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Why Mathematics is Boring (2007) [pdf]

86 points by danbst 5y ago

bluedays 5y ago
I believe this is actually a really great point. For instance, I didn't know that it was originally Brahmagupta who started using symbols in math. His initial use of symbols to represent numbers involved using color names like "blue" and "green". More interestingly before using symbols as variables in math Egyptians were capable of doing Quadratic equations without these variables. If I were teaching math today I would probably teach this. I would try to do algebra without using variables, and then I would use funny words like "blue" as Brahma Gupta use to. I think that this would probably stop a lot of the questions like "why are there letters in math!?"
- cabalamat 5y ago
  > I think that this would probably stop a lot of the questions like "why are there letters in math!?" Wouldn't people just instead ask "Why are there colours in maths?"? I don't think that this: blue + blue = 2*blue Is any more meaningful or edifying than: x + x = 2*x
  - pca006132 5y ago
    Yeah, I think the problem is not about variable names but giving meaning to these abstract objects. Beginners often have difficulty understanding abstract concepts and specific (concrete) examples are helpful for them to learn. I think 'Funny words' alone will not be helpful, or even detrimental to beginners' understanding as they may be distracted by unrelated concepts (e.g. what do you mean by adding blues together in this example?).
  - bluedays 5y ago
    Not if you show a really long example with algebra where you are not using any variables, and show how it can be simplified with variables.
- javajosh 5y ago
  I don't recall where I read this, but a young Feynman motivated himself to learn trig by imagining that he had been challenged by a mysterious stranger to answer riddles. For example (I'm making this up) "You only have a protractor and the ability to measure your paces - tell me the height of that flagpole!" The operation, then, is to pace out a distance from the base of flagpole, sight down the protractor to the top of the flagpole to get an angle, and compute. (Since tan(y/x) = a, arctan(a) = y/x, y=x * arctan(a)). So the motivation was imaginary and concrete. And it's dramatic, because there's an obstacle, a chance of failure, and a chance for glory. I can't help but see a parallel with magicians, who can dazzle us because they are willing to go further than most of us, in terms of practice. In the same way, math gives you the ability to dazzle with surprising answers, to do a lot with a little.
- elliekelly 5y ago
  I think they’re doing this more in schools. I was helping a first grader with homework recently and I was surprised the math worksheet was essentially simple algebra but with a shape or a little picture to represent the variable instead of a letter: > 3 + circle = 7 > 10 - cloud = 4 It seems like a perfectly reasonable exercise for a first grader. Clouds and puppy faces and circles. But I have to admit if I’d seen “x” in place of the symbols I’m not sure I would’ve thought it was so reasonable.
  - taneq 5y ago
    The introductory arithmetic I've seen recently even kind of flips it around and defines subtraction in terms of algebra. "7 - 5 = ?" is presented as "5 plus what equals 7?"
    - whatshisface 5y ago
      That's how subtraction is actually defined, so that's not a bad idea.
abdullahkhalids 5y ago
If you read any science papers, they start with a clear introduction with the aims, claims, importance and novelty of the work. A lot of math papers (but not all) just start off with a dry statement of what the theorems being proved are, and jump right into the proofs. I always wonder why editors don't understand the importance of these things and don't enforce them.
- enriquto 5y ago
  I for one welcome the "dryness" of mathematical writing. It feels clean, like reading a story without distracting ads. A beautiful advice that I received as a student was to write mathematics as a series of definitions, propositions and proofs. No text is allowed to exist outside of these three. In practice it is difficult to enforce, but it is helpful to keep this as an aim.
  - taneq 5y ago
    As a non-mathematician this seems to be lacking a little something. Would you not want your paper to provide some indication of what you're attempting to communicate and why? Or is that information (as the joke goes) possible, and therefore trivial, to infer?
    - enriquto 5y ago
      A great deal of maths is that you get to use theorems for purposes that they were not intended for. Presenting the theorem in a "pure" form thus allows to approach it without pre-conceptions. I love the analogy with cooking recipes in another comment. Math papers are like recipe books, deliberately devoid of their social context. The same recipe may mean different things to different cooks, even contradictory! Having a clean, neutral description of the recipe allows both cooks to safely refer to the exact same recipe, without endorsing contexts that they may find odious. I agree that knowing the context in which a recipe was created, and the contexts where it has been used, is very useful. But it would be extremely annoying to have this explanation interleaved with the recipe description itself. This information is best kept separate. Then, at the beginning of the recipe you can have a list of links to cooks who have written different things about it.
GDC7 5y ago
To me it's the language. You have to learn a whole new alphabet and signs. This is done for the sake of quick communication between mathematicians, but it's necessary to make a study and see the pros and cons. While it's true that it makes communication faster and straightforward it keeps so many people outside of the field. Maybe the field would benefit to go more towards philosophy and logic, explaining it with words.
- _Microft 5y ago
  The idea that unfamiliar symbols and alphabets are a huge problem for the accessibility of math is common. As physicist I do not agree. Math is hard. It's damned difficult. Symbols and alphabets are the least of your concerns when dealing with a math paper. I know a lot of these symbols by name, I sometimes understand the notation or could familiarize myself with it but the math itself? Nope, no chance, usually. If one cannot deal with the symbols, there is no chance in hell one could deal with the ideas.
  - andrewjl 5y ago
    One can make the same argument for doing arithmetic using Roman numerals.
  - amcoastal 5y ago
    I'll disagree. I read many papers with mathematics in them, and I get a lot of the concepts but the symbology used doesn't make sense to me so its hard for me to understand what is exactly going on. The sentence after the equation that explains each symbol is necessary for me, and many others as well. Not everyone has taken 8 math classes to know each kroniger delta by heart.
    - wiz21c 5y ago
      except that mathematicians like to use shortcuts notation everywhere, shortcuts that only them understand... For example P(A|B,C) ?= P(A|B;C). Moreover, mathematician seems driven by a frugal principle. They try to condense their though in the smallest number of symbols. To me it's like writing a Perl program with the shortest amount of text. Of course, the result is right, but it's super hard to understand.
erichahn 5y ago
It's boring but usually everything is well-defined and hence well-understandable. Not the case for most CS papers.
- cormacrelf 5y ago
  So many of the difficulties in reading many papers (well, disciplines) boil down to adding links to things the first time you mention them. Good academic writers do this, bad ones don’t. The gist of this paper is that assuming all knowledge prior to your development is very limiting, not only because far fewer people can read it, but because every time you don’t introduce knowledge you also skip over part of a story. Very well, but I think you are correct that CS’ problem is pretty much isolated to the first point, because nearly all papers get to talk about real world applications of the research and that’s the story covered. A big problem for CS papers, particularly in PL (programming language research), seems to be heavy reliance on assumed knowledge of Greek letters and notation in the very field-specific way you like to use them. People would understand your paper if only they had read your previous three, which alluded to what you might have meant by these Greek letters, but only by figuring out the two citations they each have in common. If you are bad at pronouncing Greek letters and it’s a PDF so you can’t copy them, you can’t even google what you see. Even if you could, it wouldn’t help. Notation is ten times harder to search for. (I have never, ever had this problem reading a law paper, not even slightly, not even once.) There’s an interesting demo here from Will Crichton about how to prepare better documents for conveying understanding in PL. He has a thing to show you the “read as” on hover. https://twitter.com/wcrichton/status/1442891297333800966 https://twitter.com/wcrichton/status/1442891297333800966 https://willcrichton.net/nota https://willcrichton.net/nota
decasteve 5y ago
Research Math is an inside joke between friends. The jokes fall flat unless you know the same people and attend the same parties.
greenail 5y ago
The title should be "why mathematics papers are boring, how to spice them up with narrative", that is what the article is about.
- rramadass 5y ago
  >how to spice them up with narrative Oh, Lord; NO! I would like see all Human Narratives/Unnecessary frivolities/Assume-reader-is-a-Idiot language banished from the Teaching of ALL Maths/Science. What we we need is a focus on the direct teaching of Principles along with their Real World Applications.
greenail 5y ago
I find that the older I get the more I appreciate math. I did find it boring when I was younger. I'm not sure if it is for folks like mathologer and 3brown1blue. I tend to think visually and they do a wonderful job in that area. I don't recall anyone presenting math like they do when I was in school in the 1980's.
canada_dry 5y ago
An aside: his UC Riverside page is full of interesting stuff: https://math.ucr.edu/home//baez/README.html https://math.ucr.edu/home//baez/README.html
- chmaynard 5y ago
  Dr. Baez is a brilliant mathematical physicist, but web design and publishing is not his strong suit.
  - civilized 5y ago
    I think the page looks and is great. One column of plain text and pictures. Minimal, loads quickly, easy to understand and navigate. It doesn't have any of the pointless bloat of most "modern" web design, and is all the better for it.
paulpauper 5y ago
It's hard to understand , not because it's boring, but because it's inherently hard and opaque. If mathematicians tried to make complicated topics easier to understand the papers would be 10x as long.
1cvmask 5y ago
Mathematics is not boring to those interested in it and pursuing it. It is a universal language. Adding unnecessary complexity will take away from its "purenesss" and terseness. Language is not a barrier to entry. You will then be graded on incomplete formulas but great storytelling. Let's leave the storytelling to all the other fields of life.
graycat 5y ago
Ah, in math writing, it's easy enough to say more and be not boring and at times be at least interesting, inviting, even exciting! Let's have some examples!! (1) Dimension. So, suppose we are in the first class in linear algebra: "Maybe you have heard that the real line has 1 dimension, is 1 dimensional, the plane is 2 dimensional, and the space we live in is 3 dimensional. Well, that's all true enough, but in linear algebra we do better and have more: For one, we get to say clearly what is meant by dimension, that, in particular, why the line, plane, and space are 1, 2, 3 dimensional. For much more, for any positive integer n we have n-dimensional space. Next, in linear algebra n-dimensional space is a relatively easy generalization of what we already know well in dimensions 1, 2, 3. Why might we care? For example, we know well what distance is in dimensions 1, 2, 3, and distance in n dimensions is a straight forward generalization. In dimensions 2 and 3, we understand angle, and also that carries over to n dimensions. For more, with computing it is common to have a list of, say, 15 numbers. Well, for just one benefit, with linear algebra we get to regard that list as a point in n = 15 dimensional space, and doing so lets us do some powerful things with representing and approximating that list." So, we get some sense of previews of coming attractions and some invitation to higher dimensions. (2) Optimization. "There is a subject, with a lot of development just after WWII, called linear programming (LP). The programming is in the English sense of operational planning as in war logistics and planning as was crucial in WWII. The linear is the same as in linear algebra. The main goal, point of LP is to find how to exploit the freedom we have in doing the operations, the work to be done, to get the work done as fast or cheaply as possible, that is, to find an optimal way to do the work. So, the subject LP is part of optimization. There have been some Nobel prizes from applications of LP and other math of optimization to economics. There have been applications of LP to feed mixing, oil refinery operation, management of large projects, and parts of transportation." (3) The Simplex Algorithm. "Maybe in high school algebra you saw the topic of systems of linear equations. Well, it is fair to say that the standard way to solve such a system is Gauss elimination due to C. F. Gauss. The idea is simple: Multiplying one of the equations by some non-zero number and adding the resulting equation to another of the equations does not change the set of solutions. So, doing that in a slightly clever way results in the system of equations with a lot of zeros, about half all zeros, so that the set of solutions is obvious just by inspection. Then for linear programming, in practice the main solution technique is the simplex algorithm, and it is just but done with optimization in mind." (4) Completeness. A rational number can be written as p/q for integers p and q. We will see, easily, that the rational numbers are not up to carrying the load, are not up to doing the work we need done. So we need a more powerful system of numbers -- we need the real numbers. Here is a really simple place the rational numbers fail to do what we want: At times we consider square roots. E.g., the square root of 9 is 3. Well, what is the square root of 2? Suppose that square root were a rational number, i.e., so that (p/q)^2 = 2 Then we have p^2 = 2q^2 so that the left side has an even number of factors of 2 while the right side has an odd number. Tilt. Bummer! That can't be. That's a contradiction. So, there is no rational number that is the square root of 2. So, for something really simple, just finding a square root, the rational numbers fail us, can't carry the load or do the work. The real numbers will let us find the square root of 2 and much more. With the real numbers we get what we call completeness. A joke, basically correct, is that calculus is the elementary consequences of the completeness property of the real numbers. Then we generalize: Banach space is a complete normed linear space. Hilbert space is a complete inner product space. The Fourier transform works because of completeness. So, we move on and see how the real numbers are complete ...."
Koshkin 5y ago
Reading math can be boring (often it's not), but solving problems never is. (Math is not a spectator sport.) I also hear people say programming is boring. This is absurd.
- MrLeap 5y ago
  Maybe off topic, but solving problems is a fraction of the joy I get from programming. Expression and personal power over reality are where I get the joy. A painter can create world and share a feeling. An author can manifests a memory. A musician can transmit a human experience without language. Human imaginings about magic are immemorial. Math describes reality. Math also can describe an extrapolation further. Programming can manifest from the descriptive language of math into the real. It can use it as a pigment for a new kind of picture. Programming helps with everything below, and it enables new things slightly above. By "above" I mean the layers of abstraction. A programmer isn't an painter, but a programmer/painter has an additional axis of art. Programming is our species apex of material transcendence. I don't believe it's the top of the pile, but I have no conception for what's above. Its capacity for encapsulation seems to grow as the dreams do. Programming grows to predict, programming grows to create. Everything is just a new library, a new framework, a new environment. How long before its limits are found, so we can find the next epiphanies?
  - I_AM_A_SMURF 5y ago
    Math does not describe reality, Physics does. Math is just pure thought. If anything, Math is an abstraction of how we think about things, but not the things themselves.
    - Koshkin 5y ago
      This is incorrect. Mathematics also describes reality, just a different aspect of it. It has never been “pure thought.” Hence its usefulness in science and engineering.
bedobi 5y ago
I always struggled (and still struggle) with math. A couple of years ago, randomly browsing YouTube, I came across this home made video asking how they figured out the distance to the moon before modern technology. The host starts out small scale showing he can calculate the distance to things in his back yard using trigonometry and then scales it up to the moon. My mind was blown, because no one ever told me that. It was simple, anyone could understand it. When I was in school, all I was told was to memorize abstract formulae like calculating the length of sides of triangles based on angles and known length of one side. It was never contextualized to any actual, let alone interesting or fascinating, applications.
- mydeskistoosm 5y ago
  You can't post something like this and not post the video.
  - cma 5y ago
    This isn't it, but Terrance Tao does the entire cosmic distance ladder: https://www.youtube.com/watch?v=7ne0GArfeMs https://www.youtube.com/watch?v=7ne0GArfeMs
  - bedobi 5y ago
    Pretty sure it was this https://www.youtube.com/watch?v=ohdysfFWO4w&list=PLpH1IDQEoE8QWWTnWG5cK4ePCqg9W2608&index=3 https://www.youtube.com/watch?v=ohdysfFWO4w&list=PLpH1IDQEoE... or this https://www.youtube.com/watch?v=bcn0ycIHKog&list=PLpH1IDQEoE8QWWTnWG5cK4ePCqg9W2608&index=4 https://www.youtube.com/watch?v=bcn0ycIHKog&list=PLpH1IDQEoE...
- mymythisisthis 5y ago
  Here is a fun textbook I found on astronomy http://gron.ca/math/dupuis_1910/dupuis.pdf http://gron.ca/math/dupuis_1910/dupuis.pdf that is written in a more pragmatic style.
- Jensson 5y ago
  Most math textbooks contextualize it like that, so I guess yours did too. Just that in school kids almost never care about that, they just want to pass the tests and therefore ignore all contextualization and just remember the minimum possible amount required to solve test questions. So likely you already saw those things many times before and forgot since you didn't find it important back then. That is the main struggle for many math teachers, they try to do all these fun and interesting explanations, and the kids just ignore it and go directly for the formulas and forgets everything else. the problem seems easy to solve until you have experienced trying to apply it yourself to a real class of kids needing the material for real grades. It can be done, but a teacher who could do it could make way more money in entertainment etc, since that is what is required to get kids to pay attention.
cevi 5y ago
The author says that mathematicians find math outside their own field to be boring and difficult to understand. As a mathematician, I think he's rather missing the point: - mathematics is boring to everyone right up until the moment you need it. Then suddenly it becomes very interesting. The way mathematicians typically read papers is not by randomly picking through recent submissions to the arxiv and dutifully reading everything they come across. Instead, they stumble on a hard problem in their own research which they don't know how to solve, and they search to see if anyone else has worked on it before. The paper you would have discarded as pointlessly abstract or ridiculously overspecialized just yesterday suddenly reads like a riveting novel today. No amount of creative writing tips would have made it any more interesting to you yesterday - unless the writers happened to anticipate the exact reason you would end up becoming interested in it ahead of time.
- vostok 5y ago
  > mathematics is boring to everyone right up until the moment you need it. Then suddenly it becomes very interesting. That might be true for higher level math, but anything at the graduate or undergraduate level has been already curated to be interesting.
civilized 5y ago
Meanwhile, on the front page of science.com: "NF-κB activation in cardiac fibroblasts results in the recruitment of inflammatory Ly6Chi monocytes in pressure-overloaded hearts" Sometimes papers are technical and don't need to pretend that they are telling an exciting story of interest to a general audience. It isn't just a math issue.
- abecedarius 5y ago
  https://www.science.org/doi/10.1126/scisignal.abe4932 https://www.science.org/doi/10.1126/scisignal.abe4932 This does a considerably better job of context/interest than the math example did.
  - civilized 5y ago
    It's a lot easier when all you have to do is say stuff like "heart failure is bad". What's the mathematician supposed to do, say "group theory is cool and important"?
    - rsj_hn 5y ago
      The thing is, mathematicians understand how cool and important it is, and that's enough. You can't really explain it to someone else -- it's like trying to explain how cool and important a piece of music is to a deaf person who doesn't know music. They see the conductor waving and say "well, that's boring." All you can do is explain "there is a whole world of beauty and meaning there. I'm sorry you can't experience it, but it's there."
    - abecedarius 5y ago
      The "Mathematics is Boring" author is a mathematician who seems really enthusiastic about math. He's not asking here for mathematicians to punch up their papers for nonmathematicians; he's asking them to give a bit better context for all the other mathematicians beyond the dozen others in the same sub-sub-subspecialty. This Science paper's intro/abstract sets it out for scientists, rather than for biologists in whatever subspecialty this thing is.
paulpauper 5y ago
what is the opinion of 3 brown,1 blue
hikerclimber1 5y ago
It isn’t.
omginternets 5y ago
I follow the author on Twitter, and really enjoy his exposes of mathematical concepts. I highly recommend him: @johncarlosbaez
rramadass 5y ago
Mathematics is NOT Boring; the teaching of Maths divorced of Real-World Applications is what is Boring. An over-emphasis on Formalism/Abstraction is what is killing people's interest in Maths/Sciences. The Teaching of all Maths/Sciences should always start with a Real-World motivating example and then introduce the Maths as necessary to Solve it. In this context see V. I. Arnold's essay; On Teaching Mathematics - https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html Quote from the above article: * Attempts to create "pure" deductive-axiomatic mathematics have led to the rejection of the scheme used in physics (observation - model - investigation of the model - conclusions - testing by observations) and its substitution by the scheme: definition - theorem - proof. It is impossible to understand an unmotivated definition but this does not stop the criminal algebraists-axiomatisators. * What is a group? Algebraists teach that this is supposedly a set with two operations that satisfy a load of easily-forgettable axioms. This definition provokes a natural protest: why would any sensible person need such pairs of operations? "Oh, curse this maths" - concludes the student (who, possibly, becomes the Minister for Science in the future). * We get a totally different situation if we start off not with the group but with the concept of a transformation (a one-to-one mapping of a set onto itself) as it was historically. A collection of transformations of a set is called a group if along with any two transformations it contains the result of their consecutive application and an inverse transformation along with every transformation.
LambdaTrain 5y ago
I think many people who start finding mathematics interesting at an older age and blame the math education in young age missed that their intellectual capability also strengthen with their age. The whole point of something being "interesting" is that this thing is possible to be understood but not that easily.

86 points by danbst 5y ago

I believe this is actually a really great point. For instance, I didn't know that it was originally Brahmagupta who started using symbols in math. His initial use of symbols to represent numbers involved using color names like "blue" and "green". More interestingly before using symbols as variables in math Egyptians were capable of doing Quadratic equations without these variables. If I were teaching math today I would probably teach this. I would try to do algebra without using variables, and then I would use funny words like "blue" as Brahma Gupta use to. I think that this would probably stop a lot of the questions like "why are there letters in math!?"

cabalamat 5y ago
> I think that this would probably stop a lot of the questions like "why are there letters in math!?" Wouldn't people just instead ask "Why are there colours in maths?"? I don't think that this: blue + blue = 2*blue Is any more meaningful or edifying than: x + x = 2*x
- pca006132 5y ago
  Yeah, I think the problem is not about variable names but giving meaning to these abstract objects. Beginners often have difficulty understanding abstract concepts and specific (concrete) examples are helpful for them to learn. I think 'Funny words' alone will not be helpful, or even detrimental to beginners' understanding as they may be distracted by unrelated concepts (e.g. what do you mean by adding blues together in this example?).
- bluedays 5y ago
  Not if you show a really long example with algebra where you are not using any variables, and show how it can be simplified with variables.
javajosh 5y ago
I don't recall where I read this, but a young Feynman motivated himself to learn trig by imagining that he had been challenged by a mysterious stranger to answer riddles. For example (I'm making this up) "You only have a protractor and the ability to measure your paces - tell me the height of that flagpole!" The operation, then, is to pace out a distance from the base of flagpole, sight down the protractor to the top of the flagpole to get an angle, and compute. (Since tan(y/x) = a, arctan(a) = y/x, y=x * arctan(a)). So the motivation was imaginary and concrete. And it's dramatic, because there's an obstacle, a chance of failure, and a chance for glory. I can't help but see a parallel with magicians, who can dazzle us because they are willing to go further than most of us, in terms of practice. In the same way, math gives you the ability to dazzle with surprising answers, to do a lot with a little.
elliekelly 5y ago
I think they’re doing this more in schools. I was helping a first grader with homework recently and I was surprised the math worksheet was essentially simple algebra but with a shape or a little picture to represent the variable instead of a letter: > 3 + circle = 7 > 10 - cloud = 4 It seems like a perfectly reasonable exercise for a first grader. Clouds and puppy faces and circles. But I have to admit if I’d seen “x” in place of the symbols I’m not sure I would’ve thought it was so reasonable.
- taneq 5y ago
  The introductory arithmetic I've seen recently even kind of flips it around and defines subtraction in terms of algebra. "7 - 5 = ?" is presented as "5 plus what equals 7?"
  - whatshisface 5y ago
    That's how subtraction is actually defined, so that's not a bad idea.

abdullahkhalids 5y ago

If you read any science papers, they start with a clear introduction with the aims, claims, importance and novelty of the work. A lot of math papers (but not all) just start off with a dry statement of what the theorems being proved are, and jump right into the proofs. I always wonder why editors don't understand the importance of these things and don't enforce them.

enriquto 5y ago
I for one welcome the "dryness" of mathematical writing. It feels clean, like reading a story without distracting ads. A beautiful advice that I received as a student was to write mathematics as a series of definitions, propositions and proofs. No text is allowed to exist outside of these three. In practice it is difficult to enforce, but it is helpful to keep this as an aim.
- taneq 5y ago
  As a non-mathematician this seems to be lacking a little something. Would you not want your paper to provide some indication of what you're attempting to communicate and why? Or is that information (as the joke goes) possible, and therefore trivial, to infer?
  - enriquto 5y ago
    A great deal of maths is that you get to use theorems for purposes that they were not intended for. Presenting the theorem in a "pure" form thus allows to approach it without pre-conceptions. I love the analogy with cooking recipes in another comment. Math papers are like recipe books, deliberately devoid of their social context. The same recipe may mean different things to different cooks, even contradictory! Having a clean, neutral description of the recipe allows both cooks to safely refer to the exact same recipe, without endorsing contexts that they may find odious. I agree that knowing the context in which a recipe was created, and the contexts where it has been used, is very useful. But it would be extremely annoying to have this explanation interleaved with the recipe description itself. This information is best kept separate. Then, at the beginning of the recipe you can have a list of links to cooks who have written different things about it.

GDC7 5y ago

To me it's the language. You have to learn a whole new alphabet and signs. This is done for the sake of quick communication between mathematicians, but it's necessary to make a study and see the pros and cons. While it's true that it makes communication faster and straightforward it keeps so many people outside of the field. Maybe the field would benefit to go more towards philosophy and logic, explaining it with words.

_Microft 5y ago
The idea that unfamiliar symbols and alphabets are a huge problem for the accessibility of math is common. As physicist I do not agree. Math is hard. It's damned difficult. Symbols and alphabets are the least of your concerns when dealing with a math paper. I know a lot of these symbols by name, I sometimes understand the notation or could familiarize myself with it but the math itself? Nope, no chance, usually. If one cannot deal with the symbols, there is no chance in hell one could deal with the ideas.
- andrewjl 5y ago
  One can make the same argument for doing arithmetic using Roman numerals.
- amcoastal 5y ago
  I'll disagree. I read many papers with mathematics in them, and I get a lot of the concepts but the symbology used doesn't make sense to me so its hard for me to understand what is exactly going on. The sentence after the equation that explains each symbol is necessary for me, and many others as well. Not everyone has taken 8 math classes to know each kroniger delta by heart.
  - wiz21c 5y ago
    except that mathematicians like to use shortcuts notation everywhere, shortcuts that only them understand... For example P(A|B,C) ?= P(A|B;C). Moreover, mathematician seems driven by a frugal principle. They try to condense their though in the smallest number of symbols. To me it's like writing a Perl program with the shortest amount of text. Of course, the result is right, but it's super hard to understand.

erichahn 5y ago

It's boring but usually everything is well-defined and hence well-understandable. Not the case for most CS papers.

cormacrelf 5y ago
So many of the difficulties in reading many papers (well, disciplines) boil down to adding links to things the first time you mention them. Good academic writers do this, bad ones don’t. The gist of this paper is that assuming all knowledge prior to your development is very limiting, not only because far fewer people can read it, but because every time you don’t introduce knowledge you also skip over part of a story. Very well, but I think you are correct that CS’ problem is pretty much isolated to the first point, because nearly all papers get to talk about real world applications of the research and that’s the story covered. A big problem for CS papers, particularly in PL (programming language research), seems to be heavy reliance on assumed knowledge of Greek letters and notation in the very field-specific way you like to use them. People would understand your paper if only they had read your previous three, which alluded to what you might have meant by these Greek letters, but only by figuring out the two citations they each have in common. If you are bad at pronouncing Greek letters and it’s a PDF so you can’t copy them, you can’t even google what you see. Even if you could, it wouldn’t help. Notation is ten times harder to search for. (I have never, ever had this problem reading a law paper, not even slightly, not even once.) There’s an interesting demo here from Will Crichton about how to prepare better documents for conveying understanding in PL. He has a thing to show you the “read as” on hover. https://twitter.com/wcrichton/status/1442891297333800966 https://twitter.com/wcrichton/status/1442891297333800966 https://willcrichton.net/nota https://willcrichton.net/nota

decasteve 5y ago

Research Math is an inside joke between friends. The jokes fall flat unless you know the same people and attend the same parties.

greenail 5y ago

The title should be "why mathematics papers are boring, how to spice them up with narrative", that is what the article is about.

rramadass 5y ago
>how to spice them up with narrative Oh, Lord; NO! I would like see all Human Narratives/Unnecessary frivolities/Assume-reader-is-a-Idiot language banished from the Teaching of ALL Maths/Science. What we we need is a focus on the direct teaching of Principles along with their Real World Applications.

greenail 5y ago

I find that the older I get the more I appreciate math. I did find it boring when I was younger. I'm not sure if it is for folks like mathologer and 3brown1blue. I tend to think visually and they do a wonderful job in that area. I don't recall anyone presenting math like they do when I was in school in the 1980's.

canada_dry 5y ago

An aside: his UC Riverside page is full of interesting stuff: https://math.ucr.edu/home//baez/README.html https://math.ucr.edu/home//baez/README.html

chmaynard 5y ago
Dr. Baez is a brilliant mathematical physicist, but web design and publishing is not his strong suit.
- civilized 5y ago
  I think the page looks and is great. One column of plain text and pictures. Minimal, loads quickly, easy to understand and navigate. It doesn't have any of the pointless bloat of most "modern" web design, and is all the better for it.

paulpauper 5y ago

It's hard to understand , not because it's boring, but because it's inherently hard and opaque. If mathematicians tried to make complicated topics easier to understand the papers would be 10x as long.

1cvmask 5y ago

Mathematics is not boring to those interested in it and pursuing it. It is a universal language. Adding unnecessary complexity will take away from its "purenesss" and terseness. Language is not a barrier to entry. You will then be graded on incomplete formulas but great storytelling. Let's leave the storytelling to all the other fields of life.

graycat 5y ago

Ah, in math writing, it's easy enough to say more and be not boring and at times be at least interesting, inviting, even exciting! Let's have some examples!! (1) Dimension. So, suppose we are in the first class in linear algebra: "Maybe you have heard that the real line has 1 dimension, is 1 dimensional, the plane is 2 dimensional, and the space we live in is 3 dimensional. Well, that's all true enough, but in linear algebra we do better and have more: For one, we get to say clearly what is meant by dimension, that, in particular, why the line, plane, and space are 1, 2, 3 dimensional. For much more, for any positive integer n we have n-dimensional space. Next, in linear algebra n-dimensional space is a relatively easy generalization of what we already know well in dimensions 1, 2, 3. Why might we care? For example, we know well what distance is in dimensions 1, 2, 3, and distance in n dimensions is a straight forward generalization. In dimensions 2 and 3, we understand angle, and also that carries over to n dimensions. For more, with computing it is common to have a list of, say, 15 numbers. Well, for just one benefit, with linear algebra we get to regard that list as a point in n = 15 dimensional space, and doing so lets us do some powerful things with representing and approximating that list." So, we get some sense of previews of coming attractions and some invitation to higher dimensions. (2) Optimization. "There is a subject, with a lot of development just after WWII, called linear programming (LP). The programming is in the English sense of operational planning as in war logistics and planning as was crucial in WWII. The linear is the same as in linear algebra. The main goal, point of LP is to find how to exploit the freedom we have in doing the operations, the work to be done, to get the work done as fast or cheaply as possible, that is, to find an optimal way to do the work. So, the subject LP is part of optimization. There have been some Nobel prizes from applications of LP and other math of optimization to economics. There have been applications of LP to feed mixing, oil refinery operation, management of large projects, and parts of transportation." (3) The Simplex Algorithm. "Maybe in high school algebra you saw the topic of systems of linear equations. Well, it is fair to say that the standard way to solve such a system is Gauss elimination due to C. F. Gauss. The idea is simple: Multiplying one of the equations by some non-zero number and adding the resulting equation to another of the equations does not change the set of solutions. So, doing that in a slightly clever way results in the system of equations with a lot of zeros, about half all zeros, so that the set of solutions is obvious just by inspection. Then for linear programming, in practice the main solution technique is the simplex algorithm, and it is just but done with optimization in mind." (4) Completeness. A rational number can be written as p/q for integers p and q. We will see, easily, that the rational numbers are not up to carrying the load, are not up to doing the work we need done. So we need a more powerful system of numbers -- we need the real numbers. Here is a really simple place the rational numbers fail to do what we want: At times we consider square roots. E.g., the square root of 9 is 3. Well, what is the square root of 2? Suppose that square root were a rational number, i.e., so that (p/q)^2 = 2 Then we have p^2 = 2q^2 so that the left side has an even number of factors of 2 while the right side has an odd number. Tilt. Bummer! That can't be. That's a contradiction. So, there is no rational number that is the square root of 2. So, for something really simple, just finding a square root, the rational numbers fail us, can't carry the load or do the work. The real numbers will let us find the square root of 2 and much more. With the real numbers we get what we call completeness. A joke, basically correct, is that calculus is the elementary consequences of the completeness property of the real numbers. Then we generalize: Banach space is a complete normed linear space. Hilbert space is a complete inner product space. The Fourier transform works because of completeness. So, we move on and see how the real numbers are complete ...."

Koshkin 5y ago

Reading math can be boring (often it's not), but solving problems never is. (Math is not a spectator sport.) I also hear people say programming is boring. This is absurd.

MrLeap 5y ago
Maybe off topic, but solving problems is a fraction of the joy I get from programming. Expression and personal power over reality are where I get the joy. A painter can create world and share a feeling. An author can manifests a memory. A musician can transmit a human experience without language. Human imaginings about magic are immemorial. Math describes reality. Math also can describe an extrapolation further. Programming can manifest from the descriptive language of math into the real. It can use it as a pigment for a new kind of picture. Programming helps with everything below, and it enables new things slightly above. By "above" I mean the layers of abstraction. A programmer isn't an painter, but a programmer/painter has an additional axis of art. Programming is our species apex of material transcendence. I don't believe it's the top of the pile, but I have no conception for what's above. Its capacity for encapsulation seems to grow as the dreams do. Programming grows to predict, programming grows to create. Everything is just a new library, a new framework, a new environment. How long before its limits are found, so we can find the next epiphanies?
- I_AM_A_SMURF 5y ago
  Math does not describe reality, Physics does. Math is just pure thought. If anything, Math is an abstraction of how we think about things, but not the things themselves.
  - Koshkin 5y ago
    This is incorrect. Mathematics also describes reality, just a different aspect of it. It has never been “pure thought.” Hence its usefulness in science and engineering.

bedobi 5y ago

I always struggled (and still struggle) with math. A couple of years ago, randomly browsing YouTube, I came across this home made video asking how they figured out the distance to the moon before modern technology. The host starts out small scale showing he can calculate the distance to things in his back yard using trigonometry and then scales it up to the moon. My mind was blown, because no one ever told me that. It was simple, anyone could understand it. When I was in school, all I was told was to memorize abstract formulae like calculating the length of sides of triangles based on angles and known length of one side. It was never contextualized to any actual, let alone interesting or fascinating, applications.

mydeskistoosm 5y ago
You can't post something like this and not post the video.
- cma 5y ago
  This isn't it, but Terrance Tao does the entire cosmic distance ladder: https://www.youtube.com/watch?v=7ne0GArfeMs https://www.youtube.com/watch?v=7ne0GArfeMs
- bedobi 5y ago
  Pretty sure it was this https://www.youtube.com/watch?v=ohdysfFWO4w&list=PLpH1IDQEoE8QWWTnWG5cK4ePCqg9W2608&index=3 https://www.youtube.com/watch?v=ohdysfFWO4w&list=PLpH1IDQEoE... or this https://www.youtube.com/watch?v=bcn0ycIHKog&list=PLpH1IDQEoE8QWWTnWG5cK4ePCqg9W2608&index=4 https://www.youtube.com/watch?v=bcn0ycIHKog&list=PLpH1IDQEoE...
mymythisisthis 5y ago
Here is a fun textbook I found on astronomy http://gron.ca/math/dupuis_1910/dupuis.pdf http://gron.ca/math/dupuis_1910/dupuis.pdf that is written in a more pragmatic style.
Jensson 5y ago
Most math textbooks contextualize it like that, so I guess yours did too. Just that in school kids almost never care about that, they just want to pass the tests and therefore ignore all contextualization and just remember the minimum possible amount required to solve test questions. So likely you already saw those things many times before and forgot since you didn't find it important back then. That is the main struggle for many math teachers, they try to do all these fun and interesting explanations, and the kids just ignore it and go directly for the formulas and forgets everything else. the problem seems easy to solve until you have experienced trying to apply it yourself to a real class of kids needing the material for real grades. It can be done, but a teacher who could do it could make way more money in entertainment etc, since that is what is required to get kids to pay attention.

cevi 5y ago

The author says that mathematicians find math outside their own field to be boring and difficult to understand. As a mathematician, I think he's rather missing the point: - mathematics is boring to everyone right up until the moment you need it. Then suddenly it becomes very interesting. The way mathematicians typically read papers is not by randomly picking through recent submissions to the arxiv and dutifully reading everything they come across. Instead, they stumble on a hard problem in their own research which they don't know how to solve, and they search to see if anyone else has worked on it before. The paper you would have discarded as pointlessly abstract or ridiculously overspecialized just yesterday suddenly reads like a riveting novel today. No amount of creative writing tips would have made it any more interesting to you yesterday - unless the writers happened to anticipate the exact reason you would end up becoming interested in it ahead of time.

vostok 5y ago
> mathematics is boring to everyone right up until the moment you need it. Then suddenly it becomes very interesting. That might be true for higher level math, but anything at the graduate or undergraduate level has been already curated to be interesting.

civilized 5y ago

Meanwhile, on the front page of science.com: "NF-κB activation in cardiac fibroblasts results in the recruitment of inflammatory Ly6Chi monocytes in pressure-overloaded hearts" Sometimes papers are technical and don't need to pretend that they are telling an exciting story of interest to a general audience. It isn't just a math issue.

abecedarius 5y ago
https://www.science.org/doi/10.1126/scisignal.abe4932 https://www.science.org/doi/10.1126/scisignal.abe4932 This does a considerably better job of context/interest than the math example did.
- civilized 5y ago
  It's a lot easier when all you have to do is say stuff like "heart failure is bad". What's the mathematician supposed to do, say "group theory is cool and important"?
  - rsj_hn 5y ago
    The thing is, mathematicians understand how cool and important it is, and that's enough. You can't really explain it to someone else -- it's like trying to explain how cool and important a piece of music is to a deaf person who doesn't know music. They see the conductor waving and say "well, that's boring." All you can do is explain "there is a whole world of beauty and meaning there. I'm sorry you can't experience it, but it's there."
  - abecedarius 5y ago
    The "Mathematics is Boring" author is a mathematician who seems really enthusiastic about math. He's not asking here for mathematicians to punch up their papers for nonmathematicians; he's asking them to give a bit better context for all the other mathematicians beyond the dozen others in the same sub-sub-subspecialty. This Science paper's intro/abstract sets it out for scientists, rather than for biologists in whatever subspecialty this thing is.

paulpauper 5y ago

what is the opinion of 3 brown,1 blue

hikerclimber1 5y ago

It isn’t.

omginternets 5y ago

I follow the author on Twitter, and really enjoy his exposes of mathematical concepts. I highly recommend him: @johncarlosbaez

rramadass 5y ago

Mathematics is NOT Boring; the teaching of Maths divorced of Real-World Applications is what is Boring. An over-emphasis on Formalism/Abstraction is what is killing people's interest in Maths/Sciences. The Teaching of all Maths/Sciences should always start with a Real-World motivating example and then introduce the Maths as necessary to Solve it. In this context see V. I. Arnold's essay; On Teaching Mathematics - https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html Quote from the above article: * Attempts to create "pure" deductive-axiomatic mathematics have led to the rejection of the scheme used in physics (observation - model - investigation of the model - conclusions - testing by observations) and its substitution by the scheme: definition - theorem - proof. It is impossible to understand an unmotivated definition but this does not stop the criminal algebraists-axiomatisators. * What is a group? Algebraists teach that this is supposedly a set with two operations that satisfy a load of easily-forgettable axioms. This definition provokes a natural protest: why would any sensible person need such pairs of operations? "Oh, curse this maths" - concludes the student (who, possibly, becomes the Minister for Science in the future). * We get a totally different situation if we start off not with the group but with the concept of a transformation (a one-to-one mapping of a set onto itself) as it was historically. A collection of transformations of a set is called a group if along with any two transformations it contains the result of their consecutive application and an inverse transformation along with every transformation.

LambdaTrain 5y ago

I think many people who start finding mathematics interesting at an older age and blame the math education in young age missed that their intellectual capability also strengthen with their age. The whole point of something being "interesting" is that this thing is possible to be understood but not that easily.