So you want to study mathematics(susanrigetti.com) |
So you want to study mathematics(susanrigetti.com) |
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.
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.
I knew consistency was important, and did not place much importance on "brilliance", although that attribute has been showered upon me since I was little.
What I did not know, and still don't know is that studying only for a few minutes or half an hour regularly will make me good at something as advanced math.
You seem to know your stuff and I like your approach and attitude, and I will do now what I do very rarely and upon serious consideration- take a leap of faith.
I will start doing math everyday for at least half an hour, and I will see how that goes.
I will let you know after a few months.
> [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."
:)
In other words, I do not see how you are dealing with the "baggage" of learning mathematics beyond name-dropping it. In my opinion, the mysticism is the baggage. And then the rest of the blogpost reads like a conventional curriculum within the conventional academic regime with which we associate that baggage.
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.
This is in the site guidelines: https://news.ycombinator.com/newsguidelines.html.
If you didn't intend to come across like an interrogator trying to back an opponent into the corner, then your comment needed to be written quite differently.
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.
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!
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/
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.
For 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.
Kudos if the author is this talented.
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.
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 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.
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.
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.
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)
to my program plan: - A couple teaching courses (including one for roughly grades 5-8)
- Calc I-IV
- Statistics (one without calc, one with)
- Linear algebra
- Discrete
- Geometry
- Number theory
- History of math (apparently not just a history class, I haven't taken it yet)
- Abstract algebra and into to topologyA 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.
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.
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.
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
The Math 140 series have only become more serious courses in the last 10 - 15 years or so IIRC. The 240 series was generally where to go for serious mathematical logic courses (and generally is where you would go after e.g. a first 140 series course in set theory anyways).
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.
Applying mathematical facts isn't math, it's calculation at best.
That's what most programmers already do. They throw things together in the hope they work without proof they got all edge cases covered.
And it works pretty good, because often the edge cases aren't that bad.
* Axioms
* Substitution
* Modus Ponens
* Universal Quantification
Induction or proof by contradiction are just special cases of this.
But yeah, geometry for introducing proofs is difficult, because it is so easy to confuse visual intuition with proof. At the very least, you need a capable teacher who knows the difference. But nobody expects children to understand it all from the get go. A healthy struggle to disentangle intuition and proof, and then to entangle them again later on once you know the difference, that's the path to understanding mathematics.
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.
That's more to do with the space of the coefficients, I think.
> Saying one can look fundamental stuff up elsewhere means the book is lacking.
A list of distributions is not fundamental to statistics. Discovering a new distribution doesn't meaningfully expand the fundamentals of the field of statistics: all the basic principles are the same. Anyone familiar with those basic principles can pick up a new distribution quite easily – and can probably derive new distributions when they're needed.
I don't think that phrase is a thing in linear algebra.
I've gotten a PhD in mathematics - I know both these fields quite well. I stand by my assessment of linear algebra - I've been using pieces form it nearly daily for decades.
>Anyone familiar with those basic principles can pick up a new distribution quite easily – and can probably derive new distributions when they're needed.
And a student trying to learn where and why stats is useful should be taught a wide set of distributions so they see the nuances while they're learning. Sure, you can provide only a Gaussian, but when the student leaves completely ignorant of all the places a gaussian fails and what are some choices relevant to different situations, you can failed to teach them the fundamentals, which includes enough nuance to see when and where to apply what distribution.
>A list of distributions is not fundamental to statistics.
You may as well claim all of stats is not fundamental - just learn math principles. You can look up anything in stats and derive it yourself once given the concept if you're good at math fundamentals. Surely with enough math skills, and zero stats, you can derive all the stats knowledge needed without needing to ever see any stats in a book.
But that's a bad way to go about teaching people useful skills.
A student learning about distributions needs some examples. This book has none. You can argue all you want, but this book is crap for learning the topics the OP claimed it covers.
Find a textbook for beginning students that does not cover a multitude of distributions. Either every single author, usually writing from decades of experience, is wrong, or you are.
I think the authors have the right approach.
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.
Probability theory is applied analysis. For example a measure density is a derivation of a measure with respect to another measure (Radon–Nikodym derivative). Or how do you often assign probabilities to measurable set: Lebesgue-Stieltjes integral.
So you better have a very good understanding of calculus before starting with probability and statistics.
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.
For science-oriented high school curriculums, calculus is necessary, and should only be optional in a few cases where the student already really knows what they want to do.
By the time a student wants to specialize at say 20, they should have already had some foundational ability for that specialty, otherwise they wouldn't be able to compete internationally. For people who don't end up using calculus, it's a waste, but there is a trade-off. As a result, much of the high school curriculum is to build a broad foundation to prepare for multiple possible specializations.
Maybe it would help them not get scammed by keto diets and unsafe drugs and such though.
In the land of the blind the guy with one eye walks everyone off a cliff.
I also think you could and should start by teaching probability and statistics and using that to introduce calculus as and when it shows up naturally, rather than teaching calculus in the abstract first and then showing the applications much later.
What sort of memorizations do you have in mind that you no longer need to memorize once you know calculus?
To work around the fact that many of us were still in precalc, our teacher just taught us the power rule, the relationship between slopes of tangent lines, etc., without diving into the "why" of why those things worked.
That said, yeah maybe for the most basic of university of physics the whole "derive it on the fly" strategy works, I guess? But when you get to more advanced courses like mechanics of materials, you'll do yourself the favor and take to memorizing at least a few of the commonly used equations.
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.
Real analysis, complex analysis, topology, and number theory are there (topology and number theory are both listed as electives since most math programs categorize them as such). Graph theory, functional analysis, differential geometry, probability, and statistics are almost always either electives or graduate courses.
It’s funny, because most of the things you mention as “real math” are things that many math undergraduates don’t learn (not until graduate school at least) but that physics students learn as undergraduates (differential geometry, measure theory, functional analysis, etc.).
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.
There ought to be some way to encourage late high school/early college students to "survey" the field without necessarily taking full courses in these topics. This could also give some earlier understanding in how the different fields relate - for example you could present a toy graph theory problem within linear algebra as a matrix problem, later presenting the same problem in graph theory section and walk through it using graph representation. I think high school courses struggle with memorability precisely because most units are taught basically in a vacuum.
Regardless though, the point of the survey course wouldn't be to remember details so much as to find topics of interest for further study/be generally aware of their existence in case a relevant problem comes up in the future.
And, from that perspective (with which I agree), calculus itself is just another instance of trying to turn a non-linear problem into a problem in linear algebra!
Linear algebra became a lot more interesting once you had cheap computers and Matlab.
The Math in my 4 year engineering degree was structured something like this:
First Semester Year 1 two math courses: Calculus 1, Linear Algebra
Second Semester Year 1 two math courses: Calculus 2, Sequences and Series (this one was probably least useful all I remember from this 15+ years later is Taylor Series and Binomial Theorem)
First Semester year 2 two math courses: Differential Equations, Statistics for Engineers.
From second semester year 2 onwards there were no more discrete math classes this was where the degree really specialized into various engineering streams, Mech Eng, Chem Eng, Civil etc. Had their own courses. I studied Materials Engineering some courses were shared with other eng students (For example I shared Thermodynamics with Mechanical Engineering students) but others such as Non-ferrous metallurgy were pretty deeply specialized.
A lot of subject used built on earlier math (Fluid Dynamics was backed up by lots of differential equations for example, stuff like Gamma Function would come up in a lot of places. Solid Mechanics had a lot of integrals second moment of area etc.), Linear Algebra I can remember from Fracture mechanics and crack propagation (Stress and Strain tensors etc.)
Have a Master's in Material Science and still don't remember a lot of Linear Algebra specifically.
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.
The true foundational classes in the typical undergraduate mathematics curriculum are logic and abstract algebra. People rarely start with them, because the usual way of teaching mathematics is applications before foundations. You learn linear algebra before abstract algebra, proofs before formal logic, and axiomatic probability before measure theory.
And there is definitely such thing as too much linear algebra. Once upon a time, I wanted a decent mathematical background for theoretical CS and continued (at least) until the first graduate-level class in most major topics. Graduate linear algebra was "foundations without applications" for me, as I've never worked on anything building on it.
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?
I'd argue that without a deep understanding of rank and nullity you do not understand linear algebra. Sure, you can lean to move symbols and compute simple things, but that is not much of an understanding.
I'd guess you have not moved into deeper things - then you'll realize you're missing fundamental understanding of linear algebra needed to move on.
It's like claiming one has a fundamental understanding of calculus by being able to solve high school level integrals, but really doesn't understand deep relations between the main ideas of calculus. Sure you can write things down, but there is a major difference between that level of knowledge and what I'd call a fundamental understanding of calculus.
>span of the coefficients
????
Care to link to the thing you're misnaming? I have no idea what you're talking about, and I even googled the phrase.
This is what I mean by using a proper book to learn from.
>Aren't we discussing one?
Nope - the book in question is not offered as a beginner course stats anywhere I am aware of. Care to show one? Calling it one then claiming it is evidence of one is simply circular logic.
>Is that what the book does?
So you have not looked at the book yet are arguing what kind of book it is? That about sums this up.
A linear transformation of some n-tuple (let's call it a) to some m-tuple (let's call it b) can be thought of as the sum of an element-wise multiplication of each item of a by an m-tuple coefficient (let's call the coefficient c_i). So b = ∑ a_i × c_i.
The coefficients may or may not be linearly independent. The span (https://en.wikipedia.org/wiki/Linear_span) of these coefficients is a useful property of the linear transformation; I consider it a more fundamental, more useful concept than your "rank".
The concept of “rank” drops out of the (misnamed) “Gaussian Elimination” algorithm, and there are lots of theorems that involve it (probably because it was discovered early on in the development of this field), but it seems a rather complicated and unintuitive concept, to me. If you're using “rank”, you need to use a lot of theorems and lemmas and conversions between different representations of things that just don't seem necessary. I am happy to be corrected.
> Nope - the book in question is not offered as a beginner course stats anywhere I am aware of.
That doesn't mean it's not a textbook for beginners. It looks like one, to me.
> So you have not looked at the book yet are arguing what kind of book it is?
It's over 500 pages long. I've scrolled through it a bit, and nothing jumped out as obviously wrong; I haven't read it. (Though I did see a few different distributions named, hence my confusion.)
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.
I've used one once. I wanted to make a volume control that I thought sounded subjectively even across the range and gave the right amount of control. So I used a polynomial regression calc on a random website and gave it some data points I wanted it to go through.
But that's not actually doing a regression, just knowing it exists and computers can do them.
Is there any math that has any use whatsoever unless you know a whole lot of it and plan to do some technical projects? I thought the whole point of learning any math(Even arithmetic, since phones exist now) is just so you can learn other more advanced math, and maybe someday be a solid state chemist or something.
Statistics lets you read a scientific paper, but you don't actually need to understand it, unless you're actually going to be reviewing their raw data. Most everyday people just trust the p values and move on to wondering about confounders they forgot.
Seems like you could cover all the statistics people will actually use it about 4 hours.
Perhaps because they never took statistics.
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.
This does not only extend to literature. I have had similar experiences with religion (which I thought of as utterly useless and potentially dangerous, as it "deactivates critical thinking and creates sheep-people which will follow whatever their shepherd/priest/guru tells them"), creative arts (both painting as well as music), and the basic sciences (biology, chemistry, physics).
Now, I don't genetically engineer the stuff in my garden, but understanding Mendel was useful. I don't speak in iambic pentameters, but I can appreciate when it is being used as a stylistic choice. I am in no way a church-going, devout Christian, but I have found meaning in some of the deeper wisdom enshrined in the Bible (and the Koran, and the Bhagavad Gita, and about half a dozen Sutras.)
Would I have come to that if I didn't have primers in school? Maybe. But the primers certainly helped.
On the other hand, I didn't learn plumbing in school, or laying electrical wires, or "doing my taxes", but these are things I can simply have someone do for me who is a lot better equipped and trained to do so, or - for the small stuff - I can figure them out on the fly.
I'm willing to believe that some people like Shakespeare, but it's a small minority. You can tell by the number of people who read Shakespeare for pleasure - is that number larger or smaller than the number of people who read JK Rowling? Why should we teach the entertainment that a small number of people prefer in schools? I believe the only reason we actually do is tradition.
You mention that you can simply have someone learned in plumbing or sundry skill do those tasks for you. I can do one better. I can simply have no one read Shakespeare for me and I can not read it at all and nothing is lost. That is, of course, because unlike plumbing or laying wire there is no reason to need Shakespeare.
It's good that you enjoy Shakespeare, but some people enjoy plumbing. Plus, plumbing has a practical purpose, unlike Shakespeare. There is no real reason to teach Shakespeare, other than tradition, and people trying to seem smart or educated. There are many other subjects that make a much stronger case for deserving to be in school curriculum.
I read a quote somewhere that goes something like "A society that separates warriors and scholars will have an army led by fools and thinking done by cowards." Similar logic, with different vocabulary, applies, I think, to a society where scholars can't do manual labor.
I'm trying to remember when I had one of those last!
I actually don't love the idea of cheat-sheets but maaaybe if it's a cumulative final I could see it being helpful? If you're taking chapter exams on some material, I think you ought to have worked enough problems so that formulas/constants are drilled into your head. But I guess too, where do you draw the line at engineering appendices? I sure don't have the MoI of every common body memorized.
Well, sure, but, if you don't, then what's the point of punishing you? As a teacher, frankly, I'd be happy to have my students bring in any static resources they wanted to consult—I say 'static' to emphasise not, e.g., consulting a cheating site, although it's fine with me if they've pre-compiled solutions in advance to any problems they think might be interesting or important—except that (1) I think that would encourage bad study habits, and, more importantly, (2) it would be unwieldy in a packed classroom to try to have adjacent people juggling multiple textbooks, notebooks, etc.
In fact, I loved the freedom to give extended-time, fully open-book, open-note exams during the fully remote classes. I wish I could still do that; if cheating weren't so endemic under those conditions, then I would.
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...
> Calling an m-tuple a coefficient and then calling the column span the "coefficient span" does not show understanding of the material.
So… I'm not wrong; I just don't know the words? Sounds like I actually do understand the mathematical concepts. (Maybe I'd find the words easier if mathematicians named stuff sensibly… and yes, I know I'm using English to write that sentence. I'm a hypocrite, but that's no excuse for the naming conventions in abstract algebra and category theory.)
> No, it is a fundamental property of the linear transformation. Gaussian elimination is but one of many ways to compute it.
So why does every textbook, every lecture, and Wikipedia talk incessantly about Gaussian elimination (an algorithm for inverting matrices!) when talking about rank? Rank's only useful as a property of the vector spaces, so why treat it like a separate concept?
> It is also true that the row span = the column span = the rank, which is also not immediately obvious.
The size of the row / column span is the rank, surely? Unless I'm misunderstanding the terminology.
And that's because we have several different concepts to describe the same property, for no reason that I can see. That relationship was immediately obvious to me as soon as I worked out what “rank” was, because I'd done my own investigations into linear algebra before I ever got taught it. (Investigations that I wouldn't've thought to do had I not known about the concepts of linear functions, multi-variate functions and inverses, so I'm not claiming to have independently invented linear alegbra.) There's no way they could be different, because it's about the linear independence (which regions of n- or m-dimensional space are reachable by a linear combination of the… you don't want me to say “coefficients”).
Likewise, I don't think we should be using “row” and “column” to describe linear mappings. “We sometimes write numbers in a grid to represent the function” isn't fundamental.
> And no one calls that m-tuple a coefficient.
But it is a coefficient, if you write the transformation the way I wrote it. Why is that wronger than talking about the “rows” and “columns” of a function? Other than convention, of course.
> None of this is in the book above. Zero.
To be fair, the book does purport to teach statistics; it uses, but does not claim to teach, basic linear algebra. A solid grasp of linear algebra is a prerequisite to understanding the book, so if you understand the book you probably understand linear algebra.
Column space and row space are completely sensible.
>And that's because we have several different concepts to describe the same property, for no reason that I can see.
You do not see. If you think column space and row space are the same thing, then that's completely wrong. They have the same dimension, which is a theorem, but they are not the same space.
>But it is a coefficient,
So is everything, which is why calling this a coefficient, when there is a better word, is useless.
If you have columns m1, m2, m3, m4, and form the linear combination a1m1 + a2m2 +... Then the ai are also coefficients. And they're much more like what people call coefficients since they're scalars. If you want to call the mi the coefficients, what are you calling the ai? Numbers? Integers? Crawdads?
The mi are vectors, they are column vectors, linear combinations of them form a subspace, and the things multiplied by them to form the subspace are called coefficients.
So yes, you can call them coefficients, but you may as well call them numbers, or pointy-things, or anything else you make up, and no one will be able to talk with you, since you insist on doing things in a manner that makes your work unintellgible.
>Likewise, I don't think we should be using “row” and “column” to describe linear mappings. “We sometimes write numbers in a grid to represent the function” isn't fundamental.
... and you're off the deep end again. I'm glad you invented close but not correct linear algebra, that you missed so many important relations, that you use words in the manner you believe they should be, and on and on.
Of course, your methods clearly must be better than centuries of mathematicians - you should publish a book and clear it up for everyone.
>So why does every textbook
It's baffling to me how hard you push at simply learning. Pick up one of those textbooks I mentioned, and look at every page indexed to rank, and look at how it's used.
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.
It becomes more and more important the deeper you go into math, and that is probably the biggest reason it is so important here. The concept of rank is the tip of an iceberg going through everything above linear algebra: Hilbert spaces, operator theory, exact sequences, homology, cohomology, topology, and on and on and on.
I'm done. You don't care to learn. You want to keep claiming your made up vocabulary and sloppy terms that miss important issues are superior. You're far too stubborn to educate. Go do it yourself.
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.