Trouble with Confusing Notations in Mathematics?

[Telos]

Does anyone else have trouble with confusing and inconsistent standards of notation in mathematics?

I am an economics undergrad, and I can't help but think that math would be 1000 times easier if there was just better notation for it. Does anyone know of any radical attempts to create new standards? Or, does anyone know of any learning materials than can help students such as myself "get over" this kind of problem?

To illustrate, a fundamental example is functional notation.

$$f(x) = 5x$$
does not reduce to
$$f = 5$$

It's a mental obstacle for me to think "oh yeah, that 'f' is not a variable." And I have to continuously think it, because my mind is automatically trying to reduce the equation. It drives me nuts. I hope this rudimentary example doesn't make it seem like I'm whining.

Anyways, thank you for your help! And I look forward to many stimulating discussions on physics (I'm trying to learn the math from Penrose's Road to Reality)

 

[cepheid]

Unfortunately, the function notation is so standard, that you'd be hard pressed to find any math student (myself included) who thinks there's anything the matter with it. I was introduced to this notation high school and got used to it pretty fast (after working with it for a few months). Nobody ever thinks "multiplication" when they see f(something), and my teacher took pains to make the distinction when he first introduced it. To help you out...when you multiply two variables in algebra...you just stick them next to each other: e.g. xy

So why the needless parentheses with f(x)? because f is NOT a variable, and this is not multiplication.

f(x) (read f of x) is a single entity, as those brackets should hopefully clue you in from now on. I hope this helps in that particular case.

 

[mruncleramos]

Maybe writing functions as Mappings from one set to another would help.

 

[Telos]

Thank you, cepheid. I know, my teacher said the same thing.

I don't think multiplication on the surface, but it's like there's an animal instinct telling me to. And it's a shame, because that animal instinct helped me immensely while in grade school... I did math problems quickly, easily, and efficiently, without even thinking about it. One math teacher felt compelled to give me extra work because I excelled so quickly. And then came functions.

Maybe my problem is more aesthetic?

I find it disconcerting because mathematics is supposed to be this bastion of rigorous tools for certainty and unchallengeable deduction. And beauty. But to me much of math looks ambiguous - like a self-destructive chimaera of pieces that have just been grafted on one by one over the years. I'd rather not just rote myself in overlooking certain inconsistencies.

In your opinion, do you think notations like these might turn students away from math?

Maybe writing functions as Mappings from one set to another would help.

Can you give me an example?

 

[cepheid]

To be honest, I think your problem is just aesthetic...since you seem to be going with your immediate instincts regarding what a mathematical symbol should mean (based on prior experience with things having a similar appearance), rather than consciously thinking about what it has been defined to mean.

I admit that math notation can get cumbersome, and some notations are better than others, depending on the situation. But what really matters is that in a given context the notation is consistent, so that everyone reading it can understand what it means. I've noticed math texts often take the pains to define everything from beforehand. You spoke of actual inconsistencies in your post, and I'm a bit dubious. Can you think of anything off hand?

 

[Hurkyl]

Actually, in the expression "f(x)", f is a variable -- it's a variable that denotes, say, a general real valued function, as opposed to x which denotes, say, a general real number.

Interestingly enough, some experts prefer writing fx instead of f(x). (In various contexts, writing xf or even (x)f sometimes occurs) However, a prerequisite for such comfort is the understanding that the real numbers aren't the only game in town. In particular, variables can denote things other than real numbers, and operations like + and * can denote things other than the arithmetic of real numbers.

What is standardized is that letters like f, g, and h are usually reserved as variables denoting functions and letters like a, b, c, x, y, and z are usually reserved as variables denoting real numbers. (At least in the context with which your familiar)

 

[Telos]

"Context." That's the magic word. There are no inconsistencies if everything fits in its appropriate context.

Is it too idealistic to ask for a standard notation in all contexts? For instance, that "f" will always and forever be used to describe a function?

When I was taught grade school algebra, our teacher said very simplistic things, like "letters are just numbers that we don't know." Well, that's just not true with respect to abstract mathematics, and my brain has solidified into looking at letters as stand-ins for numbers.

I guess I'm expecting there to be some other kind of symbol (maybe a letter from a different alphabet) to describe functions. A similar problem occurs with complex numbers and "i," although that's no where near as bad since an imaginary number is still technically a number. Yeah, it's definitely aesthetic lol.

Hurkyl, thank you. You described it very well. I guess I just need to "unlearn" what my teacher told me. :)

 

[Data]

Yes, it is too much to ask, because different notations suit themselves to different contexts. If you train yourself to do so, it is not that difficult to switch between different meanings for the same notation without much effort(and it is necessary, if you are dealing with many areas of mathematics or physics at the same time).

I often catch myself writing $${\cal {L}} \{V, \ V \}$$, when I mean $${\cal {L}} (V, \ V)$$, though!

 

[Telos]

Okay. I thought it would be reasonable in the Platonic sense. Since math is supposed to exist timelessly and independently of our thoughts on it, what's wrong with giving its symbols timeless and independent meaning?

Does the necessary ambiguity of some symbols lead us towards constructivism?

[Edit: I hope that question isn't too off-topic]

 

[Data]

Well, nothing would be wrong with it. But it would make a lot of things a lot harder to deal with.

As you stated earlier, the symbols are only ambiguous when you don't specify their context, since they are all defined for each context in which they are used.

 

[Telos]

Math really is like a language, then. I can't get upset over the multiple meanings of a parenthesis anymore than I can get upset over the multiple meanings of the word "fruit."

Doesn't do much for the Platonist view, then.

(sorry for turning this discussion towards philosophy)

 

[mathwonk]

actually he is correct that the notation f(x) is inconsistent, and requires an inner knowledge of the context not to be confused with multiplication.

Mathematica for example is unable to make this association accurately, hence the rules for communicating with that program require us to use square brackets for functional notation and round ones for multiplication.

e.g. Sin[x] in stead of Sin(x). This would probably solve the difficulties many students have with this as well, but we are stubborn traditionalists and do not like to admit there is a problem if we have ourselves solved it personally.

 

[Data]

That's true as long as you allow y(x) to also mean multiplication. Which usually is allowed. There's not really a reason to though.

 

[Data]

Maple, on the other hand, automatically assumes f(x) refers to a function of x, for example.

 

[matt grime]

Sadly there are too many things in mathematics for each to have a unique (and small) symbol. Plus, we often like to use things that are similar to other things exactly because they *do* remind you of something else. And don't you think that other cultures may have an issue if you were to claim the primacy of western orthography?

Probably, at the point you're talking about you can safely assume that anything labelled by f,g or h is a function, and the letters x,y,z are (real or complex) numbers. In the mathematical alphabet of course, the letter after z is w... m,n are usually integers, as are p, and q, though these usually are primes, or powers of primes.

I listened to a talk last year where the lecturer made a good case as to how the evolution of notation has affects the evolution of mathematics itself. The case in point was the realization that one could write functions as diagrams

$$X \rightarrow^f Y$$

I doubt that will work, but it's supposed to be an arrow from X to Y with the f written above it.

 

[Hurkyl]

For instance, that "f" will always and forever be used to describe a function?

Unfortunately, no. For example, I often see the letters around "f" used to denote elements of groups, or fields, and greek letters are used for functions. (Though, the elements of groups are often functions...)

However, one usually sees a letter "defined" before it's used, or otherwise made patently obvious from how it's used. For example, a proof might start:

Let "F" be a field, and let $f \in F$...

So, we know that the letter "f" is denoting a general element of "F", and "F" denotes a general field.


Or, one might have defined the letter "T" to be a linear operator on Rⁿ, in which case once we see T(v) we can infer that v was meant to represent a general element of Rⁿ.

 

[mathwonk]

actually it is impossible to always try to assume that f is a function and x is a variable or input value. i.e. in the expression f(x), either f or x can be the function.

I.e. a number x, yields a function called "evaluation" on the space of all real valued functions defined on R. thus from this point of view, f is the input, x is the function, and x(f) = f(x) is the output.

this comes up as soon as one starts to study functional analysis, and dual spaces, and is inescapable. It also raises the issue of "variance", both co and contra.

In Hurkyl's example above for instance, if T is a linear operator on R^n, and v is a vector in R^n, the expression T(v) still does not reveal which is the function.

I.e. sending v to T(v) gives a function from R^n to R^n, while sending T to T(v) gives a function from Hom(R^n,R^n) to R^n.

Thus as Hurkyl says, from knowing the nature of T, one can infer from the expression T(v) that v belongs to R^n, but one can still not always know which function is meant, whether T-->T(v), or v-->T(v).

Both are important, and which is meant must be specified. (Of course he was assuming there that T had been so specified.)

This situation arises in the simple concept of a dot product <x,y> where x is a function on y, namely <x,.>, and y is a function on x namely <.,y>.

Similarly the only thing absolute that can be said above, is there is a natural pairing

Hom(R^n,R^n) x R^n -->R^n.

Either factor in the pairing consist of functions on the other factor.

This is the difference between the two views of tensors that continually compete on these pages. The traditionalists cling to a view of tensors as some kind of huge dot product, and spend time contracting, or evaluating the expressions.

The modernist thinks in terms of functions on mutually dual spaces.

I.e. if T is a tangent space, and T* its dual, there is a natural pairing of TxT*-->R. If one artificially defines a dot product, called a riemannian metric, on T, then one also has a (non natural) pairing TxT-->R, by which one can identify T with T* (non naturally).

Some people fail to observe that this artifice can never serve to alter the nature of elements of T*, i.e. their variance.

Quite naturally the pairing TxT*-->R yields maps T-->Hom(T*,R), (taking v to evaluation at v)

or T*-->Hom(T,R), but these maps are essentially the identity.

Similarly one has natural evaluation plus multiplication pairings TxTxT*xT*-->R, and associated maps T*xT*-->Bil(TxT,R), i.e. one can view elements of T*xT* as a type of tensor. (Some traditionalists argue over whether these tensors are of type (2,0) or type (0,2), whereas to a modernist such notational conventions are of course arbitrary.)

These relations are all tautological, but become quite complicated when represented by giant arrays of indices.

In all cases, the intrinsic point of view is merely that T(v) can be viewed either as v-->T(v) or T-->T(v).

 

[Telos]

mathwonk, that's really interesting about Mathematica. Do you think the evolution of notebook programs like Mathematica will drive the selection process for acceptable standards of notation? And thanks for that second post, even though some of it went over my head. :)

matt grime,

And don't you think that other cultures may have an issue if you were to claim the primacy of western orthography?

Yes! That's exactly what I'm talking about. There are plenty of symbols for us to construct a truly cosmopolitan and "culturally objective" system.

Sadly there are too many things in mathematics for each to have a unique (and small) symbol.

Are there more concepts in mathematics than there are letters and symbols in all the world's alphabets (excluding the overly complex characters in languages like Chinese)?

Again, thank you for all your responses. I apologize if my questions seem trivial, but I can't help but feel I'm on to something - that the reason why so many people see math as complicated and confusing is because it isn't written with much of a referable standard. (i.e. it has no dictionary)

 

[mathwonk]

well my post was too long. As my teacher Lynn Loomis put it:

when you look at f(p), how do you know f is the function? why couldn't p be the function? I.e. why don't we define p(f) to be f(p)?


That blew my mind.

 

[master_coda]

Again, thank you for all your responses. I apologize if my questions seem trivial, but I can't help but feel I'm on to something - that the reason why so many people see math as complicated and confusing is because it isn't written with much of a referable standard. (i.e. it has no dictionary)

Math is complicated and confusing. A more consistent notation won't be able to change that significantly.

Of course, it's still worthwhile to try for a more consistent notation, but you do have to be realistic. Trying to get everyone to use the same notation is like trying to get everyone to use the same language. Even if you could come up with a universal standard that everyone could agree upon, people would still ignore it when obeying the standard got in the way of trying to express something.

 

[arildno]

Telos:
I share your view that the arbitrary notation choices put off several people from even trying to learn/understand maths.

However, there exists a subtle danger in trying to reify/standardize notation:
When a given symbol is used repeatedly to denote the same "thing", that symbol may acquire the "feel" (from habit/convention) of BEING that "thing" itself, rather than a short-hand reference to it.

That is, your attention becomes fixed upon the external symbol, rather than, say, the definition of what you're using.

If, however, maths as it is written today, uses a plethora of notation conventions, then, in order to follow a specific argument, you need to focus on the actual definitions the author is employing.

This has the advantage of sharpening your attention to the assumptions&logics of the author's argument, rather than to its "scriptural" shape.

 

[honestrosewater]

Are there more concepts in mathematics than there are letters and symbols in all the world's alphabets (excluding the overly complex characters in languages like Chinese)?

Do numbers count as concepts? :tongue2: Acutally, that's a clue- incorporating position into your system reduces the number of symbols you need.

 

[Telos]

Wow, arildno. That was awesome. I think that's just the mindset I need to carry me over these kinds of frustrations. That makes perfect sense.

Thank you for your help.

Thank you all for your help.

 

[Icebreaker]

Off topic a bit: we're running out of variables. We've used up the latin ones, and the greek alphabet is used everywhere, and now we're starting with Hebrew ones. I foresee that we'll be using Chinese characters in math some time in the future -- that's a **** load of them :rofl:

 

[Hurkyl]

However, there exists a subtle danger in trying to reify/standardize notation:
When a given symbol is used repeatedly to denote the same "thing", that symbol may acquire the "feel" (from habit/convention) of BEING that "thing" itself, rather than a short-hand reference to it.

I've heard an interesting story related to this.

A professor wanted to emphasize to his calculus students that using "f" for a function and "x" for real number was merely conventional, and to shock them, he proceeded to write down this definition:

x(f) := f^2

And then work out its derivative, dx/df.

As it so happened, the professor was so confused that he had difficulty working out this simple derivative!

Off topic a bit: we're running out of variables.

Don't worry, we still have a few fonts and decorations we can fiddle with before needing to add in a new alphabet!

 

[Zurtex]

I've heard an interesting story related to this.

A professor wanted to emphasize to his calculus students that using "f" for a function and "x" for real number was merely conventional, and to shock them, he proceeded to write down this definition:

x(f) := f^2

And then work out its derivative, dx/df.

As it so happened, the professor was so confused that he had difficulty working out this simple derivative!

Haha that's a good one, I'll need to remember that one. By brain had to do a lap before I could do that one.

 

[mathwonk]

this fixation on the "usual meaning": of a symbol causes me great difficulty in teaching my class. I.e. I often start over with what I want a letter to mean, but students often want it to continue to have the same meaning it had earlier.


For example suppose you prove the mean value theorem for a function f on an interval [a,b], i.e. if f is continuous on [a,b] and differentiable on (a,b), then at some popint c in (a,b) we must have f'(c)[b-a] = f(b)-f(a).

OK now you want to rpove the corollary that a function f continuosu on [a,b] with derivative zero on (a,b) is constant.

So you say well let c,d be any two points of [a,b], and I will prove that f(c) = f(d). Well you are already in trouble. If you try to apply the theorem to the interval [c,d], some people will not be able to let the symbols c,d polay the roles that were played before by a and b. And they will think c is the point where the dertivative was evaluated.

So you have to stop and go back and change the names of the new interval to some other letters, and you do not have any more available since people today where i live dislike greek letters,...

 

[Telos]

Off topic a bit: we're running out of variables.

Have you seen Omniglot?

http://www.omniglot.com/

We're set for a couple centuries, I think.

 

[arildno]

I've heard an interesting story related to this.

A professor wanted to emphasize to his calculus students that using "f" for a function and "x" for real number was merely conventional, and to shock them, he proceeded to write down this definition:

x(f) := f^2

And then work out its derivative, dx/df.

As it so happened, the professor was so confused that he had difficulty working out this simple derivative!

A great story, Hurkyl!
That reminds me of an occasion when I was a student, and took a course in real analysis :

Our lecturer had use for the constant function equal to 1, and brazenly defined it as 1(x).
A collective shudder went through the classroom, and one of the brightest gasped:
"You CAN'T call a function 1(x)!"

The lecturer then threw his hands up in mock resignation, and exclaimed:
"So, what should I call my function then; 2, perhaps?"
I thought I learned a lot that day; in particular in how we ought to choose notations intelligently, rather than to slavishly follow conventions..

 

[matt grime]

this fixation on the "usual meaning": of a symbol causes me great difficulty in teaching my class. I.e. I often start over with what I want a letter to mean, but students often want it to continue to have the same meaning it had earlier.


For example suppose you prove the mean value theorem for a function f on an interval [a,b], i.e. if f is continuous on [a,b] and differentiable on (a,b), then at some popint c in (a,b) we must have f'(c)[b-a] = f(b)-f(a).

OK now you want to rpove the corollary that a function f continuosu on [a,b] with derivative zero on (a,b) is constant.

So you say well let c,d be any two points of [a,b], and I will prove that f(c) = f(d). Well you are already in trouble. If you try to apply the theorem to the interval [c,d], some people will not be able to let the symbols c,d polay the roles that were played before by a and b. And they will think c is the point where the dertivative was evaluated.

So you have to stop and go back and change the names of the new interval to some other letters, and you do not have any more available since people today where i live dislike greek letters,...

Using the same letter in different ways can cause confusion, but spare a thought for the student who got onto a degree program in maths and when faced with

f(x)=x

asked me if the x on the left hand side was the same x as the one on the right.

And, for, Telos, there are far more things in mathematics than we could ever write down, even if we agreed a notation for them.

And, back to one of mathwonk's posts, how about a little Wittgenstein:

When a student sees on the board 2,4,6,8... and is asked to continue the sequence, what does it mean when he says "i've got it" and writes down 10,12,14,.. what is "got it", why does the thing he "gets" agree with what you or I get?

So, why when we see f(p) do we indeed "get" that f is a funtion of p.

 

[TrevorHackman]

Like Masta Coda said, trying to get everyone to use the same notation is like trying to get everyone to use the same language. Plus some notations are better than others for only certain situations. I had the same problem as you with f(x) not being f multiplied by x. f and x are never constantly a function and a variable of a real number. What I find is best is to never consider a(b) as a being multiplied by b but always when only the second variable is in parenthesis to understand the first as a function of the second. Represent multiplication with both parenthesis, a dot, or no parenthesis in the case of variables.
(f)(x)=fx=f*x= 5x
f=5
Part of notation is the ease of writing and communicating it. I still find it confusing how ordered pairs and interval notation can look exactly alike with parenthesis. (a,b) could mean the point or the interval. The keyboards inability to neatly express many things in math also annoys me.