for the image of a linear transform. It's absolutely fundamental. It's why when describing some linear algebra thing, one usually starts with something like "Let V be a n-dimensional real vector space" or similar. We rarely (bordering on never) write "Let V be a vector space spanned by the following vectors", and we never write "the coefficient span of the transform".

>The concept of “rank” drops out of the (misnamed) “Gaussian Elimination” algorithm

No, it is a fundamental property of the linear transformation. Gaussian elimination is but one of many ways to compute it. And no matter what choices you use in your elimination, you will always get the same rank. This is super important - that Gaussian elimination gives knowledge of the linear transformation that is independent of choice of basis or of steps performed. Changing of either vector space changes the numbers in the matrix, and different Gaussian elimination steps may have differing intermediate steps, but the rank is the rank is the rank.

It is also true that the row span = the column span = the rank, which is also not immediately obvious. These are theorems proven in basic linear algebra, and fundamental to claiming to understand basic linear algebra.

That rank shows up in Gaussian elimination is not some artifact or unique thing to Gaussian elimination. Since rank shows up everywhere, when it also shows up in Gaussian elimination shows that Gaussian elimination is doing something fundamental - it is but one of many, many ways that rank pops up over and over in linear algebra.

Rank (and nullity) are absolutely fundamental.

And no one calls that m-tuple a coefficient.

None of this is in the book above. Zero.

A final nice point to illustrate, here [1] is the index to Strang's Introduction to Linear Algebra. Rank occurs more than any just about every other entry in the index.

Here's Serge Lang's book [2]. After the word dimension, rank occurs the most in the index.

Book after book shows that after the concept of dimension, rank is probably the most important concept in linear algebra.

I could go on and on. You cannot claim to understand linear algebra without know how pervasive and useful rank is.

[1] https://math.mit.edu/~gs/linearalgebra/linearalgebra5_Index....

[2] http://www.math.nagoya-u.ac.jp/~richard/teaching/f2014/Lin_a...

You were the one who wrote column space = row space…?

> If you want to call the mi the coefficients, what are you calling the ai?

I'm calling them the coefficients because they're “part of the function” and don't change. They're the thing you'd naturally call a coefficient. Coefficient isn't as broad a term as you seem to think it is; In f(x) = x² + x + 3, the coefficients are 3, 1 and 1; not 1, x and x².

> You should publish a book and clear it up for everyone.

As soon as I have any actually original work, I plan to. But linear algebra isn't a specialism of mine, so I doubt I ever will.

And no, I don't think the terminology I've used here is an improvement over the status quo; I'm not a complete imbecile. I just don't get why terminology is considered more important than understanding in mathematics education, and why it's practically impossible to use different words for things even when you do have an improvement.)

> That you continue to debate this point is astounding and willful ignorance. I've given you why it's important. I've shown that in textbooks it indexed second only to dimension. I've given at least 5 places it shows up. I've explained how it's the dimension of extremely important items.

If you think any of those are counterarguments, you were never addressing what I was trying to say.

> You want to keep claiming your made up vocabulary and sloppy terms that miss important issues are superior.

Where did I claim this? I believe I said I was “not wrong” (a weaker label than “correct”), and I identified some problems I have with the existing terminology, but I don't think I ever said my (inconsistent, ad-hoc) terminology was better.