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Mathematical notation is great at facilitating formal manipulations. This is its critical feature, and without it we would get stuck at the level of ancient mat

Mathematical notation is great at facilitating formal manipulations. This is its critical feature, and without it we would get stuck at the level of ancient mathematics. This is the reason it was invented a few hundred years ago in the first place. That said, I find that notation is often abused in texts as a mere substitute for the normal human language which, while allowing to compress the text, does in fact nothing to help the reader better understand what is being said but rather looks like a crazy mess of characters and other marks in a multitude of fonts, styles and sizes the only purpose of which seems to be to cause an eye strain.

cheez 6y ago
Looks like we read the same books.
dhosek 6y ago
I remember, as a callow college freshman waiting in the hallway of the math department to be able to go into a classroom reading an article which talked about mathematical writing and the first thing it said was to prefer English text over mathematical symbols in numerous cases (e.g., writing "For all $x$ in the Reals" over $\forall x\in\mathbb{R}$). As someone who was highly skilled in TeX at a time when such skills were still fairly rare (many colleges, if they even had TeX available on their time share systems, were still using the am fonts instead of the cm fonts as the latter had only been introduced two years earlier), I remember being upset that my backslash skills were thus denigrated.
- jimhefferon 6y ago
  When I teach the intro to proofs class I require that they learn LaTeX. Some students like the availability of symbols so much that they go a bit nuts. Something like this sentence: $\forall x\in\mathbb{R}$ $\exists y$ that is $>$ the number $x+1$. Sigh.
  - madcaptenor 6y ago
    This isn't exclusive to LaTeX; I think students in general just think that the symbols make things "more mathematical". I remember feeling this way briefly when I was first exposed to things like $\forall$ and $\exists$, and it wore off. This was in the late nineties; in theory I could have had access to a typesetting system but I was writing things by hand.
    - jayjader 6y ago
      As a former maths student, I loved the "compressibility" of using symbols for things like "forAll" and "thereExists". They're quicker to write and allow fitting in more information in less space, two qualities which become highly useful when taking tests in limited amounts of time on sheets of paper with limited space (albeit needing more paper to fit your test answers on is more arguable in it's negativity). For a (even more) subjective point of view, as part of my studies I had oral practicals every 2 weeks or so where I was basically standing at a whiteboard along with 2 classmates, each solving problems given to us by an examiner for ~1 hour. It's simply easier for a lot of students to draw a symbol that's legible from 5 meters away than some words without making them huge.
watersb 6y ago
Thus: 1. Write a good narrative and then 2. Write good code Which is to say that a mathematics text is an instance of literate programming.
- madcaptenor 6y ago
  It's not surprising, then, that Knuth introduced it.
mmhsieh 6y ago
there are some academic disciplines which are extremely guilty of this. they are the ones that are insecure about their status as a science.
logicslave 6y ago
"does in fact nothing to help the reader better understand what is being said" For a mathematician it is the opposite, they just wish it would be written with symbols so that they could know precisely what the book is trying to say
- ukj 6y ago
  We have symbolic expression in programming languages already. It helps somewhat, but it isn’t a panacea. We still have to comment our code and add context.
- Smaug123 6y ago
  I don't think this is true in general. It may be true for a novice, who needs all the available help to keep them rigorous (but even then, there is definitely room for reading-to-build-intuition), but symbols definitely slow you down while you translate them.
  - quietbritishjim 6y ago
    Do you really think > speed = distance / time is less clear than > speed is the ratio of distance over time ? To me the first equation is genuinely easier to read for the purposes of understanding, not just for formal manipulation. For larger equations the difference is only more stark, not less. I have a maths PhD so your comment about symbols being for novices doesn't apply. I suppose the beauty of the first equation is that the objects (speed, distance, time) are visually very distinct from the operations and relationships. In the second form, there's a bit of a word soup so you need to "manually" parse the sentence rather than letting your eyes (really the visual cortex) do that bit of the processing for you.
    - auggierose 6y ago
      The problem is when it is not quite clear what the symbolic notation stands for. With division, that tends to be less of a problem.
nsonha 6y ago
Compressing does help it. To trained eyes a couple of lines is easier to read and less ambiguous than one or 2 pages of English explanation, which is hard to follow for both trained and untrained eyes if you're gonna encouter a bunch of them while reading a proof.
- prox 6y ago
  To me it seems language can also be interpreted as more ambiguous, where as well defined symbols and notation are far more rigorous in definition and use.
  - foerbert 6y ago
    Except you have instances of misuse of those well-defined symbols and notations. Without any accompanying language it can be very difficult to not only determine that there is an error, but then also figure out how to fix it. And other times in some fields there is a lack of well-defined symbols and notations. I don't recall the specifics, but in game theory there is some notation that is ambiguous. There are two methods or forms of representation, and the field is split on which version the notation stands for. It's also frequently not mentioned explicitly in papers, so you have to dig around a bit to see if you can find a usage that makes the meaning clear. I don't know why we can't just have both.
    - nsonha 6y ago
      my main argument is that humans only have so much sort term memory so compression is crucial for readability. Reading a proof is different than reading a book because you have to retain the whole thing with details in your mind not just vague ideas.
      - foerbert 6y ago
        I find myself completely unable to understand that. I suspect we use very different methods. Notation doesn't stay with me once my eyes move past it. The notation is merely there to inform me of what is happening, and as long as I understand the meaning it doesn't matter if it was one symbol or a whole paragraph.
cryptica 6y ago
Math symbols are a minor issue for me. What confuses me the most are descriptions of mathematical concepts. For example, Wikipedia describes a 'field' like this: "In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do." It doesn't make sense to me. What does it mean if an operation 'is defined' on a set? Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set? But if that was the case then "behave as the corresponding operations on rational and real numbers do" would mean that the fields would always need to be of infinite size (have an infinite number of elements) wouldn't it? Because if the field had a limited number of elements and you added the last two (highest) elements together, the property which requires that the result also be present in the same set could not be met because the result would be greater than the highest element in that set... The problem is that if you start with a highly abstracted math concept and you dig through all the links and definitions of sub-concepts, they all have huge gaps like this... So when you try to combine all the definitions together to make sense of that original highly abstracted concept, you end up with tens or hundreds of possible interpretations. But in fact, Math should only have 1 interpretation for each concept so this is a very bad situation to be in. I think math definitions should be more elaborate and repetitive if necessary. They should not try to sound terse and clever. They should not assume that the reader can fill in the gaps. The most rational readers will not be able to fill in the gaps because rational people know the dangers of making assumptions.
- lapinot 6y ago
  You're quoting the introduction of the article, which is notoriously a fuzzy abstract in all the wikipedia articles about mathematical concepts. Let's quote the actual (textual) formal definition (sec. 1.1) as it would stand in a textbook: > Formally, a field is a set F together with two binary operations on F called addition and multiplication. A binary operation on F is a mapping F × F → F, that is, a correspondence that associates with each ordered pair of elements of F a uniquely determined element of F. The result of the addition of a and b is called the sum of a and b, and is denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, and is denoted ab or a ⋅ b. These operations are required to satisfy the following properties, referred to as field axioms. In these axioms, a, b, and c are arbitrary elements of the field F. [...] Still, i don't know how to say it in another way but you probably don't have much experience in mathematics, even the first quote is arguably quite accurate. > Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set? Yes, unless told otherwise an operation is an internal binary operation, it's really the most common form. When it is not the output is notable enough to be specified. > But if that was the case then "behave as the corresponding operations on rational and real numbers do" would mean that the fields would always need to be of infinite size wouldn't it? Because if the field had a limited number of elements and you added the last two (highest) elements together [...] No, the important point here is your use of "highest". A set by default only has equality (and mappings, in and out) but no order relationship. So the most conservative interpretation of "behave as the corresponding operations on rational" would be to only include stuff that can be written using the 4 operations and equality, not ordering. --- Maybe i'm biaised by the fact that i know what a field is, but still, this particular intro is also how i would present a field: give the most common example and say which operations it has. It sure can create false intuitions like yours about the size, but this will always be the case when we use non-normalized language.
throwaway_pdp09 6y ago
> but rather looks like a crazy mess of characters and other marks in a multitude of fonts You're projecting (in the psychological rather than set-theoretic way).

cheez 6y ago

Looks like we read the same books.

dhosek 6y ago

I remember, as a callow college freshman waiting in the hallway of the math department to be able to go into a classroom reading an article which talked about mathematical writing and the first thing it said was to prefer English text over mathematical symbols in numerous cases (e.g., writing "For all $x$ in the Reals" over $\forall x\in\mathbb{R}$). As someone who was highly skilled in TeX at a time when such skills were still fairly rare (many colleges, if they even had TeX available on their time share systems, were still using the am fonts instead of the cm fonts as the latter had only been introduced two years earlier), I remember being upset that my backslash skills were thus denigrated.

jimhefferon 6y ago
When I teach the intro to proofs class I require that they learn LaTeX. Some students like the availability of symbols so much that they go a bit nuts. Something like this sentence: $\forall x\in\mathbb{R}$ $\exists y$ that is $>$ the number $x+1$. Sigh.
- madcaptenor 6y ago
  This isn't exclusive to LaTeX; I think students in general just think that the symbols make things "more mathematical". I remember feeling this way briefly when I was first exposed to things like $\forall$ and $\exists$, and it wore off. This was in the late nineties; in theory I could have had access to a typesetting system but I was writing things by hand.
  - jayjader 6y ago
    As a former maths student, I loved the "compressibility" of using symbols for things like "forAll" and "thereExists". They're quicker to write and allow fitting in more information in less space, two qualities which become highly useful when taking tests in limited amounts of time on sheets of paper with limited space (albeit needing more paper to fit your test answers on is more arguable in it's negativity). For a (even more) subjective point of view, as part of my studies I had oral practicals every 2 weeks or so where I was basically standing at a whiteboard along with 2 classmates, each solving problems given to us by an examiner for ~1 hour. It's simply easier for a lot of students to draw a symbol that's legible from 5 meters away than some words without making them huge.

watersb 6y ago

Thus: 1. Write a good narrative and then 2. Write good code Which is to say that a mathematics text is an instance of literate programming.

madcaptenor 6y ago
It's not surprising, then, that Knuth introduced it.

mmhsieh 6y ago

there are some academic disciplines which are extremely guilty of this. they are the ones that are insecure about their status as a science.

logicslave 6y ago

"does in fact nothing to help the reader better understand what is being said" For a mathematician it is the opposite, they just wish it would be written with symbols so that they could know precisely what the book is trying to say

ukj 6y ago
We have symbolic expression in programming languages already. It helps somewhat, but it isn’t a panacea. We still have to comment our code and add context.
Smaug123 6y ago
I don't think this is true in general. It may be true for a novice, who needs all the available help to keep them rigorous (but even then, there is definitely room for reading-to-build-intuition), but symbols definitely slow you down while you translate them.
- quietbritishjim 6y ago
  Do you really think > speed = distance / time is less clear than > speed is the ratio of distance over time ? To me the first equation is genuinely easier to read for the purposes of understanding, not just for formal manipulation. For larger equations the difference is only more stark, not less. I have a maths PhD so your comment about symbols being for novices doesn't apply. I suppose the beauty of the first equation is that the objects (speed, distance, time) are visually very distinct from the operations and relationships. In the second form, there's a bit of a word soup so you need to "manually" parse the sentence rather than letting your eyes (really the visual cortex) do that bit of the processing for you.
  - auggierose 6y ago
    The problem is when it is not quite clear what the symbolic notation stands for. With division, that tends to be less of a problem.

nsonha 6y ago

Compressing does help it. To trained eyes a couple of lines is easier to read and less ambiguous than one or 2 pages of English explanation, which is hard to follow for both trained and untrained eyes if you're gonna encouter a bunch of them while reading a proof.

prox 6y ago
To me it seems language can also be interpreted as more ambiguous, where as well defined symbols and notation are far more rigorous in definition and use.
- foerbert 6y ago
  Except you have instances of misuse of those well-defined symbols and notations. Without any accompanying language it can be very difficult to not only determine that there is an error, but then also figure out how to fix it. And other times in some fields there is a lack of well-defined symbols and notations. I don't recall the specifics, but in game theory there is some notation that is ambiguous. There are two methods or forms of representation, and the field is split on which version the notation stands for. It's also frequently not mentioned explicitly in papers, so you have to dig around a bit to see if you can find a usage that makes the meaning clear. I don't know why we can't just have both.
  - nsonha 6y ago
    my main argument is that humans only have so much sort term memory so compression is crucial for readability. Reading a proof is different than reading a book because you have to retain the whole thing with details in your mind not just vague ideas.
    - foerbert 6y ago
      I find myself completely unable to understand that. I suspect we use very different methods. Notation doesn't stay with me once my eyes move past it. The notation is merely there to inform me of what is happening, and as long as I understand the meaning it doesn't matter if it was one symbol or a whole paragraph.

cryptica 6y ago

Math symbols are a minor issue for me. What confuses me the most are descriptions of mathematical concepts. For example, Wikipedia describes a 'field' like this: "In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do." It doesn't make sense to me. What does it mean if an operation 'is defined' on a set? Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set? But if that was the case then "behave as the corresponding operations on rational and real numbers do" would mean that the fields would always need to be of infinite size (have an infinite number of elements) wouldn't it? Because if the field had a limited number of elements and you added the last two (highest) elements together, the property which requires that the result also be present in the same set could not be met because the result would be greater than the highest element in that set... The problem is that if you start with a highly abstracted math concept and you dig through all the links and definitions of sub-concepts, they all have huge gaps like this... So when you try to combine all the definitions together to make sense of that original highly abstracted concept, you end up with tens or hundreds of possible interpretations. But in fact, Math should only have 1 interpretation for each concept so this is a very bad situation to be in. I think math definitions should be more elaborate and repetitive if necessary. They should not try to sound terse and clever. They should not assume that the reader can fill in the gaps. The most rational readers will not be able to fill in the gaps because rational people know the dangers of making assumptions.

lapinot 6y ago
You're quoting the introduction of the article, which is notoriously a fuzzy abstract in all the wikipedia articles about mathematical concepts. Let's quote the actual (textual) formal definition (sec. 1.1) as it would stand in a textbook: > Formally, a field is a set F together with two binary operations on F called addition and multiplication. A binary operation on F is a mapping F × F → F, that is, a correspondence that associates with each ordered pair of elements of F a uniquely determined element of F. The result of the addition of a and b is called the sum of a and b, and is denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, and is denoted ab or a ⋅ b. These operations are required to satisfy the following properties, referred to as field axioms. In these axioms, a, b, and c are arbitrary elements of the field F. [...] Still, i don't know how to say it in another way but you probably don't have much experience in mathematics, even the first quote is arguably quite accurate. > Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set? Yes, unless told otherwise an operation is an internal binary operation, it's really the most common form. When it is not the output is notable enough to be specified. > But if that was the case then "behave as the corresponding operations on rational and real numbers do" would mean that the fields would always need to be of infinite size wouldn't it? Because if the field had a limited number of elements and you added the last two (highest) elements together [...] No, the important point here is your use of "highest". A set by default only has equality (and mappings, in and out) but no order relationship. So the most conservative interpretation of "behave as the corresponding operations on rational" would be to only include stuff that can be written using the 4 operations and equality, not ordering. --- Maybe i'm biaised by the fact that i know what a field is, but still, this particular intro is also how i would present a field: give the most common example and say which operations it has. It sure can create false intuitions like yours about the size, but this will always be the case when we use non-normalized language.

throwaway_pdp09 6y ago

> but rather looks like a crazy mess of characters and other marks in a multitude of fonts You're projecting (in the psychological rather than set-theoretic way).