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"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

"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.
    - quietbritishjim 6y ago
      * I find it quite unusual in practice for genuinely new symbolic notation to be used by an author. Maybe that just reflects the fields I read about most (information theory, Bayesian modelling, harmonic analysis). * Usually you don't come across a journal article or even blog post with a single isolated equation. So any new or unusual notation can be explained once and reused many times. * Even if you did have an isolated equation with unusual notation, I still think it's more clear to define the notation and then show the equation rather than spelling it out it words! (I'm sure you could find terrible counterexamples!) The visual benefit seems so great to me that it would be worth splitting it into two parts. A bit like a diagram or a graph can make things clearer even if it needs a bit of explanation.
      - throwaway744678 6y ago
        > * I find it quite unusual in practice for genuinely new symbolic notation to be introduced by an author. This sentence seems to contradict the rest of your comment; did you mean "I find it quite unusual in practice for genuinely new symbolic notation to NOT be introduced by an author."?
        quietbritishjim 6y ago
        Thanks, that was ambiguous and the way you read it wasn't what I intended. I meant it was unusual for an author to use a new symbolic notation at all, without saying anything about whether they define it in those cases where they do use something new. I've edited "introduced" to "used".
      - nicoburns 6y ago
        It doesn't need to be genuinely new to be confusing. It just needs to be unfamiliar to the reader. At the very least, one should point the reader to a resource where they can read about the notation.
  - erezsh 6y ago
    That's not a genuine example. Most academic texts I've read will use something like s = d/t And sometimes not even explain what the components mean, because obviously 't' stands for time. That's what all their lecturers used, so why bother explain it. Some will even make up their own notation: s = d(t) And somewhere will say "f(x) in this article describes an inverse multiplicative relation", without explaining that it's actually simple division, so other academics won't find them too obvious or boorish. Some even take it a step further and use random greek letters (without exhausting the English alphabet first ofc.), where small gamma and big gamma mean completely different and unrelated things.
    - HPsquared 6y ago
      That sounds more an issue of unclear naming, than with notation itself. Edit: Though I agree, mathematical notation lends itself to using single letters to denote objects. This can of course be problematic. I'm a fan of 'pseudocode' - somewhere between natural language and rigorous notation - where possible.
    - swilliamsio 6y ago
      >s = d/t Slightly worse than that. That equation is always written as v = s/t. v represents velocity and s represents distance, for some reason.
      - Sharlin 6y ago
        I would assume that the "s" comes either from Latin spatium or German Strecke.
      - abenga 6y ago
        This is (velocity) = (displacement) / (time). s is used because of the Latin word _spatium_ for space. If I remember correctly, there was a difference between distance and displacement. (displacement is "net").
        quietbritishjim 6y ago
  - gjulianm 6y ago
    But that example is a little bit artificial, isn't it? A lot of mathematical concepts are more complex than that and sometimes symbols are not the best option. Say, for example the definition of Hausdorff space, in words and symbols: - Any two distinct points in the space have disjoint neighbourhoods. - ∀x,y ∈ X with x ≠ y, ∃U,V ⊂ X s.t. x ∈ U, y ∈ V and U ∩ V = ∅. Another example would be Navier-Stokes equations, where they're much easier to understand in words than in symbols. Symbols are ok when you don't have to search too much to see what they mean and when the idea you're trying to transmit with them is relatively simple, but trying to build complicated phrases and definitions with symbols, for me, ends up being a mess. edit: fixed hausdorff space definition, points should be distinct
    - toxik 6y ago
      Curiously, you have been ambiguous in the mathematical notation. Counterproof: let x = y.
      - nicoburns 6y ago
        Unfortunately that's not uncommon in mathematical texts either!
      - Bootvis 6y ago
        IMO It should be "Any two distinct points in..." in English as well. Precisions is hard!
      - gjulianm 6y ago
        You are indeed right!
    - andrepd 6y ago
      You are comparing apples to oranges. Your English definition is missing the definition of neighborhood. To actually be an accurate comparison you'd need to add ", where a neighborhood is a set that contains an open ball that contains the point." But now, you're also missing the definition of open ball...
  - contravariant 6y ago
    The first one isn't really symbolic though. So really you should compare: - speed is the derivative of position with respect to time. and - Let x(t) be the position of an object at time t then its speed v(t) is: v(t) = (dx/dt)(t) Also note that most of the time I'm just putting the symbols after the word explaining what it means, while this does allow me to use the symbolic notation for differentiation it doesn't really make the first part any shorter. Also it would be a mistake to introduce speed by just 1 specific formula (even if I didn't specify the types of the object involved) since speed is a far more general concept.
  - Smaug123 6y ago
    I'm certainly not saying that symbols are always bad. Rather, I am attempting to argue that the parent is false in asserting that "mathematician[s]… just wish it would be written with symbols so that they could know precisely what the book is trying to say". The Hausdorff space example is a good one: it's very easy to say what a Hausdorff space is, but if you have to spell it out formally then the definition is kind of big and ugly.
- jvvw 6y ago
  I've got a PhD in maths and can translate symbols into concepts in my brain much quicker than words into concepts in general. Also writing symbolically forces a rigour on the writer. I've been trying to read some semi-mathematical stuff written by scientists but non-mathematicians recently and it is painful trying to figure out what they really mean!
  - Smaug123 6y ago
    Sure, but the problem is that it was written by non-mathematicians, not that they were not using symbols. That's kind of what I was trying to say: symbols help you be rigorous, but they slow you down, and often the complete text-on-the-page doesn't actually need all the rigour that's forced on you by the symbols.
- ChrisLomont 6y ago
  As a math PhD I disagree. I can read math notation far faster and more accurately than ambiguous English. We don’t translate symbols. If anything, when reading English we have to translate into symbols. For example, reading 5-7, I don’t have to translate the - symbol to the word “subtract”. I know this is -2. And I don’t translate the - in that to the word “negative,” and certainly not to the word “subtract”. And it’s vastly faster to agree 5-7=-2 is correct than “5 subtract 7 equals negative 2.” Symbols are how mathematicians think.
  - Smaug123 6y ago
    I think you've got the same misconception as quietbritishjim above. Start stacking quantifiers, and the symbols get hairy much faster than the English does.
    - ChrisLomont 6y ago
      Maybe you're not used to reading quantifiers. I also find them much easier, faster, and more accurate to read, because it's from practice. Sure you can take simple English statements, and write them with quantifiers, and claim the English is simpler. But going the other way, expressing complex items in English, is a non-starter. English is far too sloppy, whereas the quantifier version is mathematically precise. Try converting something professional, such as Godel's incompleteness proofs, into English. Without precise quantifiers you'd quickly get lost, make mistakes, and take forever to get anywhere. For example, look at page 17 of the proof [1], at AG(6)(a) (after the "Thus"), where there is a long statement in logic. Convert to English (near impossible, certainly not possible without something like parentheses) and tell me it's easier for a logician to read. It's not. As written it's concise and parseable without confusion for a logician. The simple English sentence case is a tiny part of what mathematicians do. [1] https://web.yonsei.ac.kr/bkim/goedel.pdf https://web.yonsei.ac.kr/bkim/goedel.pdf
      - Smaug123 6y ago
        Look, do you really want me to wade through sixteen pages to discover the notation first (written by someone who doesn't know about \langle and \rangle, either)? I would consider it bad argumentative form of the same water as Euler's apocryphal "does God exist" debate with Diderot. At the very least, you will struggle to persuade me that the use of \wedge is easier to understand than the English word "and" with line breaks. Also you've picked a specific example where the objects of study are these long strings of symbols. Of course any paper worth its salt is going to use them - they're literally the things that the paper is there to examine. It's the metamathematical statements in this context, not the quotation of the formulas under study, that I want to replace with English.
- commandlinefan 6y ago
  Hm - well, I'm not a mathematician, but I am a programmer and I know that there are many, many times when I've seen somebody try to describe an algorithm in what ends up being incomprehensible English followed by a code example that actually clears up what it was they were 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.
  - quietbritishjim 6y ago
    * I find it quite unusual in practice for genuinely new symbolic notation to be used by an author. Maybe that just reflects the fields I read about most (information theory, Bayesian modelling, harmonic analysis). * Usually you don't come across a journal article or even blog post with a single isolated equation. So any new or unusual notation can be explained once and reused many times. * Even if you did have an isolated equation with unusual notation, I still think it's more clear to define the notation and then show the equation rather than spelling it out it words! (I'm sure you could find terrible counterexamples!) The visual benefit seems so great to me that it would be worth splitting it into two parts. A bit like a diagram or a graph can make things clearer even if it needs a bit of explanation.
    - throwaway744678 6y ago
      > * I find it quite unusual in practice for genuinely new symbolic notation to be introduced by an author. This sentence seems to contradict the rest of your comment; did you mean "I find it quite unusual in practice for genuinely new symbolic notation to NOT be introduced by an author."?
      - quietbritishjim 6y ago
        Thanks, that was ambiguous and the way you read it wasn't what I intended. I meant it was unusual for an author to use a new symbolic notation at all, without saying anything about whether they define it in those cases where they do use something new. I've edited "introduced" to "used".
    - nicoburns 6y ago
      It doesn't need to be genuinely new to be confusing. It just needs to be unfamiliar to the reader. At the very least, one should point the reader to a resource where they can read about the notation.
- erezsh 6y ago
  That's not a genuine example. Most academic texts I've read will use something like s = d/t And sometimes not even explain what the components mean, because obviously 't' stands for time. That's what all their lecturers used, so why bother explain it. Some will even make up their own notation: s = d(t) And somewhere will say "f(x) in this article describes an inverse multiplicative relation", without explaining that it's actually simple division, so other academics won't find them too obvious or boorish. Some even take it a step further and use random greek letters (without exhausting the English alphabet first ofc.), where small gamma and big gamma mean completely different and unrelated things.
  - HPsquared 6y ago
    That sounds more an issue of unclear naming, than with notation itself. Edit: Though I agree, mathematical notation lends itself to using single letters to denote objects. This can of course be problematic. I'm a fan of 'pseudocode' - somewhere between natural language and rigorous notation - where possible.
  - swilliamsio 6y ago
    >s = d/t Slightly worse than that. That equation is always written as v = s/t. v represents velocity and s represents distance, for some reason.
    - Sharlin 6y ago
      I would assume that the "s" comes either from Latin spatium or German Strecke.
    - abenga 6y ago
      This is (velocity) = (displacement) / (time). s is used because of the Latin word _spatium_ for space. If I remember correctly, there was a difference between distance and displacement. (displacement is "net").
      - quietbritishjim 6y ago
- gjulianm 6y ago
  But that example is a little bit artificial, isn't it? A lot of mathematical concepts are more complex than that and sometimes symbols are not the best option. Say, for example the definition of Hausdorff space, in words and symbols: - Any two distinct points in the space have disjoint neighbourhoods. - ∀x,y ∈ X with x ≠ y, ∃U,V ⊂ X s.t. x ∈ U, y ∈ V and U ∩ V = ∅. Another example would be Navier-Stokes equations, where they're much easier to understand in words than in symbols. Symbols are ok when you don't have to search too much to see what they mean and when the idea you're trying to transmit with them is relatively simple, but trying to build complicated phrases and definitions with symbols, for me, ends up being a mess. edit: fixed hausdorff space definition, points should be distinct
  - toxik 6y ago
    Curiously, you have been ambiguous in the mathematical notation. Counterproof: let x = y.
    - nicoburns 6y ago
      Unfortunately that's not uncommon in mathematical texts either!
    - Bootvis 6y ago
      IMO It should be "Any two distinct points in..." in English as well. Precisions is hard!
    - gjulianm 6y ago
      You are indeed right!
  - andrepd 6y ago
    You are comparing apples to oranges. Your English definition is missing the definition of neighborhood. To actually be an accurate comparison you'd need to add ", where a neighborhood is a set that contains an open ball that contains the point." But now, you're also missing the definition of open ball...
- contravariant 6y ago
  The first one isn't really symbolic though. So really you should compare: - speed is the derivative of position with respect to time. and - Let x(t) be the position of an object at time t then its speed v(t) is: v(t) = (dx/dt)(t) Also note that most of the time I'm just putting the symbols after the word explaining what it means, while this does allow me to use the symbolic notation for differentiation it doesn't really make the first part any shorter. Also it would be a mistake to introduce speed by just 1 specific formula (even if I didn't specify the types of the object involved) since speed is a far more general concept.
- Smaug123 6y ago
  I'm certainly not saying that symbols are always bad. Rather, I am attempting to argue that the parent is false in asserting that "mathematician[s]… just wish it would be written with symbols so that they could know precisely what the book is trying to say". The Hausdorff space example is a good one: it's very easy to say what a Hausdorff space is, but if you have to spell it out formally then the definition is kind of big and ugly.
jvvw 6y ago
I've got a PhD in maths and can translate symbols into concepts in my brain much quicker than words into concepts in general. Also writing symbolically forces a rigour on the writer. I've been trying to read some semi-mathematical stuff written by scientists but non-mathematicians recently and it is painful trying to figure out what they really mean!
- Smaug123 6y ago
  Sure, but the problem is that it was written by non-mathematicians, not that they were not using symbols. That's kind of what I was trying to say: symbols help you be rigorous, but they slow you down, and often the complete text-on-the-page doesn't actually need all the rigour that's forced on you by the symbols.
ChrisLomont 6y ago
As a math PhD I disagree. I can read math notation far faster and more accurately than ambiguous English. We don’t translate symbols. If anything, when reading English we have to translate into symbols. For example, reading 5-7, I don’t have to translate the - symbol to the word “subtract”. I know this is -2. And I don’t translate the - in that to the word “negative,” and certainly not to the word “subtract”. And it’s vastly faster to agree 5-7=-2 is correct than “5 subtract 7 equals negative 2.” Symbols are how mathematicians think.
- Smaug123 6y ago
  I think you've got the same misconception as quietbritishjim above. Start stacking quantifiers, and the symbols get hairy much faster than the English does.
  - ChrisLomont 6y ago
    Maybe you're not used to reading quantifiers. I also find them much easier, faster, and more accurate to read, because it's from practice. Sure you can take simple English statements, and write them with quantifiers, and claim the English is simpler. But going the other way, expressing complex items in English, is a non-starter. English is far too sloppy, whereas the quantifier version is mathematically precise. Try converting something professional, such as Godel's incompleteness proofs, into English. Without precise quantifiers you'd quickly get lost, make mistakes, and take forever to get anywhere. For example, look at page 17 of the proof [1], at AG(6)(a) (after the "Thus"), where there is a long statement in logic. Convert to English (near impossible, certainly not possible without something like parentheses) and tell me it's easier for a logician to read. It's not. As written it's concise and parseable without confusion for a logician. The simple English sentence case is a tiny part of what mathematicians do. [1] https://web.yonsei.ac.kr/bkim/goedel.pdf https://web.yonsei.ac.kr/bkim/goedel.pdf
    - Smaug123 6y ago
      Look, do you really want me to wade through sixteen pages to discover the notation first (written by someone who doesn't know about \langle and \rangle, either)? I would consider it bad argumentative form of the same water as Euler's apocryphal "does God exist" debate with Diderot. At the very least, you will struggle to persuade me that the use of \wedge is easier to understand than the English word "and" with line breaks. Also you've picked a specific example where the objects of study are these long strings of symbols. Of course any paper worth its salt is going to use them - they're literally the things that the paper is there to examine. It's the metamathematical statements in this context, not the quotation of the formulas under study, that I want to replace with English.
commandlinefan 6y ago
Hm - well, I'm not a mathematician, but I am a programmer and I know that there are many, many times when I've seen somebody try to describe an algorithm in what ends up being incomprehensible English followed by a code example that actually clears up what it was they were trying to say.