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Mathematicians are chronically lost and confused (2014)

250 points by Halienja 10y ago

reachtarunhere 10y ago
As an undergrad who recently became serious about math (thanks to its importance in areas I am interested in) this is very inspiring. I have been trying to grok mathematics for some time and sometimes being too frustrated with problems I can't handle. I have experienced the phenomena of giving up on something and coming back to it and finding it trivial. This is exactly what I needed.
kinai 10y ago
Does anybody know a good guide on where to begin? Resources are not the issue here, but usually the overwhelming amount of information regarding all those topics and areas of mathematics. I was always very interested but got discouraged rather quickly, even after a semester at university. So far my favorite access to math was through philosophy.
- LionessLover 10y ago
  a) get started on something - if you can't decide, roll the dice and pick something at random b) ignore everything else until you are done c) repeat Don't worry that at your pace it will take ages - very soon you will develop an idea how to rank what you should look at next. If you don't, go back to a). Obviously it's useless to think about it too much when your knowledge about a subject consists mostly of holes and gaps, so first gather data (a). I can't quite see how concrete your question is, but try Khan Academy. Follow the suggested order if you have no preferences.
  - demonshalo 10y ago
    pretty much this. The last math class I took was in high school and I dropped out of college because my CS degree required me to do a metric fuckton of math, which I hated at the time. A few years ago I got interested and started working on Integer Factorization. It is not that likely that I will solve that problem but in the past few years I have learned a lot more math by attempting to solve that problem that I learned in college! Moral of the story: Just have fun and try to solve some problem(s).
  - kinai 10y ago
    But some areas require knowledge base of others
    - LionessLover 10y ago
      Which you won't find out unless you start - somewhere, anywhere. How are you going to find out when you just sit there and think about something you know nothing about? Wait for sudden enlightenment? Math is one off the easiest subjects to get started, soooooo many starter books and by now even great online courses (again: Khan Academy, for the basics, probably many more). So much guidance. There is so much out there - free and easily accessible thanks to the Internet, the problem isn't "how do I get started" but "what resource do I use to get started".
- visarga 10y ago
  I read a lot of math in ML papers, but I don't do any exercises, and maybe that's why I am not sure of myself. A very gradual collection of problems pertaining to linear algebra, calculus and statistics would be great. Something like Khan Academy, but the key point is to be gradual, so as not to terrify me before I get through it. Ideally such a training ground could guide students through problems based on exercise logs collected from other students, by creating an optimized curriculum. Anyone knows where to look?
- mazsa 10y ago
  http://us.metamath.org http://us.metamath.org
- laichzeit0 10y ago
  Step 1: Read Lockhart's Lament: https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament.... Step 2: Download the Book of Proof: http://www.people.vcu.edu/~rhammack/BookOfProof/ http://www.people.vcu.edu/~rhammack/BookOfProof/ You read through it and do all the odd numbered exercises (the solutions are at the end of the book). Step 3: Get a book called Real Mathematical Analysis by Charles Pugh and you work through that and attempt as many problems as you can, with a view not to rush through it, but to expand your mind through each problem. Step 4: Pick any of these books that interest you the most and do the same: - Calculus by Spivak - Algebra: Chapter 0 by Paolo Aluffi - Linear Algebra Done Right by Axler By then you should have enough mathematical maturity to know what to do next.
  - nextos 10y ago
    I'd also recommend Hubbard & Hubbard for a beautiful and beginner-friendly mix of algebra and analysis. My preferred starter kit is Rudin plus Halmos or Axler, but treating Rudin as a summary. So a helper would be needed, like Counterexamples in Analysis. This is what Math 55 used to do.
    - tgb 10y ago
      I started with Hubbard & Hubbard. It is truly wonderful (and based on Spivak's other calculus book), but I couldn't imagine learning it without either significant background or a formal class. Even with a 2 semester class devoted to it in college, I ended up not understanding the most advanced concepts (eg. differential forms) until years later when it finally came up again in graduate-level courses. Of course, it was really an excuse to learn to think mathematically, not to learn calculus on manifolds.
    - pmiller2 10y ago
      I don't really recommend Rudin for a true beginner at all (unless, by "as a summary," you mean not really digging into the proofs themselves, in which case any good analysis book will do). Rudin will always try to take the most elegant route to the theorem, regardless if that route goes anywhere near where the rest of the text has been. The result, for me, has been that many of his proofs seem to just meander about for a little while until, at the very end, you arrive at the theorem. It's a bit like driving to work on auto pilot, and just as disconcerting to me.
- 505 10y ago
  Since philosophy was your favourite, you might enjoy the novels of Greg Egan. As far as I can tell, he puts good mathematics in his stories.
  - tgb 10y ago
    I recently read my first Greg Egan and enjoyed it (Quarantine). Got any recommendations for specifically the ones he put good math into?
    - 505 10y ago
      Perhaps Incandescence, though it's less about mathematics and more about a society working out the physics of its environment. His website has more maths on it than I recall being in the novels.
    - stuxnet79 10y ago
      Schild's Ladder is terrific.
- vinchuco 10y ago
  I made a 3-page overview of calculus as an extra section of some notes for a class I'm teaching. Mostly to see if it could be done. Here it is. https://www.dropbox.com/s/yp8ijkzir4h8rz9/Calculus%20in%20e%2C%203%2C%20or%20pi%20pages.pdf?dl=0 https://www.dropbox.com/s/yp8ijkzir4h8rz9/Calculus%20in%20e%... I'm REALLY sorry for the picture quality. I'll use a proper scanner later. My phone's camera and app are terrible.
  - vinchuco 10y ago
    In case you check back on this, I made a better resolution copy. Hopefully it's useful to someone :) https://drive.google.com/open?id=0B2FKX2JZ061dTzRNTjl6SFJVaVk https://drive.google.com/open?id=0B2FKX2JZ061dTzRNTjl6SFJVaV...
- notarudeperson 10y ago
  If I were you, I'd start here (free book on the basics of math without which it's very easy to drown in the so called proof-based math): http://www.people.vcu.edu/~rhammack/BookOfProof/ http://www.people.vcu.edu/~rhammack/BookOfProof/
- riazrizvi 10y ago
  The Second Edition of Courant & Robin's 'What is Mathematics?', revised by Ian Stewart is good. This is from the preface: " In short, it wanted to put the meaning back into mathematics. But it was meaning of a very dif- ferent kind from physical reality, for the meaning of mathematical ob- jects states "only the relationships between mathematically 'nndefined objects' and the rules governing operations with them." It doesn't matter what mathematical things are: it's what they do that counts. " https://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192/ref=sr_1_1?ie=UTF8&qid=1465574831&sr=8-1&keywords=what+is+mathematics https://www.amazon.com/Mathematics-Elementary-Approach-Ideas...
- feklar 10y ago
  Where to begin depends on what your goal is, mine was to be able to complete Sussman's SICM book (and later, his Functional Differential Geometry book). The book I used to do this was Advanced Calculus by Loomis & Sternberg because it covers classical mechanics, potential theory, differentiable (Banach) manifolds, differential equations, (multi)linear algebra, fundamental theorum of calculus and the Fourier transform. The exercises are not very difficult compared to a lot of other texts (Spivak's Calculus) so this is an accessible math book as I don't have any formal math training. I also liked the Mathematical Preliminaries crash course in the Art of Programming Vol 1 because it led me to looking into probability which has turned out to be an infinite rabbit hole of discovery. (a grad students highly opinionated list) https://www.amazon.com/gp/richpub/listmania/fullview/1F85VWNRTMY2T/ref=cm_pdp_lm_all_itms https://www.amazon.com/gp/richpub/listmania/fullview/1F85VWN...
- ivan_ah 10y ago
  I am totally biased, but I think my book is a good introduction to math fundamentals (high school) and calculus: https://minireference.com/ https://minireference.com/ preview: https://minireference.com/static/excerpts/noBSguide_v5_preview.pdf https://minireference.com/static/excerpts/noBSguide_v5_previ... As an added bonus, the book also covers mechanics (Physics 101), which is really important to understand to build up you general modelling skills. (Physics is all about coming up with math models to describe reality.)
friendly_chap 10y ago
I feel the same way when solving tasks in my day job. The thing I tell to young people learning programming/tech that I hope they don't get frustrated easily, because they will spend every day of their life feeling rather stupid and confused, never knowing when will they discover a solution for a particular problem. This is something that was a great source of stress early in my career.
- Swizec 10y ago
  It's a game of educated guessing. Experience helps.
  - friendly_chap 10y ago
    Indeed, but the more experienced I get the harder/bigger problems I work on, so that balances it out. I still feel rather stuck on a daily basis, but I accept the situation and try to deal with it with a calm focus.
- minipci1321 10y ago
  I always thought that in the end, one has got to like facing these hard problems every day on the job in order to thrive in this field (I am in tech too). This is something I try to detect in candidates in job interviews I conduct. I noticed people can be roughly split in two categories: -- ones who spend a couple of years dealing with this sort of job activities, then want out by any means, on to something different; -- others enjoy it more and more as they gain expertise, start having more and more fun -- get more expertise, and with results (which necessarily come up), the right to pick and choose the subjects and so the hard problems that come with them. I happen to belong to the second category (and it has been a while I stay in the same domain), and I don't believe someone in that category would feel stupid and confused. it rather feels like investigation every time. Though over the years, I've become wary of the stress always associated with being put on such issues -- a last-minute demo for a trade show that doesn't work, a customer who has escalated to upper management, a delivery which is hopelessly late... -- with all attention of the management attracted. I keep telling myself that a soldier has got to participate in war campaigns, not be doing paperwork at the headquarters, and literally force myself in last years. But I don't recall I have ever felt stupid or confused, even in the beginning of the career.
te_platt 10y ago
This reminds me of the book "The Perfect Wrong Note". The book is focused on learning to play music but the principles it teaches apply to learning just about anything. The core message to not be afraid of mistakes during practice. Little kids fall over when they learn to walk, you'll have moments of confusion learning new things. There's a time to get things done well, like playing at a recital or releasing production code. There also needs to be time to practice and part of practicing is the expectation that there will be mistakes.
laingc 10y ago
To me, this is about "mathematical maturity". My observation is that many programmers, especially those who have come of age by working in startups, tend to value ability and sometimes experience over formal education. This is a result, I believe, of noticing that they can outperform many people who have a classical education, and also seeing that many of the people to whom they look up also do not have much in the way of formal education. However, I truly believe that Mathematics is a discipline that is very hard to engage with outside of formal education - or at least nobody has really found a great model for doing so yet. Learning Mathematics in a classical, structured way really does change the way you think. I notice a substantial difference even between those of my colleagues who entered industry straight after their Masters or even Bachelors, and those who completed Doctorates or even held postdoctoral positions. In my opinion, it is this lack of mathematical maturity that makes the switch from general programming to scientific programming more challenging than the converse.
- okoksowhatis 10y ago
  This is very well-written. The prosody of it illuminates the topic you're speaking about. Being so rare, I wonder what makes you different.
- agentultra 10y ago
  I disagree that mathematics itself is difficult to engage with outside of academia. There are some academic mathematicians who think mathematics has to be hard because it was difficult for them to understand. There are others for whom sharing their insights is more important than searching for new ones. I think that the set of mathematicians together have done a rather good job in the 20th century of producing mathematical literature and pedagogy that laypersons and the polymaths can understand. I skipped university when I was young due to circumstance. I still had a love for maths but couldn't pursue formal training. When I came around to programming as a career instead of odd-loner-hobby it was as you say. I was quite impressed with my ability to code circles around CS graduates who, for all their theory and training, weren't prepared for "coding in the real world." I never stopped studying mathematics. I've recently been going through temporal logics. If only more programmers were aware of the beauty of the proof of weak fairness. And its implications in system specifications. Where I can see formal training being useful is exposing you to subjects you were not aware of in short order. My experience has been more akin to the mansion metaphor. Where I encounter limitations or difficulties in my programming I look to maths for the right tool to simplify the task and ensure it is correct. So I agree about "mathematical maturity." I just don't believe it is exclusively acquired in an academic setting. I think some people have an inclination or awareness that pushes them to think this way.
vecter 10y ago
I have a great personal story that highlights how long the journey of understanding mathematics is. I took linear algebra my freshman year of college. It was the non-math major course, so it didn't require proofs. I got an A+ in the class. Not just an A, an A+. I was able to obtain such a high grade by taking tons of practice tests, and since the actual tests were basically mildly veiled calculations, I just had to map the question to the right calculation. So for instance, if after a little interpretation, I figured out that the question was asking for me to calculate the singular value decomposition of a given matrix, I would mindlessly compute, check my algebra, and move on. However, it was very clear to me by the end of the course that I didn't really understand what the heck linear algebra was about. Five years later, I started a job as an algorithmic trader. One of the first things my boss wanted to do was to do a Principal Component Analysis (PCA) of bond price movements. This is a very common thing to do. I didn't know what PCA was, but I read a short paper he gave me and I was able to grok it. After reading that paper and actually performing the PCA (which by the way was basically one line of R code), I finally came to understand the core essence of linear algebra, which is the idea of linear transformations. I was able to connect the equation Ax=lambdax to the geometry of what an eigenvector meant. Through a little more reasoning, I realized that every real matrix corresponded to a linear transformation of that space via a rotation, a reflection, a stretching, a shearing, etc. At that point, all of the mindless calculations I had been doing half a decade earlier instantly clicked, and I was enlightened. This was literally half a decade later after I "aced" my linear algebra class. I know that it seems absolutely ridiculous that I could "score so well" in a math class yet so clearly miss the core idea behind the entire class, but that's been my experience with math for as long as I can remember. You start by doing the calculations and just getting comfortable with them. Some arbitrary time later, you have an insight and suddenly everything is so crystal clear and trivial that you wonder how you could even not have understood it before. Oh, and even to this day, I don't understand what singular values actually are. Something to do with a mapping from the row space to the column space, blah blah. I'm sure if I spent an hour to read about them and picture the geometry, I could figure it out, but I just haven't gotten around to doing it.
- lordnacho 10y ago
  Wow, that's oddly similar to my story. LinAlg in college, nothing but a blur of matrices. Algo trading for work. Now there's a reason to do it, it makes sense. About SVD, btw, it is another path to PCA, one that solves certain problems that PCA does not. https://jeremykun.com/2016/04/18/singular-value-decomposition-part-1-perspectives-on-linear-algebra/ https://jeremykun.com/2016/04/18/singular-value-decompositio...
auvrw 10y ago
tao's stages of math is relevant https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/there%E2%80%99s...
deleted 10y ago
[deleted]
MikeNomad 10y ago
Shouldn't the year the article was written (2014) be included in the title?
l3robot 10y ago
What a great blog post! Totally agree with him. And, personnaly, it is why i'm having so much fun doing maths. Everytime it's a new exploration, a new challenge. My best math teacher I had was seeing math with this philosophy in mind and his class was like discovering new lands every time. I'm sure that if we explained in a way that failing a math problem is as normal and challenging that failing a Mario Bros Level, more people would be in peace with it.
gauruvbose 10y ago
"mathematicians don’t work like this"? Sure they do. Reading textbooks is normal. As a mathematician, this post is foreign.
Koshkin 10y ago
One of the ways to acquire a taste for mathematics is to try solving elementary but challenging problems, such as those included in MathCamp's qualifying quizzes: http://www.mathcamp.org/prospectiveapplicants/quiz/pastquizzes.php http://www.mathcamp.org/prospectiveapplicants/quiz/pastquizz....
jondubois 10y ago
My favourite quote about Mathematics is from John von Neumann: "In mathematics you don't understand things. You just get used to them." - This quote highlights precisely why I ended up choosing software engineering over maths. I'm just not very good at applying processes/methodologies which I don't fully understand. For example, I wasn't very good with linear algebra until I was able to visualize the equations in my head. For example, now, when I think about the equation 'f(x) = ax^2 + bx + c' - I can see that this represents the set of all possible quadratic equations and I can roughly visualize what that looks like on a cartesian plane (well it would turn the whole plane black because there would be an infinite number of graphs). Then if I choose any three points on that crowded cartesian plane, I can visualize that among this infinite set of curves, one of them passes through all three points. Thinking about it in that way allows me to make sense of Gauss-Jordan Reduction and other mathematical processes related to linear algebra. Programming is much easier for me because I can visualize the results instantly on a computer - I don't need someone else to explain it to me. Any uncertainty can be quickly resolved by simply running some code.
- pixl97 10y ago
  >Any uncertainty can be quickly resolved by simply running some code. Until you have to interact with a black box of someone else's code. You can only be certain that the data you sent that particular time works, not that all possibilities of valid data work.
  - jondubois 10y ago
    I guess with programming, you can never get 100% certainty when it comes to correctness of your code - Unless you perform formal analysis which is impractical for most use cases.
    - chipsy 10y ago
      You might not succeed at 100%, but you can remove "classes of error" by adopting particular styles or techniques backed by formal analysis. That's one of the biggest appeals of compiler technology - it can encode an understanding of patterns proven to detect failure, and in so doing lower your resulting bug count.
pm24601 10y ago
All they have to do is read this article: https://news.ycombinator.com/item?id=11874395 https://news.ycombinator.com/item?id=11874395
riazrizvi 10y ago
Beautiful article. Love the advice at the end!: "What’s much more useful is recording what the deep insights are, and storing them for recollection later. Because every important mathematical idea has a deep insight, and these insights are your best friends. They’re your mathematical “nose,” and they’ll help guide you through the mansion."
- abc_lisper 10y ago
  I can't think of a better argument to goad programmers into writing code comments, especially for elegant code. For some reason, programmers think good code should be self explanatory. It doesn't have to - if your insight is outside the flow (or the argument) of your code. I see a lot of Java code written like this; if the programmer's job is to multiply two numbers, they add number a number b times, and say comments are not needed!
j2kun 10y ago
Author here. I can't help but plug my mailing list for a book I'm writing, called "A Programmer's Introduction to Mathematics." Cheers, and thanks for reading! https://jeremykun.com/2016/04/25/book-mailing-list/ https://jeremykun.com/2016/04/25/book-mailing-list/
jonstokes 10y ago
A mathematician was walking home from campus one day, and as he walked he was pondering a particularly thorny problem. At one point, he snapped out of his reverie and looked around and realized that he had no idea where he was. He saw a young boy playing with a ball in a yard, and figured maybe the boy could tell him the way home. So he says to the boy, "young man, do you know where Prof. So-and-so lives?" The boy looked at him and said, "Dad, what's wrong with you?"
- selimthegrim 10y ago
  This story is usually told about Norbert Wiener and his daughter.
  - jonstokes 10y ago
    Good to know.
justifier 10y ago
any discipline where you are attempting to answer yet to be answered questions leaves you in a state of chronic confusion and lacking direction math the same
pjlegato 10y ago
> Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark... What are you supposed to do if you like math and the idea of grokking it, but you also have a job and a family and can't afford to spend six months contemplating each room in the mansion?
- dragonwriter 10y ago
  > What are you supposed to do if you like math and the idea of grokking it, but you also have a job and a family and can't afford to spend six months contemplating each room in the mansion? Learn to come to grips with the idea that the universe isn't always going to support your mutually exclusive preferences?
  - pjlegato 10y ago
    Yep, that's probably the only option, huh.

250 points by Halienja 10y ago

As an undergrad who recently became serious about math (thanks to its importance in areas I am interested in) this is very inspiring. I have been trying to grok mathematics for some time and sometimes being too frustrated with problems I can't handle. I have experienced the phenomena of giving up on something and coming back to it and finding it trivial. This is exactly what I needed.

kinai 10y ago

Does anybody know a good guide on where to begin? Resources are not the issue here, but usually the overwhelming amount of information regarding all those topics and areas of mathematics. I was always very interested but got discouraged rather quickly, even after a semester at university. So far my favorite access to math was through philosophy.

LionessLover 10y ago
a) get started on something - if you can't decide, roll the dice and pick something at random b) ignore everything else until you are done c) repeat Don't worry that at your pace it will take ages - very soon you will develop an idea how to rank what you should look at next. If you don't, go back to a). Obviously it's useless to think about it too much when your knowledge about a subject consists mostly of holes and gaps, so first gather data (a). I can't quite see how concrete your question is, but try Khan Academy. Follow the suggested order if you have no preferences.
- demonshalo 10y ago
  pretty much this. The last math class I took was in high school and I dropped out of college because my CS degree required me to do a metric fuckton of math, which I hated at the time. A few years ago I got interested and started working on Integer Factorization. It is not that likely that I will solve that problem but in the past few years I have learned a lot more math by attempting to solve that problem that I learned in college! Moral of the story: Just have fun and try to solve some problem(s).
- kinai 10y ago
  But some areas require knowledge base of others
  - LionessLover 10y ago
    Which you won't find out unless you start - somewhere, anywhere. How are you going to find out when you just sit there and think about something you know nothing about? Wait for sudden enlightenment? Math is one off the easiest subjects to get started, soooooo many starter books and by now even great online courses (again: Khan Academy, for the basics, probably many more). So much guidance. There is so much out there - free and easily accessible thanks to the Internet, the problem isn't "how do I get started" but "what resource do I use to get started".
visarga 10y ago
I read a lot of math in ML papers, but I don't do any exercises, and maybe that's why I am not sure of myself. A very gradual collection of problems pertaining to linear algebra, calculus and statistics would be great. Something like Khan Academy, but the key point is to be gradual, so as not to terrify me before I get through it. Ideally such a training ground could guide students through problems based on exercise logs collected from other students, by creating an optimized curriculum. Anyone knows where to look?
mazsa 10y ago
http://us.metamath.org http://us.metamath.org
laichzeit0 10y ago
Step 1: Read Lockhart's Lament: https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament.... Step 2: Download the Book of Proof: http://www.people.vcu.edu/~rhammack/BookOfProof/ http://www.people.vcu.edu/~rhammack/BookOfProof/ You read through it and do all the odd numbered exercises (the solutions are at the end of the book). Step 3: Get a book called Real Mathematical Analysis by Charles Pugh and you work through that and attempt as many problems as you can, with a view not to rush through it, but to expand your mind through each problem. Step 4: Pick any of these books that interest you the most and do the same: - Calculus by Spivak - Algebra: Chapter 0 by Paolo Aluffi - Linear Algebra Done Right by Axler By then you should have enough mathematical maturity to know what to do next.
- nextos 10y ago
  I'd also recommend Hubbard & Hubbard for a beautiful and beginner-friendly mix of algebra and analysis. My preferred starter kit is Rudin plus Halmos or Axler, but treating Rudin as a summary. So a helper would be needed, like Counterexamples in Analysis. This is what Math 55 used to do.
  - tgb 10y ago
    I started with Hubbard & Hubbard. It is truly wonderful (and based on Spivak's other calculus book), but I couldn't imagine learning it without either significant background or a formal class. Even with a 2 semester class devoted to it in college, I ended up not understanding the most advanced concepts (eg. differential forms) until years later when it finally came up again in graduate-level courses. Of course, it was really an excuse to learn to think mathematically, not to learn calculus on manifolds.
  - pmiller2 10y ago
    I don't really recommend Rudin for a true beginner at all (unless, by "as a summary," you mean not really digging into the proofs themselves, in which case any good analysis book will do). Rudin will always try to take the most elegant route to the theorem, regardless if that route goes anywhere near where the rest of the text has been. The result, for me, has been that many of his proofs seem to just meander about for a little while until, at the very end, you arrive at the theorem. It's a bit like driving to work on auto pilot, and just as disconcerting to me.
505 10y ago
Since philosophy was your favourite, you might enjoy the novels of Greg Egan. As far as I can tell, he puts good mathematics in his stories.
- tgb 10y ago
  I recently read my first Greg Egan and enjoyed it (Quarantine). Got any recommendations for specifically the ones he put good math into?
  - 505 10y ago
    Perhaps Incandescence, though it's less about mathematics and more about a society working out the physics of its environment. His website has more maths on it than I recall being in the novels.
  - stuxnet79 10y ago
    Schild's Ladder is terrific.
vinchuco 10y ago
I made a 3-page overview of calculus as an extra section of some notes for a class I'm teaching. Mostly to see if it could be done. Here it is. https://www.dropbox.com/s/yp8ijkzir4h8rz9/Calculus%20in%20e%2C%203%2C%20or%20pi%20pages.pdf?dl=0 https://www.dropbox.com/s/yp8ijkzir4h8rz9/Calculus%20in%20e%... I'm REALLY sorry for the picture quality. I'll use a proper scanner later. My phone's camera and app are terrible.
- vinchuco 10y ago
  In case you check back on this, I made a better resolution copy. Hopefully it's useful to someone :) https://drive.google.com/open?id=0B2FKX2JZ061dTzRNTjl6SFJVaVk https://drive.google.com/open?id=0B2FKX2JZ061dTzRNTjl6SFJVaV...
notarudeperson 10y ago
If I were you, I'd start here (free book on the basics of math without which it's very easy to drown in the so called proof-based math): http://www.people.vcu.edu/~rhammack/BookOfProof/ http://www.people.vcu.edu/~rhammack/BookOfProof/
riazrizvi 10y ago
The Second Edition of Courant & Robin's 'What is Mathematics?', revised by Ian Stewart is good. This is from the preface: " In short, it wanted to put the meaning back into mathematics. But it was meaning of a very dif- ferent kind from physical reality, for the meaning of mathematical ob- jects states "only the relationships between mathematically 'nndefined objects' and the rules governing operations with them." It doesn't matter what mathematical things are: it's what they do that counts. " https://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192/ref=sr_1_1?ie=UTF8&qid=1465574831&sr=8-1&keywords=what+is+mathematics https://www.amazon.com/Mathematics-Elementary-Approach-Ideas...
feklar 10y ago
Where to begin depends on what your goal is, mine was to be able to complete Sussman's SICM book (and later, his Functional Differential Geometry book). The book I used to do this was Advanced Calculus by Loomis & Sternberg because it covers classical mechanics, potential theory, differentiable (Banach) manifolds, differential equations, (multi)linear algebra, fundamental theorum of calculus and the Fourier transform. The exercises are not very difficult compared to a lot of other texts (Spivak's Calculus) so this is an accessible math book as I don't have any formal math training. I also liked the Mathematical Preliminaries crash course in the Art of Programming Vol 1 because it led me to looking into probability which has turned out to be an infinite rabbit hole of discovery. (a grad students highly opinionated list) https://www.amazon.com/gp/richpub/listmania/fullview/1F85VWNRTMY2T/ref=cm_pdp_lm_all_itms https://www.amazon.com/gp/richpub/listmania/fullview/1F85VWN...
ivan_ah 10y ago
I am totally biased, but I think my book is a good introduction to math fundamentals (high school) and calculus: https://minireference.com/ https://minireference.com/ preview: https://minireference.com/static/excerpts/noBSguide_v5_preview.pdf https://minireference.com/static/excerpts/noBSguide_v5_previ... As an added bonus, the book also covers mechanics (Physics 101), which is really important to understand to build up you general modelling skills. (Physics is all about coming up with math models to describe reality.)

friendly_chap 10y ago

I feel the same way when solving tasks in my day job. The thing I tell to young people learning programming/tech that I hope they don't get frustrated easily, because they will spend every day of their life feeling rather stupid and confused, never knowing when will they discover a solution for a particular problem. This is something that was a great source of stress early in my career.

Swizec 10y ago
It's a game of educated guessing. Experience helps.
- friendly_chap 10y ago
  Indeed, but the more experienced I get the harder/bigger problems I work on, so that balances it out. I still feel rather stuck on a daily basis, but I accept the situation and try to deal with it with a calm focus.
minipci1321 10y ago
I always thought that in the end, one has got to like facing these hard problems every day on the job in order to thrive in this field (I am in tech too). This is something I try to detect in candidates in job interviews I conduct. I noticed people can be roughly split in two categories: -- ones who spend a couple of years dealing with this sort of job activities, then want out by any means, on to something different; -- others enjoy it more and more as they gain expertise, start having more and more fun -- get more expertise, and with results (which necessarily come up), the right to pick and choose the subjects and so the hard problems that come with them. I happen to belong to the second category (and it has been a while I stay in the same domain), and I don't believe someone in that category would feel stupid and confused. it rather feels like investigation every time. Though over the years, I've become wary of the stress always associated with being put on such issues -- a last-minute demo for a trade show that doesn't work, a customer who has escalated to upper management, a delivery which is hopelessly late... -- with all attention of the management attracted. I keep telling myself that a soldier has got to participate in war campaigns, not be doing paperwork at the headquarters, and literally force myself in last years. But I don't recall I have ever felt stupid or confused, even in the beginning of the career.

te_platt 10y ago

This reminds me of the book "The Perfect Wrong Note". The book is focused on learning to play music but the principles it teaches apply to learning just about anything. The core message to not be afraid of mistakes during practice. Little kids fall over when they learn to walk, you'll have moments of confusion learning new things. There's a time to get things done well, like playing at a recital or releasing production code. There also needs to be time to practice and part of practicing is the expectation that there will be mistakes.

laingc 10y ago

To me, this is about "mathematical maturity". My observation is that many programmers, especially those who have come of age by working in startups, tend to value ability and sometimes experience over formal education. This is a result, I believe, of noticing that they can outperform many people who have a classical education, and also seeing that many of the people to whom they look up also do not have much in the way of formal education. However, I truly believe that Mathematics is a discipline that is very hard to engage with outside of formal education - or at least nobody has really found a great model for doing so yet. Learning Mathematics in a classical, structured way really does change the way you think. I notice a substantial difference even between those of my colleagues who entered industry straight after their Masters or even Bachelors, and those who completed Doctorates or even held postdoctoral positions. In my opinion, it is this lack of mathematical maturity that makes the switch from general programming to scientific programming more challenging than the converse.

okoksowhatis 10y ago
This is very well-written. The prosody of it illuminates the topic you're speaking about. Being so rare, I wonder what makes you different.
agentultra 10y ago
I disagree that mathematics itself is difficult to engage with outside of academia. There are some academic mathematicians who think mathematics has to be hard because it was difficult for them to understand. There are others for whom sharing their insights is more important than searching for new ones. I think that the set of mathematicians together have done a rather good job in the 20th century of producing mathematical literature and pedagogy that laypersons and the polymaths can understand. I skipped university when I was young due to circumstance. I still had a love for maths but couldn't pursue formal training. When I came around to programming as a career instead of odd-loner-hobby it was as you say. I was quite impressed with my ability to code circles around CS graduates who, for all their theory and training, weren't prepared for "coding in the real world." I never stopped studying mathematics. I've recently been going through temporal logics. If only more programmers were aware of the beauty of the proof of weak fairness. And its implications in system specifications. Where I can see formal training being useful is exposing you to subjects you were not aware of in short order. My experience has been more akin to the mansion metaphor. Where I encounter limitations or difficulties in my programming I look to maths for the right tool to simplify the task and ensure it is correct. So I agree about "mathematical maturity." I just don't believe it is exclusively acquired in an academic setting. I think some people have an inclination or awareness that pushes them to think this way.

vecter 10y ago

I have a great personal story that highlights how long the journey of understanding mathematics is. I took linear algebra my freshman year of college. It was the non-math major course, so it didn't require proofs. I got an A+ in the class. Not just an A, an A+. I was able to obtain such a high grade by taking tons of practice tests, and since the actual tests were basically mildly veiled calculations, I just had to map the question to the right calculation. So for instance, if after a little interpretation, I figured out that the question was asking for me to calculate the singular value decomposition of a given matrix, I would mindlessly compute, check my algebra, and move on. However, it was very clear to me by the end of the course that I didn't really understand what the heck linear algebra was about. Five years later, I started a job as an algorithmic trader. One of the first things my boss wanted to do was to do a Principal Component Analysis (PCA) of bond price movements. This is a very common thing to do. I didn't know what PCA was, but I read a short paper he gave me and I was able to grok it. After reading that paper and actually performing the PCA (which by the way was basically one line of R code), I finally came to understand the core essence of linear algebra, which is the idea of linear transformations. I was able to connect the equation Ax=lambdax to the geometry of what an eigenvector meant. Through a little more reasoning, I realized that every real matrix corresponded to a linear transformation of that space via a rotation, a reflection, a stretching, a shearing, etc. At that point, all of the mindless calculations I had been doing half a decade earlier instantly clicked, and I was enlightened. This was literally half a decade later after I "aced" my linear algebra class. I know that it seems absolutely ridiculous that I could "score so well" in a math class yet so clearly miss the core idea behind the entire class, but that's been my experience with math for as long as I can remember. You start by doing the calculations and just getting comfortable with them. Some arbitrary time later, you have an insight and suddenly everything is so crystal clear and trivial that you wonder how you could even not have understood it before. Oh, and even to this day, I don't understand what singular values actually are. Something to do with a mapping from the row space to the column space, blah blah. I'm sure if I spent an hour to read about them and picture the geometry, I could figure it out, but I just haven't gotten around to doing it.

lordnacho 10y ago
Wow, that's oddly similar to my story. LinAlg in college, nothing but a blur of matrices. Algo trading for work. Now there's a reason to do it, it makes sense. About SVD, btw, it is another path to PCA, one that solves certain problems that PCA does not. https://jeremykun.com/2016/04/18/singular-value-decomposition-part-1-perspectives-on-linear-algebra/ https://jeremykun.com/2016/04/18/singular-value-decompositio...

auvrw 10y ago

tao's stages of math is relevant https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/there%E2%80%99s...

deleted 10y ago

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MikeNomad 10y ago

Shouldn't the year the article was written (2014) be included in the title?

l3robot 10y ago

What a great blog post! Totally agree with him. And, personnaly, it is why i'm having so much fun doing maths. Everytime it's a new exploration, a new challenge. My best math teacher I had was seeing math with this philosophy in mind and his class was like discovering new lands every time. I'm sure that if we explained in a way that failing a math problem is as normal and challenging that failing a Mario Bros Level, more people would be in peace with it.

gauruvbose 10y ago

"mathematicians don’t work like this"? Sure they do. Reading textbooks is normal. As a mathematician, this post is foreign.

Koshkin 10y ago

One of the ways to acquire a taste for mathematics is to try solving elementary but challenging problems, such as those included in MathCamp's qualifying quizzes: http://www.mathcamp.org/prospectiveapplicants/quiz/pastquizzes.php http://www.mathcamp.org/prospectiveapplicants/quiz/pastquizz....

jondubois 10y ago

My favourite quote about Mathematics is from John von Neumann: "In mathematics you don't understand things. You just get used to them." - This quote highlights precisely why I ended up choosing software engineering over maths. I'm just not very good at applying processes/methodologies which I don't fully understand. For example, I wasn't very good with linear algebra until I was able to visualize the equations in my head. For example, now, when I think about the equation 'f(x) = ax^2 + bx + c' - I can see that this represents the set of all possible quadratic equations and I can roughly visualize what that looks like on a cartesian plane (well it would turn the whole plane black because there would be an infinite number of graphs). Then if I choose any three points on that crowded cartesian plane, I can visualize that among this infinite set of curves, one of them passes through all three points. Thinking about it in that way allows me to make sense of Gauss-Jordan Reduction and other mathematical processes related to linear algebra. Programming is much easier for me because I can visualize the results instantly on a computer - I don't need someone else to explain it to me. Any uncertainty can be quickly resolved by simply running some code.

pixl97 10y ago
>Any uncertainty can be quickly resolved by simply running some code. Until you have to interact with a black box of someone else's code. You can only be certain that the data you sent that particular time works, not that all possibilities of valid data work.
- jondubois 10y ago
  I guess with programming, you can never get 100% certainty when it comes to correctness of your code - Unless you perform formal analysis which is impractical for most use cases.
  - chipsy 10y ago
    You might not succeed at 100%, but you can remove "classes of error" by adopting particular styles or techniques backed by formal analysis. That's one of the biggest appeals of compiler technology - it can encode an understanding of patterns proven to detect failure, and in so doing lower your resulting bug count.

pm24601 10y ago

All they have to do is read this article: https://news.ycombinator.com/item?id=11874395 https://news.ycombinator.com/item?id=11874395

riazrizvi 10y ago

Beautiful article. Love the advice at the end!: "What’s much more useful is recording what the deep insights are, and storing them for recollection later. Because every important mathematical idea has a deep insight, and these insights are your best friends. They’re your mathematical “nose,” and they’ll help guide you through the mansion."

abc_lisper 10y ago
I can't think of a better argument to goad programmers into writing code comments, especially for elegant code. For some reason, programmers think good code should be self explanatory. It doesn't have to - if your insight is outside the flow (or the argument) of your code. I see a lot of Java code written like this; if the programmer's job is to multiply two numbers, they add number a number b times, and say comments are not needed!

j2kun 10y ago

Author here. I can't help but plug my mailing list for a book I'm writing, called "A Programmer's Introduction to Mathematics." Cheers, and thanks for reading! https://jeremykun.com/2016/04/25/book-mailing-list/ https://jeremykun.com/2016/04/25/book-mailing-list/

jonstokes 10y ago

A mathematician was walking home from campus one day, and as he walked he was pondering a particularly thorny problem. At one point, he snapped out of his reverie and looked around and realized that he had no idea where he was. He saw a young boy playing with a ball in a yard, and figured maybe the boy could tell him the way home. So he says to the boy, "young man, do you know where Prof. So-and-so lives?" The boy looked at him and said, "Dad, what's wrong with you?"

selimthegrim 10y ago
This story is usually told about Norbert Wiener and his daughter.
- jonstokes 10y ago
  Good to know.

justifier 10y ago

any discipline where you are attempting to answer yet to be answered questions leaves you in a state of chronic confusion and lacking direction math the same

pjlegato 10y ago

> Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark... What are you supposed to do if you like math and the idea of grokking it, but you also have a job and a family and can't afford to spend six months contemplating each room in the mansion?

dragonwriter 10y ago
> What are you supposed to do if you like math and the idea of grokking it, but you also have a job and a family and can't afford to spend six months contemplating each room in the mansion? Learn to come to grips with the idea that the universe isn't always going to support your mutually exclusive preferences?
- pjlegato 10y ago
  Yep, that's probably the only option, huh.