[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.
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