[HN Gopher] Ask HN: Math books that made you significantly bette...
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Ask HN: Math books that made you significantly better at math?
Do you have any special math books that you hold close to your
heart because of the value they delivered specifically to you and
your mathematical thinking and skills?
Author : optbuild
Score : 395 points
Date : 2023-01-19 13:01 UTC (9 hours ago)
| omershapira wrote:
| Donald Sarason's "Complex Function Theory". There are bigger more
| complete books on complex analysis. There are even ones that are
| more appealing, like "Visual Complex Functions" by Elias Wegert.
|
| Sarason's book is only 160 pages long, with legible text and
| clear examples. It covers the length of an undergraduate
| university class, and explains Holomorphic functions perfectly.
| The proofs are crystal clear, and so are the motivations. I
| haven't seen a better introduction.
| bmitc wrote:
| While I have a ton of favorite math books, the two books that I
| felt really helped me the relevant subject are:
|
| * _An Introduction to Manifolds_ by Loring Tu
|
| * _The Elements of Integration and Lebesgue Measure_ by Robert G.
| Bartle.
|
| These two books were instrumental to my studying for my
| qualifying exams.
| curious16 wrote:
| Not a book. But a course by Prof Keith Devlin on Coursera called
| Introduction to Mathematical Thinking.
| AlanYx wrote:
| [dead]
| wannabebarista wrote:
| I jumped into the first analysis class (using baby Rudin)
| completely unprepared and this course saved me!
| Ian_Macharia wrote:
| A Programmer's Introduction to Mathematics by Jeremy Kun
| keiferski wrote:
| I would not call myself great at math - I struggled with it in
| school, in fact - but in recent years I've begun "correcting" my
| lack of mathematical knowledge. The single best decision I've
| made is to first start with the philosophy of mathematics. Maybe
| it's because my background is in philosophy, but I also think
| that for certain people like myself, understanding _what_ math is
| makes me far more interested in understanding _how_ it works,
| rather than just doing context-less calculations using formulas I
| don't know the history or deeper purpose of. When I learned math
| in school, it was entirely cut off from any of these deeper
| questions.
|
| Here's a good starting point for philosophy of mathematics :
|
| https://plato.stanford.edu/entries/philosophy-mathematics/
| mxkopy wrote:
| In that vein I highly recommend Foundations and Fundamental
| Concepts of Mathematics by Howard Eves. I think it might be a
| little dated, but it gives an amazing overview of the most
| important developments in mathematics that were relevant at the
| time. It's less focused on practice (though there are some
| problems) and more on the history and motivation behind the
| ideas. This book introduced me to axiomatics, non-Euclidean
| geometry, quaternions, and abstract algebra in my senior year
| of high-school.
| mohamez wrote:
| History of mathematics as well, it will give you a very subtle
| entry into the minds of mathematicians and the motivation
| behind their theorems.
|
| This will surely make you more appreciative of subjects and
| concepts you are learning.
| keiferski wrote:
| Absolutely, and there are some really interesting
| personalities in the history of mathematics. Newton and
| Galois come to mind.
| Aaronstotle wrote:
| I have a very similar background, did my undergrad in
| Philosophy and feel that I need to learn some basics. Do you
| have any pointers on where to move after this?
| bombcar wrote:
| Reading _Euclid 's Elements_ and _Newton 's Principia_ really
| helped me get an intuitive feel for geometry and calculus. They
| may not be entirely easy (at least the second) without some
| commentary, but well worth the study.
| pen2l wrote:
| While it's laudable that you sought those texts and profited
| from them, I worry about what others might take away from
| this. When I was young I knew some geniuses who highly spoke
| of Principia and how it gave them great insights. And the
| teenager me said, okay cool, I'll have a go!
|
| The problem is that it's in Latin and quite impenetrable.
|
| We have some geniuses here and they would no doubt be able to
| take away a lot from these texts, but for you normals out
| there: don't optimize too much, you're quite alright in
| taking the normal approach of just taking a class at a
| community college, doing the exercises the teacher assigns,
| etc.
| bombcar wrote:
| I didn't read Euclid in Greek or Newton in Latin; there are
| quite good translations available - even free!
|
| In general I find that if someone is insisting that you
| study the philosophy of someone in their original language,
| they don't have a good enough translation yet.
| sverona wrote:
| _Physics for Mathematicians_ by Spivak is basically the
| _Principia_ updated to modern prose and rigor.
| troupe wrote:
| It might make sense to read a translation in a language you
| understand. Many of the books that are considered classics
| are specifically because they ARE accessible. That doesn't
| necessarily mean that they are easy, but there is a big
| difference between reading Euclid and learning how to
| create mathematical proofs, and taking a class focused on
| calculating the area of various shapes or determine angles.
|
| I haven't read Mathematical Principles of Natural
| Philosophy (the English title) but I have read Euclid and
| it definitely doesn't require a genius to understand. here
| is an online edition with great illustrations:
|
| https://www.c82.net/euclid/
| lordgrenville wrote:
| From what I've heard, Euclid is fairly accessible and was
| for centuries the standard geometry textbook for
| children; the _Principia_ is incredibly daunting, and
| Newton even admitted that he made it extra confusing on
| purpose to deter readers who weren 't already experts.
| krmboya wrote:
| I've been flirting with the idea of working through all of
| Leonard Euler's publications (as a life goal). Many of them
| are still not translated from Latin, so there's a
| possibility I may have to learn it.
|
| Anyone knows how long it would take to learn enough latin
| to undertake such a task?
| bombcar wrote:
| If you're trying to understand a domain work (such as
| Euler's) you could probably get a working knowledge in a
| month of strong study, a year of off-and-on.
|
| I bet you could start this week if you used machine
| translations as a crutch.
|
| I'd start working with a publication that exists in Latin
| _and_ a good translation, so you can compare your work.
| krmboya wrote:
| Thank you! Sounds much less intimidating.
|
| My fear with machine translations is that subtle errors
| here and there might throw me off in something like math
| where things are precisely stated.
|
| A year of part time study sounds doable though
| bombcar wrote:
| That's the advantage of it being a particular mathematic
| domain, you'll learn the terms relatively quickly and be
| able to catch errors in the math parts; the prose is
| where you will need the machine.
|
| In fact, you'll find that many philosophers will just use
| the Latin words directly, and not bother translating them
| - Latin _qua_ jargon if you will.
|
| Once you've learned the various forms of "is" (sum, very
| irregular) you can kinda survive reading without
| conjugations, just like this sentence can be worked out:
|
| he to go to store and to buy cheese yesterday
| macrolocal wrote:
| Ditto for Euclid. Doing this early in life pays huge
| dividends.
| eddsh1994 wrote:
| This is sort of like recommending the art of computer
| programming as a way to learn how to code, isn't it? Starting
| very far down the stack if you're working through a 2000 year
| old book in Ancient Greek!
| geenew wrote:
| Elements was a school textbook for 2000 years, up until
| ~100 years ago. It's a fine book to use for self-study.
|
| Edit: Also, to state the obvious, it's been translated into
| English
| bombcar wrote:
| Perhaps, but the Elements is quite understandable, and
| surprisingly enough it's been translated -
| https://www.barnesandnoble.com/s/Euclid%20elements
| leephillips wrote:
| Some schools still teach geometry from the _Elements_. It
| doesn't matter how old the book is. Mathematics is
| timeless.
| shrimp_emoji wrote:
| Is that entire course just wading through the wreckage wrought
| by Godel?
| college_physics wrote:
| Math always seemed a bit arbitrary to me. Why and how did all
| those fields and branches develop, why are some so much more
| intuitive than others etc. What helped me cope with that
| challenge (and ultimately be a better learner / user of
| mathematics) was digging into the history of mathematics. Many
| great books in that genre but a very influential one for me was
| the _Concise History of Mathematics_ by Dirk Jan Struik
| javajosh wrote:
| _> Math always seemed a bit arbitrary to me._
|
| That's because _math is fundamentally arbitrary_. This
| realization came to me late in life. Math was always presented
| as some aspect objective reality. However over time I 've come
| to understand it as software for human brain. Using this math
| or that to describe something is often simply a matter of
| taste! Very similar to programming, in fact. Your study of
| history gives context to the question of _utility_ different
| maths "packages" for certain problems, but does not invalidate
| your original impression.
| ska wrote:
| I think "arbitrary" gives the wrong flavor here. Math is
| fundamentally curated.
|
| It's easy to create something new, because you get to play
| with the rules - but unless it is in some sense deep,
| effective, and usually elegant [1] it won't stick around.
|
| [1] this is a bit acculturated.
| Mimmy wrote:
| Linear Algebra Done Right by Sheldon Axler for the following
| reasons:
|
| - I was revisiting a topic in greater depth, which is a common
| theme in university-level math courses.
|
| - It is a rigorous book, written in the style of definition,
| proposition, theorem, etc.
|
| - It was the first math book where the exercises don't just
| reinforce what you learned in the chapter, but teach you new
| material (another common theme in advanced math textbooks).
|
| - Linear Algebra is arguably the most important math subject
| these days.
| mohamez wrote:
| Linear Algebra Done Right by Sheldon Axler is indeed a good
| book if you are looking for a rigorous proof based book to
| learn linear algebra.
|
| Here [1] you can find Sheldon Axler himself explaining the
| topics of the book in his YouTube channel! How wonderful is
| that!
|
| Here [2] you can find the solutions to the exercises in the
| book.
|
| This [3] Lectures might help as well, among the books this
| course follow is Algebra Done Right.
|
| Good luck learning the subject of Linear Algebra you'll have
| fun doing so.
|
| [1]
| https://www.youtube.com/playlist?list=PLGAnmvB9m7zOBVCZBUUmS...
|
| [2] http://linearalgebras.com/
|
| [3] http://nptel.ac.in/courses/111106051/
| sgdpk wrote:
| I have to second this. It's very well written and presents a
| clear view of what Linear Algebra is. Although it might be best
| used as a second book in Linear Algebra (depending on your
| preparation).
| acrodrig wrote:
| The enjoyment of Math
| (https://www.barnesandnoble.com/w/enjoyment-of-math-hans-
| rade...). Best book ever. Made me fall in love with math as a
| teenager.
| sampo wrote:
| I was reading this book, when the ideas of function spaces,
| functions as vectors, functions as elements of vector spaces,
| functional analysis clicked on me.
|
| I am not sure if this book is particularly good or better than
| other books. (Well, it still looks like a very gentle
| introduction to the topic.) But as per your question, this was
| the book at the right time for me.
|
| "Fourier Series and Orthogonal Functions" by Harry S. Davis.
| https://www.amazon.com/dp/0486659739/
| mannerheim wrote:
| Baby Rudin for analysis.
|
| Dummit and Foote for algebra.
| wannabebarista wrote:
| As far as the canonical texts of math goes, Dummit and Foote is
| excellent. I'm also fond of the first half of Munkres'
| Topology.
| time_to_smile wrote:
| I highly recommend working through Claude Shannon's _Mathematical
| Theory of Communications_ [0]. It 's originally a paper but was
| later restructured as a book, in either form it works quite well.
|
| The reason I recommend it is because it shows mathematical
| reasoning that is easy to follow and relevant to your daily life.
| It's real math, but very easy to read through and understand. If
| your unfamiliar this paper is where the very idea of "bits" comes
| from.
|
| One of the most important things in the paper for non-
| mathematicians to see is that the definition Information Entropy
| is derived simply from the mathematical properties Shannon
| desires it to have.
|
| This is important because I find that one of the biggest
| questions people ask about mathematical formula and idea is "What
| does this mean? Why is it this way?" without realizing that math
| is really not engineering nor physics. When deriving his
| definition of Information, Shannon simply states that information
| should have the following x,y... properties and then goes on to
| show that the now standard definition of information meets all
| these criteria.
|
| In mathematics it is very often the case that only _after_ an
| idea is created to we start realizing the applications. This is
| quite different than science where a model is only adopted if it
| correctly describes a physical process.
|
| Work through the paper and you will have worked through the
| mathematical underpinnings of the information age and will likely
| have understood most of it pretty well.
|
| 0.
| https://people.math.harvard.edu/~ctm/home/text/others/shanno...
| enriquto wrote:
| The classical stuff is great:
|
| * _Geometry and the imagination_ by Hilbert and Cohn-Vossen
|
| * _Methods of mathematical physics_ by Courant and Hilbert
|
| * _A comprehensive introduction to differential geometry_ by
| Spivak (and its little brothers _Calculus_ and _Calculus on
| manifolds_ )
|
| * _Fourier Analysis_ by Korner
|
| * Arnold's books on ODE, PDE and mathematical physics are
| breathtakingly beautiful.
|
| * _The shape of space_ by Weeks
|
| * _Solid Shape_ by Koenderink
|
| * _Analyse fonctionnelle_ by Brezis
|
| * Tristan Needhams "visual" books about complex analysis and
| differential forms
|
| * _Information theory, inference, and learning algorithms_ by
| MacKay (great book about probability, plus you can download the
| .tex source and read the funny comments of the author)
|
| And finally, a very old website which is full of mathematical
| jewels with an incredibly fresh and clear treatment:
| https://mathpages.com/ ...I'm in love with the tone of these
| articles, serious and playful at the same time.
| blahblahblah10 wrote:
| Out of this list, the books I am familiar with, are great
| (Hilbert-Courant, Spivak, Korner's books). At the same time,
| even with extensive mathematical training, I haven't read them
| from start to finish. I wouldn't even like to say "read". For
| someone who's not used to mathematical reading, some of these
| books require careful study. That means generating examples to
| understand results (theorems), trying your own conjectures,
| proving things yourself etc. Over time, one becomes familiar
| with most/all the material in a book but the knowledge might
| have been acquired through various books (and courses) over
| time.
|
| Also, mathematics is a massive field. The first question would
| be what kinds of mathematics would you like to get better at.
| There are great books in analysis. If you are starting out with
| a solid calculus knowledge, try Abbott's Understanding Analysis
| [1] or Duren's Invitation to Classical Analysis [2]. For
| asymptotic methods in PDEs, try Bender and Orszag [3] which is
| a wonderful book. But again, this might not be your cup of tea
| at all and there are more abstract or formal books like
| Rudin's.
|
| If you want to approach fields without a lot of machinery,
| graph theory books by Bollobas are great (but difficult). See
| his Modern Graph Theory book [4] as an example.
|
| For linear algebra, one of my favorites (but it was after I
| already learned the subject) is Trefethen's Numerical Linear
| Algebra book [5]. Another beautiful topic is at the
| intersection of linear algebra and combinatorics. See Babai and
| Frankl's lectures freely available online.
|
| Then there are wonderful topics in geometry. A massive mountain
| to climb would be algebraic geometry. For one starting point,
| see [6]. Differential geometry (Spivak's multi-volume work or
| Needham's differential forms book) is another wonderful area. I
| would recommend Crane's discrete differential geometry course
| at Carnegie Mellon [7] if you want a concrete introduction.
|
| You might want to demystify a topic you have heard about. E.g.
| Galois theory and the unsolvability of quintic equations. You
| could look at [8] which guides your way through wonderful
| problems.
|
| We haven't even touched huge swathes of mathematics including
| anything topological or number theory. Even within the topics
| mentioned above, once you start, your journey will take a life
| of its own and you'll encounter multiple books and papers
| opening up new sub-fields.
|
| The only approach that worked well for me in the past was to
| get completely consumed by what one topic one was studying.
| This meant not getting distracted by multiple topics. Once one
| enters the workforce, this is very hard (or at least has been
| for me). Without knowing someone, it's hard to recommend
| anything but the advantage with topics like graph theory and
| combinatorics is that one needs less machinery (as opposed to
| something like algebraic geometry). These fields lead you to
| interesting problems very rapidly and one can wrestle with them
| part-time.
|
| [1] https://www.amazon.com/Understanding-Analysis-
| Undergraduate-...
|
| [2] https://www.amazon.com/Invitation-Classical-Analysis-
| Applied...
|
| [3] https://www.amazon.com/Advanced-Mathematical-Methods-
| Scienti...
|
| [4] https://www.amazon.com/Modern-Graph-Theory-Graduate-
| Mathemat...
|
| [5] https://www.amazon.com/Numerical-Linear-Algebra-Lloyd-
| Trefet...
|
| [6] https://www.amazon.com/Algebraic-Geometry-Approach-
| Mathemati...
|
| [7] https://www.cs.cmu.edu/~kmcrane/Projects/DDG/
|
| [8] https://www.amazon.com/Through-Exercises-Springer-
| Undergradu...
| VyseofArcadia wrote:
| Seconding all of Spivak's books. My favorite treatment of
| differential geometry.
| salusinarduis wrote:
| I've had this idea of starting back at basics and relearning
| math from the beginning since I never "really" learned it
| besides memorizing and skirting my way through it in school.
|
| Do you know a good path or book that's suitable for that?
| vmilner wrote:
| I'm reading and like Thomas Garrity's "All the mathematics
| you missed (but need for graduate school)" which is this but
| for people who did a bachelors degree but missed certain
| areas (or forgot them).
|
| https://www.amazon.co.uk/All-Math-You-Missed-
| Graduate/dp/100...
| vmilner wrote:
| Something else I've found extremely useful in getting into
| maths topics is the Princeton Companion to Mathematics - it
| doesn't have exercises but gives excellent overview essays
| of a wide range of maths topics - expensive to buy (mine
| was a present) but should be available in academic
| libraries, say.
|
| https://www.amazon.co.uk/Princeton-Companion-Mathematics-
| Tim...
| trenchgun wrote:
| I bought and read it (more like, skimmed) and liked it a
| lot too.
|
| Gives a bird's eye view of math very nicely. Even from a
| skimming it was very useful to help me understand the gaps
| I have, and the shape of those gaps, and partially filling
| them.
| friedman23 wrote:
| I'm doing this and am starting with Linear Algebra on MIT OCW
| (taught by Gilbert Strang). My current plan is to relearn
| Linear Algebra, Calculus, Probability, and Statistics and
| actually focus on retaining the knowledge in my memory using
| something like SRS learning. I think planning past that is
| pointless since by the time I'm done I will have a better
| ability to plan my future coursework.
| _fullpint wrote:
| Going back through Discrete would probably be a good idea
| as well.
| Philip-J-Fry wrote:
| I had the same idea.
|
| I bought myself a Remarkable 2 and signed up to Khan Academy.
| Now I'm revising algebra basics and I plan to go as advanced
| as Khan Academy lets me.
|
| I was really bad at maths in school (UK A Levels). But I'm a
| successful software developer today. I felt like knowing more
| advanced maths could make me a better developer and not feel
| intimidated by a lot of the things I see.
|
| I'm actually enjoying it as well. Maths isn't just something
| I have to do to get out of school, now it's something I want
| to do. And it gives me the same satisfaction as solving
| puzzles like sudoku.
|
| I'd recommend it to anyone. The Remarkable 2 is actually
| really nice to write on too, since I want to store my notes
| digitally. And I make so many mistakes when writing, so undo
| is great.
| erksa wrote:
| Thinking of doing the same thing. The last math class I had
| was at 16, and the most advanced classes was on binary, so
| not very complex stuff. I've mostly been winging it for
| another 16 years and seemingly picked up things here and
| there. But math is definitely been trial and error, and I
| definitely do not know the lingo in math, which I'm now
| starting to feel is a big disadvantage.
|
| also: the remarkable 2 is great, we have one, but the
| screen broke and the refurbished replacement arrived with a
| screen that's not functioning correctly at all, making it a
| unusable device. Good reminder to reach out to them again.
|
| Thanks!
| DoingIsLearning wrote:
| If you don't mind the question how is the Remarkable
| helping you in this. Just to avoid the clutter of paper? Or
| does it some how OCR you're handwriting?
| MrMan wrote:
| I have a remarkable 2 and a Microsoft Surface Pro,
| intended to de-clutter math coursework. Both work, but I
| found that the real estate on the remarkable was too
| limited, even though its a great device, so I tried the
| Surface Pro. I can fit just about any sized work onto it,
| and you can endlessly scroll down which was something I
| couldn't figure out on the remarkable. It makes doing
| math easy or at least takes away some housekeeping which
| I find really distracting. And saving and organizing work
| and being able to import and export files is a bonus.
| AlanYx wrote:
| For what it's worth, version 3.0 of the RM's software now
| allows endlessly scrolling down.
| Philip-J-Fry wrote:
| I hate the clutter of paper and how hard to organise it
| is for me. Plus, how messy I am writing on paper and
| crossing things out all the time.
|
| The Remarkable, at least for me, is good because I can
| organise my notebooks into folders by certain math
| lessons or concepts. And I can undo any mistakes, so my
| notes are clean. Even if I am quickly working something
| out I am clean it up and make it a good note for future
| me. The feel of writing on it is much nicer as well
| versus my laptop's pen or my iPad's pen.
|
| Also, I like that it basically just does notes. There's
| no Android bullshit, it's just no nonsense note taking.
| Some competitor tablets have Android and all that
| baggage.
|
| It can OCR you writing, but I don't know how good that
| would be for math.
|
| The Remarkable isn't the only tablet that can do this,
| but it's the one I bought because I like the style and
| the simplicity of the software.
| strangattractor wrote:
| A similar solution might be https://getrocketbook.com/
| RocketBook. It can be a $0 dollar solution if you
| download their app and then print out the free PDF pages
| that are pre-formatted.
| meken wrote:
| I did this same exact thing back in 2010. I used khan academy
| for it. Started with positive and negative numbers,
| arithmetic, through trig and algebra.
|
| I like khan academy back in 2010 because all the videos were
| in one place and you could see everything right there in
| front of you
| chriswarbo wrote:
| Norman Wildberger's YouTube channels are the most thorough
| I've seen ( https://www.youtube.com/@njwildberger and
| https://www.youtube.com/@WildEggmathematicscourses ).
|
| There are hundreds of videos, organised in playlists, from
| undergraduate lectures (
| https://www.youtube.com/playlist?list=PL55C7C83781CF4316 )
| and research seminars (
| https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF )
| all the way to basic fundamentals like how to think about
| counting (e.g. https://www.youtube.com/watch?v=Puk-
| ipOTiD4&list=PL5A714C94D... )
|
| The reason I find them fascinating is that Wildberger doesn't
| agree with some of the conventional approaches, in particular
| with the use of infinity and taking limits. This leads him
| down interesting paths (e.g. Rational Trigonometry and
| Algebraic Calculus), which (a) show the _process_ of
| mathematics (exploring, making definitions, building up in
| different directions, etc.), whilst (b) remaining mostly
| grounded and approachable (e.g. no appeals to inscrutable
| lemmas from abstract research areas).
|
| For example, he's recently been making videos about
| "multisets" (computer scientists would call them Bags), their
| arithmetic (where "adding" is union, and "multiplying" is
| pairwise/cartesian product of the elements), and how this
| generalises: from an algebra containing only empty bags
| (trivial, but self-consistent; behaves like zero), to bags of
| zeros (behaves like natural number arithmetic), to bags of
| natural numbers (behaves like polynomial arithmetic), to bags
| of polynomials (behaves like polynomials in arbitrarily-many
| variables) https://www.youtube.com/watch?v=4xoF2SRp194
| jdkee wrote:
| "The reason I find them fascinating is that Wildberger
| doesn't agree with some of the conventional approaches, in
| particular with the use of infinity and taking limits."
|
| So no transfinite ordinal analysis or large cardinals? Hard
| to take him seriously.
| chriswarbo wrote:
| More than that: no Real numbers, no pi, no square root of
| 2, no sine/cosine, etc.
|
| It's similar to 'reverse mathematics' (trying to find the
| minimum set of assumptions required to prove a known
| result)
| mindcrime wrote:
| _Do you know a good path or book that 's suitable for that?_
|
| I've been using Professor Leonard's Youtube video series[1]
| mostly, along with some of those "workbook" type books by
| Chris McMullen, and a variety of books with titles like "1001
| solved problems in $SUBJECT", "The Humongous Book of $SUBJECT
| problems", and the like. The nice thing about Professor
| Leonard is that he has videos on everything starting from
| pre-algebra, middle-school math, up through Differential
| Equations. Note that his diff-eq class isn't _quite_ complete
| but he just announced he 's about to start recording new
| videos to finish that, and he's also going to be starting a
| Linear Algebra sequence. And he's a great lecturer who does a
| really good job of explaining things and making them
| understandable.
|
| I also use Khan Academy sometimes, and stuff on Youtube from
| The Math Sorcerer[2]. Oh, and of course there is
| 3blue1brown[3], whose videos are also useful. And for Linear
| Algebra I've been using Gilbert Strang's OCW videos[4] on
| Youtube.
|
| FWIW, I've evolved the way I study math, and what I do now
| works for me, even though it's 100% _not_ the way you 'd
| ordinarily see suggested. That is, I watch math videos fairly
| passively and don't work problems at the same time and treat
| it like being in a class per-se. I _used_ to do the thing of
| treating it like a class, pausing the video to work examples,
| and what-not, and that does work. But it 's very slow and
| tedious.
|
| Now, I just watch the videos, acknowledging that I won't
| absorb everything and that I also need to work problems for
| long-term retention. So now what I do is watch passively to a
| certain point (which I determine fairly subjectively) _then_
| I stop with the videos for a while, pick up a textbook or one
| of those "workbook" type books I mentioned earlier, and work
| problems for a while. Then I review the parts that I find
| myself struggling with. I'm also just now starting to add
| "creating Anki cards" as something I do during that second
| pass.
|
| Once I start getting a decent Anki deck built up, I'll be
| reviewing that regularly as well to help build retention. I
| only create cards for things that seem amenable to rote
| memorization, and TBH, I'm still working on figuring out what
| things are best to include, and how to structure those cards.
| What I don't intend to do is include specific problems where
| all I'd be doing is memorizing the answer to a problem. So
| far it's just formulas and things are are very obvious
| candidates to be memorized, and "algorithm" things like the
| "chain rule" from calculus, and similar.
|
| [1]: https://www.youtube.com/@ProfessorLeonard
|
| [2]: https://www.youtube.com/@TheMathSorcerer
|
| [3]: https://www.youtube.com/c/3blue1brown
|
| [4]: https://www.youtube.com/playlist?list=PLE7DDD91010BC51F8
| tomca32 wrote:
| I think Khan Academy is pretty much made for this.
| hosh wrote:
| I restarted with Kalid Azad's _Math, Better Explained_. It
| approached math first by intuition, and then worked out to
| refine it with proofs.
| rcarr wrote:
| I've been doing a similar thing with Brilliant and really
| enjoying it. It feels like every course is orientated around
| teaching maths from a problem solving perspective so you
| actually get why you're learning stuff rather than teachers
| just trying to brute force things into your head which
| unfortunately seems to be the default at schools nowadays.
| purpleblue wrote:
| What do you mean by basics? How far back do you want to go?
| Algebra? Art of Problem Solving is what my kids use, it's
| pretty good and thorough.
| Ishmaeli wrote:
| Susan Fowler Rigetti posted a self-study curriculum last
| March: https://www.susanrigetti.com/math
|
| I don't know how good it is, but her earlier entries on
| Physics and Philosophy were well-received.
|
| HN thread: https://news.ycombinator.com/item?id=30591177&p=2
| markb139 wrote:
| I had a similar thought back in 2014. I had only studied the
| maths required for various engineering courses I'd taken.
|
| So, I decided I wanted to study maths for the maths. I was in
| the fortunate position of being able to self fund myself
| through the Open University (uk based) Maths and Statistics
| BSc. One module at a time I'm now on my last module. There
| many things I'd studied before (calculus, sequences) and many
| new to me (group theory, graph theory)
| orm wrote:
| 3Blue1Brown videos seem like a good resource to use along any
| book. My experience as a math major (in the distant past) is
| that the kind of visualization the author shows you is also
| something you want to imitate in your head when you are
| learning new concepts. I find things I learned in this level
| tended to stick in my head 10+ years later, other stuff less.
| orm wrote:
| I should mention focusing on doing a few interesting
| problems, rather than many not so interesting ones, is also
| one way to help yourself understand more deeply.
| bee_rider wrote:
| Lots of easy problems is a good way to build up muscle
| memory, though. IMO the brute-force method of, say, Saxon
| Math really makes sense for things like basic elementary
| school algebra and probably intro calculus, where the
| student is sort of learning the math equivalent of how to
| walk. Not sure where the switch over ought to be, though.
| TjZkxkxeky wrote:
| I like Saxon math for kids. It implements spaced
| repetition with their exercises, so the kids actually
| retain what was taught.
| skyde wrote:
| we tried "Saxon math", Singapore math dimensions, and
| Beast Academy.
|
| And my impression was that Saxon Math was the worst. What
| I mean by worst is that it just make you practice an
| algorithm by doing lot of repetition but doesn't force
| you to have a deep understanding or problem solving
| skill.
| bee_rider wrote:
| Saxon math worked out for me, although we didn't shop
| around as far as I remember, so maybe Singapore would
| have worked fine as well.
|
| My experience is that I didn't really feel like I was
| memorizing an algorithm. Because the problem set includes
| assignments from all of the old sets, it is hard to
| memorize all of the algorithms. So you instead memorize
| the different moves that are allowed and have a general
| idea of what types of moves might be useful.
|
| I dunno. I went on to do engineery stuff as an undergrad
| rather than pure math stuff, it seems like a good match
| because engineering problems are also often in the "no
| need to be super clever, just don't mess up" vein, so it
| could be just a lucky match. This is what I mean by
| muscle memory -- I'll use the famous names theorems when
| necessary but sometimes you just need to bash the math
| until the thing you want is on that side of the equal
| sign and the other stuff is on the other side.
|
| I think anything that results in
|
| 1) actually reading some textbook
|
| 2) actually working through problems for a couple hours a
| week
|
| will compare well to the typical US math education pretty
| well anyway.
| nopeopenope wrote:
| Shankar Basic Training in Mathematics
| kdavis wrote:
| * _4-Manifolds and Kirby Calculus_ by Andras I. Stipsicz and
| Robert E. Gompf
|
| * _Differential Manifolds_ by Antoni A. Kosinski
|
| * _Introduction to Smooth Manifolds_ by John M. Lee
| agentultra wrote:
| _How To Solve It_ by G. Polya
| _https://press.princeton.edu/books/paperback/9780691164076/ho..._
|
| _A Logical Approach to Discrete Math_ by David Greis and Fred
| Schneider,
| https://link.springer.com/book/10.1007/978-1-4757-3837-7
|
| I'm self-taught so for me it was learning how to write proofs
| that gave me a big boost in being able to branch out into
| different area of interest and not give up. :)
| super256 wrote:
| For the German speakers:
|
| Something that really helped me with my mathematic modules at
| university: Lothar Papula's "Mathematik fur Ingenieure und
| Naturwissenschaftler" [1]. If you get the stuff in this book
| right, you're set for life.
|
| [1] https://www.amazon.com/Mathematik-f%C3%BCr-Ingenieure-
| Naturw...
| amai wrote:
| I recommend Lang: Mathematische Methoden der Physik
|
| https://link.springer.com/book/10.1007/978-3-662-49313-7
| okaleniuk wrote:
| Also, Geometry for Programmers but only because I wrote it. I had
| to update my skills significantly while gathering material and
| doing all the experiments. Not sure if reading the book would
| have the same effect :-)
| aught wrote:
| A decade of the berkeley math circle and concrete mathematics
| epistemer wrote:
| [dead]
| mythhouse wrote:
| spivak calculus
| skyde wrote:
| On a similar note: Does anyone have book they would recommend to
| teach Algebra or pre-algebra to young kid?
| rahimnathwani wrote:
| Not a book, but I recently had my son look at the pre-algebra
| track brilliant.org, and it looks nice. I'm not sure whether
| there's a paywall.
| samuel2 wrote:
| If you are into numerical optimization, a nice source of
| intersting problems and examples (that e.g. contradict the
| intuition) can be found in
|
| Mathematical Tapas: Volume 1 and Vol. 2.
| troupe wrote:
| From Mathematics to Generic Programming - Stepanov & Rose
|
| Godel, Escher, Bach: an Eternal Golden Braid - Hofstadter
|
| Euclid's Elements
| michaericalribo wrote:
| In my experience, the best way to get better at math is to do a
| lot of it. Find some book that's "good enough" for some topic
| you're interested, and work many many problems from the book.
| You'll learn about the topic, but more importantly you'll learn
| problem solving skills. I recommend working the problem until
| you're sure the answer is right -- in grad school problem sets
| didn't have answers you could check, and full understanding was
| necessary to get the problem sets correct.
|
| For me, a watershed book was Introduction to Analysis by
| Rosenlicht [1]. Proof-based, very "mathy", small and compact (so
| to speak) but with a massive scope. A great introduction to a
| really important topic, and it'll put your brain through its
| paces.
|
| Again, I recommend working nearly every problem.
|
| [1] https://www.amazon.com/Introduction-Analysis-Dover-Books-
| Mat...
| agumonkey wrote:
| If the HN crowd is up for it, I'd love similar book threads for
| physics, chemistry and biology.
| pncnmnp wrote:
| During my undergraduate studies, I loved "Discrete Mathematics
| and Applications" by Kenneth Rosen. I really enjoyed reading
| through the various examples and biographies of famous
| mathematicians included in each chapter.
|
| For those looking to delve into discrete mathematics, I highly
| recommend the lecture notes from L. Lovasz and K. Vesztergombi
| (Yale University, Spring 1999) and from Eric Lehman, Tom
| Leighton, and Albert Meyer (MIT, 2010).
| jytug123 wrote:
| On a similar subject I recall Concrete Mathematics by Donald
| Knuth being my favourite book from school.
| stormdennis wrote:
| K A Stroud Engineering Mathematics is probably the book that
| helped me most. 31 chapters each composed of about 60 problems.
| The problems are progressive and contain explanations of the new
| concepts that they contain. All the answers are at the back.
| kingkongjaffa wrote:
| I used this book and the sister book "advanced engineering
| mathematics" for my bachelors and masters degree in mechanical
| engineering.
|
| It is probably the best pair of books ever written for what I
| like to call "plug and chug" maths. Strouds books cover the
| whole of engineering mathematics.
|
| I turned to them for calculus in first year, and for Fourier
| and Laplace in my final year and masters.
|
| But it will not teach you how to solve and develop proofs.
| Kaizeras wrote:
| Mathematik fur Ingenieure und Wissenschaftler I, II and III from
| Lothar Papula (in German). The solutions are detailed, making it
| perfect for self-studying.
|
| Book of Proof by Richard Hammack. A great introduction to proofs
| in mathematics. The book is available free online [0], but also I
| bought the physical version because I really enjoyed it.
|
| [0]https://jdhsmith.math.iastate.edu/class/BookOfProof.pdf
| oogway8020 wrote:
| I just started reading Book of Proof by Richard Hammack and I
| agree it's an amazing book
| mindcrime wrote:
| Maybe not "made me better at math" per-se, but definitely "made
| me more enthusiastic about math":
|
| _The Universe Speaks in Numbers_ [1] by Graham Farmelo
|
| I found this very motivating and insightful, in terms of
| developing even more of an appreciation for how much math
| underpins other branches of science. Not that that is a novel
| insight by any means... but the details of the incidents where
| breakthroughs in mathematics allowed further advances in physics,
| etc. and looking at the "back and forth" between the domains,
| that was wildly interesting to me. Reading this book definitely
| helped motivate me to get serious about committing more time /
| focus to studying mathematics.
|
| I also enjoyed the "counterpoint" book by Sabine Hosenfelder,
| _Lost in Math_ [2]. I think these two books complement each other
| nicely.
|
| Then the handful of additional (no pun intended) books that jump
| to mind would be:
|
| - _How Mathematicians Think_ by William Byers[3]
|
| - _How to Think Like a Mathematician_ by Kevin Houston[4]
|
| - _Discrete Mathematics with Applications_ [5] by Susanna Epp
|
| - _How Not To Be Wrong_ [6] by Jordan Ellenberg
|
| - _Introduction to Mathematical Thinking_ [7] by Keith Devlin
|
| - _How to Measure Anything_ [8] by Douglas Hubbard
|
| [1]: https://www.amazon.com/Universe-Speaks-Numbers-Reveals-
| Natur...
|
| [2]: https://www.amazon.com/Lost-Math-Beauty-Physics-
| Astray/dp/15...
|
| [3]: https://www.amazon.com/How-Mathematicians-Think-
| Contradictio...
|
| [4]: https://www.amazon.com/How-Think-Like-Mathematician-
| Undergra...
|
| [5]: https://www.amazon.com/Susanna-S-Epp-Mathematics-
| Application...
|
| [6]: https://www.amazon.com/How-Not-Be-Wrong-
| Mathematical/dp/0143...
|
| [7]: https://www.amazon.com/Introduction-Mathematical-Thinking-
| Ke...
|
| [8]: https://www.amazon.com/How-Measure-Anything-Intangibles-
| Busi...
| lsandov1 wrote:
| * 'The Joy of X' by Steven Strogatz. I really like this book, on
| my second read now.
| annowiki wrote:
| A Programmer's Introduction to Mathematics https://pimbook.org/
|
| It introduces math from a mathematician's point of view (complete
| with proofs, etc.) rather than rote memorization and exercises,
| but it does so from the perspective of a programmer.
| dfan wrote:
| This is different from the other answers, but it does answer your
| question: When I was a kid I had tons of math and logic puzzle
| books. Two I remember specifically are "Aha! Insight" and "Aha!
| Gotcha" by Martin Gardner. Decades later, when a math problem
| comes up in my work, I have an apparently unusual ability to cut
| to the heart of it ("by symmetry, we must have X" or "looking at
| this extreme case, we must have Y" or "this looks like a special
| case of Z" sort of things) instead of starting by soldiering
| through equations, and I credit a lot of that to all the puzzle-
| solving I did as a kid.
| wannabebarista wrote:
| I had a similar experience with Raymond Smullyan's books,
| particularly The Godelian Puzzle Book:
| https://www.raymondsmullyan.com/books/the-godelian-puzzle-
| bo.... Recreational math is quite underrated.
| macrolocal wrote:
| A Mathematical Mosaic is a little-known gem here.
| hintymad wrote:
| For what level and in which area? Books like _Methods of
| mathematical physics_ can be both too hard and irrelevant to your
| needs. For starters,
|
| To become a better problem solver with high-school level maths:
| - Polya's How to Solve It. - Books of your choice about
| math contests. - Concrete Maths. I understand that this
| book is taught in college, but it requires very little advanced
| maths, and its techniques are hugely useful for high school
| students too.
|
| To hone my intuitions. I learned it the hard way that college
| maths were different from high school math: in high school, my
| teachers painstakingly drilled intuitions into us with very
| targeted explanations and tons of well designed exercises. In
| college, we won't get such luxury. So, it's really up to us to
| understand mathematical concepts intuitively before diving into
| technical details. For that matter, the following books helped me
| a lot: - The visual series. Visual Complex
| Analysis and Visual Group Theory, for instance - Pinter's A
| book of Abstract Algebra - Strichartz's The Way of Analysis
| - Linear Algebra Through Geometry by Wermer. The book offers a
| comprehensive geometric interpretation to linear algebra
| concepts. It's especially helpful for me to understand quadratic
| forms.
|
| To understand Analysis better. This area is vast, so I'll skip
| recommendations of excellent text books: -
| _Counterexamples in Analysis_. Those counterexamples in Analysis
| play a huge role in helping me truly appreciate the intricacies
| of Analysis. Similarly, books like _Counterexamples in
| Probability and Real Analysis_ are of great help too. -
| The Way of Analysis by Robert S. Strichartz. This books is
| AMAZING for laymen like me. You'd want someone to *explain* how
| concepts emerge, and how intuitions evolve.
|
| To become good at maths by doing maths, so the following books
| used to help me a lot: - Problems and Proofs in
| Real Analysis - Putnam and Beyond. I still suck at maths,
| but those well designed problems in Putnam really taught me how
| to seek insights in higher maths. - Piotr's Problems in
| Mathematical Analysis. But really, any problem books that
| challenge you will do. I'd recommend you find problem sets from
| the website of university courses. They cover essential
| techniques, and will not be as overwhelming as the books.
| wannabebarista wrote:
| Measure and Category by John Oxtoby. This book studies duality
| results between different notions of "small" sets in measure
| theory and topology. It's the first (and to some extent the only)
| math book where things just clicked and I didn't feel like I was
| drowning in a sea of notation and ideas. Here are some more
| thoughts on it: https://bcmullins.github.io/Top-Books-2019.
| enriquto wrote:
| I'm so happy to hear that! I've always loved this little book
| (even if it's completely independent of the math needed for my
| work).
|
| In a similar spirit, but with a much more geometrical flavor,
| there is Evans-Gariepy.
| bluenose69 wrote:
| Jeffreys, Harold, and Bertha Swirles Jeffreys. Methods Of
| Mathematical Physics. Cambridge At The University Press, 1950.
| http://archive.org/details/methodsofmathema031187mbp.
| cccybernetic wrote:
| The best resource I've found is this random, somewhat obscure
| website (though I've learned that it has grown in popularity)
| called Paul's Online Notes. The professor has a real knack of
| pedagogy, and the problems are perfectly structured in terms of
| their difficulty. His explanations are clear and without jargon,
| and it goes from algebra to diff eq.
|
| A note: this isn't a resource for higher-level, proof based
| maths. It will give you a solid foundation and a pragmatic
| understanding to build upon. Very useful for STEM.
|
| Link: https://tutorial.math.lamar.edu
| downboots wrote:
| Seconded.
| agucova wrote:
| This page got me through my engineering calculus I, II and III,
| linear algebra and ODEs courses.
|
| I'm eternally grateful ;)
| ask_b123 wrote:
| I used this a lot while in school, very useful!
| siavosh wrote:
| Best "math book" for kindling interest in mathematical thinking
| in an elementary school kid?
| alimw wrote:
| There are some Lockhart books mentioned on this page.
| yodsanklai wrote:
| I wonder how good you can get at maths just by casually reading
| books. You need to work on problems for hours and hours to get a
| grasp on the theories. Programming is different in the sense that
| it's something people routinely do as a hobby because it's quite
| fun and addictive. But maths? maybe if you have already strong
| foundations you can pick up a new topic and develop your culture.
| But I doubt one can get these foundations without actually
| graduating in maths as it's an extremely strong commitment.
| mindcrime wrote:
| _I wonder how good you can get at maths just by casually
| reading books. You need to work on problems for hours and hours
| to get a grasp on the theories._
|
| Maybe I'm unique in this regard, but I always took it as sort
| of implied that "reading a math book" entails "reading the book
| and working (at least some of) the exercises".
|
| The tricky part is once you get to math where you can't
| trivially check your answer by "substituting back in" or "using
| a calculator" or whatever. Doing proofs, for example. Without a
| teacher, how do you know if your proof is correct? So far the
| only thing I've really found to do for that is to post on
| MathOverflow or one of the "learn math" related sub-reddits.
| I've often wondered if learning to use an automated theorem
| prover / proof assistant of some sort would be helpful, but
| that's such a huge undertaking in its own right...
| ag315 wrote:
| I don't think you can get good at doing calculations without
| practicing the calculations, but reading books that discuss the
| higher-level aspects of math and the philosophical
| underpinnings can help you look at it in a different way that
| may inspire more interest as well as an easier time grasping
| the difficult parts.
| posed wrote:
| I second this strongly, you can't get better at math just by
| reading books. You need to hone your problem solving skills,
| you need to fight with the problems, have the mindset of a
| warrior, a conqueror, only then you'll get the juice out of it
| and have a clear understanding of the subject. I'll suggest
| starting with Concrete Mathematics by Donald Knuth, it's a
| beautiful book that catches the essence of mathematics.
|
| Art of problem solving(https://www.amazon.in/Art-Problem-
| Solving-Basics/dp/09773045...) is also a great start,
| especially if you don't have much experience.
| rs999gti wrote:
| > You need to work on problems for hours and hours to get a
| grasp on the theories.
|
| Very true. For maths, it is drill to win.
| samuel2 wrote:
| Probably the most elegant math book I have ever seen is
| Probabilty theory a graduate course by Achim Klenke. A very nice
| exposition into the abstract, measure theoretic prob. thoery (but
| it assumes some prior knowledge).
| WillAdams wrote:
| A pair which I am most of the way through:
|
| _Make: Geometry: Learn by coding, 3D printing and building_
| https://www.goodreads.com/en/book/show/58059196
|
| and
|
| _Make: Calculus: Build models to learn, visualize, and explore_
| https://www.goodreads.com/book/show/61739368-make
|
| I'd really like to find a similar book on conic sections --- my
| next major project seems to need them, and when I tried to solve
| it using trigonometry alone, I wound up 7 or 8 levels deep in
| triangles and wasn't much more than half-way to where I needed to
| be.
| kxyvr wrote:
| 1. Principles of Mathematical Analysis by Walter Rudin (baby
| Rudin) - I'd studied real analysis in the past, but this book is
| direct and rigorous and provided a good framework to move forward
| into things like functional analysis in a way that I was not
| prepared for with other books.
|
| 2. Differential Equations and Dynamical Systems by Lawrence Perko
| - Solidified for me how dynamic systems behaved and were solved.
| Very much helped my understanding of control theory as well.
|
| 3. A Concise Introduction to the Theory of Integration by Daniel
| Stroock - Helped solidify concepts related to Lebesgue
| integration and a rigorous formulation of the divergence theorem
| in high dimensions.
|
| 4. Convex Functional Analysis by Kurdilla and Zabarankin - Filled
| in a lot of random holes missing in my functional analysis
| knowledge. Provides a rigorous formulation of when an
| optimization formulation contains an infimum and whether it can
| be attained. Prior to this point, I often conflated the two.
| hyperific wrote:
| Calculus Made Easy (1910) simply for the quote at the beginning:
| "What one fool can do, another can."
|
| I did horribly in math because I figured it was hard and just
| accepted I'd never be good at it. That quote somehow managed to
| dissolve my mental block.
| djrockstar1 wrote:
| Had to do a Calculus course in uni despite not having taken any
| calc or pre-calc in high school. "Precalculus Mathematics in a
| Nutshell" and "Calculus Made Easy" were complete lifesavers.
| adamsmith143 wrote:
| A lot of comments about textbooks that helped in specific topics
| but I don't think that answers the spirit of OP's question. Sure
| working through ANY Linear Algebra textbook is going to improve
| your Linear Algebra skills.
|
| In the spirit of OP's question:
|
| How to Solve it by G. Polya
|
| Solving Mathematical Problems by Terrence Tao
|
| Introduction to Mathematical Thinking by Keith Devlin
|
| Are all amazing, How to Solve it in particular is an all time
| classic.
| fghorow wrote:
| For my tastes, Strang's Linear Algebra book is a winner!
|
| I'm somewhat surprised the nobody has mentioned it yet...
| thanatos519 wrote:
| Mathematics, a Human Endeavor: A Book for Those Who Think They
| Don't Like the Subject by Harold R. Jacobs
| theusus wrote:
| > Discrete Mathematics with Applications by Susanna S. Epp
|
| Fantastic book for Discrete Mathematics with lucid explanation
| and good exercises. The other one would be concrete mathematics.
| adamddev1 wrote:
| I'm working through this right now and really enjoying it.
| LAC-Tech wrote:
| what a coincidence, so am I.
|
| Though I realise I've spent over a week on section 2.2 so
| "working through it" may be a bit of a generous term.
| mindcrime wrote:
| Epp's book is my favorite Discrete Math book by far. Her
| writing style is very clear and easy to follow. And as you say,
| there are good exercises. And if you buy an older edition, used
| copies can be had for a (relatively) reasonable price.
| codr7 wrote:
| If I could pick one, that would be How to Solve it by George
| Polya.
| downboots wrote:
| Algebra Baldor !
|
| College Algebra Heineman
|
| Discrete Math Rosen !
|
| Linear algebra D lay
|
| Calculus Stewart
|
| Nonlinear Dynamics Strogatz +
|
| Combinatorics Mazur +
|
| Statistics *
|
| ESLR +
| racl101 wrote:
| I had a print of Euclid's Elements as a kid.
|
| My mom was really into mathematical proofs and I being a huge
| loser kid with no friends naturally took to this book as well.
| krmboya wrote:
| I have not yet become significantly better, and not a math book,
| but I recently read A mathematicians Lament by Paul Lockhart and
| it resonated so much with me that I plan to take another stab at
| math different from how it is taught in school.
|
| Waiting to get my hands on his book 'Measurement' and approach it
| more like art.
|
| If what he says is true, perhaps many who would have turned out
| great at math are locked out by how it's taught in school.
|
| For now, I have a test subject of one :)
| leephillips wrote:
| What he says is true. If you teach math, you should read it. If
| you're like all the other people I know who teach math, you
| will ignore it.
| binarymax wrote:
| "Mathematical Notation: A Guide for Engineers and Scientists"[0]
| really changed my abilities with being able to read papers and
| decipher what was going on. I had university math experience but
| it was a long time ago. When I started reading papers for
| algorithms later in my career I couldn't get past the notation.
| Once the symbols are explained, as a programmer, I was able to
| grok so much more. This should be on everyone's shelf.
|
| [0] https://a.co/d/gQmDIo7
| slicktux wrote:
| Plus one for this! I bought two copies of the referenced
| book...and for the exact same reasons; I'm a programmer and
| being able to explain my algorithms using mathematical notation
| helps validate a program as well as troubleshoot a program...
| An oldie but goodie is "Mathematics for the million"
| fouronnes3 wrote:
| As a programmer I really wish math notation was more rigorous:
| less ambiguity, more explicit typing, no implicit variables,
| etc. So much of it would never pass code review. We programmers
| figured out that code should be optimized for readability, not
| writtability ; I wish mathematicians did too.
| willhslade wrote:
| To pick a nit, Iverson did this with APL and I still think
| it's brilliant. Executable math. It's time will come, I hope.
| yig wrote:
| Our project IHeartLA is a language with syntax designed to
| closely mimic conventionally-written linear algebra, while
| still ensuring an unambiguous, compilable interpretation:
| https://iheartla.github.io/
| tdba wrote:
| This is amazing, thank you! The symbol replacement seems
| really slick
| https://github.com/iheartla/iheartla/wiki/Keywords-And-
| Speci...
| larve wrote:
| Mathematics only clicked and became fun for me when I started
| using Wolfram Mathematica, because I could fairly easily mess
| around with the formulas I saw in books until I understood
| the types and arguments and what is an index vs a reference
| to some unnamed convention of the field.
| breck wrote:
| This is a project of mine that I take attempts at here and
| there.
|
| Another great book on this topic is "History of Mathematical
| Notations" https://www.amazon.com/History-Mathematical-
| Notations-Dover-...
| [deleted]
| time_to_smile wrote:
| I find one of the biggest mistakes programmers have about
| mathematical notation is that it's somehow just a terse,
| badly implemented programming language. But this is a very
| poor understanding of what mathematical notation is doing.
|
| I think this error in thinking comes from the fact that Sigma
| notation can often be trivially implemented as a for loop.
|
| Programming languages are designed to describe a specific
| _computation_ , whereas mathematical notation is typically
| trying to describe an idea (one that might not even have a
| implementation!) Notation only sometimes and coincidentally
| describes _computation_ as well.
|
| The ambiguity, implied variables etc are an essential part of
| mathematical notation in the same way it is in common spoken
| language. Mathematical notation exists to help abstract and
| work out very hairy ideas, and often that ambiguity is
| necessary to show connections.
|
| > code should be optimized for readability, not writtability
|
| Mathematical notation _is_ readable if you 're literate in
| it. It takes lots of practice to become fluent in it, but
| once you become more familiar it's much easier to read than
| text (which is why it's used in the first place).
| Mathematical notation is an extension of mathematical
| writing, not computational implementation.
|
| Reading mathematical notation is _much_ closer to reading
| poetry than reading code.
| User23 wrote:
| > I find one of the biggest mistakes programmers have about
| mathematical notation is that it's somehow just a terse,
| badly implemented programming language. But this is a very
| poor understanding of what mathematical notation is doing.
|
| No, we think that because proofs and programs are
| isomorphic[1]. It's not a mistake: traditional mathematical
| notation provably is a terse badly implemented programming
| language. Actually it's worse than that, because oftentimes
| it doesn't even parse. Now I'm not going to say I can't on
| some level see the appeal. After all I think Perl is a lot
| of fun to code in.
|
| Naturally, its adherents are practiced at making a virtue
| out of its defects. Who wants to admit they dedicated
| considerable brainpower to doing something in a
| fundamentally suboptimal way? That doesn't really matter
| though. As Mathematica and other tooling shows, the
| formalists have already won and now it's just a matter of
| mopping up the stragglers, or waiting for them to age out.
| This isn't terribly surprising to those who know the basics
| of the history of mathematics. It took something on the
| order of two centuries before Recorde's innovation of the
| equal sign was generally accepted.
|
| [1] https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_corr
| espon...
| rowanG077 wrote:
| Couldn't agree more. Mathematician are masters at whipping up
| random notations and then not adhering to it rigorously.
| Higher level math would be an order of magnitude easier with
| machine checked syntax.
| dmitrybrant wrote:
| Totally agree! I remember raising my eyebrow at ambiguities
| in math notation as early as high school, before I could
| articulate them as such. One specific example is the
| convention of cos^-1(x) referring to the inverse cosine of x,
| instead of the multiplicative inverse of cos(x). Similarly,
| the convention of cos^2(x) referring to the square of cos(x)
| instead of a nested cos(cos(x)). It's madness, and totally
| avoidable.
| User23 wrote:
| It can be made to be. Dijkstra came up with a nice and
| rigorous notation he used for his own proofs[1]. That page
| also includes some slightly spicy takes on why things are as
| they are. I agree that this is an area where the broader
| mathematical field has much to learn from computing science.
| The unforgiving nature of computing automata really drove
| that innovation. Meanwhile one can afford to be sloppy when
| one is trying to convince some other mathematician with a sky
| high IQ.
|
| [1] https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13x
| x/E...
| blevin wrote:
| I did not expect that to read so similarly to a good doc on
| coding conventions as it did.
| TimTheTinker wrote:
| Wow, I wish I had known about this book (and had a license to
| Mathematica) when I was in college. I always got hung up on the
| notation and my inability to visualize the concept.
| MrMan wrote:
| take math classes, start a level lower than you think you need
| to, do lots of practice problems
| moomin wrote:
| Baby Rudin opened my eyes to what you can do with maths.
| lvl102 wrote:
| The answer is always Elementary Number Theory by David Burton. I
| read this book since 7th grade.
| [deleted]
| ABeeSea wrote:
| The classic, How to Solve It by Polya.
|
| A lot of the advice seems obvious in retrospect but being
| systematic about a problem solving framework is enormously
| helpful.
| jsenn wrote:
| * The Art of Probability by Hamming. An opinionated, slightly
| quirky text on probability. Unlike the text used in my university
| course its explanations were clear and rigourous without being
| pedantic. The exercises were both interesting and enlightening.
| The only book in this list that taught skills I've actually used
| in the real world.
|
| * Calculus by Spivak. This was used in my intro calculus course
| in university. It's very much a bottom-up, first-principles
| construction of calculus. Very proof-based, so you have to be
| into that. Tons of exercises, including some that sneakily
| introduce pretty advanced concepts not explicitly covered in the
| main text. This book, along with the course, rearranged by brain.
| Not sure how useful it would be for self-study though.
|
| * Measurement by Lockhart. I haven't read the whole thing, but
| have enjoyed working through some of the exercises. A good book
| for really grokking geometric proofs and understanding
| "mathematical beauty", rather than just cranking through
| algebraic proofs step by step.
|
| * Naive Set Theory by Halmos. Somewhat spare, but a nice, concise
| introduction to axiomatic set theory. Brings you from nothing up
| to the Continuum Hypothesis. I read this somewhere around my
| first year in university and it was another brain-rearranger.
| bitforger wrote:
| Arithmetic by Lockhart is also a gem.
| nextos wrote:
| These are good recommendations, but I think beginners tend to
| burn out due to the lack of a structured program and/or
| exercise solutions. The simplest structured program I can think
| of that satisfies both is:
|
| * _Basic Mathematics_ by Lang. Covers basic algebra and
| geometry at high school level.
|
| Then one of these two, depending on your interests, or both:
|
| * _Vector calculus, linear algebra and differential forms_ by
| Hubbard and Hubbard. Takes you through linear algebra, single-
| variable calculus and multiple variable calculus. Analysis is
| discussed in an appendix. All proofs have a constructive bias,
| so it 's very algorithmic and natural for a CS-minded student.
|
| * _Program = Proof_ by Mimram. Discusses logic and computation,
| and takes you from the basics to depedent type theory and
| beyond. Uses OCaml and Agda. Freely available at
| https://www.lix.polytechnique.fr/Labo/Samuel.Mimram/teaching...
| readingnews wrote:
| Books do not make you better at math. Working math problems makes
| you better at math. Go ahead, down vote all you want.
|
| Reading about running does not make you a better runner. You can
| watch 1000 marathons, sprinters, Olympians. You may get _ideas_
| for running, but it will never make you a better runner. To be a
| better runner you have to do it. To be a better
| programmer/mathematician/physicist/whatever, you need to go work
| at it.
|
| I suppose I am just taking action against how the question is
| written, but I see a lot of people seemingly hoping that "if they
| just found the correct book, tutorial, or video, they would be
| better". A lof of those people are my students. When I ask how
| many problems they have worked, I typically always get the same
| response. Zero, or the bare minimum.
| algon33 wrote:
| Problem sets are useful for improving your mathematical
| ability, as is looking at an expert's attempt at a problem
| after your attempt. Both are a core part of reading books well.
| So when someone asks for books to improve their mathematical
| ability, a likely interpretation is just "they want books with
| good problem sets, clear presentation and elegance". Another
| interpretation is "what's a book which, when after I intrepret
| all the individual sentences into a vague impression, will make
| me good at maths?"
|
| Your answer is somewhat helpful in the latter sort of world
| where OP didn't know how to read a textbook (which is
| unfortunately common) but not in the former sort. You seem to
| be venting though, which is understandable. But venting with a
| side of helpful content would be even better.
|
| For instance, advising OP on how to to read a maths book
| (generate content yourself, check dependancies, connect things
| to what you know etc.) or suggest books which contain advise
| like this alongside their main content (I think Tao's analysis
| texts might do this?)
| mcculley wrote:
| If I were a student, eager to solve math problems and become
| better at math, no book would be better than any other book at
| presenting and guiding me to problems that would improve my
| understanding?
| enriquto wrote:
| > Books do not make you better at math. Working math problems
| makes you better at math.
|
| But math books are often full of exercices that you are
| supposed to do! Actually reading a math book means that you try
| to anticipate the proofs before reading them, and you work out
| all the details and do all the exercices. Reading a math book
| and working math problems are essentially the same thing.
| fumeux_fume wrote:
| So many great responses recommending all kinds of books and
| then there's... this. Lol. Such a jaded/cynical teacher thing
| to say, but sure, nothing is a substitute for what you get out
| of putting the effort into doing. For me, the biggest hurdle to
| succeeding at college-level math was a lack of motivation due
| to the meaninglessness of most of the content.
| philip-b wrote:
| Reading math textbooks typically involves solving a lot of
| problems, so you saying it like it's books OR problem solving
| doesn't make sense.
| amai wrote:
| Polya: How to solve it
|
| https://en.wikipedia.org/wiki/How_to_Solve_It
| sfpotter wrote:
| Honestly, unless you're very gifted, a book on its own is not
| going to be enough to really develop your skills. You need to
| work interactively with a teacher. Getting a degree in math is a
| good start, but even then it will be limited if you don't work
| with other people, go to office hours, form relationships with
| professors---embed yourself in the culture, so to speak. I think
| getting engaged with an online community like MathOverflow or
| similar could be a substitute for this.
|
| IMO, programming is "easier" to learn on your own for a few major
| reasons:
|
| 1) The sorts of things most people are interested in building
| just aren't unforgiving intellectually in the same way that math
| is.
|
| 2) You have a compiler to check if you're right, and your code
| will often still work even if it's "wrong" (not as efficient as
| it could be, has unwanted side effects, etc.). In some sense the
| compiler is a bit like a teacher this way.
|
| 3) With programming, you can upload and make it available for
| free, and whether it's legit or not is largely disconnected from
| your pedigree or how "correct" it is. This makes programming far
| more accessible. This makes sense considering that programming is
| primarily a practical tool. On the other hand, mathematics is
| primarily a field of scientific inquiry and is judged by
| different standards. If you learn a bit of math, try to write a
| paper, submit it to the arXiv... well, people will probably think
| you're a crank.
|
| On the other hand, if you're just interested in math for the love
| of the game... you can certainly pick up a book and read it,
| maybe work some problems, but I think at this point it's quite
| easy to fool yourself into thinking you understand more than you
| actually do. I guess there's no real harm in being a charlatan,
| but probably the average person is interested in having some kind
| of real relationship with mathematics that they can be confident
| has a firm foundation. I'm very skeptical most people can truly
| pull this off by just reading books and not actually going to
| school.
|
| ---
|
| As an aside, I think the fetishization of math in programming
| communities is very interesting...
| mindcrime wrote:
| _Honestly, unless you 're very gifted, a book on its own is not
| going to be enough to really develop your skills. You need to
| work interactively with a teacher._
|
| Wouldn't that depend on what level of skill we're talking
| about? I believe that most people who need to learn just, say,
| high-school algebra/geometry/trigonometry/etc. can probably do
| so with "just books" if they are motivated.
|
| Heck, I'd even venture that most anybody who is motivated
| enough can learn at least through Calc III on their own using
| online resources. I'm going through Calc III right now
| actually, using a combination of books and online videos, and
| there hasn't been anything that has struck me as particularly
| challenging so far. This after only making it to Calc I before
| I dropped out of school "back in the day".
|
| _Getting a degree in math is a good start, but even then it
| will be limited if you don 't work with other people, go to
| office hours, form relationships with professors---embed
| yourself in the culture, so to speak._
|
| I think these things would be critical if one wants to _become
| a mathematician_. But for somebody who just wants to learn more
| math for the purpose of reading (non math) research papers
| (maybe in machine learning to pick an obvious, if possibly
| trite, example), solving problems in everyday life or at work,
| or possibly even for doing research in another (non math) field
| and then writing up their research, I would think of all of
| those things as being "nice, sure, if you have access, time,
| money etc. But not even close to absolutely essential".
|
| Unfortunately the OP didn't really say what their goal is, so
| it's hard to say what advice makes the most sense for them.
|
| Also unfortunate is that for probably most people who are
| working career professionals, there likely isn't enough free
| time available to go back and do an actual math degree in
| school. So learning on ones own from books, videos, etc. is
| probably the only viable choice.
| sfpotter wrote:
| I basically agree with what you're saying.
|
| My response was based on the reader saying something about
| "mathematical thinking" in their post. In my experience,
| programmers talking about "mathematical thinking" usually
| have math insecurity and are referring to more advanced
| topics, thinking they need to consult tomes of deep
| mathematical wisdom to correct this deficiency (wrong). Could
| be totally off base, of course. But note some of the other
| replies here, suggesting _very_ sophisticated books. Are they
| appropriate for a random HN poster who is soliciting random
| book suggestions to improve their "mathematical
| thinking"...? Seems unlikely to me...
|
| One caveat, though:
|
| _... Also unfortunate is that for probably most people who
| are working career professionals, there likely isn 't enough
| free time available to go back and do an actual math degree
| in school. So learning on ones own from books, videos, etc.
| is probably the only viable choice._
|
| I'm not so sure about this. If, like you say, someone is in a
| job where they need to read some papers with some math in
| them, they should be organizationally near someone who can
| help them out. If possible, I think a better strategy (better
| even than getting a degree) would be to find these people and
| develop a relationship with them to the point where you can
| ask them questions. They should then be able to explain any
| unclear notation, unfamiliar (simple) concepts, etc. Possibly
| some of that person's advice might come in the form of "watch
| a Khan Academy video on Topic X". But this will be a far more
| productive use of time than self-directed learning in this
| case.
|
| On the other hand, if they _aren 't_ near anyone with those
| skills but they're reading these papers... something is
| probably amiss. For example, if they're reading a machine
| learning paper which requires a significant knowledge of
| "engineering math" (Calc 3, linear algebra, etc.), and there
| is _no one_ with that knowledge nearby... having them read
| that paper is probably a waste of time from an organizational
| perspective.
|
| There's also the question of why they're reading that paper
| when they don't have those basic mathematical skills. Without
| those skills, it is unlikely they will be able to do very
| much that is useful with it. If they want to implement the
| algorithm because they think it will be suitable for some
| task, I would argue that without those skills they are not in
| a good position to be able to accurately assess whether the
| algorithm will perform well. Part of what you learn in an
| advanced degree is how to read a paper---i.e., how to sniff
| out the bull shit, what to be wary of, etc.
| mindcrime wrote:
| _I 'm not so sure about this. If, like you say, someone is
| in a job where they need to read some papers with some math
| in them, they should be organizationally near someone who
| can help them out. If possible, I think a better strategy
| (better even than getting a degree) would be to find these
| people and develop a relationship with them to the point
| where you can ask them questions._
|
| Fair point.
|
| One could also form a math "study circle" of some sort if
| you are in an area with enough mathematically oriented
| folks to find people to participate. I did this briefly and
| it was a valuable thing. It kind of fell apart for
| different reasons, but I could see doing it again at some
| point.
| Py-o7 wrote:
| It's interesting. My first instinct was to disagree with this
| post, but on reflection I think I mostly agree with it. A
| couple useful mental models are (i) deliberate practice and
| (ii) train-validation-test(/out of sample) sets from machine
| learning
|
| Your point (2) about compiler/interpreter in programming giving
| you rapid objective feedback is spot on and a vital component
| for deliberate practice that most people don't think on. You
| can kind of get this in math, in particular when you have some
| familiarity with the subject matter so the machinery isn't "too
| abstract" for you to sort through. (I.e. you should be able to
| confirm whether your proof/answer is accurate the vast majority
| of the time.) This is much trickier for first exposure to a
| subject though and the checking effort is on you, not the
| compiler.
|
| The biggest issue I've seen with people self studying or in
| small math groups is your final (non-aside) paragraph which is
| perhaps more a psychological problem than and aptitude problem.
| When things get tough there's an enormous temptation to delude
| yourself to think you understand something that you are
| clueless about. The typical, schoolroom, way of mitigating this
| is via a final exam and you can check your grade at the end of
| the class; this gets typically gets short circuited in self
| guided study. Exams, btw, are essentially validation data sets
| you compare your math knowledge/model against. (We can call
| them 'test' sets if you prefer). The most important step really
| is repeatedly seeing how your knowledge works out of sample
| i.e. on 'new' stuff that comes out of the wood works and
| math.stackexchange is a perfect place for this when dealing
| with undergrad to mid-grad level problems. I do this all the
| time to get a sense of my understanding of a new subject I've
| recently acquired. But most people refuse this final step.
| People will tell me its 'too hard' and 'takes too much time'
| (meanwhile they start a new math book) but I strongly suspect
| it's in large part due to cognitive dissonance. (Another kind
| of out of sample test comes up when working on a subject matter
| that uses something you just "learned" as a pre-req, though
| there's a recursive element here and at some point they
| basically need to interact with 3rd parties.)
|
| I suppose my relatively minor quibble is how much effectiveness
| depends on being "very gifted" [in some sort of math specific
| sense] vs understanding the basics of self-learning and being
| psychologically aware (astute?) enough to not go into denial.
| Insert quote from Feynman or whomever about how easy it is to
| fool yourself.
| SoftTalker wrote:
| Generally agree. There are books, and some are better than
| others, but unless you have both the passion and the aptitude,
| there is no book that will magically make everything
| understandable. If you struggle with math, it's probably you
| and not the book you are using.
|
| Edit: just to add, I struggle with math, lest there is any
| misinterpretation of my perspective. I went through phases
| where I thought I just needed to find the right books or the
| right teachers who could explain it in a way that meshed with
| my "learning style," but ultimately concluded I just don't have
| a very strong innate ability in the subject.
| makr17 wrote:
| Freshman year of undergraduate math required How
| to Solve it -- Polya The Art of Problem Posing -- Brown and
| Walter
|
| I'm not sure it made me any _better_ at math, but I did always
| enjoy How to Lie With Statistics -- Huff
| aquafox wrote:
| I can recommend Teschls book on ODEs, and it's completely free:
| https://www.mat.univie.ac.at/~gerald/ftp/book-ode/index.html
|
| And if you like something very applied: Modern Statistics for
| Modern Biology https://www.huber.embl.de/msmb/
| angvp wrote:
| Baldor's series of books (Spanish)
| brainzap wrote:
| something with gamedev probably
| dqpb wrote:
| Not a book, but I loved this:
|
| * The Natural Number Game
| https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_g...
| zrkrlc wrote:
| Vector Calculus, Linear Algebra, and Differential Forms: A
| Unified Approach
|
| It's a rigorous but chatty textbook in the style of Spivak but
| written by someone who is sensitive to applied maths. I would not
| have survived my astrophysics classes without it.
|
| (Not to mention it's where I first saw this really intuitive way
| of doing matrix multiplication:
| https://blogs.ams.org/mathgradblog/2015/10/19/matrix-multipl...)
| ericmay wrote:
| I'm going to cheat and combine a couple of books into "one book".
| The Manhattan Prep GMAT test prep math books were really good for
| me for everyday life. I learned a lot of shortcuts and quick
| heuristics to use and got better at estimating after going
| through those books. It doesn't help me "get better" in an
| academic sense, but those books pay dividends every day for me.
| lamontcg wrote:
| Div Grad Curl and All That
|
| https://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161
| actinium226 wrote:
| Elementary Differential Equations by Boyce and DiPrima
|
| I'm not sure I'd say that it made me significantly better at
| math, but I keep coming back to it time and again, and usually
| via very different paths.
| Llamamoe wrote:
| A better question: What books made you significantly better at
| math that are also genuinely FUN to learn from?
|
| I find that my biggest barrier to learning math has always been
| how unengaging, excessively contrived, and unfun the learning
| material has been.
| ABeeSea wrote:
| David Bressoud's 4 books on calculus/analysis are the most
| engaging math books I've ever read. He uses the history of
| mathematics to drive the narrative and it's an enlightening
| approach. However, this means certain theorems often taught in
| undergrad analysis are delayed until his "graduate" book and
| his graduate book would not be sufficient for passing many
| universities analysis quals. But the context and history he
| gives is fantastic.
| wannabebarista wrote:
| I always recommend Bressoud's measure theory/integration text
| as a supplement the first graduate course in analysis. His
| discussion of weird and pathological sets of reals such as
| fat Cantor sets are really helpful for building intuition.
| kache_ wrote:
| linear algebra done right - sheldon axler
| thehappypm wrote:
| The Art of Approximation gave me far more intuition than any
| class
| DrNosferatu wrote:
| The classic "Advanced Engineering Mathematics" by Erwin Kreyszig
| was absolutely 'good enough'. Maybe even more important - for me:
| it was one of the easiest books to follow and digest during my
| undergrad. See it as a solid base for other heavier/purer titles.
| mohamez wrote:
| Any good book about the history of mathematics that will teach
| you a natural historical development of concepts to reach more
| generalizations.
|
| History of mathematics will give you a very subtle entry into the
| minds of mathematicians and the motivation behind their theorems.
|
| This will surely make you more appreciative of subjects and
| concepts you are learning.
| mrazomor wrote:
| The Mathematical Experience --
| https://www.goodreads.com/book/show/1113522.The_Mathematical...
|
| It's touching many areas. For some it explains how they were
| developed or the controversy around them (e.g. the definition
| and use of infinity).
| mharig wrote:
| Foundations and Fundamental Concepts of Mathematics, by Howard
| Eves
| d0m wrote:
| Quantum Mechanics: The Theoretical Minimum --> Great book to
| learn about quantum mechanics and as a side effect math
| scruple wrote:
| Calculus Made Easy and Probability Through Problems. I'm not sure
| that I'd have gotten through either my university Calculus
| courses or Probability and Statistics without these two books. I
| used them as supplementary material to the course textbooks and
| homework. They both have a style that is approachable and helped
| me build an intuition for the material unlike anything else I
| found.
| tylerstorm wrote:
| I second this suggestion for Calculus Made Easy by Thompson.
| It's become a bit of a classic...was published in like 1915.
| Super unique approach to teaching calculus. It's an excellent
| supplement...lots of good insights. It may be particularly good
| for people who believe they're bad at math. His style may
| convince people otherwise.
|
| Also, Vibrations and Waves, by AP French. Granted, this is a
| physics book, but I appreciate his style so much. He makes use
| of a lot of geometric methods to solving problems. It definitly
| expanded my math horizons! His other books are good too.
| apohn wrote:
| >Probability Through Problems
|
| First time I'm hearing about this one, thanks for the
| recommendation. Unlike Calculus or even a typical one semester
| Statistics course, probability is one of those topics where you
| need to see a lot of problems to really grok anything. The only
| way is to see a lot of solved problems and think about why
| that's the right answer.
|
| Even highly recommended books (e.g. by Blitzstein) don't have
| enough solved problems, so it's nice to there's a problem
| focused book out there.
| steppi wrote:
| Some favorites below. Books 0-3 are accessible. The remaining
| books are more difficult but I'd highly recommend them to math
| students.
|
| 0. Jan Gullberg, _Mathematics, From the Birth of Numbers_. A
| highly accessible popular survey on different branches of higher
| mathematics. I read this over the Summer between high school and
| starting my undergraduate degree. It 's what made me want to
| study math. Previously I'd wanted to be a guitar player, but had
| to find a new ambition after an injury left me unable to play.
|
| 1. The high school mathematics series by Israel Gelfand.
| _Algebra_ , _Trigonometry_ , _The Method of Coordinates_ , and
| _Functions and Graphs_. I didn 't have much mathematics
| background in high school, but working through these really
| solidified my grasp on the basics.
|
| 2. George Polya. _How to Solve it_. A short book giving excellent
| high level advice on mathematical problem solving.
|
| 3. George E. Andrews, _Number Theory_. I worked through this
| freshman year contemporaneously with my first proof based class
| on simple logic and set theory. A very beautiful and accessible
| introduction to basic number theory. The combinatorial /geometric
| proofs of Fermat's Little Theorem and Wilson's Theorem are
| lovely. It also includes a very nice proof of Chebyshev's theorem
| on the asymptotic density of primes and even the Rogers-Ramanujan
| identities for integer partitions.
|
| 4. Vladimir Arnold, _Ordinary Differential Equations_ : Undergrad
| ODE classes are often taught in a cookbook fashion and if so,
| don't offer much enlightenment. This book explains what's going
| on at geometrical level. I didn't appreciate ODEs until I read
| this. See https://www.uni-
| muenster.de/Physik.TP/~munsteg/arnold.html for Arnold's views on
| teaching mathematics.
|
| 5. E.C. Titchmarsh _The Theory of Functions_ : Recommended by my
| undergraduate advisor because he noticed that I liked reading
| older books. It contains sections on complex analysis and real
| analysis with measure theory, but I've only read the complex
| analysis sections. It's not for everyone, if I recall correctly,
| there is not a single picture, but it is very lively and has a
| lot of material you won't find in a standard complex analysis
| book, including Dirichlet series. Excellent as a supplement to a
| standard complex analysis book.
|
| 6. George Polya. _Mathematics and Plausible Reasoning_. An
| excellent expansion on Polya 's ideas on _How to Solve it_. While
| the goal is to seek rigorous proofs, to get there it 's powerful
| to be able to think based on intuition, heuristics, and plausible
| reasoning. A lot of math exposition is theorem/proof based and
| doesn't help develop these skills. In a similar vein, see also
| Terence Tao's classic post _There 's more to mathematics than
| rigour and proofs_ https://terrytao.wordpress.com/career-
| advice/theres-more-to-....
|
| 7. H.S.M Coxeter, _An Introduction to Geometry_. A book of very
| beautiful classical geometry. Something typically not touched on
| at all in a typical mathematics curriculum.
| belter wrote:
| If you have a Love of Books and Math Books in particular, you
| can't miss this playlist by The Math Sorcerer:
| https://www.youtube.com/playlist?list=PLO1y6V1SXjjM-1azbCNYq...
| sthu11182 wrote:
| I've really enjoyed the Math Sorcerer's overview of various
| math books. Such a breath of topics and levels.
| phonebucket wrote:
| A lot of recommendations depend on what you're trying to learn.
|
| But I've enjoyed the following texts to a larger extent than
| others:
|
| - Algebra: Chapter 0 (Aluffi)
|
| - Real Mathematical Analysis (Pugh)
|
| - Mathematics and its History (Stillwell)
|
| - An Introduction to Manifolds (Tu)
|
| - Gauge Fields, Knots and Gravity (Baez)
|
| - A First Look at Rigorous Probability Theory (Rosenthal)
|
| - All of Statistics (Wasserman)
|
| There are some authors I trust and am happy to buy so long as the
| topic vaguely interests me: VI Arnold, Tristan Needham and John
| Stillwell.
|
| I really like the list put out by @enriquto in a separate
| comment, but I've avoided duplicating those recommendations in
| the list above.
| wannabebarista wrote:
| Great list! I want to call out John Stillwell's Reverse
| Mathematics as a fun and accessible introduction to the field.
| anthomtb wrote:
| _The Art of Problem Solving_ :
| https://artofproblemsolving.com/store/book/aops-vol1
|
| Yes, it is targeted towards middle and high school students. Yes,
| I read it and (more importantly) worked through most of the
| problems in my mid-30's. It is great if, like me, you
| coasted/crammed through your early mathematics education and
| never felt like you dialed in the fundamentals. It is also great
| if, like me, you needed some pen-on-paper practice and did not
| know where to start.
| Buttons840 wrote:
| All the Math You Missed by Thomas A. Garrity. I have not read it,
| but it looks interesting and is on my list. It is aimed at new
| graduate students who need a quick refresher that is still
| detailed enough to be useful for postgraduate math.
| ummonk wrote:
| It's unclear what kind of mathematical thinking and skills you
| want. "How to solve it," by Polya perhaps?
| imranq wrote:
| Counter-intuitively, reading Tim Ferris and his DSSS approach
| made me much better at math
|
| Deconstruct: Break down the math you want to know into big
| problems and concepts. Pick a math-related goal that is
| Measurable and Time-Bound
|
| Selection: What the 20% of math concepts, that if made really
| strong, would solve 80% of math problems
|
| Sequencing: What order of material should you study for maximal
| progress
|
| Stakes: Find some incentive to complete the problem. Some nice
| view of mathematical terrain, as part of a masters program,
| applications to another field, a prize, a cookie. Anything that
| motivates you to actually make progress towards the goal
|
| This approach helped me learn a bunch of high level math like
| abstract algebra, analysis, linear algebra, etc.
| homerowilson wrote:
| I'm coming from an applied math perspective. A few of my
| favorites (and ones I find myself regularly referring to) are:
|
| Matrix Analysis by Horn and Johnson (perhaps the best end-of-
| chapter problem sets of any math book I've encountered!)
|
| Matrix Computations by Golub and Van Loan
|
| Elements of Statistical Learning by Hastie, Friedman, Tibshirani
|
| Functional Analysis by Reed and Simon
| nhatcher wrote:
| Lot's of good books already here! In the same spirit as Polya's
| book "Thinking mathematically" by J. Mason, L. Burton and K.
| Stacey I learned a lot in the early days with Demidovich book on
| 5000 problems on mathematical analysis. Tom Apostol books on
| calculus, but for me his book on analytical number theory.
| Alongside alan baker's thin book on number theory. Gilbert Strang
| book(s) on linear algebra. Rudin book on functional analysis. Oh
| Hardy's book on divergent series! Ian Steward books on
| transcendental numbers and Galois theory. Elements of algebraic
| topology by Munkres is a fantastic book. So many books are
| invaluable to me in teaching not only math but mathematical
| thinking.
|
| I guess if you want to learn thinking but not necessarily math
| "thinking mathematically" above mentioned is your friend.
| matthewmorgan wrote:
| Came here to recommend Thinking Mathematically (text only) by
| J. Mason. If only every high school maths teacher had this
| book, the world would be a better place
| d_tr wrote:
| _A Course in Modern Mathematical Physics: Groups, Hilbert Space
| and Differential Geometry_ by Peter Szekeres
|
| My intro to abstract math... Wide range of topics, very clearly
| written and very well structured. Sets, groups, vector spaces,
| tensors, topology, differential geometry, lie groups and more.
|
| _An Introduction to Category Theory_ by Harold Simmons
|
| Very enjoyable read. You cannot go wrong with this as your first
| book on the subject.
| thrownawaydad wrote:
| Not a book, but an animated short:
| https://en.wikipedia.org/wiki/Donald_in_Mathmagic_Land
|
| "If you want to build a ship, don't drum up the people to gather
| wood, divide the work, and give orders. Instead, teach them to
| yearn for the vast and endless sea." --Antoine de Saint-Exupery
| jspx wrote:
| A book of abstract algebra - Charles C. Pinter. Each chapter is a
| few pages of explanation, and the rest you solve yourself by
| doing exercises that introduce aspects of the theory step by
| step.
| legends2k wrote:
| Not strictly math books but these three books turned me to live
| mathematics and appreciate for what it is:
|
| [1]: 3D Math Prime for Games for Graphics and Game Development;
| https://www.gamemath.com/ (free to read online) [2]: Essential
| Mathematics for Games and Interactive Applications;
| https://www.essentialmath.com/book.htm [3]: Mathematics for 3D
| Game Programming and Computer Graphics;
| https://www.mathfor3dgameprogramming.com/
| s-xyz wrote:
| Optimization by Jan Brinkhuis
|
| https://books.google.com/books/about/Optimization.html?id=UW...
| javier123454321 wrote:
| Not a direct answer, but I once read that the best book about a
| technical topic is the third book you read on it. Often you'll
| see things in comments sections like: "I have heard this
| explained so many times by others, but this explanation finally
| clicked!". The assumption is that that's the case because the
| explanation is better, rather than assuming it's the case because
| you've struggled with the material before and you're still going
| at it.
| northstart001 wrote:
| What if by Randall Munroe Not rigorous but it changed my
| perspective that was instilled in me in middle school. Otherwise
| Feynman Notes changed me academically. Easier to pickup math
| through physics if you arent looking for deeply pure avenues.
| miguelmurca wrote:
| Knuth's "Concrete Mathematics" is fantastic, precisely because
| it's very applicable.
| guhsnamih wrote:
| Statistics by Freedman, Pisani and Purves. Don't know if I got
| better but loved the real world examples and cartoons. Does not
| have too many pre-requisites. Each section presents a tiny
| concept which is followed by plenty of exercises that have
| answers at the end. The furthest I got in a book in recent days,
| Math or not.
| madcaptenor wrote:
| I taught from this book (it wasn't my choice, it was the
| standard book where I was teaching). It's really good for
| intuition, but because it doesn't use standard notation I think
| it might have done a disservice to students who were going to
| go on to learn more.
| jldugger wrote:
| My pandemic project in 2020 was to finally read through the
| used copy I bought a decade ago. I agree it was really useful
| at building foundational intuitions. And that it doesn't use
| professional jargon which sometimes makes Stats Wikipedia's
| "death by integrals" approach a dense barrier to entry.
|
| For example, the book uses "the box model" all over the book
| but is not used anywhere else, and every else uses the phrase
| "i.i.d" which is not used in the book.
|
| Still, it's been really useful at my job in reasoning about
| timeseries data from Prometheus, especially in canary
| analysis. Far more useful than the whirlwind tour of
| distributions my 1 semester "Statistics for Engineers" course
| in college undertook.
| fsloth wrote:
| "Road to reality" by Roger Penrose is an interesting book as a
| refresher and review if the content is otherwise within familiar
| territory.
| mindcrime wrote:
| To add one more thing: the "thing" that has helped me most lately
| isn't a specific book or video or anything, but rather simply
| committing to spending 1 hour every day on math. I even set up a
| Google Calendar task to remind me of this every. single. day.
|
| And so far this year I haven't missed a day yet. Now what
| constitutes that hour can vary. It can be watching math videos,
| it can be solving problems on paper, and I might even let myself
| count futzing around with numerical computing stuff or something
| at some point. In practice so far it's basically always either
| watching videos, reading books, or doing exercises (from books).
|
| I won't claim that everybody _must_ do this, or that you need to
| commit 1 hour every day. Maybe 30 minutes would be fine. Or maybe
| some people who can spare the time would be well served to commit
| 2 hours a day. Who knows? But having _some_ kind of routine
| strikes me as something that most people would probably find
| valuable.
| kevinventullo wrote:
| This is a bit of an odd suggestion, but I learned the basics of
| category theory from the appendix to Weibel's "An Introduction to
| Homological Algebra".
|
| I'm not sure why, but I think the fact that it's an _appendix_
| meant the author had no motivation to inflate the content
| unnecessarily. So it's more like a pamphlet; only about 30 pages
| IIRC, and it's really just the bare-bones definitions and facts.
| The full-on textbooks dedicated to category theory have way too
| much superfluous content IMO, unless your aim is to be a
| researcher in that field specifically.
| enriquto wrote:
| This happens a lot! For example, at the appendix of an advanced
| book on PDE (e.g. Evans') you find a three-page summary of main
| definitions and results in integration theory and L^p spaces.
| Or at the appendix of a book on differential geometry (e.g. do
| Carmo's) you find a succinct compendium of elementary
| differential calculus, explained in the most efficient way.
| These kind of condensed summaries, or _fascicules de resultats_
| , are rarely found on books that deal with the subject matter
| directly.
| geocrasher wrote:
| "How to use Calculators" by R. U. Kiddinme
| jjgreen wrote:
| _Analysis Now_ , Gert Pedersen
| https://link.springer.com/book/10.1007/978-1-4612-1007-8
| boothby wrote:
| No specific book, but generic advise about how to _use_ a math
| book. Homework, homework, homework. Read the whole thing, but
| focus on the exercises. Do every exercise as soon as you can
| manage: don 't wait until you've read the whole chapter -- once
| you get confused and stumped, the lesson of the chapter becomes
| _urgent_ and I find that sharpens my attention.
| wheelinsupial wrote:
| I would add understanding the reasons for definitions, how they
| fit together with theorems, lemmas, corollaries, proofs, and
| some basics of the format of proofs. You can find good
| explanations through Google.
|
| For more applied or computational branches of math, I'd also
| add how to check your answers by using numerical methods or a
| computer algebra system if possible.
| wallscratch wrote:
| Going to echo the suggestions for the Art of Problem Solving
| books, particularly I recommend the contest books (vol 1 or 2).
| Several very talented people have said to me that these books
| taught them how to think. Maybe a bit exaggerated, but they're
| very good.
| gibrown wrote:
| Nonlinear Dynamics and Chaos by Strogatz
|
| Chaos theory and deterministic systems are a fascinating vantage
| point for thinking about the dynamics of large computer systems.
| Thinking of them as stochastic systems is sometimes useful, but
| most of the systems are actually just operating in unstable
| periodic processes which are much closer to being a chaotic
| system rather than a stochastic system. This influences how I
| think about testing and debugging large distributed systems.
|
| I will say, I'm not sure I could have learned it well without a
| class and a good professor. The author has a number of books
| though and is a professor at Cornell.
| dorchadas wrote:
| Absolutely loved this book. Had a class on it in my applied
| maths and theoretical physics masters and it was hands-down my
| favourite.
| sambapa wrote:
| Not one book, but this:
| https://github.com/TalalAlrawajfeh/mathematics-roadmap
| Jimmc414 wrote:
| Some that stand out
|
| "Concrete Mathematics: A Foundation for Computer Science" by
| Knuth, Graham, and Patashnik - solid foundation in mathematical
| concepts and techniques, and it helped me develop a deeper
| understanding of mathematical notation and problem-solving.
|
| "Introduction to the Theory of Computation" by Michael Sipser -
| introduced me to the theoretical foundations of computer science,
| and it helped me develop a strong understanding of formal
| languages, automata, and complexity theory.
|
| "A Course in Combinatorics" by J.H. van Lint and Wilson -
| provided a comprehensive introduction to combinatorics, and it
| helped me develop a strong understanding of combinatorial
| techniques and their applications.
|
| "The Art of Problem Solving" by Richard Rusczyk - This book is a
| comprehensive guide to problem-solving, with a focus on
| mathematical problem-solving strategies. It helped me develop my
| problem-solving skills and learn how to think critically about
| mathematical problems.
| bick_nyers wrote:
| +1 for The Art of Problem Solving
| rcpt wrote:
| Apostol's Calculus changed my life when I was 18
| lcuff wrote:
| Several people who have mentioned "How to Solve It" by George
| Polya. It's been decades since I've looked at it, but a favorite
| Polya quote of mine is " "The open secret of real success is to
| throw your whole personality into your problem."
|
| I can't remember if the book addresses this, but for myself my
| inability to tolerate frustration really impeded my ability to
| work on any mathematical challenge for decades.
| mohamez wrote:
| I think you guys might find this list I found long ago very
| useful when deciding on a mathematics book you want to read.
|
| This is an introduction written by the original author of the
| list:
|
| "Somehow I became the canonical undergraduate source for
| bibliographical references, so I thought I would leave a list
| behind before I graduated. I list the books I have found useful
| in my wanderings through mathematics (in a few cases, those I
| found especially unuseful), and give short descriptions and
| comparisons within each category. I hope that this list may serve
| as a useful "road map" to other undergraduates picking their way
| through Eckhart Library. In the end, of course, you must explore
| on your own; but the list may save you a few days wasted reading
| books at the wrong level or with the wrong emphasis.
|
| The list is biased in two senses. One, it is light on foundations
| and applied areas, and heavy (especially in the advanced section)
| on geometry and topology; this is a consequence of my interests.
| I welcome additions from people interested in other fields. Two,
| and more seriously, I am an honors-track student and the list
| reflects that. I don't list any "regular" analysis or algebra
| texts, for instance, because I really dislike the ones I've seen.
| If you are a 203 student looking for an alternative to the awful
| pink book (Marsden/Hoffman), you will find a few here; they are
| all much clearer, better books, but none are nearly as gentle. I
| know that banging one's head against a more difficult text is not
| a realistic option for most students in this position. On the
| other hand, reading mathematics can't be taught, and it has to be
| learned sometime. Maybe it's better to get used to frustration as
| a way of life sooner, rather than later. I don't know." - by
| original author.
|
| [List] https://www.ocf.berkeley.edu/~abhishek/chicmath.htm
| lower wrote:
| _There exist only two kinds of modern mathematics books: ones
| which you cannot read beyond the first page and ones which you
| cannot read beyond the first sentence._ -- Chen Ning Yang
| Penyngton wrote:
| Since I haven't seen many discrete maths books, he's my list:
|
| Beginner: NL Biggs, Discrete Mathematics, Oxford University Press
|
| Intermediate: PJ Cameron, Combinatorics: Topics, Techniques,
| Algorithms, Cambridge University Press
|
| Advanced: JH van Lint & RM Wilson, A Course in Combinatorics,
| Cambridge University Press
| Bootvis wrote:
| Meta comment: might be good to add the level of mathematical
| maturity needed to enjoy the book.
| 082349872349872 wrote:
| Skimming over the replies, they range from arithmetic to
| algebraic geometry and measure theory!
|
| Along the lines of _fascicules de resultats_ , I find talks are
| a good way to get a _coup d 'oeil_ for a field: people giving a
| talk tend to take a direct approach to what they want to
| introduce, hitting only the salient points. But that yields
| enough keywords to then consult any relevant texts.
| 6gvONxR4sf7o wrote:
| _Probabilistic Graphical Models_ by Koller & Friedman. In
| anything statistics and ML related, being able to deal with
| complicated probabilitistic things that are all related is really
| useful. This book gives you that toolkit. It's a "strong
| foundations" kind of book, rather than a bunch of methods you'll
| use directly.
| vippy wrote:
| Velleman's How to Prove It greatly helped my ability to construct
| set theoretic proofs, which better prepared me for Spivak's
| calculus and Baby Rudin. Hamkins' Proof and the Art of
| Mathematics is designed as a a good, less set-theory heavy,
| introduction to proof writing that leads more naturally to
| analysis. OpenStax books are FREE.
| strls wrote:
| Surprised to see Velleman's book so far down. It taught me that
| proofs are fun and do not in general require clever tricks. As
| a bonus, it provided plenty of practice with foundational
| objects such as sets, relations and functions. All this made me
| much better at doing mathematics and prepared to texts in real
| analysis, CS, algebra.
| JoelMcCracken wrote:
| I have been going through Velleman, and it has been
| significantly helping me understand various CS papers and
| books, for example, I struggled understanding through proof
| outlines in PFPL, but working through just part of this book
| has helped.
|
| I have had life things interfere with my learning now for the
| past month or so, but I hope to get back to it soon.
| jordibc wrote:
| For sure I got significantly(?) better with classics like Spivak,
| Apostol, Rudin.
|
| "Real and Complex Analysis" by Rudin, and the two books both
| named "Calculus" from Spivak and Apostol. But also from Apostol
| his more concise and far-reaching "Mathematical Analysis". And
| from Spivak his small gem "Calculus On Manifolds" made quite a
| dent on me.
|
| Other than more "classic math" books, I also wanted to mention
| two outliers that I found eye-opening and generally awesome:
|
| * Street-Fighting Mathematics, by Mahajan
| (http://streetfightingmath.com/). Intuitive, useful and fun.
|
| * Geometric Algebra for Physicists, by Doran and Lasenby. I found
| the power and elegance of geometric algebra mesmerizing, and even
| if this book is also about physics and there may be more
| appropriate math-only books about geometric algebra, this is the
| one that made it for me.
| dorchadas wrote:
| > * Geometric Algebra for Physicists, by Doran and Lasenby. I
| found the power and elegance of geometric algebra mesmerizing,
| and even if this book is also about physics and there may be
| more appropriate math-only books about geometric algebra, this
| is the one that made it for me.
|
| I've tried to read several of them, and, sadly, I feel most
| geometric algebra books fail at explaining it. It's a shame as
| it's part of what kindled my interest in pure mathematics and I
| still feel I'm nowhere nearer understanding it despite working
| through several other mathematics textbooks, including just
| plain algebra. But, it did spark my interest and now I've moved
| on to other interesting topics, though Geometric Algebra is
| still my white whale.
| okaleniuk wrote:
| Calculus on Manifolds by Spivak. Brilliant. And relatively thin
| too.
| LanceH wrote:
| The Time/Life book "Mathematics" published 1969. I was in 2nd
| grade, liked mathematics and saw a book with that name and
| pictures. It was a high level survey of a lot of mathematical
| concepts, explaining things in a way I could understand at that
| age, but also in a way that wouldn't be talking down to me today.
| haskellandchill wrote:
| I love math books with great exercises, I just wish I could code
| them up in a theorem prover and solve and store my proofs that
| way. I've tried a bunch of tools but haven't found a language or
| workflow that really meets the needs for computer assisted study
| of mathematics.
| kbelder wrote:
| I don't think there is any book I've read as an adult that was
| _particularly_ special. If I wasn 't already good at math, I
| wouldn't be reading these books in the first place. Not that
| there weren't good and helpful books, but I wouldn't say any of
| them were revolutionary to me.
|
| But I'd like to mention two books I read as a child which had a
| life-altering effect. They probably wouldn't do any good for an
| adult, but might really help your kids... Unfortunately, I don't
| remember the specific titles or authors (I was probably around 10
| yrs old). The first was _similar_ to this book:
|
| "Speed Math for Kids: The Fast, Fun Way To Do Basic
| Calculations." This gave all sorts of advice and tips to quickly
| do math in your head... simple things, mostly. For example, to
| multiply by 18 just double, multiply by 10, and subtract 10%; or
| how it's frequently faster to multiply numbers by moving from
| most significant digits to least, which is opposite of how we're
| taught; or how to quickly estimate square roots. This really
| didn't teach new concepts, but by making routine and tiresome
| math operations faster and easier, it made the entire field more
| enjoyable to engage with.
|
| The second book was a guide to slide rulers, and I couldn't even
| find a similar book on Amazon. But learning advanced slide ruler
| techniques can trigger an epiphany; you learn mathematical
| relationships, how you can transform how numbers are represented.
| It was the first time I really saw an elegant structure behind
| the math.
| chkhd wrote:
| * The Language of Mathematics: Utilizing Math in Practice by
| Baber https://www.amazon.com/Language-Mathematics-Utilizing-
| Math-P... really helped me "get it", as I always found
| programming natural but math hard. This one is written by a CS
| professor and it really makes all the difference.
|
| * How to Solve it by Polya https://www.amazon.com/How-Solve-
| Mathematical-Princeton-Scie... and How to Prove it by Velleman
| https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/...
| helped strengthen that understanding.
|
| * This year I am trying to master https://www.amazon.com/Methods-
| Mathematics-Calculus-Probabil... which focuses on how to "connect
| the dots".
|
| * I am using Geometry and the Imagination by Hilbert
| https://www.amazon.com/Geometry-Imagination-AMS-Chelsea-Publ...
| as an attempt to "immerse" myself in Geometry. I just love this
| book.
| ForHackernews wrote:
| The Art and Craft of Problem Solving
| https://archive.org/details/the-art-and-craft-of-problem-sol...
|
| This is only math book I've ever read that teaches the mindset
| needed to work mathematics problems, rather than mathematical
| concepts or techniques.
| awelxtr wrote:
| Significantly better I don't know but when I was a child I was
| given Der Zahlenteufel. Ein Kopfkissenbuch fur alle, die Angst
| vor der Mathematik haben (The Number Devil) and I liked it very
| much
| hackerbrother wrote:
| Rudin's Principles of Mathematical Analysis has a really special
| place in my heart. Chapter 3 is great- it's a great reference for
| derivations of a lot of fundamental identities about limits used
| in undergrad calculus.
|
| Chapter 4 is a great place to learn about topology for the first
| time.
|
| In general, it kicks up the mathematical rigor you're used to a
| notch. Seeing ">" defined as "not <" really blew my mind when I
| first read it! "<" is just something that satisfies some axioms,
| like anything else in math.
| r-zip wrote:
| Wouldn't ">=" be "not <"?
| hackerbrother wrote:
| Yup.
| fnordpiglet wrote:
| Advanced engineering mathematics by Kreyszig
|
| Advanced Engineering Mathematics, 10Ed, Isv
| https://a.co/d/axcq9nk
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