[HN Gopher] So you want to study mathematics ___________________________________________________________________ So you want to study mathematics Author : musgravepeter Score : 255 points Date : 2022-03-07 18:05 UTC (4 hours ago) (HTM) web link (www.susanrigetti.com) (TXT) w3m dump (www.susanrigetti.com) | tunesmith wrote: | How do you like to solve math problems in this day and age? | | 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. | bnbond wrote: | I find paper and pencil works pretty well. | paulpauper wrote: | You don't need so many books. many of the old texts will cover | many important college-level concepts in a single source. | | 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 | ChrisLomont wrote: | At a first glance that book completely fails for an undergrad | lin alg course, and looks weak in other areas too. | | Examples: the words nullity and kernel don't appear, rank of a | matrix is not in it, and, well, every topic I can think of for | undergrad linear algebra is simply not in the book. | | It's equally bad for stats: no mention of many common | distributions a student would learn for example. | wizzwizz4 wrote: | Is that such a bad thing? I'm half-convinced that the "rank | of a matrix" is just an artefact of a particular algorithm | for inverting matrices. And distributions aren't everything; | I can look up any distribution I want on Wikipedia, just as | soon as I need it, but a proper foundation in what statistics | _means_ is much harder to come by. (I have enough of a | foundation to know when it 's being taught very wrong, but | not enough to actually be very useful in day-to-day life.) | philomathdan wrote: | The curriculum guides Susan Rigetti provides are an amazing | resource for self-study. And the fact that she worked through all | of this is truly inspiring. | | 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. | Jun8 wrote: | Susskind's Theoretical Minimum is fantastic for physics self | study: https://theoreticalminimum.com/ | philomathdan wrote: | Another good one. Thanks! | da39a3ee wrote: | I am fairly confident that Susan Rigetti is a future president of | the USA. In addition to becoming somewhat well-known as a | household name early on in her adult life, she has achieved so | many difficult and impressive things (publishing multiple books, | studying physics and philosophy at graduate level, working for a | top-tier tech company, taking down the CEO of a top-tier tech | company and damaging the company's reputation, being asked to | work for the New York Times, publishing curricula in graduate | Physics, graduate Philosophy, and undergraduate mathematics). | Furthermore, she seems to have a gift for or knack with the | public eye. | rscho wrote: | POTUS is a very low bar... | potbelly83 wrote: | As a math PhD I have to say the only way you're going to learn | mathematics is if you actually have a pressing need to do so. | i.e. You have a project at work that needs some math, you have a | hobby that needs some math. In this case you just learn what you | need. Just learning math for its own sake outside of a University | STEM track is just too hard (I wouldn't be able to do it and I've | tried). | JimTheMan wrote: | I think the question needs to be asked, "What am I hoping to | get from this?" | | If its tools to help you solve some problem, great. If its | because you find it fun, great. | | But if its because you feel you want to 'better yourself' or | perhaps feel like it would somehow prove your intellectual | worth, you probably won't get a lot out of it. | treeman79 wrote: | Was programming a 3-axis machine in early college back in 90s. | After a few months I was mostly re-inventing trigonometry. The | I actually took trig later on. That would have been super handy | at time. | kurthr wrote: | I hesitate to contradict someone who has gone through the whole | Math PhD process, but I have to say that the best | mathematicians that I know treat the problems they're working | on as games, or riddles to be solved... and they've taught | their kids (and others) this same method of thinking about | these problems. There's a huge mental tool set, and often a lot | of grinding to get to a solution, but it's just a game (and | there are often more elegant solutions)! | | I enjoy provoking interest in complex numbers and exponentials | among precocious teens, but I've never been more humbled than | having a Galois theory joyously explained to me by an 11 year | old. (p.s. I'm an engineer so I follow your technique). | 300bps wrote: | To me the problem isn't learning it, it's retaining it. | | 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!! | [deleted] | Grambo wrote: | I experienced something like this recently. I struggled to | grasp linear algebra when I took it in undergrad and as a | result was always intimidated by the subject. Now, years later, | I'm taking a course in graphics and naturally needing to | relearn linear algebra and suddenly everything just clicks. | zinclozenge wrote: | This is true. When I started my first job, I tried casually | learning more from where I stopped after I finished school. My | motivation slowly waned as I realized I vastly preferred | playing video games than watching lectures and trying to do | some problem sets. | exdsq wrote: | Correct me if I'm wrong, but I assume you're looking at this | through the wrong lens. As a PhD you've learnt something hard | at a considerable depth but this isn't what I, or most people, | hope to achieve by learning maths themselves. It's normally to | a far lower depth that's far more achievable like Calculus I or | some aspects of number theory or knowing what all the funky | symbols such as Summation mean. | parsd wrote: | If you're interested in both mathematics and physics, does it | make sense to learn both concurrently? If yes, what areas | complement each other? Or is there no overlap to warrant | concurrent study of the essentials? By essentials I mean what a | college student must know, or really anyone who pursues self- | education without a background in these areas. Beautiful website, | by the way! | susanrigetti wrote: | Check out my physics guide: | https://www.susanrigetti.com/physics. It has both the physics | core curriculum AND the math essentials you need to know in | order to understand the physics essentials. (And thank you!) | C-x_C-f wrote: | The website mentions some good courses, personally I love Richard | Borcherds' YouTube channel[1] for both undergraduate and graduate | courses. No frills, exceptionally clear, (mostly) bite-sized | lectures that cover a good range of material (especially in | geometry). | | 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... | brimble wrote: | Are there any "math for people who just want to use it" tracks in | math pedagogy? I don't care a bit about proving any of it's true, | or even reading others proofs of same. "Recognize which tool to | apply, then apply tool", all focused on real-world use (so, yes, | it wouldn't be "real" mathematics). That's the math education I'd | like--try as I might, I just can't make myself care even a little | about math for math's sake. | | 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. | dwohnitmok wrote: | > I want a mathematics education designed for all those kids | (likely a large majority?) who spent math from about junior | high on wondering, aloud or to themselves, why the hell they | were spending so much time learning all this. One that puts | that question front and center and doesn't teach a single thing | without answering it really well, first. | | The problem is that the answer will depend heavily from person | to person and from field to field and often the most sensible | answers require a mathematical maturity that creates a chicken- | and-egg kind of difficulty. | | Do you care about engineering? Well you'll need some calculus | for that. Do you care about prediction modeling? Well there's | some stats you'll need for that. Do you care about finding | patterns in the world? Well there's abstract algebra for that. | Do you care about reasoning itself? Have fun with mathematical | logic. But because humans have different motivations, there's | no one-size-fit-all motivation-based approach. | | This is especially painful for mathematics because I think most | people who have learned some amount of pure mathematics will | relate heavily to what helpfulclippy says in a sibling comment: | "However, one thing that has been VERY applicable is | proofwriting. Although math proofs are far more rigorous than | most real world stuff, the discipline I learned in writing | proofs has carried over into pretty much everything from | programming (will this algorithm work every time?) to executive | decisions (why, specifically, should we believe X?)." | | The skills of rigorous and abstract thinking that pure | mathematics provides is both nearly-universally helpful, but | also simultaneously as a result very difficult to motivate. | "This will help you think better across everything you do" is | lofty-sounding, but generally not a convincing sell unless | someone is already curious. But it's true that being able to | wrap one's mind around pure abstraction (after rattling off a | rigorous definition for an abstract question: Question: "But | what is X _really_? " Answer: "X is just that. No more, no | less.") has ramifications for all that one does. | | And the most painful part of all of this is if you try to start | by teaching the wonders of pure mathematics instead of all the | messy, boring rote stuff, students' eyes are liable to glaze | over even more because of the aforementioned chicken-and-egg | issue with mathematical maturity. | | In this way it's similar to trying to motivate someone to read | and write. The key that unlocks that interest for everyone is | going to be different and it's very hard to explain the near- | universal benefits that reading and writing bring to one's way | of thinking (but I can always just have a computer transcribe | it or read it aloud to me!) without some inherent curiosity in | them. | joatmon-snoo wrote: | My strong opinion as someone who majored in math is that, at | least within the US, the standard calculus requirement should | be replaced with statistics. So much more useful and so much | more important as an adult. | | The analytical type of thinking that proof-writing is certainly | useful, but you can make much the same argument of many other | curricula, and besides, it's not like most intro calc courses | even do any proofs. The vast majority of them, I would assert, | are simply pre-med weed-out courses. | | I still remember freshman year, showing up to the standard | intro calc course, and dropping it as quickly as I could in | favor of my uni's equivalent of Math 55 (i.e. the hardest u/g | intro math course) because of how asinine I found the | content... | jldugger wrote: | > My strong opinion as someone who majored in math is that, | at least within the US, the standard calculus requirement | should be replaced with statistics. So much more useful and | so much more important as an adult. | | 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. | jpz wrote: | You can't understand much about statistical inference without | calculus. | | It's all integrals over pdf's. A lot of integration by parts | and other things in the core curriculum. | dwohnitmok wrote: | FWIW (and I know you're not making this claim) I don't | believe Math 55 is a good example of how mathematics should | be taught at large, nor do I think calling it an intro class | accurately conveys what it is. It's effectively the | compression of an entire undergraduate degree into a single | freshman course. I say "effectively" because the material | covered depends heavily on who's teaching it, but certainly | anyone coming out of Math 55 can pick up any undergraduate | math content trivially. For example, undergraduates who have | taken it are generally explicitly prohibited in course | descriptions from taking further undergraduate mathematics | classes (because it would be free credit for retreading the | same ground) and can only take graduate courses from then on | out. | | 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. | C-x_C-f wrote: | > For example, undergraduates who have taken it are | generally explicitly prohibited in course descriptions from | taking further undergraduate mathematics classes (because | it would be free credit for retreading the same ground) and | can only take graduate courses from then on out. | | 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 | dwohnitmok wrote: | I wonder if the pamphlet's changed or I'm misremembering. | I distinctly remember Math 123 and Math 114 both | explicitly excluding Math 55 (It's also worth stating | that Math 122 and Math 113 are definitely not freshman | courses). But regardless I'd be very surprised to learn a | Math 55 student took Math 114 or Math 123. I would also | be surprised (although less so) to learn of a Math 55 | student in the 130 series. | | 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). | Koshkin wrote: | > _within the US_ | | All fifty states or just continental? | Siira wrote: | Statistics needs calculus as a prerequisite. Heck, a non- | introductory treatment of probabilities needs measure theory. | Bostonian wrote: | I wonder how much statistics someone can understand without | calculus. For example, how do you explain what a continuous | probability density function (such as the Gaussian) is | without calculus? | pmyteh wrote: | I teach introductory quantitative research methods to | communication undergrads, most of whom have no calculus. | (Which is essentially optional in the UK as it's taught at | A Level after most students have specialised away from | maths). | | 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. | pjbeam wrote: | My prob and stats courses required the calc sequence (US | based math undergraduate). | Melatonic wrote: | Not sure I agree with that - you also have to take into | account that calculus is a requirement for many other | sciences. I suppose you could just of course make it a pre- | req for things like that but I found that in highschool | rarely did they go that deep. Physics becomes a hell of a lot | easier with basic calculus for example. | | 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. | JadeNB wrote: | > 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. | | What sort of memorizations do you have in mind that you no | longer need to memorize once you know calculus? | ghaff wrote: | Pretty much all the classic Newtonian formulae you | memorize in high school physics classes can be derived | (relatively) easily once you bring calculus in. A general | problem with high school classes in particular--still | somewhat true in certain university classes but much | fewer--is that there are dependencies across classes. So | you end up with a lot of "Memorize this thing because you | don't have an advanced enough background to understand | why $XYZ is the case." | mmmpop wrote: | In my high school experience, this was avoided if you had | the desire and aptitude to just take Honors Physics | instead. | | 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. | p1necone wrote: | Knowing the basics of integration/derivation makes all | sorts of very common concepts in physics much more | intuitive - velocity, acceleration, area etc. | wizzwizz4 wrote: | s = ut + 1/2at2 | zozbot234 wrote: | Calculus is also a requirement for statistics and | probability. You simply cannot meaningfully talk about the | latter without referencing the former. | wbsss4412 wrote: | You can certainly make significant headway with some | basic combinatorics and basic tables for things like the | normal distribution, no calculus needed. This is in fact | how most people currently learn statistics. | geebee wrote: | I'm really wary of this. House wiring would be more useful | than Shakespeare. That's not to knock house wiring, or | statistics. I'd love to know more about both. | | 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? | SamReidHughes wrote: | Well, AP statistics covered the area under a probability | distribution function and people seemed to understand that | -- you look up the answer in a table (or use the TI-83 | function). Presumably they'd do the same for a cumulative | distribution function. | jll29 wrote: | Calculus and linear algebra seem to _totally_ dominate the | curriculum in most (all?) countrie. | | 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!). | edflsafoiewq wrote: | Calculus and linear algebra continue to dominate in applied | mathematics. "How can I turn this into a problem in linear | algebra?" is probably the most fruitful mathematical | technique that has ever existed. | JadeNB wrote: | > "How can I turn this into a problem in linear algebra?" | is probably the most fruitful mathematical technique that | has ever existed. | | 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! | Koshkin wrote: | True: the differential at a point is a linear map; the | integral is a linear form (on the vector space of | functions). | ghaff wrote: | Interestingly the ascent of linear algebra is a | _relatively_ recent thing. I have an engineering degree but | while I was certainly exposed to basic matrix stuff, never | took a linear algebra class. When I was an undergrad, you | took differential equations in addition to basic calculus | for engineering. (You needed for system dynamics among | other things but it was very cookbook.) | | Linear algebra became a lot more interesting once you had | cheap computers and Matlab. | eftychis wrote: | Anything besides a cursory layperson's outlook on a lot of | these topics (besides basic logic and history/philosophy of | math -- although not sure how you would teach the last | two/what you have in mind as curriculum) requires calculus | and/or linear algebra. There is a reason they say you can | never learn too much linear algebra. | | 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. | dwohnitmok wrote: | > meta-mathematics? Topology? Logics? History of | mathematics? Philosophy of mathematics? Combinatorics? | Number theory? Discrete mathematics? Graph theory? | | 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. | staindk wrote: | Super agree. | | 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. | ghaff wrote: | There are different types of stats. I took a stats course | in my grad engineering program. Not being the best person | at math I think I probably struggled with the math | sufficiently to get distracted from the concepts. When I | breezed through a stats course when getting an MBA I | understood the concepts much better--though the professor | was almost certainly better as well. | dhosek wrote: | Or the math covered in finite math classes which is a mix of | combinatorics, stats, probability and linear algebra. I | remember doing my time in the math tutorial room during my | grad school days and helping the kids from the business calc | classes who were learning math they weren't going to use from | books written by people who didn't understand the domain that | they were trying to teach math for. Seriously, what business | use is there for f(x) = x^1.3 or the indefinite integral | thereof? | ghaff wrote: | When I tutored in business school--mostly related to math- | related things (which itself says something about the level | of math knowledge among those who weren't engineering | undergrads)--pretty much the only calculus that came into | play was finding maxima and minima of curves in economics | which was both simple differentials and mostly pretty | academic anyway. A little later I did some more complex | optimization problems but that was done by software (LINDO | at the time) anyway. | hardwaregeek wrote: | I dunno, at that point, are you really doing math? Math is | proof. Now, granted, nobody expects you to know exactly how | everything is proved. But there is an expectation that you can | prove most of the stuff. Otherwise, it's very easy to stray | from the truth into the plausible-sounding, but incorrect. | | That said, you're absolutely correct that more justification | and motivation is important. So much of math can be taught with | problems from physics, computer science, etc. Perhaps a good | book for you would be Concrete Mathematics by Knuth? I haven't | read it but people swear by it. | cptaj wrote: | I just want a place I can go to weekly for 15min to practice | math. It keeps track of what I know and gives me exercises of | stuff so I can retain the skills. | | I've forgotten 90% of the math stuff I learned | robotresearcher wrote: | How about a textbook? At 15 mins a week, a couple of books | would last a year. | Buttons840 wrote: | Khan Academy. Finish a course, retake the course exam once a | year afterwards, fill in knowledge gaps as needed. Doing this | should keep all undergraduate math fresh in your mind. | Siira wrote: | The proofs can help immensely in remembering theorems and | knowing their needed assumptions and application area. | Mainstream math education is also already light on difficult | proofs in introductory (undergraduate) texts. You can always | skip the proofs anyhow. | Jach wrote: | Yes, all tracks of math involve people using math. Proof | writing is a part of use for a lot of it. There are also very | many tools within the use of proof writing where a lot of | education is around learning a bunch of tools so that you can | better "recognize which tool to apply, then apply tool", in a | future real-world use where you want or need to prove | something. | | I'm a little snarky, but you have a broken idea of what math | is. It's not even your fault. I don't even claim an unbroken | idea for myself, I went through public education too, though I | do think it's less broken. Somehow compulsory education has | managed to get near universal basic literacy, but seems to have | failed on whatever equivalent some sibling comments have hinted | at exists for math or at least mathematical reasoning. A lot of | algebra work taught for junior high can be understood as just a | foundation to be able to understand later things (though you | can of course use some of it directly as taught without having | to learn more for every-day things like some boy scouting | activities, or helping with putting together a garden or a | fence, or programming). 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... | narrator wrote: | If you just want to learn math for purely practical reasons, | Khan Academy (http://www.khanacademy.org) is great. It might | have added more lessons, but when I went through it, it went up | through 1st year college calculus. | | The thing that makes it VASTLY better than most self study math | programs or books is that there are hundreds of exercises that | you can do, and see if you got the right answer. If you didn't, | it will in most cases explain how to do the problem so you can | try again with a completely different problem, so you're not | just memorizing the answers. | | Another thing that makes it great is you can do a little bit a | day, start and stop, and come back to it and it will remember | your progress and where you left off. | | Khan is also a gifted teacher. Unlike a lot of math teachers, | he has great pronunciation and handwriting and you can watch | his lessons as many times as needed. | macrolime wrote: | Where are there exercises? | | 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... | LolWolf wrote: | Sure, here's a fun one ! | | https://web.stanford.edu/~boyd/vmls/ | | (I'd even replace Strang's "linear algebra" recommendation with | this book.) Imo, proofs are useful in so far as they are | enlightening (e.g., the proof that a problem has a minimum is | often useful in so far as it tells you how to solve it!) but in | many cases they are less so. | | Math is pretty fun, though, proofs and all, and I'd recommend | trying your hand at it as a cool little side hobby! It can | often help with "clarity of thought" :) (In many cases, proofs | are just one or two lines that tell you something interesting, | too, not page-long arguments that are mostly definitions | chasing.) | Koshkin wrote: | > _" math for people who just want to use it"_ | | It's called "engineering mathematics." (The books by Stroud | would be an excellent choice here.) | susanrigetti wrote: | If you have a solid background in calculus, I'd recommend | Zill's Advanced Engineering Mathematics, which is pretty much | basic math for physicists and engineers (aka for people who | need to "use it"). | brimble wrote: | Heh, I have a long-forgotten-due-to-complete-lack-of-use- | for-15-years-straight background in calculus. Solid, it is | not. Thanks for the tip, though. | susanrigetti wrote: | In that case, I recommend starting out with Zill's | Precalculus with Calculus Previews and then working through | Stewart's Calculus: Early Transcendentals! | brimble wrote: | I'll check those out, thank you! | jackallis wrote: | i second this. there should clear distinction between academic | math and "real world usage" math. | psyklic wrote: | Having a deep (proof-based) understanding leads to more "real | world" insights. So, there may not be a clear distinction. | Koshkin wrote: | But the distinction is generally not as clear as you may | think. (1) Much of the mathematics came from real world | problems (so, in particular, one may get drawn into some kind | of mathematical research and even end up discovering new | mathematical facts). (2) Sometimes when applying mathematics | one still needs to employ deduction (derive a formula, prove | a statement one wants to rely on, etc.). | joe_the_user wrote: | Math education has become torturously miserable in the US by | moving extraordinarily slowly. You have multiple years of | working on only slightly more complex equations and concepts | and naturally people get sick of it. | | You have generations of teachers who barely know math and view | it as a punishment, teaching kids and instilling the same views | in them. | | And then you have outside 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 remain 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. | pphysch wrote: | > Are there any "math for people who just want to use it" | tracks in math pedagogy? | | Use it for _what_? That is the question. If you pursue the | _what_ , you will inevitably be exposed to genuine ways that | mathematics may be employed by it. | | The academic standard for "learning math" is like "learning | programming" by reading the C++ language/STL spec from front to | back. No one productively learns programming that way, and even | if someone did, they would hardly be well off when faced with a | real-world production C++ codebase that follows $BIGCORP's | inhouse programming style. | vharuck wrote: | I agree. If anyone wants to study "math in practice," they | should pick a science or engineering track they like. Math | will be included. | dwrodri wrote: | Swinging by to plug my personal favorite resource for | refreshing my self on Bayesian stats: | https://www.youtube.com/playlist?list=PLwJRxp3blEvZ8AKMXOy0f... | | I think statistics is by and large the most proportionally | underrated subject proportional to its utility. A good command | of stats and probability expands your power to use data to | reason about answering questions. The channel author, Ben | Lambert, has an alternative playlist where he uses some of the | techniques taught in this playlist to solve problems in | econometrics. However, a lot of what is taught here builds a | great foundation for other domains, on everything from | bioethics to data journalism to computer vision. | | Another great channel that focuses a bit more on the machine | learning side of things is StatQuest with Josh Starmer: | https://www.youtube.com/c/joshstarmer | k__ wrote: | I think, the problem is, proofs are what makes math useful. | | This might sound a bit dense, but the alternative is what 90% | of programmers do every day. | Koshkin wrote: | I think one can get far enough by applying known mathematical | facts. (Proof: elementary school math is useful.) | auggierose wrote: | The problem is, usually you need to combine known | mathematical facts to solve a problem. Now, how do you know | that you combined them properly? | | Yup. You need to know what a proof is. | eftychis wrote: | Well you can think of it like going to the gym. You don't | exactly see people doing squats during their daily life, but | you can see the results of people having a good physical core. | | This needs to be explained further during education and | motivated appropriately. We have a short-term utilitarian | perspective, and we need to take a step back at times and | recall that it takes time and lots of sculpting to transform a | wood log to a art piece. | | As you can jog everyday for fun and/or for the challenge you | can also jog to improve your physical health. And not doing | proofs is like declaring a guy can weight-lift by just watching | videos on youtube and never lifting a weight. Or a guy can | "code" without writing a line of code. | konschubert wrote: | You don't need proofs, but you still need an intuition for why | things work the way they do. | | Otherwise you will be lost as soon as you leave the textbook | territory. | | Proofs are just one way to build intuition. | | The best way to learn applied maths and get intuition is an | "Introduction to Maths for Physicists" 101 course. | helpfulclippy wrote: | I did a degree in applied math. You'd think this would be "math | you'll use," but the fact is that despite my program having a | CS concentration, most of the stuff I did was not really | applicable in practice. | | However, one thing that has been VERY applicable is | proofwriting. Although math proofs are far more rigorous than | most real world stuff, the discipline I learned in writing | proofs has carried over into pretty much everything from | programming (will this algorithm work every time?) to executive | decisions (why, specifically, should we believe X?). Obviously | in the former case I wind up doing actual proofs, and in the | latter I make strong arguments based on logical consequences of | established or presumed facts, or find flaws or gaps in | arguments that are being considered. | | I really wish I'd spent a lot more time on proofwriting than | say, vector calculus. | | Of course you may want specific math to solve real problems, | and that's a real need too! Not to diminish your point at all, | just advocating for proofs to be seen in a practical light. | auggierose wrote: | Proofs are indeed very practical. For example, I believe | Leslie Lamport mentioned somewhere that he only came up with | the final version of Paxos once he tried to prove it, and | noticed that some condition he assumed wasn't necessary at | all. | dhosek wrote: | The reasoning behind having geometry be the standard high | school sophomore math class is that that's the age where | kids would be ready to do proofs. Except that curriculum | designers seem to have forgotten this and except in honors | classes, most sophomores don't get taught proofs in | geometry and instead get a set of inert rules about shapes | that they have no use for. | whatshisface wrote: | Geometry uses a deduction-only basket of proof techniques | that don't prepare students for proofs done afterwards. I | would like to see it replaced by elementary number theory | which naturally uses a lot of induction/recursion and has | some uses for proof by contradiction. | auggierose wrote: | Geometry has really all that is needed for proofs: | | * 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. | whatshisface wrote: | The thing about geometry is that it does not take long | before you've taught those four things, and then you | start teaching stuff that is specific to plane geometry. | auggierose wrote: | There are worse things to learn than plane geometry. It's | actually a good thing to have a fixed topic to really | learn those 4 things. Because once you really understood | those 4 things, you are done, and you know everything | about proofs in general there is to know. | wanderingmind wrote: | No recommendation on probability. Thats strange given that the | author is a physicist and fundamentals of modern physics rests on | probability. My recommendation is the classic "Probability | Theory, The logic of Science by E.T.Jaynes" which is a Bayesian | formulation. | RoddaWallPro wrote: | I read a Murakami novel in high school, 1Q84. The protagonist is | a math teacher who talked about math in a way that I had never | seen before. I'd been told I was "good at math" beforehand(for | whatever that means, I'm not a fields medalist or anything), but | for ~6 months after reading that book, I was _really good_. Like, | suddenly I did not have to do any homework in my sr. year | calculus class. I loved sitting in class and watching my teacher | work through problems, and it seemingly imprinted directly into | my brain, because while doing no homework I could still ace the | exams while writing with a pen (no erasing and re-do'ing with a | pencil). All because of the way this fictional teacher from 1Q84 | talked about math. | | Has anyone else had an experience like that? (With math or other | things?) | LAC-Tech wrote: | I really don't. | | 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. | [deleted] | [deleted] | dekhn wrote: | I'm still stuck at "wait, sets can contain other sets, and sets | can contain themselves?" part of Russell's paradox, and I'm close | to retirement! | | 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. | dr_dshiv wrote: | If _doing_ math is essential to conceptual understanding and | application, could the interface of math and physics be made more | human-centered? For instance, the shift from Roman numerals to | Arabic numerals made _doing_ math easier. Based on your | experience, might it be possible to increase accessibility by | revising some of the arcane conventions of math and physics? | | See Brett Victor's 2011 proposal: http://worrydream.com/KillMath/ | andrepd wrote: | Why is it that every time any subject about mathematics comes | up there is _always_ a complaint about notation? | | Your link doesn't even exactly talk about notation, but about | pedagogy. Can you be more specific about which notation your | consider "arcane"? | Jach wrote: | My favorite accessibility-increasing tool is the computer. | Doing math shouldn't involve so much _circus math_ , i.e. doing | things just for show, since a computer does so much immediately | and accurately. We already use graphing calculators, but | there's so much more they can do, let alone actual PCs, cell | phones, and web apps. By chance in 9th grade "Intermediate | Algebra/Algebra 2" I had a teacher not wholly opposed to modern | technology and so he only had us do a small amount of those | "solve this system of equations using a 3x4 matrix by hand, | showing each matrix transformation to reach the row reduced | form, taking up some pages of paper" problems before he brought | in a classroom set of chonky TI-92 calculators and showed us | the rref() function. That Christmas I asked my mom to upgrade | me from my non-graphing scientific calculator that had served | since elementary school to a TI-89 Titanium that served me even | through college until I learned and got used to various PC | programs. The lesson that there were powerful tools around | stuck with me pretty fast though, and I wrote some programs on | the calculator for that and other classes throughout HS; in HS | physics I also had learned more programming and did a little | simulation with pygame and it was fun to enter numbers in the | program, run it, see the mass trajectory animate and show some | computed values, and then do the actual experiment and get the | same results. | | I 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). | abhisuri97 wrote: | As a fellow penn alum, I can totally vouch for Ghrist's approach | to calculus. Check out his youtube channel: | https://www.youtube.com/c/ProfGhristMath | cpp_frog wrote: | > _My goal here is to provide a roadmap for anyone interested in | understanding mathematics at an advanced level. Anyone that | follows and completes this curriculum will walk away with the | knowledge equivalent to an undergraduate degree in mathematics._ | | 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. | ratzkewatzke wrote: | I'm a fan of functional analysis, but even in my (very | competitive) undergraduate curriculum, it wasn't required for a | bachelor's in mathematics. I think Susan's guide covers most of | what the undergraduate programs I've seen require. | cpp_frog wrote: | It was for me (french school of math), that and also measure | theory. | davidmr wrote: | This is certainly not universally the case, even in very well- | regarded departments. The University of Chicago, for example, | does not require it: | http://collegecatalog.uchicago.edu/thecollege/mathematics/. | Mimmy wrote: | Going from Strang to D&F seems like a steep jump. The former is | an applied textbook for non-mathematicians and the latter is a | proof-based text for advanced undergraduate / graduate-level math | students. | | 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. | selimthegrim wrote: | Strang's latest book DE&LA is disappointing, it is linear | algebra and its applications with the abstraction taken out and | mushed together with supplementary notes from ODE | videolectures. Mattuck's ODE course is good. | Py-o7 wrote: | For many years MIT students would go from Strang in year 1 to | Artin in year 2. Artin != D&F of course though many would say | it does less hand holding than D&F | susanrigetti wrote: | Gross's review of linear algebra from his MIT algebra course | bridges the gap: http://wayback.archive- | it.org/3671/20150528171650/https://ww.... A combination of that | and then chapter 11 in D&F should cover whatever readers didn't | get from Strang. | | That being said, Axler is an excellent book. I don't know if I | would replace Strang with it, but I should add it as a | supplement to the next edition of this guide! | tptacek wrote: | Both Strang and D&F are extra-relevant for cryptography (I was | struck by how much the earliest parts of D&F --- which I | haven't gotten much further beyond --- read like the | mathematics background chapter of a cryptography book), and | I've been in study groups for both of them with non- | mathematicians that went OK. But the D&F study group fell apart | for logistical reasons, so maybe it would have hit a wall after | a couple more months. | pvg wrote: | _read like the mathematics background chapter of a | cryptography book_ | | A lot of maths-related books, especially ones intended as | textbooks will read like that in part because they aren't | kidding about the 'abstract' in the title - they're trying to | teach/re-summarize key concepts of mathematical abstraction. | It's a good and true thing to notice. | dwohnitmok wrote: | Second Axler. "Linear Algebra Done Right" is probably the pure | mathematics textbook I've most enjoyed reading ever (but be | warned you will learn very little about applied methods from it | if that's what you care about). | | Also enjoyed Artin's Algebra. | musgravepeter wrote: | +1 for Artin's Algebra. I think is very under appreciated. | cgriswald wrote: | My university course in linear algebra taught me how to | manipulate matrices. It was super uninteresting, and easy. I | aced every test, but got a B in the course, because the | professor assigned an asinine amount of homework (that I | either aced or didn't do), perhaps holding the article | author's view that: | | > 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. | dwohnitmok wrote: | > 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 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. | selimthegrim wrote: | I have been using Morris in my class. | nyc111 wrote: | I don't agree with this article, it as off-putting as the usual | math eduacation it criticizes. I wonder how one can propose a | curriculum to study math and not mention Euclid. One learns more | mathematics from this article https://mathshistory.st- | andrews.ac.uk/Extras/Russell_Euclid/ by B. Russell where he | harshly criticizes Euclid than 2 years of calculus. Newton did | not know calculus but he knew Euclid's Book 5, the book about | ratios and proportions. Euclid's 5th Book must be the starting | point for the study of math. When we say "math is the language of | nature" we really mean that nature is proportional. Ratios and | proportions are fundamental. | ouid wrote: | If you actually want to study math, you probably shouldn't touch | calculus until you've take linear algebra and a fair amount of | topology, since these are the two structures on sets that | (differential) calculus is founded upon. | | 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. | pattt wrote: | Spivak's Calculus reignited my interest and appreciation in math. | Sad to discover the author passed away quite recently. The way of | explaining principles and making you do the hard work via | problems which I believe is a must with this book, is profoundly | astonishing. There's a lot of mathematical insight packed into | those problems, it almost feels you can build up the entire high | school and the early uni curriculum from the ground up, for | instance there are a number of popular formulas you'd arrive at | and derive accidentally while working on those problems. | Furthermore it really works your brains by making sure you can | reason within the established framework and exercise great doubt. | I'm taking this book very slowly. | mathgenius wrote: | Modern calculus (analysis) was invented because people shot | themselves in the foot working with topology and wondering | exactly what is a "curve" ? I am a big fan of this approach to | learning mathematics, just forge ahead and when (if) things fall | apart then go back and fix up the foundations. To this end I | recommend a couple of books. "The Knot Book" by Adams is a very | interesting exploration in topology (without requiring all the | years of study at university before you are allowed to learn | exactly what a topology is). And in another direction, group | theory was invented because the study of symmetry gets very | tricky! But if you want to dive in anyway then have a look at | Conway's "The symmetries of things". It is a lot of fun. Most | modern group theory (or algebra) books don't actually have any | pictures of symmetric things, just endless formulas and lemmas. | If you want to be a pro, then you gotta learn that stuff, but | there's definitely pathways into higher mathematics that don't | require you to learn that. | PartiallyTyped wrote: | Speaking of group theory, I can recommend "A book of abstract | algebra". I think that it's a very approachable introduction to | the topic. As a person with a CS degree doing ML, it changed my | perspective on so many different topics, I can't recommend it | enough. | | https://www.goodreads.com/book/show/8295305-a-book-of-abstra... | daxfohl wrote: | I loved last year being able to take university courses online. I | knocked out analysis, topology, and quantum mechanics as a non | matriculated student. I'd had those books for years but never | could get through them alone. (The main thing being, you really | don't have anything to gague whether you know it well enough or | not). | | 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. | elteto wrote: | Where did you take your classes? | daxfohl wrote: | University of Washington | adamsmith143 wrote: | Texas AM has a program that gets somewhat close though it | definitely has a computational focus. Here's a list of their | recently offered courses: | https://www.math.tamu.edu/graduate/distance/openletter.html | itcrowd wrote: | Susan, I greatly appreciate this list and will definitely come | back to use it as a reference if I need a book recommendation. (I | don't think I'm the target audience, although who knows what the | future brings..) | | 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. | jerry1979 wrote: | Do we have good universal descriptors for math levels? I'm a | big fan of accessibility, and I think your idea about tweaking | language to reach a wider audience could be a big win for | increasing the article's impact. | | To update the article to include your recommendations, the | author would probably need some kind of "cross-walk" which | would map the American perspective to a more universally | understood framework. Would you happen to know what "pre- | calculus's" opposite number would be in the universal | framework? | rongenre wrote: | I have a decades-old math degree and ended up working in tech as | an engineer. Are there options, like a "Math Camp for the Middle- | aged" where I could get a chance to re-learn everything I've | forgotten? | paulpauper wrote: | You can get good or better at something with effort, but few will | ever make to leap to being great or world class at it, no matter | how hard they try. | atan2 wrote: | True! But sometikes getting better at something is all people | really want, and that's ok. I see that most of my CS students | just want to be able to not see math as an obstacle when | learning new/interesting things. | fjfaase wrote: | I am bit surprised there is nothing about graph theory in there. | Also nothing about combinatorics or knot theory to mention two | other subjects. If you want to make people dive into mathematics, | it might be a good idea to show a broad range of subjects instead | of focusing on the traditional subjects. | travisjungroth wrote: | It's amazing how different the subjects of mathematics are. | It's like the difference between a drum and flute. | | You listed some of my favorite stuff. Weirdly, when I was 11, | my math tutor told me I'd probably really like finite | mathematics. She turned out to be right. | Someone wrote: | > It's amazing how different the subjects of mathematics are. | It's like the difference between a drum and flute. | | 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) | sdenton4 wrote: | You can add 'combinatorics' to the list of primitives. | | Algebraic combinatorics (imo) encompasses related | structures in all three of combinatorics, algebra, and | geometry, though. | Buttons840 wrote: | Where's statistics? You mean to tell me I could go through all | that and come out not knowing statistics? | Koshkin wrote: | Agree, but I have a feeling that statistics is more like | (theoretical) physics, in the sense that it is "not math." | Buttons840 wrote: | Yeah, it's more application oriented and philosophical than | the pure calculation of pure math. I think it's under-taught | in schools though. I think it's more useful than calculus for | most people and should be taught before it. | ghufran_syed wrote: | I went from only having done high school math 10 years ago to | completing an MS in math and statistics at my local state | university while working in an unrelated field. I would recommend | NOT starting with calculus if you haven't done it, instead, just | learn how to do proofs - I used Chartrand "Mathematical proofs" - | You don't need to know any math beyond algebra in order to do | that most of this book. If you need to revise or learn Algebra, | then I would do Stroud "engineering math" first which is designed | for self-learners with lots of solutions and feedback. | | 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! | criddell wrote: | I looked up the Chartrand _Mathematical Proofs_ book and it 's | been a while since I had to buy a textbook, but $175 for | hardcover and $75 for paperback or ebook? That's nuts. If I | were a student today, I'd pirate that and feel absolutely no | remorse for doing so. | joe_the_user wrote: | Well, | | I feel think one can 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. | hintymad wrote: | I find it hard to believe that the author started to appreciate | physics by reading The Feynman Lectures on Physics before any | exposure to physics or even algebra, and in less than three years | went from barely knowing high school math to enjoying advanced | mathematical physics and graduate-level quantum physics. It looks | this is one-in-a-million level brilliance as learning the sheer | amount of requirement knowledge in such a short time is amazingly | challenging: analysis, functional analysis, complex analysis, | linear algebra, abstract algebra, differential equations, | mathematical statistics, and all the physics: mechanics, | electromagnetism, thermodynamics, optics, statistical mechanics, | relativity, and of course quantum physics, all in less than three | years. | | Kudos if the author is this talented. | whatshisface wrote: | I agree that this does not on its surface seem possible, but I | can think of a few explanations. | | 1. I recently spent a week on one section of one chapter of a | math book. I was able to follow it within an hour on the level | of "these are the rules and this is the sequence of their | application," but I have stuck with it since then because I | wanted to understand it well enough that the proof they chose | to use would seem obvious to me. If you saw "understanding | math" like the peak of a mountain, you'd get there a lot more | quickly, but if you want to try out every permutation of every | device and condition anything can take forever. | | 2. Algebra seems simple in retrospect, and my teenage self was | kind of dumb. Maybe with my complete adult brain I'd be able to | finish highschool starting from scratch in a few months. | Evidence to that point is the pacing of college remedial math | classes. Maybe, to a certain extent, people have an innate math | setpoint that they will snap to very quickly when given the | chance. | | 3. Intelligence is equally distributed between genders, but | most professional physicists are men, which means that for | every professor there is almost exactly one corresponding woman | who has equal potential but isn't in the system. If you heard | that the department chair at a university sat down and read a | book about topology without a lot of trouble you wouldn't be | surprised at all. In other words, it's not surprising that | someone can do this, it's surprising that someone who can do | this is not in the social bucket for people that do it, but if | you think about the other things you've heard about that, you | realize you already knew. | | I am inclined towards #3 out of all these explanations but all | may be true at once. | paulpauper wrote: | women had an advantage over men in regard to memorization. | this helps greatly at learning. | hintymad wrote: | > Intelligence is equally distributed between genders, but | most professional physicists are men, | | 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. | whatshisface wrote: | I'm not sure what objection is being made. We know that | there are lots and lots of women who could be physicists | but decided not to. You don't stop existing when you don't | get labeled, but you do start surprising people who expect | you to have been. | | > _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... | [deleted] | paulpauper wrote: | More like one in 50-100 million brilliance, and such people do | exist. It's a statistical certainty they exist. Terrance Tao | for example. | [deleted] | Py-o7 wrote: | This felt like it was written by a physicist or engineer. | | 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. | susanrigetti wrote: | guilty as charged! :) | bitexploder wrote: | As someone with a keen interest in learning Engineering part | time, I found your write ups really helpful though! I enjoy | learning math but like to have an angle towards a practical | and useful application. It keeps me a little more motivated | than pure math learning. With ADHD the concept of being able | to build cooler things always keeps me going. But somewhere | along the way of learning purely theoretical things for too | long my brain just loses interest (not enough reward), even | though I enjoy it in the moment it is hard to get to the | starting line and take the first step after a while :) | pphysch wrote: | IME (as a math-degree-haver) the value of mathematics is in | improving one's ability to mentally model and reason about | complicated _real-world phenomena_. A lot of folks lose sight of | the reality and get lost in the mysticism, especially within the | academic regime. | | > [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. | susanrigetti wrote: | You cut out the middle of that paragraph, which says: | | "Sadly, there is all sorts of baggage around learning it (at | least in the US educational system) that is completely | unnecessary and awful and prevents many people from | experiencing the pure joy of mathematics. One of the lies I | have heard so many people repeat is that everyone is either a | "math person" or a "language person" -- such a profoundly | ignorant and damaging statement. Here is the truth: if you can | understand the structure of literature, if you can understand | the basic grammar of the English language or any other | language, then you can understand the basics of the language of | the universe." | | :) | pphysch wrote: | I'm not sure what your point is. Are you implying that you | are _not_ contributing to the mystification and idealization | of mathematics? | | 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. | dang wrote: | Please don't post in the cross-examining style. We want | _curious_ conversation here. | | This is in the site guidelines: | https://news.ycombinator.com/newsguidelines.html. | pphysch wrote: | I don't follow. The author dismissively ctrl-V'd a | paragraph with no further explanation, and my response | asking for elaboration gets shadow-buried by a mod. What? | akomtu wrote: | On the mysticism note, I want to add that math is perhaps the | only subject that forces one to engage the upper "abstract" | mind. The lower mind is concerned with modeling real world | phenomenas, while the upper mind works with purely abstract | things, aka the "true reality" in mysticism. | cathrach wrote: | While I understand that the author has good intentions, I | strongly disagree with the general idea of this post, which is | that anyone can learn math through an almost entirely analysis- | focused curriculum while other topics like topology, game theory, | set theory, etc. are presented as advanced and graduate-level. | This is practically equivalent to saying that anyone can learn | history, and they should learn all about British history in | undergrad, and then graduate-level courses might teach you more | about the history of South America. | | 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! | tzs wrote: | Overall a pretty decent list, although I would suggest | considering some tweaks. | | 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". | jeffreyrogers wrote: | I'll second this. "How to Prove It" gets recommended a lot, but | I couldn't get through it. I found it terribly boring and | unmotivated. Some people can power through dry material but I'm | not one of them. I found it much easier to learn to write | proofs when they were related to topics I was interested in. | l33t2328 wrote: | Spivak is a better analysis book than Abbot. | irrational wrote: | I never got beyond algebra/geometry in High school. I think I had | to take one 100 level math class in college, but it was basically | a review of HS math. Oh, and I had to take a stats class for non- | technical people in graduate school. That was my worst graduate | class by far. But, I would like to learn some more math, like | calculus. I'm hoping to get to it when I retire in a decade or | so. | foobarbecue wrote: | "... but make sure you get the paperback or hardcover version for | readability purposes." | | As opposed to... the ebook? | musgravepeter wrote: | I've been on a Math journey since I retired a couple of years ago | and I agree with all the books mentioned that I know and look | forward to picking up some of the one I do not know. I agree baby | Rudin is essential, but I find it tough going. | | 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/ | selimthegrim wrote: | Surprised Arnol'd isn't mentioned for ODEs. | auggierose wrote: | Very pretty book (Needham's), will check it out! I think over | 20 years ago I actually attended a house party that Needham was | giving in SF. It's a small world. | voldacar wrote: | Those are good, I also really like Visual Complex Analysis | threatofrain wrote: | Consider Analysis 1/2 by Terence Tao for introduction to | analysis. | vermarish wrote: | I think learning Real Analysis from baby Rudin is like learning | Probability Theory from Wikipedia. It's so encyclopedic that if | it's your first look at real analysis, it will be too dense to | understand, but if it's your second or third look, you will | find beauty in its brevity. | susanrigetti wrote: | Agree that Baby Rudin is VERY difficult to study on its own. I | recommend only studying it alongside the other two books I | listed: Abbott's Understanding Analysis and Spivak's Calculus | (which has a solutions manual). Abbott in particular is very | straightforward (at least in comparison with baby Rudin haha) | tzs wrote: | Another point for Abbott is that it was one of the ~400 books | Springer made available for free download near the start of | the pandemic. I remember there were a few scripts here on HN | back then to grab all those books, so many here probably | already have a copy. | graycat wrote: | Calculus: I suggest just forget about "precalculus" and, instead, | just get a good calculus book and dig in. | | 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. ___________________________________________________________________ (page generated 2022-03-07 23:00 UTC)