9 ms·

Why math is painful to read

70 points by gnosis 14y ago

lutusp 14y ago
Math is such a big field that there tends to be much re-use of symbols, and the limited number of Greek letters doesn't help (no one likes multiple-Greek-letter variable names). To a mathematician reading the technical literature from his own field, there's no issue with math notation. Problems sometimes come up when a specialist in one field tries to read literature from a different specialty -- but that's true in computer programming too. This may come as a surprise, but math notation is much more uniform than computer-science notation, including all the efforts to create generic program listings. Mathematicians are reluctant to invent new symbols, but it's sometimes necessary. When I read computer program source code, I often wish the programmer had summarized the meaning of the code using math notation. I guess that puts me in a position precisely opposite that of the author.
- ahallerberg 14y ago
  For me, it's because each symbol, letter or number is given a ton of weight in mathematical equations. You can't skip over and fill in the gaps later - like reading a book, for example - you have to understand each aspect before the whole makes sense.
- btilly 14y ago
  To a mathematician reading the technical literature from his own field, there's no issue with math notation. Let me guess. You haven't learned much differential geometry? Constants vary by 2 pi depending on which definition you use. Functions might come before or after their arguments. And, of course, you leave everything that is "unambiguous" out of the notation. (Proving that it truly is unambiguous is sometimes nontrivial for the novice.) Guessing what notation a particular author is using has been known to be daunting, even for other differential geometers. In grad school I remember sitting down with homework sets and spending longer working out what the question said than I did in solving it once understood. That said, if you tried to actually write everything out explicitly, it would be impossible to read anything. Compact notation is a necessary evil - without it your brain simply can't think certain things. With it, it is hard for anyone else to figure out what you just said.
  - saraid216 14y ago
    There's something to be said for the notion of a glossary or legend... have these been considered at all? Or a notational standards body of some kind, perhaps?
    - trhtrsh 14y ago
      How many standards would you like? :-) http://mathoverflow.net/questions/33152/is-there-a-reference-containing-standard-mathematical-notations http://mathoverflow.net/questions/33152/is-there-a-reference...
      - saraid216 14y ago
        Enough that (1) someone tells me about it in 5th grade and (2) people stop complaining. I mean, seriously. The top answer is, "Wikipedia".
tokenadult 14y ago
The article "How to Write Mathematics" http://www.math.uga.edu/~azoff/courses/halmos.pdf http://www.math.uga.edu/~azoff/courses/halmos.pdf by the late Paul Halmos, a mathematician famous for his clarity of writing, suggests procedures for making mathematical papers less painful to read.
- glomph 14y ago
  Really enjoyed some of that. Thanks.
aroberge 14y ago
I could not disagree more with that post. Mathematical notation is to Python what English would be to Assembly Language: it is a high-level notation, designed see the forest rather than getting lost in the trees. Take Maxwell's equations for example. After learning vector calculus and becoming familiar with its notation, you notice how much meaning you can extract at a quick glance from the four Maxwell's equations expressed in differential form. The purpose is not to obfuscate the meaning (like the author implies when he mentions the Obfuscated C contest) but rather to reveal/encapsulate as much information as possible in a way that is easy to understand at a glance. You want to know what would happen if magnetic monopoles existed? Easy to do by just adding a few extra terms that are obvious by symmetry of the equations. And, if you go beyond vector calculus, you can see an even clearer picture by writing F = dA (equivalent to the usual two homogeneous Maxwell Equations) d*F = *J (equivalent to the remaining two)
- gnaritas 14y ago
  > Mathematical notation is to Python what English would be to Assembly Language Which is often the problem, it looks specific, but it really isn't; it's full of implicit assumptions about what the reader should know. It's not executable, because it's a language designed for being written by hand rather than executed by a computer. However, the teaching of math would greatly benefit from an explicit executable form that makes no assumptions, i.e. a programming language. Gerry Sussman makes this case, he's pretty convincing.
  - john_b 14y ago
    You think that people learning math will find it less confusing if they have no context (which, in mathematics, is largely equivalent to "assumptions") for what they are learning and have to build their mathematical knowledge from first principles? Or am I misunderstanding your post?
    - gnaritas 14y ago
      I don't think anything, I was recalling what I heard Sussman say at a conference, which he makes clear in http://en.wikipedia.org/wiki/SICM http://en.wikipedia.org/wiki/SICM. Though I saw him a good 8 years after the book, so I'm sure he'd grown some new opinions. He is of the opinion that if you can't write the algorithm, you don't understand it; forcing students to write the algorithm aids in teaching them a real understanding. He spent quite a bit of time complaining about the vagueness of mathematical notation due to its implicit assumptions for the purposes of teaching and the superiority of executable code for the task. On a side note, the speech was aimed at both engineers and programmers and the audience was a mix of both. He did a wow demo with a circuit on the overview, calculating all the voltages and resistances on the fly from chosen starting values that made all the programmers smile really big and all the engineers jaws drop. Everyone was impressed because he flew through it, but the programmers much less so because what he did was just how a programmer thinks, albeit it very fast. He was demonstrating to the engineers the benefits of thinking like a programmer. As he is both and teaches both, he clearly connected some dots that they generally don't.
michaelochurch 14y ago
Disagree strongly. Math is often the easiest and simplest thing that works for the class of problems it exists to solve. Imagine trying to explain singular value decomposition without the notion of a matrix. No group theory, no well-formed concept of a linear transformation or function. You wouldn't be able to do it. No one even had such ideas before generations of mathematical machinery had been built. I've come to the conclusion that there are lifer languages (/frameworks/technologies) and start-today languages. C, Lisp, Scala, and Haskell are lifer languages. (Now q, for those who've worked in finance, is even more of a hardcore lifer language.) You'll probably find them painful when you start out, but enjoy them a lot more once you really get them. Python, on the other hand, is much more of a start-today language. It's more modular, you can get up to speed quickly, and it's good enough the vast majority of the time. Also a start-today language is Java, with an IDE. PHP is a start-today language. This also applies to tools. Emacs or vi (both being keyboard-driven) are lifer tools. Unix is a lifer tool. IDEs are start-today tools. This isn't about "better" or "worse". It's about tradeoffs: with a lifer language, it takes at least a few months before you get it (you start off hating it, because it feels weird and limited) but it becomes immensely powerful once you understand it in a deep way. Start-today languages give you a lot of power to start, but then the learning curve flattens. What's better is unclear. The lifer languages have a lot of benefits and depth, but in fast-moving disciplines like web programming I'd prefer that the frameworks be start-today rather than lifer-oriented. Math is an extreme lifer language.
- davegauer 14y ago
  I think you make a really good point about the power of mathematical notation and conventions. However, I completely agree with the article. Do mathematical formulas have to use Greek letters rather than useful variable names like "distance" or "speed"? Does C's syntax actually allow you to express something you can't in Python? Or is it more terse for historical reasons? (And never mind that it's a good idea to use descriptive variables and function names in either language!) I think the keyboard-driven vs GUI tool analogy is very apt. vi is very powerful. Would the average user like it much if we made vi the interface for all text forms in browsers? I think sometimes it's desirable to present a friendlier form of a thing. (You obviously agree regarding web programming.) So I think the complaint is this: why is math always expressed in the tersest form? Is it really necessary? Why does math have to be restricted to the lifers!?
gjm11 14y ago
It may not be coincidental that code written by mathematicians tends to be ... terse.
nn2 14y ago
Really good point. That always annoys me with most formulas. I have to dig back several paragraphs (with no scope/indexing tool) to figure out what the individual variables mean. And a lot of papers do not even bother to explain some variable but just assume the reader already knows it. One of the earlier commenters made this mistake very prominently enthusing over Maxwell's equations. Yes of course the formula looks better shorter when you already know what it means. But that completely misses the point that it was written down to explain it to someone who doesn't already know what it means. If you already do you're the wrong target audience. Longer variable names would help. Or for online paper just have a tooltip that opens the paragraph that explains what the variable mans when you hover over it would also help a lot. Or maybe some color coding to easily find it (there was a recent link here recent explaining FFT which used this trick very successfully) But I guess most Mathematicians don't bother because they only write for a small circle of colleagues anyways.
- modarts 14y ago
  Maxwell's equations were not "written down to explain it to someone who doesn't already know what it means". They, quite simply were written down to express relationships between the motion of charge, and the behavior of electric and magnetic fields as a consequence. You have to understand the physics behind the equations (what's charge? what's an electric field) as well as the mathematical structure (what's a vector space? what's a line integral?) in order to make use of them. They are equations used to answer questions like "What happens to the strength of this magnetic field if I increase the velocity of this charged particle responsible for generating it"? They aren't meant to be explanatory (but have the beautiful side-effect of explaining how light propagates in a vacuum.)
- Xcelerate 14y ago
  I agree with you. A book that aims to TEACH should provide the most common form of the equation and provide detailed annotations of what everything is. (Wikipedia should be like this as well since I know no one is using it as a reference). For a REFERENCE, you can just list the equations in their most commonly used form.
deleted 14y ago
[deleted]
vecter 14y ago
The real reason math is hard to read is that it's the distillation of a thousand winding ideas and failed attempts that took hundreds of hours of thinking and hundreds of pages of scribbling, all condensed into the most concise possible three pages of unmotivated line-by-line proof with no context or explanation. In other words, math is a language for formalizing intuition which, when finally written, unfortunately removes all traces of intuition and leaves only the end result of the thinking, not the path that it took to arrive at the critical insight.
- yaks_hairbrush 14y ago
  That about sums it up. It's made worse by professors who have been trained to teach things in an unmotivated fashion. I had a graduate probability teacher who said (paraphrased) "one of the beautiful things about math is that you start with a definition, and you prove some very simple things with the definition... so simple that you feel you're just using circular reasoning. Then you find that your definition is equivalent to a useful property!" Actually, what happened when the stuff was developed is that someone wanted the useful property, and managed to rationalize a definition which fit. (example: group theory was done for hundreds of years before we had a definition of group. Noether noticed that a lot of folks were writing the same sorts of things in different contexts, so she distilled it down to three axioms.) For a really concrete example, consider ordered pairs. A person being coy about their intent would say that "we define an ordered pair (x,y) on a set S as a set of the form {{x},{x,y}}, where x and y are in S." It's safe to say that no one thinks of an ordered pair in that fashion. The intent was to have two ordered pairs be equal if and only if each of the two coordinates are equal. You can prove this from the above definition, but it's far more helpful to tell the audience in advance that we really want a structure with this property.
- bennyg 14y ago
  Which is a beautiful thing, no doubt. The creative process is amazing, but reading through years worth of scribbles isn't the point of the proof. Write a proof, then accompany it with a blog of sketchbook scans and a detailed walkthrough of getting there.
ColinWright 14y ago
No, that's not why math is painful to read. The reason math is painful to read is because it's either obvious, in which case it's not painful to read, or because it contains new ideas that you need to work hard on before you can grok them. There is no short cut. You can't write math like a novel, you can't read math like a novel. There are ideas that are deep and difficult, and unless you actually spend time working on the material you simply won't "get it." Time and again I have explained to people something that's comparatively simple, and easily within their grasp. I've tried different approaches, different formulations, different arcs through the material, and time and time again the person has followed every step in detail, and just not "got it." There is no enlightenment, there is no sense of what's happening. It's only after they've invested the time that they have come to a sense of what's happening, and then the light dawns. Occasionally they say things like "Why didn't you explain it like that in the first place?" Then I point out that I did, but at that stage, they weren't ready for it. It's like following a SatNav. You get to your destination having followed the turn-by-turn instructions, and yet, in truth, you really have no idea where you are. You have traveled from one local pocket to another local pocket, with no understanding of the space through which you've been. Math is painful to read because you're trying to follow the turn-by-turn directions, instead of taking control of your own understanding and working to gain knowledge and insight, scope and intuition. You won't understanding it without doing the work, and complaining that it's badly written is the smallest part of the problem.
- Semaphor 14y ago
  > The reason math is painful to read is because it's either obvious, in which case it's not painful to read, or because it contains new ideas that you need to work hard on before you can grok them. The same could be said for code that uses only single letter variables and functions. It'd be a lot easier with readable variable names.
  - kd0amg 14y ago
    Yes. In reading other people's code, I've had coding style make the difference between taking about a day and taking about half an hour (two programs to do the same thing in essentially the same way).
john_b 14y ago
It seems to me like the author approaches math looking for a concise & elegant programming language, but instead just finds...math. The notion of scope as an integral part of mathematics is a particularly interesting suggestion. Mathematics is the study of abstract, quantitative truths (and falsehoods), and these truths, being universal, have universal scope (although you can constrain some truths as special cases of more general truths, but I doubt this is what the author had in mind). To consider this a problem indicates that the author is searching for a tool rather than a way of thinking. "Math is such a cognitive overload of operators" because it succinctly describes a great deal of action. A program in a conventional language like C might take a hundred lines to describe what could be described in a handful, using some mathematical notation. If you want a programming example, just look at the average length of an APL program vs the average length of a Javascript or C program. Math is inherently hard. So is programming. Not many people can do either very well. Even fewer can do both very well. The notations of math are meant to be flexible because math is a tool for discovering new ideas and analyzing their properties. Programs are meant to accomplish some real world task. Criticizing the language of one solely from the perspective of the other is not a useful activity.
- ryankey 14y ago
  Completely agree with you, but from your experience, how many fewer do both very well? I have always been interested in both and I can't imagine having one tool without the other, as they both enhance the way that I think about things. Programming makes the concepts of variables and sigma notation even easier to understand, while math makes it far easier to solve problems. Do people actually get by in programming without math?
  - gnaritas 14y ago
    How much math do you think it took to program this website?
- kd0amg 14y ago
  The notion of scope as an integral part of mathematics is a particularly interesting suggestion. Mathematics is the study of abstract, quantitative truths (and falsehoods), and these truths, being universal, have universal scope (although you can constrain some truths as special cases of more general truths, but I doubt this is what the author had in mind). To consider this a problem indicates that the author is searching for a tool rather than a way of thinking. The scope issue is not just for theorems. In programming, scope is about what meanings get bound to what symbols, whether those symbols are types, operators, data, etc. Even in math, we can see a notion of "binding" vs. "bound" vs. "free" occurrences of a symbol: seeing something like "∀x∈N.x+k≥0" would make me think "Ok, I see what `x' is, but what's this `k'?" Treating every definition like it's at top-level makes for unreadable code and unreadable math. Ever had to search back through a document looking for the definition of some notation only to find it buried in an earlier proof of an unrelated theorem without so much as a "Definition: …" marker? I would much prefer not to have to do that again. Transferred to another domain, "Why should we need scope? What good would that do?" becomes "Why should pronouns have antecedents? What good would that do?" It is important to acknowledge that mathematical notation is also a tool for explaining the ideas one has discovered with it.
tlb 14y ago
I think for many, the problem is pacing. When you're reading text, your eyes are moving down a line at least once a second. But reading an equation takes much longer. You have to stop your eyes and ponder every piece of it. It can help to copy it down on paper.
ryankey 14y ago
Hungarian notation for formulas would be terrible. It would make it that much harder to memorize, as well as increase the difficulty of seeing patterns between various formulas. The real problem here is trying to force programming notation into mathematics notation. Programming variables emphasize what you're working with while mathematics variables emphasize ideas and relationships.
dxbydt 14y ago
Frankly, there are 2 kinds of programmers. People who do this: val result = directProduct(cyclicGroupOfDegree3, finiteAbelianGroupOfDegree7) and the second kind, people like me, who do this: // Compute the direct product of 2 cyclic groups val z = dP( cg1, cg2 ) You can easily guess that the 2nd kind are math majors. If my math professor started writing everything out in plain English like the first example, he'd never be able to cover a theorem per class. He'd still be writing the first lemma and the hour would be up. So he resorts to 1-character symbols, precise notations, and terse comments. The idea is - if the reader doesn't grok direct products or cyclic groups, he's fucked anyways, so why bother ? And if he does grok them, why impose cognitive overload by spelling it all out in great detail, just use 1-character symbols and move on. Now both these styles are in direct conflict with each other, and in the Fortran/C/C++ community during the 80s-90s ( when every respectable programmer had a copy of Numerical Recipes on their desk ), you would emulate the 2nd kind. In the 2000s & later, people got rid of their Numerical Recipes & exchanged them for "14 days to EJB" and "Learn Python in 21 days" and "Master Ruby in 7 days" and the like...the community started becoming a lot less math-y and a lot more verbose, and style 1 is back in vogue. Nowadays I get pulled up constantly in code review for using single character names....but I think this too will pass :)
- barakm 14y ago
  That's kind of a strawman argument. cyclicGroupOfDegree3 is a terrible name, unless you need to be really specific about degrees. cyclicGroup, please. (and I could imagine: cyclicGroup.degree() == 3) dP is also a terrible name. My first thought goes to derivatives. directProduct is the right name. dProduct, dProd (if you like the Numerical Recipes style, expounded below) are better than dP, but still wrong for a library. So first, let's assume directProduct is a library somewhere; maybe one you've even created. So let's reconstruct: val z = directProduct(cg1, cg2) Better, and more believable. And if the declarations of cg1 and cg2 are obvious (ie, the lines directly preceding) then you might have a case. I imagine directProduct(group1, group2) would actually be a happy medium. And if you use z in the next line or two, I'd let that slide. The thing about Numerical Recipes is that often you're taking math syntax and coding it. Often doing so requires a good deal of commenting and temporary variables. One thing the book gets very wrong is it's function declarations (the function body is a separate argument) -- at the very least, rooted in a past of 80-character-lines and before code reviews. The first example in the book: void flmoon(int n, int nph, long* jd, float* frac) ought, in a modern era, be something like: void MoonPhaseToDay(int n_cycles, int phase, long* julian_day, float* fractional_day); If for no other reason than I can have some hope at understanding the second argument, or finding it again. You'll also notice flmoon is a misnomer -- it computes more than full moons!
norswap 14y ago
I agree. I always felt that if math theorems were presented in a programming language, they would be way more understandable (especially if the good practices are respected: meaningful variable names, etc). Mathematicians love to talk about rigor, but when it comes down to write their ideas, they often have little enough of it.
- kd0amg 14y ago
  I always felt that if math theorems were presented in a programming language, they would be way more understandable …. Find out empirically! Look at theorems in some formal methods/programming languages papers and their corresponding mechanizations and see which you find more understandable. (Actually, it might be better to start with something like Software Foundations to get a gentler introduction to a programming language used for this sort of thing -- http://www.cis.upenn.edu/~bcpierce/sf/ http://www.cis.upenn.edu/~bcpierce/sf/ )
  - norswap 14y ago
    Hard to do, the one you will look at second will always be at an advantage. So you'd have to select two theorems of "equivalent difficulty", which is not easy. Also you need to select hard to understand theorems, else you'll understand both the representations easily anyway.
Jimmy 14y ago
The main reason it's hard to read math papers is because they contain complex, difficult ideas. Even professional mathematicians take a long time to read papers, and they usually won't be able to understand papers outside of their own specific field of math. I agree with the author that math papers could borrow some ideas from programming - explicitly defining all variables/functions, scoping, etc - but acquainting yourself with the conventions of the field alleviate most of those problems.
monjaro 14y ago
People seem to be pretty critical of this post, but can anyone give me a good reason why math notation shouldn't have some concept of scope? I can't see any downsides. It would certainly help me puzzle out my professor's notes when he has used f to mean three different things in as many lines.
- yaks_hairbrush 14y ago
  There is some concept of scope in math. At a fairly early level, a student learns sums and integrals. The variable of integration/summation's scope is just within the integral/sum. At a higher level, some things have scope equal to a subfield of mathematics. A gothic p, for example, represents a prime ideal in algebraic number theory. It represents a parabolic Lie algebra in Lie theory. (Actually this causes problems when you're doing Lie theory and number theory at the same time.)
ggchappell 14y ago
Math researcher here. I agree with the basic idea of this article, but not the specifics. So: notation is a wonderful thing. If someone can improve on it, then great, but modern math notation is what makes math doable. It's true that there is poor notation and poorly used notation; these things need to be fixed. Regardless, I agree that math is hard to read. I read a lot of math. It's hard. Part of this is because it involves abstract, complicated stuff that often must be understood exactly. But part is because we are not very good at writing it. Or perhaps we don't put as much effort into making it readable as we should. Being someone who knows a fair amount about programming (or so I like to think) and also writes math research papers, I've found that some of the concepts used to manage complexity in software are also applicable to math. In particular, I must agree about scope. I've made an effort in my recent writing to be careful about this issue. Another helpful idea involves how to write good comments in code. Too many mathematical proofs perform some operation without telling the reader why or giving any kind of overview. There is nothing wrong with saying, "Here's what we're going to do," and then doing it, and then saying, "Here's what we just did."
shocks 14y ago
Mathematics is only hard to read if it has been written badly or if you do not understand all the concepts involved. Source: I am a Mathematics student.
73939133 14y ago
I never posted before but this article made me so mad I had to jump into the comments! I am so glad to find everyone pointing out all the reasons the article is wrong, actually there is a highly relevant link on math.stackexchange about this particular issue: http://math.stackexchange.com/questions/24241/why-do-mathematicians-use-single-letter-variables/24246#24246 http://math.stackexchange.com/questions/24241/why-do-mathema...
gknoy 14y ago
> But are mathematicians too embroiled in some misguided quest for Huffman coding? I think that yes, they are: for centuries, mathematics has been written by hand. It's much easier to write 'f' than 'force'. on a blackboard. Moreover, when writing on paper, with ink that you have to make or buy, with a pen that wears out as you write with it, it makes sense that early mathematicians would embrace a compressed notation. Scribes even did it, so it shouldn't surprise us that mathematicians would. The compression is based on domain knowledge, and this is only problematic for people who are not intimately familiar with the domain at hand. Physics is similar. Decoding the rocket equation is only possible when you know what m1 and m0 are, or ve, and many properties, constants, and even operations are symbolized with a single greek letter. (Gradients, for example) This is very powerful, because (as others have mentioned) it allows you to easily write more-complex things, and often times you (as a mathematician or physicist) prefer to think at a higher level of abstraction. It also makes it tremendously arcane for those of us who don't even know the names of the greek letters, let alone have a sound backing in that particular field of math or physics.
- CJefferson 14y ago
  The compressed notation is not just about saving paper and ink. It's all about the amount of mental power and work you have to exert. If I'm filling a few pages with equations, if I have to write 'force' instead of 'f' all over the place, it's going to take me about 5 times longer. I'm going to only fit 1/5 of the content on each line and page, (OK, 1/5 might be excessive, but the principle holds I feel). Programming languages have similar properties. Why don't we always write integer instead of int, loop_variable instead of i, and have no compact way of writing unnamed lambda functions?
verroq 14y ago
The author has some preconceived notion of how math is supposed to be read, deduces that math is hard to read. See: How to avoid "compliation errors" when reading math. https://profiles.google.com/114134834346472219368/buzz/GPQDR3gHsYr https://profiles.google.com/114134834346472219368/buzz/GPQDR...
gfodor 14y ago
The problem with math isn't symbology or notation per se, it's that it's not evolved to take advantage of modern technology. It's an artifact of pencil-and-paper being the medium of choice for expressing ideas. We can do much better. When you are looking at an equation, you are looking at the purest distillation of an idea. Underneath that equation sits countless layers of abstraction and reasoning. Why can't I peel back these layers and see them on my computer or iPad? The equation is the iceberg tip peeking out of the water onto the paper, I want to see underneath. Let me feel around and slowly and confidently fill in the places where my understanding is fuzzy. Prevent me from moving back up to the final equation until everything underneath is fleshed out and solid in my mind. And if bits start to fade, let me quickly react to snap them back into focus. This is what we are doing mentally anyway when we flip back and forth in a math text referencing previous proofs and equations. Math needs it's "hyperlink." Anything I say here has been said better and more convincingly by Bret Victor: http://worrydream.com/KillMath/ http://worrydream.com/KillMath/ http://worrydream.com/LadderOfAbstraction/ http://worrydream.com/LadderOfAbstraction/ Though I will argue that at least in his Kill Math writings he's throwing the baby out with the bath water in a certain sense. Let's keep the symbology as a optimal way to encapsulate our knowledge for the amazing mental leverage it gives us, but give us a way to move up and down the ladder of abstraction in order to facilitate understanding.
- trhtrsh 14y ago
  Math's hyperlink: http://mathworld.wolfram.com/Angle.html http://mathworld.wolfram.com/Angle.html
  - gfodor 14y ago
    not following?
b_emery 14y ago
Meyer's matrix analysis has an excellent, and relevant, quote: The great French mathematician Pierre-Simon Laplace (1749–1827) said that, “Such is the advantage of a well-constructed language that its simpliﬁed notation often becomes the source of profound theories.”"
halayli 14y ago
It's hard to understand any programming language if you haven't taken the time to learn its syntax. Unlike teaching programming, the problem I see in teaching math is that the syntax is slipped into the lessons without making a conscious effort teaching it.
jdougan 14y ago
Math notation was't so much designed as accreted, and it shows in how awkward it can get. Dijkstra indirectly makes this point (among others) in "The notational conventions I adopted, and why" at http://www.cs.utexas.edu/~EWD/ewd13xx/EWD1300.PDF http://www.cs.utexas.edu/~EWD/ewd13xx/EWD1300.PDF
steamer25 14y ago
If someone were awesome (maybe somebody like John Resig), they could come up with a math notation framework that started terse but could expand to more detailed definitions with a click. E.g., Starts with (capital sigma) but expands to 'Sum(1..n)' and then expands again to 1 + 2 + 3 + ...n. Clicking the + changes it to 'add' which maybe links off to Wikipedia or Khan Academy. Clicking the numbers might change them to a pictograph of a group of widgets, etc.
- pixl97 14y ago
  A math IDE. I think Wolfram Alpha may do just that, then again dealing with formal maths is not my thing. An IDE makes sense if it can encapsulate the symbols and terms you're conveying. How many programmers sit down with notepad or vi and write huge programs? Yes, some do (and I have), but it's far easier with the IDE.
narrator 14y ago
Richard Feynman in one of his quantum physics lectures explains that he is going to abbreviate a few things because they are a pain to write out longhand on the chalkboard. I think this is the main reason everything is so abbreviated, it comes from the era when doing lots of manual calculations by hand called for efficiency in notation over clarity.
el_cuadrado 14y ago
> People criticize c++ because you can overload the operators to mean anything you want By the same people who struggle to read math, I guess.
pash 14y ago
No doubt about it, mathematical notation can be difficult to read. It's terse, conceptually dense, operators are commonly overloaded, etc., etc. But that's not why it's hard to read. Math is hard to read, I think, because math is hard: (1) Some core mathematical concepts are fundamentally difficult. Notions like infinity (and different kinds of infinity!) are entirely foreign to our daily experience. (2) In addition, mathematics is more abstract than most programming. Tim Gowers, in his lovely little book Mathematics: A Very Short Introduction, writes: What is the black king in chess? This is a strange question, and the most satis- factory way to deal with it seems to be to sidestep it slightly. What more can one do than point to a chessboard and explain the rules of the game, perhaps paying particular attention to the black king as one does so? What matters about the black king is not its existence, or its intrinsic nature, but the role that it plays in the game. The abstract method in mathematics, as it is sometimes called, is what results when one takes a similar attitude to mathematical objects. This sort of abstraction is, I think, abhorred by many programmers. But it's the reason single-letter variable names are common in mathematics. There's no way to describe what a mathematical object is with a name relating it to some real-world, concrete, familiar thing, as programmers often try to do. You can only describe what it does by writing down some rules it obeys.* (3) In math, concepts are inherently more hierarchical than is common in programming. As programmers, we're all familiar with building higher-level things from lower-level pieces. And we try, and often succeed, in building them up in ways that let other people use them without needing to know much of anything about the lower-level pieces. In math, there's a similar accretive process, but because the concepts are so much more abstract, it's often impossible to understand how a high-level concept behaves without understanding how its lower-level constituents behave. Mathematical concepts are conceptually hierarchical, whereas pieces of programs are often structurally hierarchical, but conceptually (nearly) orthogonal. * — (A controversial aside.) An object-oriented mindset can exacerbate the problem by demanding that we reify concepts into objects that represent what they "are". This is often impossible, hence the inscrutable object names we've all laughed and despaired at. Phil Karlton's well known witticism comes to mind: There are only two hard things in computer science: cache invalidation and naming things. Naming things is hard because, just like in math, we sometimes lack a familiar semantic reference for the concepts we deal with as programmers. Conversely, in functional languages, there's less need to think about what something "is", because the natural way of programming is to define relations (functions) on a small number of core data structures. Interestingly, single-letter variable names are also common in some functional languages, like Haskell, because it generally doesn't matter what the data is so long as you know what you can do with it. And that information is embodied in its type, expressed in its type signature, and enforced by the type-checker at compile time.
amplitwist 14y ago
Consider the solution to the quadratic equation x = (-b ± sqrt(b^2-4ac))/(2a) What is x? What is a? The wonderful thing about math is that it doesn't matter. x could be length, dollars, area, volume, mass, potatoes, lines of code, chickens, or electron-volts. But in the general case, it represents a number. Why would anyone consider it an improvement to write Given that number1 answer^2 + number2 answer + number3 = 0, answer = (-number2 ± sqrt(number2^2-4 number1 number3))/(2 number1). The extra characters convey no additional semantics. Even when presented with an equation where things have specific units, you can usually mentally figure out the units of everything else, due to conventions, previous definitions, and mental dimensional analysis. In my experience, the notation isn't so much of an issue as knowing definitions, closely followed by being able to translate those definitions into intuition. You must be able to remember them exactly, otherwise nothing makes any sense, and you must have an intuition for what they mean, otherwise you'll never get anything done. Consider some terms from a recent talk I attended that was outside my immediate areas of expertise: "solvable Lie group", "left-invariant metric", "upstairs", "double cover". I was able to understand the main idea of the talk, but my full understanding of the talk was sunk by not knowing the definition of left-invariance [1]. Since math is inherently abstract, it is hard, and there is no substitute for the hard work necessary to acquire an intuition for it. When doing high-level math, it is necessary to have a rigorous intuition for the subject, where you are able to intuitively see a path to a proof, and then are able to translate that intuition into a suitably rigorous argument. [1] To illustrate my point that knowing definitions precisely is one of the keys to understanding math, here's the definition of a left-invariant metric. Let G be a Lie group with metric <,>, L_g be left multiplication, and L_g^* denote the pullback of L_g. The metric is said to be left-invariant if L_g^* <u,v> = <u,v> for all u,v in G. It makes no sense unless you know what metrics, Lie groups, left multiplication, and pullbacks are, and you'll only shoot yourself in the foot if you can't define them precisely.
j2kun 14y ago
So this article is just the author complaining that he has no practice in reading mathematics. Would you be surprised if someone who never written a computer program before couldn't understand a complicated dynamic programming algorithm? Of course not, they have no practice or background! And for the record, computer programming is much more precise than mathematics. Most people here don't do mathematics, so they don't realize how many implicit identifications we make between objects which are very very different. In programs, correct types are tantamount to the success of a program. In mathematics, we will readily identify rings with topological spaces and never think twice. Moreover, we need to readily be able to change the rules by which we consider two different things to be identical on the fly. It is simply a different way of thinking that programmers aren't used to.
hybridthesis 14y ago
Math is the most psychedelic thing ever. It's short so you don't have to read a novel for its powerful effects to kick in.
sbi 14y ago
This is an old chestnut. Mathematical notation is terse because mathematics doesn't consist entirely of notation! Look at any book or research paper; you'll find a great deal of text surrounding equations. Comparing this to code is nonsensical. Unless you're reading something in WEB, you are much more likely to encounter code than comments. There are certainly some unfortunate conventions and cruft in mathematics, but every generation of mathematicians rewrites in part what it inherited from its forebears.
aroberge 14y ago
If we ignore the cosmological constant contribution, for each of the 10 non-zero components of the Ricci curvature tensor, the sum of the component of the Ricci curvature tensor minus half of the corresponding component of the metric tensor multiplied by the Ricci scalar is equal to four times the ratio of the circumference of an arbitrary circle divided by its radius, multiplied by Newton's gravitational constant divided by the fourth power of the speed of light in vacuum and multiplied by the corresponding component of the stress-energy tensor. Would this be clearer than the original [http://en.wikipedia.org/wiki/Einstein_field_equations http://en.wikipedia.org/wiki/Einstein_field_equations]? After all, I don't use any weird symbols and avoid greek letters... Surely the author of that blog post would be satisfied by this ... Perhaps, I should include what I mean by Ricci curvature tensor ... but first define what I mean by a tensor, including the metric tensor, and explain other words like stress-energy tensor; it should not take more than a few lines of text for it to be completely clear, right?

70 points by gnosis 14y ago

Math is such a big field that there tends to be much re-use of symbols, and the limited number of Greek letters doesn't help (no one likes multiple-Greek-letter variable names). To a mathematician reading the technical literature from his own field, there's no issue with math notation. Problems sometimes come up when a specialist in one field tries to read literature from a different specialty -- but that's true in computer programming too. This may come as a surprise, but math notation is much more uniform than computer-science notation, including all the efforts to create generic program listings. Mathematicians are reluctant to invent new symbols, but it's sometimes necessary. When I read computer program source code, I often wish the programmer had summarized the meaning of the code using math notation. I guess that puts me in a position precisely opposite that of the author.

ahallerberg 14y ago
For me, it's because each symbol, letter or number is given a ton of weight in mathematical equations. You can't skip over and fill in the gaps later - like reading a book, for example - you have to understand each aspect before the whole makes sense.
btilly 14y ago
To a mathematician reading the technical literature from his own field, there's no issue with math notation. Let me guess. You haven't learned much differential geometry? Constants vary by 2 pi depending on which definition you use. Functions might come before or after their arguments. And, of course, you leave everything that is "unambiguous" out of the notation. (Proving that it truly is unambiguous is sometimes nontrivial for the novice.) Guessing what notation a particular author is using has been known to be daunting, even for other differential geometers. In grad school I remember sitting down with homework sets and spending longer working out what the question said than I did in solving it once understood. That said, if you tried to actually write everything out explicitly, it would be impossible to read anything. Compact notation is a necessary evil - without it your brain simply can't think certain things. With it, it is hard for anyone else to figure out what you just said.
- saraid216 14y ago
  There's something to be said for the notion of a glossary or legend... have these been considered at all? Or a notational standards body of some kind, perhaps?
  - trhtrsh 14y ago
    How many standards would you like? :-) http://mathoverflow.net/questions/33152/is-there-a-reference-containing-standard-mathematical-notations http://mathoverflow.net/questions/33152/is-there-a-reference...
    - saraid216 14y ago
      Enough that (1) someone tells me about it in 5th grade and (2) people stop complaining. I mean, seriously. The top answer is, "Wikipedia".

tokenadult 14y ago

The article "How to Write Mathematics" http://www.math.uga.edu/~azoff/courses/halmos.pdf http://www.math.uga.edu/~azoff/courses/halmos.pdf by the late Paul Halmos, a mathematician famous for his clarity of writing, suggests procedures for making mathematical papers less painful to read.

glomph 14y ago
Really enjoyed some of that. Thanks.

aroberge 14y ago

I could not disagree more with that post. Mathematical notation is to Python what English would be to Assembly Language: it is a high-level notation, designed see the forest rather than getting lost in the trees. Take Maxwell's equations for example. After learning vector calculus and becoming familiar with its notation, you notice how much meaning you can extract at a quick glance from the four Maxwell's equations expressed in differential form. The purpose is not to obfuscate the meaning (like the author implies when he mentions the Obfuscated C contest) but rather to reveal/encapsulate as much information as possible in a way that is easy to understand at a glance. You want to know what would happen if magnetic monopoles existed? Easy to do by just adding a few extra terms that are obvious by symmetry of the equations. And, if you go beyond vector calculus, you can see an even clearer picture by writing F = dA (equivalent to the usual two homogeneous Maxwell Equations) d*F = *J (equivalent to the remaining two)

gnaritas 14y ago
> Mathematical notation is to Python what English would be to Assembly Language Which is often the problem, it looks specific, but it really isn't; it's full of implicit assumptions about what the reader should know. It's not executable, because it's a language designed for being written by hand rather than executed by a computer. However, the teaching of math would greatly benefit from an explicit executable form that makes no assumptions, i.e. a programming language. Gerry Sussman makes this case, he's pretty convincing.
- john_b 14y ago
  You think that people learning math will find it less confusing if they have no context (which, in mathematics, is largely equivalent to "assumptions") for what they are learning and have to build their mathematical knowledge from first principles? Or am I misunderstanding your post?
  - gnaritas 14y ago
    I don't think anything, I was recalling what I heard Sussman say at a conference, which he makes clear in http://en.wikipedia.org/wiki/SICM http://en.wikipedia.org/wiki/SICM. Though I saw him a good 8 years after the book, so I'm sure he'd grown some new opinions. He is of the opinion that if you can't write the algorithm, you don't understand it; forcing students to write the algorithm aids in teaching them a real understanding. He spent quite a bit of time complaining about the vagueness of mathematical notation due to its implicit assumptions for the purposes of teaching and the superiority of executable code for the task. On a side note, the speech was aimed at both engineers and programmers and the audience was a mix of both. He did a wow demo with a circuit on the overview, calculating all the voltages and resistances on the fly from chosen starting values that made all the programmers smile really big and all the engineers jaws drop. Everyone was impressed because he flew through it, but the programmers much less so because what he did was just how a programmer thinks, albeit it very fast. He was demonstrating to the engineers the benefits of thinking like a programmer. As he is both and teaches both, he clearly connected some dots that they generally don't.

michaelochurch 14y ago

Disagree strongly. Math is often the easiest and simplest thing that works for the class of problems it exists to solve. Imagine trying to explain singular value decomposition without the notion of a matrix. No group theory, no well-formed concept of a linear transformation or function. You wouldn't be able to do it. No one even had such ideas before generations of mathematical machinery had been built. I've come to the conclusion that there are lifer languages (/frameworks/technologies) and start-today languages. C, Lisp, Scala, and Haskell are lifer languages. (Now q, for those who've worked in finance, is even more of a hardcore lifer language.) You'll probably find them painful when you start out, but enjoy them a lot more once you really get them. Python, on the other hand, is much more of a start-today language. It's more modular, you can get up to speed quickly, and it's good enough the vast majority of the time. Also a start-today language is Java, with an IDE. PHP is a start-today language. This also applies to tools. Emacs or vi (both being keyboard-driven) are lifer tools. Unix is a lifer tool. IDEs are start-today tools. This isn't about "better" or "worse". It's about tradeoffs: with a lifer language, it takes at least a few months before you get it (you start off hating it, because it feels weird and limited) but it becomes immensely powerful once you understand it in a deep way. Start-today languages give you a lot of power to start, but then the learning curve flattens. What's better is unclear. The lifer languages have a lot of benefits and depth, but in fast-moving disciplines like web programming I'd prefer that the frameworks be start-today rather than lifer-oriented. Math is an extreme lifer language.

davegauer 14y ago
I think you make a really good point about the power of mathematical notation and conventions. However, I completely agree with the article. Do mathematical formulas have to use Greek letters rather than useful variable names like "distance" or "speed"? Does C's syntax actually allow you to express something you can't in Python? Or is it more terse for historical reasons? (And never mind that it's a good idea to use descriptive variables and function names in either language!) I think the keyboard-driven vs GUI tool analogy is very apt. vi is very powerful. Would the average user like it much if we made vi the interface for all text forms in browsers? I think sometimes it's desirable to present a friendlier form of a thing. (You obviously agree regarding web programming.) So I think the complaint is this: why is math always expressed in the tersest form? Is it really necessary? Why does math have to be restricted to the lifers!?

gjm11 14y ago

It may not be coincidental that code written by mathematicians tends to be ... terse.

nn2 14y ago

Really good point. That always annoys me with most formulas. I have to dig back several paragraphs (with no scope/indexing tool) to figure out what the individual variables mean. And a lot of papers do not even bother to explain some variable but just assume the reader already knows it. One of the earlier commenters made this mistake very prominently enthusing over Maxwell's equations. Yes of course the formula looks better shorter when you already know what it means. But that completely misses the point that it was written down to explain it to someone who doesn't already know what it means. If you already do you're the wrong target audience. Longer variable names would help. Or for online paper just have a tooltip that opens the paragraph that explains what the variable mans when you hover over it would also help a lot. Or maybe some color coding to easily find it (there was a recent link here recent explaining FFT which used this trick very successfully) But I guess most Mathematicians don't bother because they only write for a small circle of colleagues anyways.

modarts 14y ago
Maxwell's equations were not "written down to explain it to someone who doesn't already know what it means". They, quite simply were written down to express relationships between the motion of charge, and the behavior of electric and magnetic fields as a consequence. You have to understand the physics behind the equations (what's charge? what's an electric field) as well as the mathematical structure (what's a vector space? what's a line integral?) in order to make use of them. They are equations used to answer questions like "What happens to the strength of this magnetic field if I increase the velocity of this charged particle responsible for generating it"? They aren't meant to be explanatory (but have the beautiful side-effect of explaining how light propagates in a vacuum.)
Xcelerate 14y ago
I agree with you. A book that aims to TEACH should provide the most common form of the equation and provide detailed annotations of what everything is. (Wikipedia should be like this as well since I know no one is using it as a reference). For a REFERENCE, you can just list the equations in their most commonly used form.

deleted 14y ago

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vecter 14y ago

The real reason math is hard to read is that it's the distillation of a thousand winding ideas and failed attempts that took hundreds of hours of thinking and hundreds of pages of scribbling, all condensed into the most concise possible three pages of unmotivated line-by-line proof with no context or explanation. In other words, math is a language for formalizing intuition which, when finally written, unfortunately removes all traces of intuition and leaves only the end result of the thinking, not the path that it took to arrive at the critical insight.

yaks_hairbrush 14y ago
That about sums it up. It's made worse by professors who have been trained to teach things in an unmotivated fashion. I had a graduate probability teacher who said (paraphrased) "one of the beautiful things about math is that you start with a definition, and you prove some very simple things with the definition... so simple that you feel you're just using circular reasoning. Then you find that your definition is equivalent to a useful property!" Actually, what happened when the stuff was developed is that someone wanted the useful property, and managed to rationalize a definition which fit. (example: group theory was done for hundreds of years before we had a definition of group. Noether noticed that a lot of folks were writing the same sorts of things in different contexts, so she distilled it down to three axioms.) For a really concrete example, consider ordered pairs. A person being coy about their intent would say that "we define an ordered pair (x,y) on a set S as a set of the form {{x},{x,y}}, where x and y are in S." It's safe to say that no one thinks of an ordered pair in that fashion. The intent was to have two ordered pairs be equal if and only if each of the two coordinates are equal. You can prove this from the above definition, but it's far more helpful to tell the audience in advance that we really want a structure with this property.
bennyg 14y ago
Which is a beautiful thing, no doubt. The creative process is amazing, but reading through years worth of scribbles isn't the point of the proof. Write a proof, then accompany it with a blog of sketchbook scans and a detailed walkthrough of getting there.

ColinWright 14y ago

No, that's not why math is painful to read. The reason math is painful to read is because it's either obvious, in which case it's not painful to read, or because it contains new ideas that you need to work hard on before you can grok them. There is no short cut. You can't write math like a novel, you can't read math like a novel. There are ideas that are deep and difficult, and unless you actually spend time working on the material you simply won't "get it." Time and again I have explained to people something that's comparatively simple, and easily within their grasp. I've tried different approaches, different formulations, different arcs through the material, and time and time again the person has followed every step in detail, and just not "got it." There is no enlightenment, there is no sense of what's happening. It's only after they've invested the time that they have come to a sense of what's happening, and then the light dawns. Occasionally they say things like "Why didn't you explain it like that in the first place?" Then I point out that I did, but at that stage, they weren't ready for it. It's like following a SatNav. You get to your destination having followed the turn-by-turn instructions, and yet, in truth, you really have no idea where you are. You have traveled from one local pocket to another local pocket, with no understanding of the space through which you've been. Math is painful to read because you're trying to follow the turn-by-turn directions, instead of taking control of your own understanding and working to gain knowledge and insight, scope and intuition. You won't understanding it without doing the work, and complaining that it's badly written is the smallest part of the problem.

Semaphor 14y ago
> The reason math is painful to read is because it's either obvious, in which case it's not painful to read, or because it contains new ideas that you need to work hard on before you can grok them. The same could be said for code that uses only single letter variables and functions. It'd be a lot easier with readable variable names.
- kd0amg 14y ago
  Yes. In reading other people's code, I've had coding style make the difference between taking about a day and taking about half an hour (two programs to do the same thing in essentially the same way).

john_b 14y ago

It seems to me like the author approaches math looking for a concise & elegant programming language, but instead just finds...math. The notion of scope as an integral part of mathematics is a particularly interesting suggestion. Mathematics is the study of abstract, quantitative truths (and falsehoods), and these truths, being universal, have universal scope (although you can constrain some truths as special cases of more general truths, but I doubt this is what the author had in mind). To consider this a problem indicates that the author is searching for a tool rather than a way of thinking. "Math is such a cognitive overload of operators" because it succinctly describes a great deal of action. A program in a conventional language like C might take a hundred lines to describe what could be described in a handful, using some mathematical notation. If you want a programming example, just look at the average length of an APL program vs the average length of a Javascript or C program. Math is inherently hard. So is programming. Not many people can do either very well. Even fewer can do both very well. The notations of math are meant to be flexible because math is a tool for discovering new ideas and analyzing their properties. Programs are meant to accomplish some real world task. Criticizing the language of one solely from the perspective of the other is not a useful activity.

ryankey 14y ago
Completely agree with you, but from your experience, how many fewer do both very well? I have always been interested in both and I can't imagine having one tool without the other, as they both enhance the way that I think about things. Programming makes the concepts of variables and sigma notation even easier to understand, while math makes it far easier to solve problems. Do people actually get by in programming without math?
- gnaritas 14y ago
  How much math do you think it took to program this website?
kd0amg 14y ago
The notion of scope as an integral part of mathematics is a particularly interesting suggestion. Mathematics is the study of abstract, quantitative truths (and falsehoods), and these truths, being universal, have universal scope (although you can constrain some truths as special cases of more general truths, but I doubt this is what the author had in mind). To consider this a problem indicates that the author is searching for a tool rather than a way of thinking. The scope issue is not just for theorems. In programming, scope is about what meanings get bound to what symbols, whether those symbols are types, operators, data, etc. Even in math, we can see a notion of "binding" vs. "bound" vs. "free" occurrences of a symbol: seeing something like "∀x∈N.x+k≥0" would make me think "Ok, I see what `x' is, but what's this `k'?" Treating every definition like it's at top-level makes for unreadable code and unreadable math. Ever had to search back through a document looking for the definition of some notation only to find it buried in an earlier proof of an unrelated theorem without so much as a "Definition: …" marker? I would much prefer not to have to do that again. Transferred to another domain, "Why should we need scope? What good would that do?" becomes "Why should pronouns have antecedents? What good would that do?" It is important to acknowledge that mathematical notation is also a tool for explaining the ideas one has discovered with it.

tlb 14y ago

I think for many, the problem is pacing. When you're reading text, your eyes are moving down a line at least once a second. But reading an equation takes much longer. You have to stop your eyes and ponder every piece of it. It can help to copy it down on paper.

ryankey 14y ago

Hungarian notation for formulas would be terrible. It would make it that much harder to memorize, as well as increase the difficulty of seeing patterns between various formulas. The real problem here is trying to force programming notation into mathematics notation. Programming variables emphasize what you're working with while mathematics variables emphasize ideas and relationships.

dxbydt 14y ago

Frankly, there are 2 kinds of programmers. People who do this: val result = directProduct(cyclicGroupOfDegree3, finiteAbelianGroupOfDegree7) and the second kind, people like me, who do this: // Compute the direct product of 2 cyclic groups val z = dP( cg1, cg2 ) You can easily guess that the 2nd kind are math majors. If my math professor started writing everything out in plain English like the first example, he'd never be able to cover a theorem per class. He'd still be writing the first lemma and the hour would be up. So he resorts to 1-character symbols, precise notations, and terse comments. The idea is - if the reader doesn't grok direct products or cyclic groups, he's fucked anyways, so why bother ? And if he does grok them, why impose cognitive overload by spelling it all out in great detail, just use 1-character symbols and move on. Now both these styles are in direct conflict with each other, and in the Fortran/C/C++ community during the 80s-90s ( when every respectable programmer had a copy of Numerical Recipes on their desk ), you would emulate the 2nd kind. In the 2000s & later, people got rid of their Numerical Recipes & exchanged them for "14 days to EJB" and "Learn Python in 21 days" and "Master Ruby in 7 days" and the like...the community started becoming a lot less math-y and a lot more verbose, and style 1 is back in vogue. Nowadays I get pulled up constantly in code review for using single character names....but I think this too will pass :)

barakm 14y ago
That's kind of a strawman argument. cyclicGroupOfDegree3 is a terrible name, unless you need to be really specific about degrees. cyclicGroup, please. (and I could imagine: cyclicGroup.degree() == 3) dP is also a terrible name. My first thought goes to derivatives. directProduct is the right name. dProduct, dProd (if you like the Numerical Recipes style, expounded below) are better than dP, but still wrong for a library. So first, let's assume directProduct is a library somewhere; maybe one you've even created. So let's reconstruct: val z = directProduct(cg1, cg2) Better, and more believable. And if the declarations of cg1 and cg2 are obvious (ie, the lines directly preceding) then you might have a case. I imagine directProduct(group1, group2) would actually be a happy medium. And if you use z in the next line or two, I'd let that slide. The thing about Numerical Recipes is that often you're taking math syntax and coding it. Often doing so requires a good deal of commenting and temporary variables. One thing the book gets very wrong is it's function declarations (the function body is a separate argument) -- at the very least, rooted in a past of 80-character-lines and before code reviews. The first example in the book: void flmoon(int n, int nph, long* jd, float* frac) ought, in a modern era, be something like: void MoonPhaseToDay(int n_cycles, int phase, long* julian_day, float* fractional_day); If for no other reason than I can have some hope at understanding the second argument, or finding it again. You'll also notice flmoon is a misnomer -- it computes more than full moons!

norswap 14y ago

I agree. I always felt that if math theorems were presented in a programming language, they would be way more understandable (especially if the good practices are respected: meaningful variable names, etc). Mathematicians love to talk about rigor, but when it comes down to write their ideas, they often have little enough of it.

kd0amg 14y ago
I always felt that if math theorems were presented in a programming language, they would be way more understandable …. Find out empirically! Look at theorems in some formal methods/programming languages papers and their corresponding mechanizations and see which you find more understandable. (Actually, it might be better to start with something like Software Foundations to get a gentler introduction to a programming language used for this sort of thing -- http://www.cis.upenn.edu/~bcpierce/sf/ http://www.cis.upenn.edu/~bcpierce/sf/ )
- norswap 14y ago
  Hard to do, the one you will look at second will always be at an advantage. So you'd have to select two theorems of "equivalent difficulty", which is not easy. Also you need to select hard to understand theorems, else you'll understand both the representations easily anyway.

Jimmy 14y ago

The main reason it's hard to read math papers is because they contain complex, difficult ideas. Even professional mathematicians take a long time to read papers, and they usually won't be able to understand papers outside of their own specific field of math. I agree with the author that math papers could borrow some ideas from programming - explicitly defining all variables/functions, scoping, etc - but acquainting yourself with the conventions of the field alleviate most of those problems.

monjaro 14y ago

People seem to be pretty critical of this post, but can anyone give me a good reason why math notation shouldn't have some concept of scope? I can't see any downsides. It would certainly help me puzzle out my professor's notes when he has used f to mean three different things in as many lines.

yaks_hairbrush 14y ago
There is some concept of scope in math. At a fairly early level, a student learns sums and integrals. The variable of integration/summation's scope is just within the integral/sum. At a higher level, some things have scope equal to a subfield of mathematics. A gothic p, for example, represents a prime ideal in algebraic number theory. It represents a parabolic Lie algebra in Lie theory. (Actually this causes problems when you're doing Lie theory and number theory at the same time.)

ggchappell 14y ago

Math researcher here. I agree with the basic idea of this article, but not the specifics. So: notation is a wonderful thing. If someone can improve on it, then great, but modern math notation is what makes math doable. It's true that there is poor notation and poorly used notation; these things need to be fixed. Regardless, I agree that math is hard to read. I read a lot of math. It's hard. Part of this is because it involves abstract, complicated stuff that often must be understood exactly. But part is because we are not very good at writing it. Or perhaps we don't put as much effort into making it readable as we should. Being someone who knows a fair amount about programming (or so I like to think) and also writes math research papers, I've found that some of the concepts used to manage complexity in software are also applicable to math. In particular, I must agree about scope. I've made an effort in my recent writing to be careful about this issue. Another helpful idea involves how to write good comments in code. Too many mathematical proofs perform some operation without telling the reader why or giving any kind of overview. There is nothing wrong with saying, "Here's what we're going to do," and then doing it, and then saying, "Here's what we just did."

shocks 14y ago

Mathematics is only hard to read if it has been written badly or if you do not understand all the concepts involved. Source: I am a Mathematics student.

73939133 14y ago

I never posted before but this article made me so mad I had to jump into the comments! I am so glad to find everyone pointing out all the reasons the article is wrong, actually there is a highly relevant link on math.stackexchange about this particular issue: http://math.stackexchange.com/questions/24241/why-do-mathematicians-use-single-letter-variables/24246#24246 http://math.stackexchange.com/questions/24241/why-do-mathema...

gknoy 14y ago

> But are mathematicians too embroiled in some misguided quest for Huffman coding? I think that yes, they are: for centuries, mathematics has been written by hand. It's much easier to write 'f' than 'force'. on a blackboard. Moreover, when writing on paper, with ink that you have to make or buy, with a pen that wears out as you write with it, it makes sense that early mathematicians would embrace a compressed notation. Scribes even did it, so it shouldn't surprise us that mathematicians would. The compression is based on domain knowledge, and this is only problematic for people who are not intimately familiar with the domain at hand. Physics is similar. Decoding the rocket equation is only possible when you know what m1 and m0 are, or ve, and many properties, constants, and even operations are symbolized with a single greek letter. (Gradients, for example) This is very powerful, because (as others have mentioned) it allows you to easily write more-complex things, and often times you (as a mathematician or physicist) prefer to think at a higher level of abstraction. It also makes it tremendously arcane for those of us who don't even know the names of the greek letters, let alone have a sound backing in that particular field of math or physics.

CJefferson 14y ago
The compressed notation is not just about saving paper and ink. It's all about the amount of mental power and work you have to exert. If I'm filling a few pages with equations, if I have to write 'force' instead of 'f' all over the place, it's going to take me about 5 times longer. I'm going to only fit 1/5 of the content on each line and page, (OK, 1/5 might be excessive, but the principle holds I feel). Programming languages have similar properties. Why don't we always write integer instead of int, loop_variable instead of i, and have no compact way of writing unnamed lambda functions?

verroq 14y ago

The author has some preconceived notion of how math is supposed to be read, deduces that math is hard to read. See: How to avoid "compliation errors" when reading math. https://profiles.google.com/114134834346472219368/buzz/GPQDR3gHsYr https://profiles.google.com/114134834346472219368/buzz/GPQDR...

gfodor 14y ago

The problem with math isn't symbology or notation per se, it's that it's not evolved to take advantage of modern technology. It's an artifact of pencil-and-paper being the medium of choice for expressing ideas. We can do much better. When you are looking at an equation, you are looking at the purest distillation of an idea. Underneath that equation sits countless layers of abstraction and reasoning. Why can't I peel back these layers and see them on my computer or iPad? The equation is the iceberg tip peeking out of the water onto the paper, I want to see underneath. Let me feel around and slowly and confidently fill in the places where my understanding is fuzzy. Prevent me from moving back up to the final equation until everything underneath is fleshed out and solid in my mind. And if bits start to fade, let me quickly react to snap them back into focus. This is what we are doing mentally anyway when we flip back and forth in a math text referencing previous proofs and equations. Math needs it's "hyperlink." Anything I say here has been said better and more convincingly by Bret Victor: http://worrydream.com/KillMath/ http://worrydream.com/KillMath/ http://worrydream.com/LadderOfAbstraction/ http://worrydream.com/LadderOfAbstraction/ Though I will argue that at least in his Kill Math writings he's throwing the baby out with the bath water in a certain sense. Let's keep the symbology as a optimal way to encapsulate our knowledge for the amazing mental leverage it gives us, but give us a way to move up and down the ladder of abstraction in order to facilitate understanding.

trhtrsh 14y ago
Math's hyperlink: http://mathworld.wolfram.com/Angle.html http://mathworld.wolfram.com/Angle.html
- gfodor 14y ago
  not following?

b_emery 14y ago

Meyer's matrix analysis has an excellent, and relevant, quote: The great French mathematician Pierre-Simon Laplace (1749–1827) said that, “Such is the advantage of a well-constructed language that its simpliﬁed notation often becomes the source of profound theories.”"

halayli 14y ago

It's hard to understand any programming language if you haven't taken the time to learn its syntax. Unlike teaching programming, the problem I see in teaching math is that the syntax is slipped into the lessons without making a conscious effort teaching it.

jdougan 14y ago

Math notation was't so much designed as accreted, and it shows in how awkward it can get. Dijkstra indirectly makes this point (among others) in "The notational conventions I adopted, and why" at http://www.cs.utexas.edu/~EWD/ewd13xx/EWD1300.PDF http://www.cs.utexas.edu/~EWD/ewd13xx/EWD1300.PDF

steamer25 14y ago

If someone were awesome (maybe somebody like John Resig), they could come up with a math notation framework that started terse but could expand to more detailed definitions with a click. E.g., Starts with (capital sigma) but expands to 'Sum(1..n)' and then expands again to 1 + 2 + 3 + ...n. Clicking the + changes it to 'add' which maybe links off to Wikipedia or Khan Academy. Clicking the numbers might change them to a pictograph of a group of widgets, etc.

pixl97 14y ago
A math IDE. I think Wolfram Alpha may do just that, then again dealing with formal maths is not my thing. An IDE makes sense if it can encapsulate the symbols and terms you're conveying. How many programmers sit down with notepad or vi and write huge programs? Yes, some do (and I have), but it's far easier with the IDE.

narrator 14y ago

Richard Feynman in one of his quantum physics lectures explains that he is going to abbreviate a few things because they are a pain to write out longhand on the chalkboard. I think this is the main reason everything is so abbreviated, it comes from the era when doing lots of manual calculations by hand called for efficiency in notation over clarity.

el_cuadrado 14y ago

> People criticize c++ because you can overload the operators to mean anything you want By the same people who struggle to read math, I guess.

pash 14y ago

No doubt about it, mathematical notation can be difficult to read. It's terse, conceptually dense, operators are commonly overloaded, etc., etc. But that's not why it's hard to read. Math is hard to read, I think, because math is hard: (1) Some core mathematical concepts are fundamentally difficult. Notions like infinity (and different kinds of infinity!) are entirely foreign to our daily experience. (2) In addition, mathematics is more abstract than most programming. Tim Gowers, in his lovely little book Mathematics: A Very Short Introduction, writes: What is the black king in chess? This is a strange question, and the most satis- factory way to deal with it seems to be to sidestep it slightly. What more can one do than point to a chessboard and explain the rules of the game, perhaps paying particular attention to the black king as one does so? What matters about the black king is not its existence, or its intrinsic nature, but the role that it plays in the game. The abstract method in mathematics, as it is sometimes called, is what results when one takes a similar attitude to mathematical objects. This sort of abstraction is, I think, abhorred by many programmers. But it's the reason single-letter variable names are common in mathematics. There's no way to describe what a mathematical object is with a name relating it to some real-world, concrete, familiar thing, as programmers often try to do. You can only describe what it does by writing down some rules it obeys.* (3) In math, concepts are inherently more hierarchical than is common in programming. As programmers, we're all familiar with building higher-level things from lower-level pieces. And we try, and often succeed, in building them up in ways that let other people use them without needing to know much of anything about the lower-level pieces. In math, there's a similar accretive process, but because the concepts are so much more abstract, it's often impossible to understand how a high-level concept behaves without understanding how its lower-level constituents behave. Mathematical concepts are conceptually hierarchical, whereas pieces of programs are often structurally hierarchical, but conceptually (nearly) orthogonal. * — (A controversial aside.) An object-oriented mindset can exacerbate the problem by demanding that we reify concepts into objects that represent what they "are". This is often impossible, hence the inscrutable object names we've all laughed and despaired at. Phil Karlton's well known witticism comes to mind: There are only two hard things in computer science: cache invalidation and naming things. Naming things is hard because, just like in math, we sometimes lack a familiar semantic reference for the concepts we deal with as programmers. Conversely, in functional languages, there's less need to think about what something "is", because the natural way of programming is to define relations (functions) on a small number of core data structures. Interestingly, single-letter variable names are also common in some functional languages, like Haskell, because it generally doesn't matter what the data is so long as you know what you can do with it. And that information is embodied in its type, expressed in its type signature, and enforced by the type-checker at compile time.

amplitwist 14y ago

Consider the solution to the quadratic equation x = (-b ± sqrt(b^2-4ac))/(2a) What is x? What is a? The wonderful thing about math is that it doesn't matter. x could be length, dollars, area, volume, mass, potatoes, lines of code, chickens, or electron-volts. But in the general case, it represents a number. Why would anyone consider it an improvement to write Given that number1 answer^2 + number2 answer + number3 = 0, answer = (-number2 ± sqrt(number2^2-4 number1 number3))/(2 number1). The extra characters convey no additional semantics. Even when presented with an equation where things have specific units, you can usually mentally figure out the units of everything else, due to conventions, previous definitions, and mental dimensional analysis. In my experience, the notation isn't so much of an issue as knowing definitions, closely followed by being able to translate those definitions into intuition. You must be able to remember them exactly, otherwise nothing makes any sense, and you must have an intuition for what they mean, otherwise you'll never get anything done. Consider some terms from a recent talk I attended that was outside my immediate areas of expertise: "solvable Lie group", "left-invariant metric", "upstairs", "double cover". I was able to understand the main idea of the talk, but my full understanding of the talk was sunk by not knowing the definition of left-invariance [1]. Since math is inherently abstract, it is hard, and there is no substitute for the hard work necessary to acquire an intuition for it. When doing high-level math, it is necessary to have a rigorous intuition for the subject, where you are able to intuitively see a path to a proof, and then are able to translate that intuition into a suitably rigorous argument. [1] To illustrate my point that knowing definitions precisely is one of the keys to understanding math, here's the definition of a left-invariant metric. Let G be a Lie group with metric <,>, L_g be left multiplication, and L_g^* denote the pullback of L_g. The metric is said to be left-invariant if L_g^* <u,v> = <u,v> for all u,v in G. It makes no sense unless you know what metrics, Lie groups, left multiplication, and pullbacks are, and you'll only shoot yourself in the foot if you can't define them precisely.

j2kun 14y ago

So this article is just the author complaining that he has no practice in reading mathematics. Would you be surprised if someone who never written a computer program before couldn't understand a complicated dynamic programming algorithm? Of course not, they have no practice or background! And for the record, computer programming is much more precise than mathematics. Most people here don't do mathematics, so they don't realize how many implicit identifications we make between objects which are very very different. In programs, correct types are tantamount to the success of a program. In mathematics, we will readily identify rings with topological spaces and never think twice. Moreover, we need to readily be able to change the rules by which we consider two different things to be identical on the fly. It is simply a different way of thinking that programmers aren't used to.

hybridthesis 14y ago

Math is the most psychedelic thing ever. It's short so you don't have to read a novel for its powerful effects to kick in.

sbi 14y ago

This is an old chestnut. Mathematical notation is terse because mathematics doesn't consist entirely of notation! Look at any book or research paper; you'll find a great deal of text surrounding equations. Comparing this to code is nonsensical. Unless you're reading something in WEB, you are much more likely to encounter code than comments. There are certainly some unfortunate conventions and cruft in mathematics, but every generation of mathematicians rewrites in part what it inherited from its forebears.

aroberge 14y ago

If we ignore the cosmological constant contribution, for each of the 10 non-zero components of the Ricci curvature tensor, the sum of the component of the Ricci curvature tensor minus half of the corresponding component of the metric tensor multiplied by the Ricci scalar is equal to four times the ratio of the circumference of an arbitrary circle divided by its radius, multiplied by Newton's gravitational constant divided by the fourth power of the speed of light in vacuum and multiplied by the corresponding component of the stress-energy tensor. Would this be clearer than the original [http://en.wikipedia.org/wiki/Einstein_field_equations http://en.wikipedia.org/wiki/Einstein_field_equations]? After all, I don't use any weird symbols and avoid greek letters... Surely the author of that blog post would be satisfied by this ... Perhaps, I should include what I mean by Ricci curvature tensor ... but first define what I mean by a tensor, including the metric tensor, and explain other words like stress-energy tensor; it should not take more than a few lines of text for it to be completely clear, right?