8 ms·

This is a lovely (edit: having been in similar shoes, also terrifying-in-the-moment) example of a broader problem in teaching mathematics: the language we use t

by wcarey 6y ago

This is a lovely (edit: having been in similar shoes, also terrifying-in-the-moment) example of a broader problem in teaching mathematics: the language we use to describe mathematical reasoning is a natural language, like English or Latin, and therefore full of the sorts of bizarre irregularities you'd find in a natural language. Mathematics is also a language about rigorously and precisely defined objects. The conceptual shear between those two things is murder for lots of students. Just like Tacitus omits his verbs (!), when we describe fractions we often omit the implicit definition of the whole. Turns out that's a problem for many students. It's a bit like trying to learn a context-dependent programming grammar with an inconsistent API, but worse, because it's your first "mathematical" language so you're also trying to learn what the abstract objects the language manipulates are. Some other lovely examples: 3(5) means three times five. 3(x) means three times x. 35 means three times ten plus five. 3x means three times x. x(3) means that x is the name of a function taking, in this instance, 3 as its input. x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}. \sin{30}. Radians or degrees? Probably the writer means degrees, but there's no way to tell. There are many more.

mkl 6y ago
I think x(3) would more commonly mean x times 3. f(3) would be function application. Even more context dependence.
- wcarey 6y ago
  Right. Of course, if you write x(3), other mathematicians should frown at you because you're making bad notational choices. It's a bit like explaining to students that the real way to know which 3rd declension nouns are i-stems in Latin is to say the genitive plural both ways. The one that doesn't sound wrong is correct. But you have to have a lot of time in the language for that to work.
  - mkl 6y ago
    Well unnecessary parentheses are often used to indicate a substitution has happened. E.g. in a topic I just taught I would write things like ∫_{y=0}^3 x dy = [xy]_{y=0}^3 = x(3) - x(0) = 3x. In context I think it's perfectly clear and a good notational choice.
    - wcarey 6y ago
      And so it is. edit: I suppose, what I'm trying to get at, perhaps too glibly, is that audience matters terribly much in mathematical writing. In the same way that Latin students don't start with Tacitus or Sallust, famous for their idiosyncratic grammar, math students shouldn't jump into the full context-dependent mess of the notation that experienced mathematicians use. But I think we often thrown them in unintentionally because we're so used to it.
      - jacobush 6y ago
        You used too little context when arguing with an audience who isn't used to that style of writing...
  - C4stor 6y ago
    Also, there are actual quite reasonable rules to know which 3rd declension nouns are i-stems, so it doesn't seem too right too just say "follow your gut". Btw, I wouldn't abuse latin comparisons on an american forum, I don't think it's quite in the culture ^^
- jerf 6y ago
  x(3) is a function "x" being applied to the number 3, because "three times x" is 3x. Of course context would help resolve this if you have a function named x or not, or a variable named x or not, and if you have both a function and a variable named x, well, you worked for your confusion and you have obtained it; congratulations. :)
- jki275 6y ago
  I'm sure some would take it that way as "x" isn't commonly used in mathematical notation as a function name, the norm is to use f, g, etc. However, the notation is relatively clear notwithstanding. It almost has to be x as a function with an input of 3.
- wycy 6y ago
  I think x(3) would more commonly imply a function of 3, since you'd virtually always otherwise just write 3x.
  - qubex 6y ago
    This is one of the reasons I annoy people by following Wolfram’s convention in Mathematica of using square braces to denote arguments passed to a function: f(x)=fx=f×x while f[x] means “apply the function f to the argument x”. An unusual convention it may well be, but at least it’s one devoid of ambiguity.
    - mkl 6y ago
      Not at all. f[x] is clearly the xth component of the array f :-D
      - qubex 6y ago
        On the blackboard in the mathematics department it’s not. But I get your point. I have also heard the objection that [x] is a 1×1 matrix whose only element is ‘x’.
        OscarCunningham 6y ago
        Square brackets can also denote the floor function (especially in old works where '⌊' and '⌋' were typographically unavailable).
      - gowld 6y ago
        An array is merely a function from N to some set.
    - pwdisswordfish2 6y ago
      It's not devoid of ambiguity, because f[A] usually means the image of the set A under f.
  - mkl 6y ago
- Wowfunhappy 6y ago
  If you meant "function", would you generally write 𝑓(3) instead of f(3)? (I have no mathematics background past K-12)
acqq 6y ago
I also remember having constant confusion of what were assumed associations in written bigger formulas especially involving the functions and the practice of not writing the braces, and before that, if 3 1/5 meant three times 1/5 or 3 wholes and 1/5th. I thought it was only on these low levels of educations that people settle on such confusing practices. Later on the university I've discovered that it's the same on all levels. The notations used outside of programming are simply very context-dependent, and it's pity that that dependency is not made clear more often. Also, some practices directly come from some historical uses, and different paths to the current notations still end up to the similarly looking but different meanings.
twic 6y ago
> x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}. That notation for inverse functions is truly appalling. I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading.
- abnry 6y ago
  But the inverse function is the "multiplicative inverse" in the group of functions with composition as "multiplication". In that way, it makes a ton of sense. It's only a problem because you are mixing together two group operations.
- StavrosK 6y ago
  > I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading. Maybe they did.
- DavidSJ 6y ago
  They’re both inverses: one with respect to multiplication, the other with respect to function composition. In abstract algebra, we observe that there are many types of “products”: multiplication, addition, function composition, composition of rotations, matrix multiplication, etc. A common, unified “power” notation for repeatedly taking the product of a single element, or of its inverse, has some value. There is some ambiguity, since multiplication of functions can also often make sense. This is usually resolved by context, or by being explicit where the context isn’t clear.
  - lisper 6y ago
    > one with respect to multiplication, the other with respect to function composition And, of course, multiplication is a function so one is just a specific instance of the other. Infix notation is another blight on the mathematical notation landscape. The amount of human effort that has been put into figuring out how to parse a+b*c is staggering. All of this confusion could have been avoided if we'd just started with s-expressions in the first place.
Spivak 6y ago
I think it's important to know that this really isn't true. Maths at all times is a subjective language. Maths notation is imprecise "intentionally confused", or just ad-hoc defined all the time. When you see something like. f(x) = summation(x^n, n=0, 10) We conveniently ignore that this polynomial is defined at x=0 despite 0^0 not making any sense by ad-hoc defining 0^0 = 1 in this context.
- wcarey 6y ago
  What isn't really true?
- ColinWright 6y ago
  > We conveniently ignore that this polynomial is defined at x=0 despite 0^0 not making any sense by ad-hoc defining 0^0 = 0 in this context. What do you think the polynomial is? I ask, because in all situations similar to this that I've encountered it made sense to define 0^0 as 1, not as 0. If you genuinely have a case where 0^0 = 0 makes consistent sense then I'd be interested in understanding it. So, what do you think the summation actually is when expanded?
  - Spivak 6y ago
    Sorry you're right, it should be 0^0 = 1. An example where 0^0 = 0 occurs when dealing with areas. The measure of the real line in the plane is zero but it's also a rectangle with sides (0, inf) and we define the area of a rectangle to be l*w. You usually see this written as inf × 0 = 0 but you sometimes you see the interpretation as 1/0 × 0 = 0^0 = 0. And you know this is a an ad-hoc definition because you're not allowed to algebraically manipulate it at all.
- grodeni 6y ago
  actually 0^0 is usually chosen equal to 1... which probably still supports your argument.
  - Spivak 6y ago
    Corrected! Thank you!
Denvercoder9 6y ago
> \sin{30}. Radians or degrees? Probably the writer means degrees, but there's no way to tell. I once had a professor that insisted that sin(30) meant sin(30 pi) in radians, with the pi being implicit. Unsurprisingly, it was the worst class I've ever taken.
- wcarey 6y ago
  Absolutely terrifying. How did he communicate with other peer mathematicians?
  - Denvercoder9 6y ago
    I honestly don't know - it was an intro level physics class.
  - gowld 6y ago
    Mathematicians understand how to read definitions of notation. If all of your trig is fractions of pi, writing pi redundantly everywhere is not useful.
    - wcarey 6y ago
      Is that commonly done?
      - renewiltord 6y ago
        It is not. Though it wouldn't be too hard to adapt, I think, if it was noted at the beginning of a text. For it to take, it would have to power some influential result, sort of like how the summation convention in tensor notation works (something that infuriated me as a kid).
    - saagarjha 6y ago
      …but that's not a fraction of pi?
      - 6y ago
billfruit 6y ago
Another is sin^2(x) meaning (sin(x))^2, but by a more intuitive reading it should mean sin(sin(x)). I don't know what exactly is being gained by the usual notation: surely better clarity should be preferred over the time/effort saved in writing the extra pair of paranthesis, and I would prefer it be written as (sin(x))^2 always. Thing is mathematics, and mathematical pedagogy seems hardly concerned with such rampant notation confusion plaguing much of maths. Perhaps moving to some type of machine-readable notation will be better for consistency and avoiding of much notational confusion.
- abdullahkhalids 6y ago
  Does f^2(x) in any scenario mean (f(x))^2? Usually the square of a function, with named arguments is written as f(x)^2. So, not clear on the confusion.
  - bonoboTP 6y ago
    Yes, in trigonometry sin^2(x) is commonly written to mean (sin(x))^2
- saagarjha 6y ago
  When actually working with trigonometric identities, using parentheses typically gets unwieldily very quickly. Same reason sin(x) often becomes sin x.
  - billfruit 6y ago
    Then perhaps we should drop all parens and move to prefix notation.
mamcx 6y ago
So, have been any sufficient effort in design a new syntax of maths that could help the regular folk? If we learn new syntax(langs) all the time then push for a new syntax(math) could not be that bad of a idea...
- wcarey 6y ago
  There are some gentle efforts in that direction (tau radians for example), but mathematical notation is old (except the parts that are new). And it's really hard to strike the right balance between concision and ambiguity.
abdullahkhalids 6y ago
Why can't we have mathematics devoid of these ambiguities? One reasons is that humans have small working memories, and novice mathematics students have even smaller ones. The ambiguity of notation, while confusing, once mastered, allows us to write shorter expressions, whose meaning we resolve from context, and which become both easier to write and to understand. A second reason is that while mathematical logic is rigorous and precise, unequal mathematical objects are similar to each - even objects that on first glace seem nothing alike. For instance, the two types of inverses you mention, or sets and linear spaces, or groups and tetrahedrons. And because of the diffuse nature of mathematical objects, it is inevitable that the same notation will be used for unequal objects. Because, it advances our human understanding of mathematics to use the same notation for two unequal but similar objects. This second reason, once understood, is one threshold between the mechanical mastery of the intermediate student and almost artistic use of mathematics by the advanced student.
- spenczar5 6y ago
  I think it's a lot less beautiful. I think it has to do more with historical accidents. We've stumbled our way forward in mathematics; there is no grand plan that unifies our notation. Just look at calculus for a prime example - df/dx and f'(x) come from two different lineages and get used interchangeably; the df/dx notation can be intensely misleading when students think that (for example) they should be able to use normal fraction rules.
  - abdullahkhalids 6y ago
    Of course, there is no grand plan, and yes, a lot of notation is historical accidents. But my point is that attempting to "fix" it will probably yield a better notation, but not the "perfect" notation. As an aside, df/dx treated a fraction is used as soon as second or third semester of uni when they learn how to solve differential equations using separation of variables, and physicists/chemists start using total derivatives for thermodynamics. I am not aware of notation different from df/dx where these subjects would be just as clear.
- wcarey 6y ago
jiggawatts 6y ago
Allow me to introduce modern theoretical physics, where comments like "X is obvious in context" never is, except to the person that wrote it. Don't worry though, they helpfully add that "When X isn't what it seems in context, this will be called out". Okay. "Sometimes it may not be." Wat? Of course, every individual researcher, or at least each individual Physics department has their own conventions, and the conventions are critical to the meaning. It's like the programming paradigm where the naming of functions invokes "magic glue" instead of using strongly typed interfaces. It's unbelievably confusing to the uninitiated. If you're trying to "cut across" a bunch of theories being worked on by different groups, you basically have no hope. Everybody ends up being super specialised not only to a specific sub-field of study, but to a specific research group. I just watched a 1 hour lecture on extending GR by some physicist last night. He was reading the equations out loud, and at one point he was making noises like the following non-stop for about 2 minutes: "Eta mu nu, one minus one zeta nu mu, mu nu eta zeta one". It was ludicrous.

by wcarey 6y ago

mkl 6y ago

I think x(3) would more commonly mean x times 3. f(3) would be function application. Even more context dependence.

wcarey 6y ago
Right. Of course, if you write x(3), other mathematicians should frown at you because you're making bad notational choices. It's a bit like explaining to students that the real way to know which 3rd declension nouns are i-stems in Latin is to say the genitive plural both ways. The one that doesn't sound wrong is correct. But you have to have a lot of time in the language for that to work.
- mkl 6y ago
  Well unnecessary parentheses are often used to indicate a substitution has happened. E.g. in a topic I just taught I would write things like ∫_{y=0}^3 x dy = [xy]_{y=0}^3 = x(3) - x(0) = 3x. In context I think it's perfectly clear and a good notational choice.
  - wcarey 6y ago
    And so it is. edit: I suppose, what I'm trying to get at, perhaps too glibly, is that audience matters terribly much in mathematical writing. In the same way that Latin students don't start with Tacitus or Sallust, famous for their idiosyncratic grammar, math students shouldn't jump into the full context-dependent mess of the notation that experienced mathematicians use. But I think we often thrown them in unintentionally because we're so used to it.
    - jacobush 6y ago
      You used too little context when arguing with an audience who isn't used to that style of writing...
- C4stor 6y ago
  Also, there are actual quite reasonable rules to know which 3rd declension nouns are i-stems, so it doesn't seem too right too just say "follow your gut". Btw, I wouldn't abuse latin comparisons on an american forum, I don't think it's quite in the culture ^^
jerf 6y ago
x(3) is a function "x" being applied to the number 3, because "three times x" is 3x. Of course context would help resolve this if you have a function named x or not, or a variable named x or not, and if you have both a function and a variable named x, well, you worked for your confusion and you have obtained it; congratulations. :)
jki275 6y ago
I'm sure some would take it that way as "x" isn't commonly used in mathematical notation as a function name, the norm is to use f, g, etc. However, the notation is relatively clear notwithstanding. It almost has to be x as a function with an input of 3.
wycy 6y ago
I think x(3) would more commonly imply a function of 3, since you'd virtually always otherwise just write 3x.
- qubex 6y ago
  This is one of the reasons I annoy people by following Wolfram’s convention in Mathematica of using square braces to denote arguments passed to a function: f(x)=fx=f×x while f[x] means “apply the function f to the argument x”. An unusual convention it may well be, but at least it’s one devoid of ambiguity.
  - mkl 6y ago
    Not at all. f[x] is clearly the xth component of the array f :-D
    - qubex 6y ago
      On the blackboard in the mathematics department it’s not. But I get your point. I have also heard the objection that [x] is a 1×1 matrix whose only element is ‘x’.
      - OscarCunningham 6y ago
        Square brackets can also denote the floor function (especially in old works where '⌊' and '⌋' were typographically unavailable).
    - gowld 6y ago
      An array is merely a function from N to some set.
  - pwdisswordfish2 6y ago
    It's not devoid of ambiguity, because f[A] usually means the image of the set A under f.
- mkl 6y ago
Wowfunhappy 6y ago
If you meant "function", would you generally write 𝑓(3) instead of f(3)? (I have no mathematics background past K-12)

acqq 6y ago

I also remember having constant confusion of what were assumed associations in written bigger formulas especially involving the functions and the practice of not writing the braces, and before that, if 3 1/5 meant three times 1/5 or 3 wholes and 1/5th. I thought it was only on these low levels of educations that people settle on such confusing practices. Later on the university I've discovered that it's the same on all levels. The notations used outside of programming are simply very context-dependent, and it's pity that that dependency is not made clear more often. Also, some practices directly come from some historical uses, and different paths to the current notations still end up to the similarly looking but different meanings.

twic 6y ago

> x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}. That notation for inverse functions is truly appalling. I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading.

abnry 6y ago
But the inverse function is the "multiplicative inverse" in the group of functions with composition as "multiplication". In that way, it makes a ton of sense. It's only a problem because you are mixing together two group operations.
StavrosK 6y ago
> I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading. Maybe they did.
DavidSJ 6y ago
They’re both inverses: one with respect to multiplication, the other with respect to function composition. In abstract algebra, we observe that there are many types of “products”: multiplication, addition, function composition, composition of rotations, matrix multiplication, etc. A common, unified “power” notation for repeatedly taking the product of a single element, or of its inverse, has some value. There is some ambiguity, since multiplication of functions can also often make sense. This is usually resolved by context, or by being explicit where the context isn’t clear.
- lisper 6y ago
  > one with respect to multiplication, the other with respect to function composition And, of course, multiplication is a function so one is just a specific instance of the other. Infix notation is another blight on the mathematical notation landscape. The amount of human effort that has been put into figuring out how to parse a+b*c is staggering. All of this confusion could have been avoided if we'd just started with s-expressions in the first place.

Spivak 6y ago

I think it's important to know that this really isn't true. Maths at all times is a subjective language. Maths notation is imprecise "intentionally confused", or just ad-hoc defined all the time. When you see something like. f(x) = summation(x^n, n=0, 10) We conveniently ignore that this polynomial is defined at x=0 despite 0^0 not making any sense by ad-hoc defining 0^0 = 1 in this context.

wcarey 6y ago
What isn't really true?
ColinWright 6y ago
> We conveniently ignore that this polynomial is defined at x=0 despite 0^0 not making any sense by ad-hoc defining 0^0 = 0 in this context. What do you think the polynomial is? I ask, because in all situations similar to this that I've encountered it made sense to define 0^0 as 1, not as 0. If you genuinely have a case where 0^0 = 0 makes consistent sense then I'd be interested in understanding it. So, what do you think the summation actually is when expanded?
- Spivak 6y ago
  Sorry you're right, it should be 0^0 = 1. An example where 0^0 = 0 occurs when dealing with areas. The measure of the real line in the plane is zero but it's also a rectangle with sides (0, inf) and we define the area of a rectangle to be l*w. You usually see this written as inf × 0 = 0 but you sometimes you see the interpretation as 1/0 × 0 = 0^0 = 0. And you know this is a an ad-hoc definition because you're not allowed to algebraically manipulate it at all.
grodeni 6y ago
actually 0^0 is usually chosen equal to 1... which probably still supports your argument.
- Spivak 6y ago
  Corrected! Thank you!

Denvercoder9 6y ago

> \sin{30}. Radians or degrees? Probably the writer means degrees, but there's no way to tell. I once had a professor that insisted that sin(30) meant sin(30 pi) in radians, with the pi being implicit. Unsurprisingly, it was the worst class I've ever taken.

wcarey 6y ago
Absolutely terrifying. How did he communicate with other peer mathematicians?
- Denvercoder9 6y ago
  I honestly don't know - it was an intro level physics class.
- gowld 6y ago
  Mathematicians understand how to read definitions of notation. If all of your trig is fractions of pi, writing pi redundantly everywhere is not useful.
  - wcarey 6y ago
    Is that commonly done?
    - renewiltord 6y ago
      It is not. Though it wouldn't be too hard to adapt, I think, if it was noted at the beginning of a text. For it to take, it would have to power some influential result, sort of like how the summation convention in tensor notation works (something that infuriated me as a kid).
  - saagarjha 6y ago
    …but that's not a fraction of pi?
    - 6y ago

billfruit 6y ago

Another is sin^2(x) meaning (sin(x))^2, but by a more intuitive reading it should mean sin(sin(x)). I don't know what exactly is being gained by the usual notation: surely better clarity should be preferred over the time/effort saved in writing the extra pair of paranthesis, and I would prefer it be written as (sin(x))^2 always. Thing is mathematics, and mathematical pedagogy seems hardly concerned with such rampant notation confusion plaguing much of maths. Perhaps moving to some type of machine-readable notation will be better for consistency and avoiding of much notational confusion.

abdullahkhalids 6y ago
Does f^2(x) in any scenario mean (f(x))^2? Usually the square of a function, with named arguments is written as f(x)^2. So, not clear on the confusion.
- bonoboTP 6y ago
  Yes, in trigonometry sin^2(x) is commonly written to mean (sin(x))^2
saagarjha 6y ago
When actually working with trigonometric identities, using parentheses typically gets unwieldily very quickly. Same reason sin(x) often becomes sin x.
- billfruit 6y ago
  Then perhaps we should drop all parens and move to prefix notation.

mamcx 6y ago

So, have been any sufficient effort in design a new syntax of maths that could help the regular folk? If we learn new syntax(langs) all the time then push for a new syntax(math) could not be that bad of a idea...

wcarey 6y ago
There are some gentle efforts in that direction (tau radians for example), but mathematical notation is old (except the parts that are new). And it's really hard to strike the right balance between concision and ambiguity.

abdullahkhalids 6y ago

Why can't we have mathematics devoid of these ambiguities? One reasons is that humans have small working memories, and novice mathematics students have even smaller ones. The ambiguity of notation, while confusing, once mastered, allows us to write shorter expressions, whose meaning we resolve from context, and which become both easier to write and to understand. A second reason is that while mathematical logic is rigorous and precise, unequal mathematical objects are similar to each - even objects that on first glace seem nothing alike. For instance, the two types of inverses you mention, or sets and linear spaces, or groups and tetrahedrons. And because of the diffuse nature of mathematical objects, it is inevitable that the same notation will be used for unequal objects. Because, it advances our human understanding of mathematics to use the same notation for two unequal but similar objects. This second reason, once understood, is one threshold between the mechanical mastery of the intermediate student and almost artistic use of mathematics by the advanced student.

spenczar5 6y ago
I think it's a lot less beautiful. I think it has to do more with historical accidents. We've stumbled our way forward in mathematics; there is no grand plan that unifies our notation. Just look at calculus for a prime example - df/dx and f'(x) come from two different lineages and get used interchangeably; the df/dx notation can be intensely misleading when students think that (for example) they should be able to use normal fraction rules.
- abdullahkhalids 6y ago
  Of course, there is no grand plan, and yes, a lot of notation is historical accidents. But my point is that attempting to "fix" it will probably yield a better notation, but not the "perfect" notation. As an aside, df/dx treated a fraction is used as soon as second or third semester of uni when they learn how to solve differential equations using separation of variables, and physicists/chemists start using total derivatives for thermodynamics. I am not aware of notation different from df/dx where these subjects would be just as clear.
wcarey 6y ago

jiggawatts 6y ago

Allow me to introduce modern theoretical physics, where comments like "X is obvious in context" never is, except to the person that wrote it. Don't worry though, they helpfully add that "When X isn't what it seems in context, this will be called out". Okay. "Sometimes it may not be." Wat? Of course, every individual researcher, or at least each individual Physics department has their own conventions, and the conventions are critical to the meaning. It's like the programming paradigm where the naming of functions invokes "magic glue" instead of using strongly typed interfaces. It's unbelievably confusing to the uninitiated. If you're trying to "cut across" a bunch of theories being worked on by different groups, you basically have no hope. Everybody ends up being super specialised not only to a specific sub-field of study, but to a specific research group. I just watched a 1 hour lecture on extending GR by some physicist last night. He was reading the equations out loud, and at one point he was making noises like the following non-stop for about 2 minutes: "Eta mu nu, one minus one zeta nu mu, mu nu eta zeta one". It was ludicrous.