15 ms·

Mathematicians are chronically lost and confused

300 points by aditgupta 13y ago

jmnicolas 13y ago
Before a few articles of this kind I never suspected there was such depth in Maths. There's already so much to learn in programming, but I'm sure I'd love to dive in Maths (without the pressure of school like "understand this or you're an idiot").
- pseut 13y ago
  Without that pressure, you won't learn it. :)
ColinWright 13y ago
I highly recommend reading this. I didn't agree 100% with everything, and you probably won't either, but it's an excellent insight into what learning and doing math is about, and what it's like. I'd love to read alternate viewpoints, but this is an excellent read.
Bahamut 13y ago
I think this is a good read, although I don't agree with all of it - I'm of the mind that there is immense value in being able to figure out difficult proofs. The process develops your logical ability.
- michaelochurch 13y ago
  I'm of the mind that there is immense value in being able to figure out difficult proofs. Absolutely. However, the rabbit hole is very deep. Many papers make leaps from one sentence to the next that, if you're not familiar with the field, can take a couple days to figure out. Even then, real world proofs are informal and therefore not air-tight. They're close enough, almost always, but there's a reason why a mathematical proof isn't considered valid unless it's lived for two years under peer scrutiny. One could drill down to formal proof in the Godelian sense, in which proofs are mere typography and can be checked mechanically, but that's not how most of real mathematics is done and, practically speaking, most of it can't be done that way and remain useful to humans (like assembly language, it's too low-level for most applications).
  - nnq 13y ago
    > most of it can't be done that way and remain useful to humans (like assembly language, it's too low-level for most applications). Sincere question (I'm not a mathematician): why can't it be done that way?! On top of an assembly language you can create a higher level language and on top of that an even higher level one, and it is airtight, it has to be or the code won't compile or will throw a runtime exception, the compiler or interpreter doesn't just "roll a dice" when it comes across and ambiguous statement! You just can't have ambiguous statements, so starting from a "precise" assembler everything else built on top can absolutely be "air tight" at the language level. (Now concerning what the program actually ends up doing (like something else than you intended), or that sometimes you trade off security for speed and get a buffer overflow, ok, these things happen, but not at the language level! usually, and when they do - like C programs exploiting undefined but known for certain targets compiler behavior this is either advanced malicious obsfucation or random rookie mistakes.) So explaining the question: why can't one build a higher level mathematical language bottom up, starting from an "assembler" of machine-checkable proof steps and building one or a few levels of higher level human-friendly languages that still map unambiguously to the lower level one? Just because mathematical language has evolved in a top down fashion, starting with describing proofs in words or symbols derived from words, and then developing more an more precise language and systems, it doesn't mean that one can't go the reverse route, bottom up, an maybe meet closer to the top in a way, so that the resulting new mathematical language will be similar enough to classical one not to scare everyone away, right? ...and the benefits seem immense! Imagine: (1) replacing years of peer review replaced by machine checking basic correcting (+ some machine testing on huge data samples, for testable proofs, just to be sure there was no bug) (2) AI expert systems bringing real contributions to math by actually discovering new proofs AND providing them in a language understandable for humans, so humans learn from them and discover new techniques EDIT+: (3) allowing the development of much more advanced theories, because just as in software you can build much larger systems once you learn how to write more "bug free" code, the actual complexity of the proof could be much larger and maybe new realms of mathematical will become accessible to human understanding once we have a "linguistic aid" to reducing the percent of faulty proofs and the time spent debugging them
    - 13y ago
- DerpDerpDerp 13y ago
  I'd agree that there is a place for complicated detail work in mathematics, but I don't think that's really what mathematics is about. It's more about finding good conjectures, and realizing which theorems and lemmas are actually worth proving, because they say something interesting about the topic. By analogy, it's certainly important to understand complex grammar, have a large vocabulary, etc in order to be a writer, and develop a good sense of word choice, but that's not really what writing is about; writing is fundamentally about the process by which we tell a story using these tools.
fidotron 13y ago
This misses the dangerous part, which is mathematicians in groups can confuse each other into accepting ideas which are basically nonsensical, especially if the counter argument relies on some obvious but intuitive observation of reality but cannot be easily formalised within their chosen framework of the moment. As a consequence of this it wouldn't surprise me if the overwhelming majority of maths was actually incoherent nonsense and that the people that understood this thought they were just very confused due to being shouted down all the time, when the really confused people are the ones oblivious to their own situation.
- tokenadult 13y ago
  I'd be a lot more worried about the danger you mention if you could give even one example of that happening, ever. What ideas are mathematicians confusing each other into accepting that are basically nonsensical?
  - fidotron 13y ago
    Cantor's conception of transfinite numbers is the one that I think has done most damage.
    - michaelochurch 13y ago
      How are transfinite numbers "nonsensical"? When you get into infinity, you have two notions of "number" that diverge. Mathematical operations on them do different things. (For example, cardinal "exponentiation" is the power set; ordinal "exponentiation" is something different and smaller.) One is size, but proper subsets can have the same size at infinity (integers, even numbers, rationals). That's where Aleph-0 (cardinality of the integers) and "c" (cardinality of the reals) come from. With cardinal infinities, you can't really do meaningful arithmetic because the field properties don't apply. "Infinity" violates the mathematical fact that x+1 != x, for example. The other notion comes from the concept of a well-ordered set, which also maps nicely to "indexes" into possibly infinite lists. With this foundation, you have more options in terms of mathematical manipulations: you can add ordinals (but not always subtract them) and, because they pertain to list operations, the traditional "field" properties aren't always commutative. That's where we get ω, ω+1, ω^2, ω^ω, ε_0 and so on. Those all have rigorous definitions. For example, ω^2 is the order type of ordered pairs of numbers with lexicographic comparison: (0, 0) < (0, 1) < ... < (0, 10^100000) < ... < (1, 0) < ... < (2, 0) < ... . ... and ω^ω is the order type of formal natural-number polynomials in one variable with lexicographic comparison: 0 < 1 < 10^100 < X < X+1 < X + 10^100 < 2*X < 10^100*X < X^2 < X^3 < X^3 + 1... Where things get messy is that the relationship between cardinal and ordinal numbers (more formally, what ordinal number has the same cardinality as the reals, or the continuum?) is, in fact, formally undecidable. (Continuum Hypothesis). That doesn't mean no one has solved it. It means there's no mathematical way to refute or prove it from ZFC, the Zermelo-Frankel set axioms plus the Axiom of Choice. The CH is neither true nor false, insofar as one can have valid mathematics with or without it. To put the above more succinctly, we know that the countable ordinals are a well-ordered set (totally ordered with a minimum) and since no set contains itself, that set is uncountable. It is, in fact, the smallest countable set (the ordinal numbers are totally ordered by the subset relation). That's called ω_1. Intuitively, we might hope that that's also the same "size" as the real numbers (we don't know of any smaller uncountable infinities, and we can't construct any). But there is no way to prove or refute whether that is true. Mathematics is valid either way; it has to "fork". It's not "nonsensical". What it is is formal. It may or may not map to the real world. You can't actually perform Banach-Tarski (Axiom of Choice hack) on an orange, nor can you store a complete Hamel basis on your hard drive. But these concepts are still useful in defining our notion of what a "set", precisely, is.
zacinbusiness 13y ago
I really enjoyed this because it captures so much of the frustration that felt early in my programming career - especially in college when I had classmates several years my junior who were (as far as I could tell) mathematics and programming wunderkinds. I also think that this is the sort of rhetoric that should be used to begin teaching children basic mathematics and more advanced concepts as well, because I still recall many of my classmates in elementary and even highschool who simply felt like failures or that they weren't smart enough to understand things because they didn't "get" it the first, or fourth, or fiftyth time.
gaius 13y ago
A svbtle article worth reading, deserves an upvote.
deleted 13y ago
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graycat 13y ago
No, the OP is giving bad advice. Reading good foundational text books carefully is darned good advice. But for solving every exercise before moving on, no, that's not a good idea. Instead, be willing to be happy solving some 90-99% of the exercises. For the rest, guess, with some evidence, that they are incorrectly stated, out of place, just too darned hard, or some such. If insist on solving 100%, then get on the Internet and look for solutions. Next, if read some foundational text books, then in each subject also read several competing text books, perhaps just one mostly but also look at least a little at the others for views from 'a different angle' that can be a big help. Why? Because likely no text book is perfect and, instead, in some places is awkward, unclear, misleading, clumsy, etc. So, views from a 'different angle' can make it much easier to learn both better and faster. His description of doing applications by just getting what really need and forgetting the rest can be done but is not so good. Instead, having a good foundation helps a lot. And, commonly for an application in an important field, there really is some good material in that field that should understand with the application. Else risk doing the application significantly less well than could have. His description from Wiles is more or less okay for doing some research but, really, not for learning. And for research, more of a 'strategic' overview, i.e., with the 'lay of the land', would be good, i.e., for publishing not just one okay, likely isolated, paper but a series of better papers that yield a nice 'contribution'.
- nbouscal 13y ago
  You misread the article. He is not recommending that students solve every exercise; he's recommending the exact opposite.
  - graycat 13y ago
    No, I'm correct: He set up an extreme straw man to knock it down. I clearly agreed that his extreme straw man is foolish. There is a common reason students fall for his straw man: They are concerned that if there is an exercise they can't work they are missing something important. My advice was, instead, for a very diligent student, to solve 90-99% of the exercises and just let go of the last few as illposed, stated in error, out of place, use the Internet, etc. To do just the "opposite" of his straw man is not good -- for solid foundational material, Halmos, Rudin, Royden, etc., the exercises are darned important. Right the Rudin exercises where have to consider uncountability are not so good. The Royden exercises on upper and lower semi-continuity are a lot of work for a little curiosity but likely won't see again. The Fleming exercise on every bounded linear functional on a intersection of finitely many closed half spaces achieves a maximum value is mis places. Etc. The abstract algebra book I had had an exercise where the student had to reinvent Sylow's theorem; a student wrote the author and got back a letter that the purpose of the exercise was to see if a student could reinvent Sylow's theorem -- bummer, misplaced exercise. I'm correct.
yomritoyj 13y ago
Mathematicians are indeed lost and confused but in a very different way from beginning students. One must put in one's dues in what Terence Tao calls the "rigorous" phase before one can become productively confused in the "post-rigorous" phase. http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/ http://terrytao.wordpress.com/career-advice/there%E2%80%99s-...
napowitzu 13y ago
This is true with many, many things. Very often it is the connections between ideas that yields the deep understanding, not the ideas themselves. Focusing too intensely on a single idea or subject results in not making connections and, consequently, not really understanding.
- helicon 13y ago
  I agree. I think most if not everything he said can be applied to learning almost any discipline. I studied physics at university and I definitely recognize the authors description.
- j2kun 13y ago
  I think that, for whatever reason, people tend to think mathematics is somehow different.
nly 13y ago
Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+). Nobody mentioned matrices, broader number theory or discrete transforms until I was in university. I studied EE not compsci. Things like algorithmic complexity I had to learn for myself and from Knuth. I'm trying to grok group theory right now to help with my understanding of crypto. Before this, it was never mentioned throughout my education, so I don't know what courses you would have had to take to learn that. The fact that I didn't even know group theory was important to crypto until after I had made the choice strikes me as a bad sign. The common theme at every level is learning cherry-picked skills, before you're even told what the branches of mathematics even are. Everything seems disjointed because you're not taught to look past the trees for the forest. Most people infact, even technical folk, go through their entire lives without knowing the forest even exists. Any idiot can point to a random part of their anatomy and posit that there's a field of study dedicated to it. The same goes for mechanics or computer science. You just can't do that with mathematics as a student. I loath academic papers. Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating. So many ideas seem closed to mortals because of the nature of mathematics.
- cs702 13y ago
  Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating. This is SO TRUE. The same thing happens to me regularly, and not just with "computer science" but with other technical fields, hard sciences, and mathematics. The purpose of most academic papers is not to explain (let alone teach!) ideas in an intuitive manner, but rather to express them in formal, correct, unambiguous terms -- that is, to make them as accurate and critique-proof as possible for publication in some journal.
minikomi 13y ago
First time I've intentionally kudos'd
acjohnson55 13y ago
I've felt this is the case for a long time. A lot of people have a smooth experience in math for years until they hit their first serious discontinuity. That could happen anywhere: times tables, fraction arithmetic, two-step equations, geometric proofs, radicals, limits, or maybe even college math. The reaction is nearly universal though. The person thinks, "holy crap, I guess I'm actually not good at math", anxiety strikes, and they freeze up. Some people find eventually find their way around this first road block, and future discontinuities in understanding become less stressful, and eventually understood to be a completely normal part the process. But the usual experience is that a person's math confidence is blown and as the math truck barrels on ahead, they never catch up. They understandably accept the identity of not being "good at math". What's missing in math pedagogy at most schools is a systematic way to deal with the discontinuities when they strike, especially that first time. We can prepare students to deal with that panic. The tough part is that the math teacher probably has 90 students on roster, but the discontinuity could hit pretty much any given lesson, for some given student. I know so many people who have come back to intermediate math later in life and breezed through it, armed with intellectual confidence gained from other fields. They look back and wonder how they came to be so intimidated by math in their younger days. We've got to give younger people the tools and knowledge for overcoming this intimidation at a younger age. We've got to kill "I'm just not good at math".
- japhyr 13y ago
  as the math truck barrels on ahead I've been teaching math to at-risk high school students for the last 10 years. I have spent more time helping students understand that they are not stupid, that something just got in the way of their learning at one point, and they never understood anything after that. I'm going to use your quote in some of these conversations now. What most of my students think: "I could never do math, I fucking hate it, and I might drop out because I will never finish my math credits. I can't do math because it's stupid and meaningless and I will never get it." What really happened to get people off track? - Some just didn't follow one topic in some early grade, nothing else made sense after that, and no teacher was prepared to get them back on track. - Parents split up, student couldn't focus in school for 6 months, they got off track. - Parent/ sibling/ significant person passed away when student was young, couldn't focus for 6 months-2 years, no way to get back on track. Any number of other external events happen, and it is perfectly reasonable for students to get off track in math. a systematic way to deal with the discontinuities when they strike, especially that first time Exactly. I would like to see every elementary school have a math specialist, who knows advanced math, to help students with their overall understanding when they get off track. Helping a kid master some mechanics does a little to get them back on track, but diagnosing misunderstandings takes more math expertise than most elementary teachers have. I could go on forever; thank you for putting some of these issues so clearly in focus.
weavie 13y ago
I started off doing a combined maths and computer science degree. With both computer science and maths you are chronically confused. The difference being with computer science it doesn't matter so much if you don't understand something, if you can get it to work you know you are on the right track. Maths is much more progressive, each proof builds on a previous one. So if you fail to understand one step you are screwed from that point on. After the first year I realised I didn't actually enjoy being permanently confused and so I ditched the maths to focus on computers. I do regret this. It didn't take long at all before I forgot all that knowledge I had spent years sweating over.
egdelwonk 13y ago
What's the best way to relearn math?
- 11001 13y ago
  Textbooks and exercises. Avoid shortcuts such as online tutorials or anything with "for hackers" in the title.
  - egdelwonk 13y ago
    I've been using Khan Academy to refresh my skills and it's been really helpful, but do you have any recommendations for textbooks? I'm 10 years out of school and haven't really needed to flex my math muscles in years.
    - 11001 13y ago
      Any particular "math" you're interested in? If you just want to get your foot in with pure math I'd recommend studying basic abstract algebra and analysis at the same time. For abstract algebra look into Hungerford (Intro, not his grad text), for analysis, maybe Rudin, or Kolmogorov and Smirnov.
    - j2kun 13y ago
      If you're past calculus, I'd say start with Sheldon Axler's Linear Algebra Done Right (it has the most immediate applications, and linear algebra is essential for almost all higher mathematics).
- dwaltrip 13y ago
  Focus as much as possible on the underlying concepts from any math topic and how they connect to other concepts. Try to boil these concepts down to the most simple, clear form that makes sense to you. Test your boiled down conceptual understanding by applying it to related exercises/problems you haven't tried before and seeing what happens. For each sub-topic, most math books give you the tools first and then teach you problems they should be used on. Read the problems first, and think how you might solve them (don't expect to figure it out, but if you do, great!). Then, go back and learn the tools, trying mostly discern the "how" and the "why" as opposed to the "what". Math is all about "how" and the "why". As some motivation, whenever a new thing clicks, it is very satisfying! :) But it definitely is a tough process. Good luck!
dalke 13y ago
"If you’re going to get anywhere in learning mathematics, you need to learn to be comfortable not understanding something." This is true for all research. And I don't mean just the physical sciences either. Historians and sociologists are also chronically "lost and confused." Otherwise it wouldn't be a topic worth of study. This is why students who are "good at X", whether it be math, German, sports, or programming, may become frustrated when they find out that "good at researching X" is a very different matter.
- j2kun 13y ago
  I think it's a good point, but I still think the kind of lost and confused in mathematics is more embarrassingly extreme. Imagine a few hundred historians trying to discern when King George I died, and after 50 years of work they conclude, "All we know for sure is that it was between the day he was born and yesterday." A startlingly large part of mathematics feels like this. And I think the reason is that "prevailing theories" mean nothing in mathematics.
  - dalke 13y ago
    That's a poor comparison. I find it hard to believe that mathematicians are still trying to decide if the set {1, 2, 3} is finite or infinite. A "startling large part" of all science fields like this. Physicists don't even know if the gravitational mass of an object is really the same as its inertial mass. Or if there are true magnetic monopoles. And that's after over a century of trying. Immunologists have barely scratched the surface of how that field works. Economists make lots of conjectures, with lots of math to back it up, but it's not a perfect predictor of the human economic system. Biological evolution still surprises us, 150 years after Darwin and nearly 100 years after the neodarwinian synthesis. Chemists still don't come close to handling some of the reactions that natural systems have figured out. And so on.
    - j2kun 13y ago
      Good point.
mathattack 13y ago
It's strange to hear mathematics described more as a search for art and structure than computation. Unfortunately most of my math education was on the computational/applied side. I'm only getting into number theory and the more esoteric math later in life for fun. As a parent I think we can't let the school system destroy our kids love of math through too much rote learning. We have to make it fun for them. (Same with music btw)
bpyne 13y ago
I wish this post was around when I finished my undergraduate degree in Mathematics. I would have taken my adviser's advice to go to grad school. At the time, I remember telling him that I feel like a barely made it through the program. Apparently I wasn't alone. Amazing the difference 25 years and the internet makes.
japhyr 13y ago
I currently teach math to at-risk students. I don't read all of these submissions about math education, but I skim the comments on most of them. The comments people make change the way I teach math. I have always done a decent job of teaching math. I focus on helping students understand concepts, even when they are focusing on mechanics. I use words like "shortcut" and "more efficient method" rather than "trick" when showing students more efficient ways to solve problems. I have students do problems and projects that relate to their post-high-school goals. But with the routines of school life, I get away from the fun of math from time to time. The comments on these submissions often remind me to go in and just tell stories about math: - "Hey everyone, did you know that some infinities are bigger than other infinities?" - "Hey everyone, do you have any idea how your passwords are actually stored on facebook/ twitter/ etc.?" - "Have any of you heard the story about the elementary teacher who got mad at their class, and told everyone to add up all the numbers from 1 to 100? One kid did it in less than a minute, do you want to see how he did it?" Thanks everyone, for sharing your perspective on your own math education, and about how you use math in your professional lives as well. Your stories help.
- countrybama24 13y ago
  No, thank you for teaching math to at-risk kids. Seriously, bravo.
- csours 13y ago
  I love stories like these. I haven't gone much past calculus, what stories can I look for that will take me farther?
  - wetmore 13y ago
    You may recall that one can use the quadratic formula to find solutions to a quadratic equation (polynomial with highest term being x^2). Did you know that (one) guy that proved there is no such formula for quintic equations (highest term x^5) died in a duel when he was only 20? [1] There is a town with a particular rule when it comes to facial hair: those who do not shave themselves are shaved by the barber. But then who shaves the barber? [2] The other poster mentioned different infinities. One "size" of infinity is called "countable infinity" and is the infinity describing the size of the natural numbers (1,2,3,...). Say we have a hotel with a countable infinite number of rooms. I've been travelling all day and I show up at the hotel, and talk to the clerk at the front desk. He tells me every room is full, but when he sees the sad look on my face he tells me not to worry - he can make room for me. He simply moves the person in room 1 to room 2, the person in room 2 to room 3, room 3 to room 4, etc... And then the first room is empty for me, and everyone still has a room. [3] [1] http://en.wikipedia.org/wiki/%C3%89variste_Galois http://en.wikipedia.org/wiki/%C3%89variste_Galois [2] http://en.wikipedia.org/wiki/Russell's_paradox http://en.wikipedia.org/wiki/Russell's_paradox [3] http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_...
chwolfe 13y ago
The entire post was enjoyable but I found the last paragraph to have the most actionable advice: 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.
vsbuffalo 13y ago
Funny, this title is the same as a recent thread on HN: https://news.ycombinator.com/item?id=7331791 https://news.ycombinator.com/item?id=7331791
- j2kun 13y ago
  That comment received so much positive feedback that I decided to write an article expanding on the idea :)
ChristianMarks 13y ago
Fair enough. I tried in my youth to solve every problem I came across. There were many I couldn't solve. It took a while before I developed the wisdom and discipline not to solve every problem no matter how long it took. By a while I mean decades. I sacrificed the possibility of family life, have stopped talking to my uncomprehending stepfather, and have kept my social interactions to an absolute minimum to pursue my consuming interest. (I mention this as a point of pride.) I find myself continually astonished by the ingenuity of solutions I probably could never have imagined after years of work. Perhaps, after a lifetime of effort that must be continually maintained, I have attained the level an entering freshman at Harvard. At this stage, I may be reduced at best to connoisseurship of some aspects of mathematics. Now for some reflections on attitudes. Mathematicians sometimes act as if they believe that expertise in mathematics transfers to expertise in mathematics education. Suppose you are a sensitive student, lacking in confidence. You open Korner's beautiful book on Fourier Analysis, and the first thing you are greeted with is "This book is meant neither as a drill book for the successful student nor as a lifebelt for the unsuccessful student." Korner does not mention other references suitable for the successful and the unsuccessful student. You take this comment to mean that Korner would let the unsuccessful student drown. There is no implication, but this is the psychological import, the implicature. Why mention the unsuccessful student at all? Why not say who the book is for, without planting this gratuitous image in the reader's mind? It would take some time to return to this book, to get past the wonder at a mind capable of such an incidental, dismissive, off-handed acknowledgement of "the unsuccessful student." You could say this is "overthinking." Such remarks, microagressions as they are termed today, "perpetrated against those due to gender, sexual orientation, and ability status", are sometimes revealed in the asides of mathematical authors [1]. And now if only mathematics educators would evaluate their students on the state of their confusion! [1] http://en.wikipedia.org/wiki/Microaggression http://en.wikipedia.org/wiki/Microaggression
api 13y ago
I completely agree about the power of math, and why programmers should learn it. There are two problems with math: (1) Math is IMHO the worst taught of all academic subjects. It's taught as if it were not a language. Math profs and books on mathematics never explain what the symbols mean. They just throw symbols at you and then do tricks with them and expect you to figure out that this symbol means "derivative" in this context. I have literally seen math texts that never explain the language itself, introducing reams of new math with no definitions for mathematical notation used. I've looked for a good "dictionary of math" -- a book that explains every mathematical notation in existence and what it means conceptually -- and have never found such a thing. It's like some medieval guild craft that is passed down only by direct lineage among mathematicians. Concepts are often never explained either. I remember struggling in calculus. The professor showed us how to do a derivative, so I mechanically followed but had no idea why I was doing what I was doing. I called up my father and he said one single sentence to me: "A derivative is a rate of change." A derivative is a rate of change. I completed his thought: so an integral is its inverse. Bingo. From then on I understood calculus. The professor never explained this, and the textbook did in such an unclear and oblique way that the concept was never adequately communicated. It's one g'damn sentence! The whole of calculus! Just f'ing say it! "A derivative is a rate of change!" (2) The notation is horrible. If math were a programming language it would be C++, maybe even Perl. There are many symbols to do the same thing. Every sub-discipline or application-area of mathematics seems to have its own quirky style of notation and sometimes these styles even conflict with each other. Yet baroque languages like C++ and Perl at least document their syntax. If you read an intro to C++ book it begins its chapter on templates by explaining both what templates are for and the fact that type<int> means "type is templated on int." Math doesn't do this. It doesn't explain its syntax. See point #1 above.
- j2kun 13y ago
  I agree wholeheartedly with how frustrating it is. I think part of the problem is that really great mathematicians are encouraged to stay as far away from teaching (and improving their teaching) as possible, and great teachers are often discouraged from pursuing more mathematics for a variety of reasons. And when I personally teach calculus I make sure to explain derivatives in the way you want in the very first day (before describing limits or anything else). As to your second point, I think notation is a big problem, but it's a bit of a straw man. With very few exceptions that I doubt you would ever find yourself in, I have never met a professor or mathematician that would not explain notation if you asked (gladly stopping in the middle of a lecture or talk to clarify). There is still a lot of it, but every mathematician who is presenting the mathematics can explain the notation to any degree of precision you could ever want, and I have very few colleagues who have never stopped someone for this reason. I think the bigger problem is trying to read mathematics by yourself, without the ability to ask questions. And even after understanding the notation, I feel programmers have bigger problems, which I've expanded more on in this post [1], the main difference between learning programming being there are simply more free and open resources for learning programming. This is probably because programmers invented the internet and filled it with their favorite content first. But one point I make is that mathematical notation is inherently ad-hoc, and the only kinds of notation that stick around are the kinds that get used ad-hoc enough times to become standard. And even then people will make up their own notation for no other reason than that it's their favorite (Physicists are really good at this, and perhaps ironically it drives mathematicians crazy). Because of that (and because notation is introduced often to be rigorous, not to explain a concept) you're unlikely to ever find such a dictionary. Sorry :( [1]: http://jeremykun.com/2013/02/08/why-there-is-no-hitchhikers-guide-to-mathematics-for-programmers/ http://jeremykun.com/2013/02/08/why-there-is-no-hitchhikers-...
trevorhartman 13y ago
Jeremy, I really appreciate this post and all the excellent content over at Math ∩ Programming. Thanks, and please keep it up!
- j2kun 13y ago
  I'm just so happy that I get to read everyone's interesting stories and thoughts in the HN comment threads! HN is really one of the highest-quality places for discussion on the web ^_^
GIFtheory 13y ago
Reminds me of this great quotation, which Oksendal places before the preface to his stochastic differential equations book: We have not succeeded in answering all our problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things. Posted outside the mathematics reading room, Tromsø University
baby 13y ago
That's what I tell people around me. Studying math is hard because it makes you feel stupid. You always feel lost, you always feel like you missed so many things when you're starting to learn a new thing, you always feel like your questions are stupid (until you get that the rest of the class is pointless as well). Especially with talented professors (Lyon 1, France, the professors there are not really good educators, but they are geniuses), they make you feel bad for not understanding things that seem so simple to them. Studying math is depressive if you take it too seriously.
beltex 13y ago
Loved this post. FYI, the Andrew Wiles quote is from the opening of an awesome BBC documentary about how he solved Fermat's Last Theorem - http://www.youtube.com/watch?v=7FnXgprKgSE http://www.youtube.com/watch?v=7FnXgprKgSE
nilkn 13y ago
Heh, I knew this title seemed awfully familiar. Here's the discussion on Hacker News which (presumably) spawned this: https://news.ycombinator.com/item?id=7331693 https://news.ycombinator.com/item?id=7331693
edtechdev 13y ago
"If you’re going to get anywhere in learning mathematics, you need to learn to be comfortable not understanding something." That's true of everything. It's fear and anxiety that prevents a lot of people from learning and trying new things. I keep trying to tell students or family members when they are learning to do stuff on the computer, just right click everything, just google anything you can think of, don't worry about it being perfect, don't worry about breaking anything. You have to hold back showing them the "answers" or else they become dependent.

300 points by aditgupta 13y ago

Before a few articles of this kind I never suspected there was such depth in Maths. There's already so much to learn in programming, but I'm sure I'd love to dive in Maths (without the pressure of school like "understand this or you're an idiot").

pseut 13y ago
Without that pressure, you won't learn it. :)

ColinWright 13y ago

I highly recommend reading this. I didn't agree 100% with everything, and you probably won't either, but it's an excellent insight into what learning and doing math is about, and what it's like. I'd love to read alternate viewpoints, but this is an excellent read.

Bahamut 13y ago

I think this is a good read, although I don't agree with all of it - I'm of the mind that there is immense value in being able to figure out difficult proofs. The process develops your logical ability.

michaelochurch 13y ago
I'm of the mind that there is immense value in being able to figure out difficult proofs. Absolutely. However, the rabbit hole is very deep. Many papers make leaps from one sentence to the next that, if you're not familiar with the field, can take a couple days to figure out. Even then, real world proofs are informal and therefore not air-tight. They're close enough, almost always, but there's a reason why a mathematical proof isn't considered valid unless it's lived for two years under peer scrutiny. One could drill down to formal proof in the Godelian sense, in which proofs are mere typography and can be checked mechanically, but that's not how most of real mathematics is done and, practically speaking, most of it can't be done that way and remain useful to humans (like assembly language, it's too low-level for most applications).
- nnq 13y ago
  > most of it can't be done that way and remain useful to humans (like assembly language, it's too low-level for most applications). Sincere question (I'm not a mathematician): why can't it be done that way?! On top of an assembly language you can create a higher level language and on top of that an even higher level one, and it is airtight, it has to be or the code won't compile or will throw a runtime exception, the compiler or interpreter doesn't just "roll a dice" when it comes across and ambiguous statement! You just can't have ambiguous statements, so starting from a "precise" assembler everything else built on top can absolutely be "air tight" at the language level. (Now concerning what the program actually ends up doing (like something else than you intended), or that sometimes you trade off security for speed and get a buffer overflow, ok, these things happen, but not at the language level! usually, and when they do - like C programs exploiting undefined but known for certain targets compiler behavior this is either advanced malicious obsfucation or random rookie mistakes.) So explaining the question: why can't one build a higher level mathematical language bottom up, starting from an "assembler" of machine-checkable proof steps and building one or a few levels of higher level human-friendly languages that still map unambiguously to the lower level one? Just because mathematical language has evolved in a top down fashion, starting with describing proofs in words or symbols derived from words, and then developing more an more precise language and systems, it doesn't mean that one can't go the reverse route, bottom up, an maybe meet closer to the top in a way, so that the resulting new mathematical language will be similar enough to classical one not to scare everyone away, right? ...and the benefits seem immense! Imagine: (1) replacing years of peer review replaced by machine checking basic correcting (+ some machine testing on huge data samples, for testable proofs, just to be sure there was no bug) (2) AI expert systems bringing real contributions to math by actually discovering new proofs AND providing them in a language understandable for humans, so humans learn from them and discover new techniques EDIT+: (3) allowing the development of much more advanced theories, because just as in software you can build much larger systems once you learn how to write more "bug free" code, the actual complexity of the proof could be much larger and maybe new realms of mathematical will become accessible to human understanding once we have a "linguistic aid" to reducing the percent of faulty proofs and the time spent debugging them
  - 13y ago
DerpDerpDerp 13y ago
I'd agree that there is a place for complicated detail work in mathematics, but I don't think that's really what mathematics is about. It's more about finding good conjectures, and realizing which theorems and lemmas are actually worth proving, because they say something interesting about the topic. By analogy, it's certainly important to understand complex grammar, have a large vocabulary, etc in order to be a writer, and develop a good sense of word choice, but that's not really what writing is about; writing is fundamentally about the process by which we tell a story using these tools.

fidotron 13y ago

This misses the dangerous part, which is mathematicians in groups can confuse each other into accepting ideas which are basically nonsensical, especially if the counter argument relies on some obvious but intuitive observation of reality but cannot be easily formalised within their chosen framework of the moment. As a consequence of this it wouldn't surprise me if the overwhelming majority of maths was actually incoherent nonsense and that the people that understood this thought they were just very confused due to being shouted down all the time, when the really confused people are the ones oblivious to their own situation.

tokenadult 13y ago
I'd be a lot more worried about the danger you mention if you could give even one example of that happening, ever. What ideas are mathematicians confusing each other into accepting that are basically nonsensical?
- fidotron 13y ago
  Cantor's conception of transfinite numbers is the one that I think has done most damage.
  - michaelochurch 13y ago
    How are transfinite numbers "nonsensical"? When you get into infinity, you have two notions of "number" that diverge. Mathematical operations on them do different things. (For example, cardinal "exponentiation" is the power set; ordinal "exponentiation" is something different and smaller.) One is size, but proper subsets can have the same size at infinity (integers, even numbers, rationals). That's where Aleph-0 (cardinality of the integers) and "c" (cardinality of the reals) come from. With cardinal infinities, you can't really do meaningful arithmetic because the field properties don't apply. "Infinity" violates the mathematical fact that x+1 != x, for example. The other notion comes from the concept of a well-ordered set, which also maps nicely to "indexes" into possibly infinite lists. With this foundation, you have more options in terms of mathematical manipulations: you can add ordinals (but not always subtract them) and, because they pertain to list operations, the traditional "field" properties aren't always commutative. That's where we get ω, ω+1, ω^2, ω^ω, ε_0 and so on. Those all have rigorous definitions. For example, ω^2 is the order type of ordered pairs of numbers with lexicographic comparison: (0, 0) < (0, 1) < ... < (0, 10^100000) < ... < (1, 0) < ... < (2, 0) < ... . ... and ω^ω is the order type of formal natural-number polynomials in one variable with lexicographic comparison: 0 < 1 < 10^100 < X < X+1 < X + 10^100 < 2*X < 10^100*X < X^2 < X^3 < X^3 + 1... Where things get messy is that the relationship between cardinal and ordinal numbers (more formally, what ordinal number has the same cardinality as the reals, or the continuum?) is, in fact, formally undecidable. (Continuum Hypothesis). That doesn't mean no one has solved it. It means there's no mathematical way to refute or prove it from ZFC, the Zermelo-Frankel set axioms plus the Axiom of Choice. The CH is neither true nor false, insofar as one can have valid mathematics with or without it. To put the above more succinctly, we know that the countable ordinals are a well-ordered set (totally ordered with a minimum) and since no set contains itself, that set is uncountable. It is, in fact, the smallest countable set (the ordinal numbers are totally ordered by the subset relation). That's called ω_1. Intuitively, we might hope that that's also the same "size" as the real numbers (we don't know of any smaller uncountable infinities, and we can't construct any). But there is no way to prove or refute whether that is true. Mathematics is valid either way; it has to "fork". It's not "nonsensical". What it is is formal. It may or may not map to the real world. You can't actually perform Banach-Tarski (Axiom of Choice hack) on an orange, nor can you store a complete Hamel basis on your hard drive. But these concepts are still useful in defining our notion of what a "set", precisely, is.

zacinbusiness 13y ago

I really enjoyed this because it captures so much of the frustration that felt early in my programming career - especially in college when I had classmates several years my junior who were (as far as I could tell) mathematics and programming wunderkinds. I also think that this is the sort of rhetoric that should be used to begin teaching children basic mathematics and more advanced concepts as well, because I still recall many of my classmates in elementary and even highschool who simply felt like failures or that they weren't smart enough to understand things because they didn't "get" it the first, or fourth, or fiftyth time.

gaius 13y ago

A svbtle article worth reading, deserves an upvote.

deleted 13y ago

[deleted]

graycat 13y ago

No, the OP is giving bad advice. Reading good foundational text books carefully is darned good advice. But for solving every exercise before moving on, no, that's not a good idea. Instead, be willing to be happy solving some 90-99% of the exercises. For the rest, guess, with some evidence, that they are incorrectly stated, out of place, just too darned hard, or some such. If insist on solving 100%, then get on the Internet and look for solutions. Next, if read some foundational text books, then in each subject also read several competing text books, perhaps just one mostly but also look at least a little at the others for views from 'a different angle' that can be a big help. Why? Because likely no text book is perfect and, instead, in some places is awkward, unclear, misleading, clumsy, etc. So, views from a 'different angle' can make it much easier to learn both better and faster. His description of doing applications by just getting what really need and forgetting the rest can be done but is not so good. Instead, having a good foundation helps a lot. And, commonly for an application in an important field, there really is some good material in that field that should understand with the application. Else risk doing the application significantly less well than could have. His description from Wiles is more or less okay for doing some research but, really, not for learning. And for research, more of a 'strategic' overview, i.e., with the 'lay of the land', would be good, i.e., for publishing not just one okay, likely isolated, paper but a series of better papers that yield a nice 'contribution'.

nbouscal 13y ago
You misread the article. He is not recommending that students solve every exercise; he's recommending the exact opposite.
- graycat 13y ago
  No, I'm correct: He set up an extreme straw man to knock it down. I clearly agreed that his extreme straw man is foolish. There is a common reason students fall for his straw man: They are concerned that if there is an exercise they can't work they are missing something important. My advice was, instead, for a very diligent student, to solve 90-99% of the exercises and just let go of the last few as illposed, stated in error, out of place, use the Internet, etc. To do just the "opposite" of his straw man is not good -- for solid foundational material, Halmos, Rudin, Royden, etc., the exercises are darned important. Right the Rudin exercises where have to consider uncountability are not so good. The Royden exercises on upper and lower semi-continuity are a lot of work for a little curiosity but likely won't see again. The Fleming exercise on every bounded linear functional on a intersection of finitely many closed half spaces achieves a maximum value is mis places. Etc. The abstract algebra book I had had an exercise where the student had to reinvent Sylow's theorem; a student wrote the author and got back a letter that the purpose of the exercise was to see if a student could reinvent Sylow's theorem -- bummer, misplaced exercise. I'm correct.

yomritoyj 13y ago

Mathematicians are indeed lost and confused but in a very different way from beginning students. One must put in one's dues in what Terence Tao calls the "rigorous" phase before one can become productively confused in the "post-rigorous" phase. http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/ http://terrytao.wordpress.com/career-advice/there%E2%80%99s-...

napowitzu 13y ago

This is true with many, many things. Very often it is the connections between ideas that yields the deep understanding, not the ideas themselves. Focusing too intensely on a single idea or subject results in not making connections and, consequently, not really understanding.

helicon 13y ago
I agree. I think most if not everything he said can be applied to learning almost any discipline. I studied physics at university and I definitely recognize the authors description.
j2kun 13y ago
I think that, for whatever reason, people tend to think mathematics is somehow different.

nly 13y ago

Secondary math education, for me in the UK, didn't deal with anything outside of elementary algebra, Euclidean geometry, some statistics, and relatively simple calculus. Nobody talked to us about imaginary or complex numbers, or bayes theorem, decision theory, or non-trivial mechanics problems until I was in college (age 16+). Nobody mentioned matrices, broader number theory or discrete transforms until I was in university. I studied EE not compsci. Things like algorithmic complexity I had to learn for myself and from Knuth. I'm trying to grok group theory right now to help with my understanding of crypto. Before this, it was never mentioned throughout my education, so I don't know what courses you would have had to take to learn that. The fact that I didn't even know group theory was important to crypto until after I had made the choice strikes me as a bad sign. The common theme at every level is learning cherry-picked skills, before you're even told what the branches of mathematics even are. Everything seems disjointed because you're not taught to look past the trees for the forest. Most people infact, even technical folk, go through their entire lives without knowing the forest even exists. Any idiot can point to a random part of their anatomy and posit that there's a field of study dedicated to it. The same goes for mechanics or computer science. You just can't do that with mathematics as a student. I loath academic papers. Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating. So many ideas seem closed to mortals because of the nature of mathematics.

cs702 13y ago
Often I find I spend days or weeks deciphering mathematics in compsci papers only to find the underlying concept is intuitive and plain, but you're forced to learn it bottom up, constructing the authors original genius from the cryptic scrawlings they left in their paper... and you realise a couple of block diagrams and a few short paragraphs could have made the process a lot less frustrating. This is SO TRUE. The same thing happens to me regularly, and not just with "computer science" but with other technical fields, hard sciences, and mathematics. The purpose of most academic papers is not to explain (let alone teach!) ideas in an intuitive manner, but rather to express them in formal, correct, unambiguous terms -- that is, to make them as accurate and critique-proof as possible for publication in some journal.

minikomi 13y ago

First time I've intentionally kudos'd

acjohnson55 13y ago

I've felt this is the case for a long time. A lot of people have a smooth experience in math for years until they hit their first serious discontinuity. That could happen anywhere: times tables, fraction arithmetic, two-step equations, geometric proofs, radicals, limits, or maybe even college math. The reaction is nearly universal though. The person thinks, "holy crap, I guess I'm actually not good at math", anxiety strikes, and they freeze up. Some people find eventually find their way around this first road block, and future discontinuities in understanding become less stressful, and eventually understood to be a completely normal part the process. But the usual experience is that a person's math confidence is blown and as the math truck barrels on ahead, they never catch up. They understandably accept the identity of not being "good at math". What's missing in math pedagogy at most schools is a systematic way to deal with the discontinuities when they strike, especially that first time. We can prepare students to deal with that panic. The tough part is that the math teacher probably has 90 students on roster, but the discontinuity could hit pretty much any given lesson, for some given student. I know so many people who have come back to intermediate math later in life and breezed through it, armed with intellectual confidence gained from other fields. They look back and wonder how they came to be so intimidated by math in their younger days. We've got to give younger people the tools and knowledge for overcoming this intimidation at a younger age. We've got to kill "I'm just not good at math".

japhyr 13y ago
as the math truck barrels on ahead I've been teaching math to at-risk high school students for the last 10 years. I have spent more time helping students understand that they are not stupid, that something just got in the way of their learning at one point, and they never understood anything after that. I'm going to use your quote in some of these conversations now. What most of my students think: "I could never do math, I fucking hate it, and I might drop out because I will never finish my math credits. I can't do math because it's stupid and meaningless and I will never get it." What really happened to get people off track? - Some just didn't follow one topic in some early grade, nothing else made sense after that, and no teacher was prepared to get them back on track. - Parents split up, student couldn't focus in school for 6 months, they got off track. - Parent/ sibling/ significant person passed away when student was young, couldn't focus for 6 months-2 years, no way to get back on track. Any number of other external events happen, and it is perfectly reasonable for students to get off track in math. a systematic way to deal with the discontinuities when they strike, especially that first time Exactly. I would like to see every elementary school have a math specialist, who knows advanced math, to help students with their overall understanding when they get off track. Helping a kid master some mechanics does a little to get them back on track, but diagnosing misunderstandings takes more math expertise than most elementary teachers have. I could go on forever; thank you for putting some of these issues so clearly in focus.

weavie 13y ago

I started off doing a combined maths and computer science degree. With both computer science and maths you are chronically confused. The difference being with computer science it doesn't matter so much if you don't understand something, if you can get it to work you know you are on the right track. Maths is much more progressive, each proof builds on a previous one. So if you fail to understand one step you are screwed from that point on. After the first year I realised I didn't actually enjoy being permanently confused and so I ditched the maths to focus on computers. I do regret this. It didn't take long at all before I forgot all that knowledge I had spent years sweating over.

egdelwonk 13y ago

What's the best way to relearn math?

11001 13y ago
Textbooks and exercises. Avoid shortcuts such as online tutorials or anything with "for hackers" in the title.
- egdelwonk 13y ago
  I've been using Khan Academy to refresh my skills and it's been really helpful, but do you have any recommendations for textbooks? I'm 10 years out of school and haven't really needed to flex my math muscles in years.
  - 11001 13y ago
    Any particular "math" you're interested in? If you just want to get your foot in with pure math I'd recommend studying basic abstract algebra and analysis at the same time. For abstract algebra look into Hungerford (Intro, not his grad text), for analysis, maybe Rudin, or Kolmogorov and Smirnov.
  - j2kun 13y ago
    If you're past calculus, I'd say start with Sheldon Axler's Linear Algebra Done Right (it has the most immediate applications, and linear algebra is essential for almost all higher mathematics).
dwaltrip 13y ago
Focus as much as possible on the underlying concepts from any math topic and how they connect to other concepts. Try to boil these concepts down to the most simple, clear form that makes sense to you. Test your boiled down conceptual understanding by applying it to related exercises/problems you haven't tried before and seeing what happens. For each sub-topic, most math books give you the tools first and then teach you problems they should be used on. Read the problems first, and think how you might solve them (don't expect to figure it out, but if you do, great!). Then, go back and learn the tools, trying mostly discern the "how" and the "why" as opposed to the "what". Math is all about "how" and the "why". As some motivation, whenever a new thing clicks, it is very satisfying! :) But it definitely is a tough process. Good luck!

dalke 13y ago

"If you’re going to get anywhere in learning mathematics, you need to learn to be comfortable not understanding something." This is true for all research. And I don't mean just the physical sciences either. Historians and sociologists are also chronically "lost and confused." Otherwise it wouldn't be a topic worth of study. This is why students who are "good at X", whether it be math, German, sports, or programming, may become frustrated when they find out that "good at researching X" is a very different matter.

j2kun 13y ago
I think it's a good point, but I still think the kind of lost and confused in mathematics is more embarrassingly extreme. Imagine a few hundred historians trying to discern when King George I died, and after 50 years of work they conclude, "All we know for sure is that it was between the day he was born and yesterday." A startlingly large part of mathematics feels like this. And I think the reason is that "prevailing theories" mean nothing in mathematics.
- dalke 13y ago
  That's a poor comparison. I find it hard to believe that mathematicians are still trying to decide if the set {1, 2, 3} is finite or infinite. A "startling large part" of all science fields like this. Physicists don't even know if the gravitational mass of an object is really the same as its inertial mass. Or if there are true magnetic monopoles. And that's after over a century of trying. Immunologists have barely scratched the surface of how that field works. Economists make lots of conjectures, with lots of math to back it up, but it's not a perfect predictor of the human economic system. Biological evolution still surprises us, 150 years after Darwin and nearly 100 years after the neodarwinian synthesis. Chemists still don't come close to handling some of the reactions that natural systems have figured out. And so on.
  - j2kun 13y ago
    Good point.

mathattack 13y ago

It's strange to hear mathematics described more as a search for art and structure than computation. Unfortunately most of my math education was on the computational/applied side. I'm only getting into number theory and the more esoteric math later in life for fun. As a parent I think we can't let the school system destroy our kids love of math through too much rote learning. We have to make it fun for them. (Same with music btw)

bpyne 13y ago

I wish this post was around when I finished my undergraduate degree in Mathematics. I would have taken my adviser's advice to go to grad school. At the time, I remember telling him that I feel like a barely made it through the program. Apparently I wasn't alone. Amazing the difference 25 years and the internet makes.

japhyr 13y ago

I currently teach math to at-risk students. I don't read all of these submissions about math education, but I skim the comments on most of them. The comments people make change the way I teach math. I have always done a decent job of teaching math. I focus on helping students understand concepts, even when they are focusing on mechanics. I use words like "shortcut" and "more efficient method" rather than "trick" when showing students more efficient ways to solve problems. I have students do problems and projects that relate to their post-high-school goals. But with the routines of school life, I get away from the fun of math from time to time. The comments on these submissions often remind me to go in and just tell stories about math: - "Hey everyone, did you know that some infinities are bigger than other infinities?" - "Hey everyone, do you have any idea how your passwords are actually stored on facebook/ twitter/ etc.?" - "Have any of you heard the story about the elementary teacher who got mad at their class, and told everyone to add up all the numbers from 1 to 100? One kid did it in less than a minute, do you want to see how he did it?" Thanks everyone, for sharing your perspective on your own math education, and about how you use math in your professional lives as well. Your stories help.

countrybama24 13y ago
No, thank you for teaching math to at-risk kids. Seriously, bravo.
csours 13y ago
I love stories like these. I haven't gone much past calculus, what stories can I look for that will take me farther?
- wetmore 13y ago
  You may recall that one can use the quadratic formula to find solutions to a quadratic equation (polynomial with highest term being x^2). Did you know that (one) guy that proved there is no such formula for quintic equations (highest term x^5) died in a duel when he was only 20? [1] There is a town with a particular rule when it comes to facial hair: those who do not shave themselves are shaved by the barber. But then who shaves the barber? [2] The other poster mentioned different infinities. One "size" of infinity is called "countable infinity" and is the infinity describing the size of the natural numbers (1,2,3,...). Say we have a hotel with a countable infinite number of rooms. I've been travelling all day and I show up at the hotel, and talk to the clerk at the front desk. He tells me every room is full, but when he sees the sad look on my face he tells me not to worry - he can make room for me. He simply moves the person in room 1 to room 2, the person in room 2 to room 3, room 3 to room 4, etc... And then the first room is empty for me, and everyone still has a room. [3] [1] http://en.wikipedia.org/wiki/%C3%89variste_Galois http://en.wikipedia.org/wiki/%C3%89variste_Galois [2] http://en.wikipedia.org/wiki/Russell's_paradox http://en.wikipedia.org/wiki/Russell's_paradox [3] http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_...

chwolfe 13y ago

The entire post was enjoyable but I found the last paragraph to have the most actionable advice: 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.

vsbuffalo 13y ago

Funny, this title is the same as a recent thread on HN: https://news.ycombinator.com/item?id=7331791 https://news.ycombinator.com/item?id=7331791

j2kun 13y ago
That comment received so much positive feedback that I decided to write an article expanding on the idea :)

ChristianMarks 13y ago

Fair enough. I tried in my youth to solve every problem I came across. There were many I couldn't solve. It took a while before I developed the wisdom and discipline not to solve every problem no matter how long it took. By a while I mean decades. I sacrificed the possibility of family life, have stopped talking to my uncomprehending stepfather, and have kept my social interactions to an absolute minimum to pursue my consuming interest. (I mention this as a point of pride.) I find myself continually astonished by the ingenuity of solutions I probably could never have imagined after years of work. Perhaps, after a lifetime of effort that must be continually maintained, I have attained the level an entering freshman at Harvard. At this stage, I may be reduced at best to connoisseurship of some aspects of mathematics. Now for some reflections on attitudes. Mathematicians sometimes act as if they believe that expertise in mathematics transfers to expertise in mathematics education. Suppose you are a sensitive student, lacking in confidence. You open Korner's beautiful book on Fourier Analysis, and the first thing you are greeted with is "This book is meant neither as a drill book for the successful student nor as a lifebelt for the unsuccessful student." Korner does not mention other references suitable for the successful and the unsuccessful student. You take this comment to mean that Korner would let the unsuccessful student drown. There is no implication, but this is the psychological import, the implicature. Why mention the unsuccessful student at all? Why not say who the book is for, without planting this gratuitous image in the reader's mind? It would take some time to return to this book, to get past the wonder at a mind capable of such an incidental, dismissive, off-handed acknowledgement of "the unsuccessful student." You could say this is "overthinking." Such remarks, microagressions as they are termed today, "perpetrated against those due to gender, sexual orientation, and ability status", are sometimes revealed in the asides of mathematical authors [1]. And now if only mathematics educators would evaluate their students on the state of their confusion! [1] http://en.wikipedia.org/wiki/Microaggression http://en.wikipedia.org/wiki/Microaggression

api 13y ago

I completely agree about the power of math, and why programmers should learn it. There are two problems with math: (1) Math is IMHO the worst taught of all academic subjects. It's taught as if it were not a language. Math profs and books on mathematics never explain what the symbols mean. They just throw symbols at you and then do tricks with them and expect you to figure out that this symbol means "derivative" in this context. I have literally seen math texts that never explain the language itself, introducing reams of new math with no definitions for mathematical notation used. I've looked for a good "dictionary of math" -- a book that explains every mathematical notation in existence and what it means conceptually -- and have never found such a thing. It's like some medieval guild craft that is passed down only by direct lineage among mathematicians. Concepts are often never explained either. I remember struggling in calculus. The professor showed us how to do a derivative, so I mechanically followed but had no idea why I was doing what I was doing. I called up my father and he said one single sentence to me: "A derivative is a rate of change." A derivative is a rate of change. I completed his thought: so an integral is its inverse. Bingo. From then on I understood calculus. The professor never explained this, and the textbook did in such an unclear and oblique way that the concept was never adequately communicated. It's one g'damn sentence! The whole of calculus! Just f'ing say it! "A derivative is a rate of change!" (2) The notation is horrible. If math were a programming language it would be C++, maybe even Perl. There are many symbols to do the same thing. Every sub-discipline or application-area of mathematics seems to have its own quirky style of notation and sometimes these styles even conflict with each other. Yet baroque languages like C++ and Perl at least document their syntax. If you read an intro to C++ book it begins its chapter on templates by explaining both what templates are for and the fact that type<int> means "type is templated on int." Math doesn't do this. It doesn't explain its syntax. See point #1 above.

j2kun 13y ago
I agree wholeheartedly with how frustrating it is. I think part of the problem is that really great mathematicians are encouraged to stay as far away from teaching (and improving their teaching) as possible, and great teachers are often discouraged from pursuing more mathematics for a variety of reasons. And when I personally teach calculus I make sure to explain derivatives in the way you want in the very first day (before describing limits or anything else). As to your second point, I think notation is a big problem, but it's a bit of a straw man. With very few exceptions that I doubt you would ever find yourself in, I have never met a professor or mathematician that would not explain notation if you asked (gladly stopping in the middle of a lecture or talk to clarify). There is still a lot of it, but every mathematician who is presenting the mathematics can explain the notation to any degree of precision you could ever want, and I have very few colleagues who have never stopped someone for this reason. I think the bigger problem is trying to read mathematics by yourself, without the ability to ask questions. And even after understanding the notation, I feel programmers have bigger problems, which I've expanded more on in this post [1], the main difference between learning programming being there are simply more free and open resources for learning programming. This is probably because programmers invented the internet and filled it with their favorite content first. But one point I make is that mathematical notation is inherently ad-hoc, and the only kinds of notation that stick around are the kinds that get used ad-hoc enough times to become standard. And even then people will make up their own notation for no other reason than that it's their favorite (Physicists are really good at this, and perhaps ironically it drives mathematicians crazy). Because of that (and because notation is introduced often to be rigorous, not to explain a concept) you're unlikely to ever find such a dictionary. Sorry :( [1]: http://jeremykun.com/2013/02/08/why-there-is-no-hitchhikers-guide-to-mathematics-for-programmers/ http://jeremykun.com/2013/02/08/why-there-is-no-hitchhikers-...

trevorhartman 13y ago

Jeremy, I really appreciate this post and all the excellent content over at Math ∩ Programming. Thanks, and please keep it up!

j2kun 13y ago
I'm just so happy that I get to read everyone's interesting stories and thoughts in the HN comment threads! HN is really one of the highest-quality places for discussion on the web ^_^

GIFtheory 13y ago

Reminds me of this great quotation, which Oksendal places before the preface to his stochastic differential equations book: We have not succeeded in answering all our problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things. Posted outside the mathematics reading room, Tromsø University

baby 13y ago

That's what I tell people around me. Studying math is hard because it makes you feel stupid. You always feel lost, you always feel like you missed so many things when you're starting to learn a new thing, you always feel like your questions are stupid (until you get that the rest of the class is pointless as well). Especially with talented professors (Lyon 1, France, the professors there are not really good educators, but they are geniuses), they make you feel bad for not understanding things that seem so simple to them. Studying math is depressive if you take it too seriously.

beltex 13y ago

Loved this post. FYI, the Andrew Wiles quote is from the opening of an awesome BBC documentary about how he solved Fermat's Last Theorem - http://www.youtube.com/watch?v=7FnXgprKgSE http://www.youtube.com/watch?v=7FnXgprKgSE

nilkn 13y ago

Heh, I knew this title seemed awfully familiar. Here's the discussion on Hacker News which (presumably) spawned this: https://news.ycombinator.com/item?id=7331693 https://news.ycombinator.com/item?id=7331693

edtechdev 13y ago

"If you’re going to get anywhere in learning mathematics, you need to learn to be comfortable not understanding something." That's true of everything. It's fear and anxiety that prevents a lot of people from learning and trying new things. I keep trying to tell students or family members when they are learning to do stuff on the computer, just right click everything, just google anything you can think of, don't worry about it being perfect, don't worry about breaking anything. You have to hold back showing them the "answers" or else they become dependent.