Help We have forgotten how to write math stuff

[micromass]

Help! We have forgotten how to write math stuff!

Let's assume for a moment that something mysterious has happened. We still know all the science and mathematics, but somehow we have forgotten all the notations and all the conventions. It is our job to invent new notations and conventions and to throw out the old ones.

So, which notations and conventions do you find really annoying, but are rooted so deep into scientific and mathematical practice that it can't be changed? And what would be the alternative?
And also, which notations and conventions do you think are actually very good?

 

[WannabeNewton]

Einstein notation stays, Dirac notation goes. Also, replace all vector calculus with exterior calculus. It's not really a notational issue but I wanted to throw that out there :)

 

[Jorriss]

I used to like dirac notation more but wbn and micro have led me into the light.

 

[jedishrfu]

well you could look at IBM's APL programming language. Prof Iverson developed it using the greek alphabet and other key symbols to make a working programmable language. Quite remarkable at the time.

IBM even went so far as to use APL to describe the operations of its arithmetic instruction set on the IBM 360/370 machines.

We used to joke that it was a write-only language because a few days after you wrote it you read figure out what it was doing.

 

[micromass]

I hate the $\subset$ notation. The logical thing would be to write $\subseteq$, unless you want proper inclusions.

 

[micromass]

I don't like the notations $f^{-1}(A)$ and $f(A)$ for inverse and forward image. I prefer the more categorical notation $f^*(A)$ or $f_*(A)$. But yeah, nobody uses this.

 

[Jorriss]

I don't like the notations $f^{-1}(A)$ and $f(A)$ for inverse and forward image. I prefer the more categorical notation $f^*(A)$ or $f_*(A)$. But yeah, nobody uses this.

Why?

 

[WannabeNewton]

On that note, I also hate the $dV$ notation for integrals e.g. $\int _{\Omega}\alpha dV$ where $\alpha$ is a scalar field. Unless one knew beforehand, this notation totally obscures the fact that integration is done using forms i.e. $\int _{\Omega}\alpha \epsilon$ would be much more appropriate as it makes clear that we are integrating using a differential form (the volume form $\epsilon$). It may seem like a minor detail but the fact that integration is done using forms is not something I've seen stressed in many of the physics texts I've seen at the appropriate level even though it is something introduced to undergraduates who take an analysis on manifolds class.

 

[micromass]

Why?

Well, if $f:X\rightarrow Y$, then $f^{-1}$ is well-defined as the inverse function (if it exists) and $f$ is defined as operating on elements of $X$. I don't like it if they start using the same notation to operate on sets as well! Furthermore, it is very confusing for newbies.

In fact, we can associate two maps with $f$:

f^*:\mathcal{P}(Y)\rightarrow \mathcal{P}(X)

and

f_*:\mathcal{P}(X)\rightarrow \mathcal{P}(Y)

These should be seen as actual and genuine maps. But the current notation doesn't do justice to the notation. Furthermore, the notation $f^*$ suggests that it is some kind of pullback. This is a very accurate view of the map in certain sense. The same with $f_*$ being a pushforward.

So I think the notation really makes more sense mathematically and it's less confusing.

 

[Jorriss]

I've seen stressed in many of the physics texts I've seen at the appropriate level even though it is something introduced to undergraduates who take an analysis on manifolds class.

Yeah, it does suck for those .1% of physics majors.

 

[micromass]

Not really a notational issue, but more a convention. I would love analysis and rigorous calculus classes to stop teaching Riemann integration. The Henstock-Kurzweil integral is far more superior. Furthermore, the integral its definition and properties are not much harder than those of the Riemann integral.

 

[WannabeNewton]

Yeah, it does suck for those .1% of physics majors.

Alright then when you see something like $\int _{\Sigma}d\star F = \int _{\partial \Sigma}\star F$ in electromagnetism, don't come back to me asking for advice on Pokemon Black and White >.>

 

[ZombieFeynman]

Einstein notation stays, Dirac notation goes. Also, replace all vector calculus with exterior calculus. It's not really a notational issue but I wanted to throw that out there :)

Why do you dislike Dirac notation?

 

[WannabeNewton]

Why do you dislike Dirac notation?

No particular reason, I'm just a GR fanboy. Team Einstein!

On an aesthetic level, I find Einstein notation very beautiful but I personally don't find Dirac notation elegant in the same way. I know I always bring up Maxwell's equations but I mean cmon how can you not marvel at the beauty of these equations especially using Einstein's notation: $\nabla^{a}F_{ab} = -4\pi j_{b}, \nabla_{[a}F_{bc]} = 0$. They look even better in terms of differential forms $dF = 0, d\star F = 4\pi \star j$ but that doesn't really help my point so let's focus on the former xD.

 

[ZombieFeynman]

No particular reason, I'm just a GR fanboy. Team Einstein!

On an aesthetic level, I find Einstein notation very beautiful but I personally don't find Dirac notation elegant in the same way.

Oh. I think that's rather poor reason to want to abolish useful notation.

 

[WannabeNewton]

Oh. I think that's rather poor reason to want to abolish rather useful notation.

Well I wasn't being serious lol - 'twas just a joke. I just don't find it aesthetically pleasing is all.

 

[micromass]

Dirac notation is only useful if they also teach rigged Hilbert spaces. Without that, it's a pretty awful notation. When I read something in Dirac notation, then I always get confused. If I then read the same thing in ordinary math notation, then I understand it immediately.

Furthermore, I think that Dirac notation tends to obfuscate domain issues. So you're more prone to errors.

 

[WannabeNewton]

I think George Jones had some excellent threads in the QM section that made light of the short comings of the notation in fact.

 

[ZombieFeynman]

Dirac notation is only useful if they also teach rigged Hilbert spaces. Without that, it's a pretty awful notation. When I read something in Dirac notation, then I always get confused. If I then read the same thing in ordinary math notation, then I understand it immediately.

Furthermore, I think that Dirac notation tends to obfuscate domain issues. So you're more prone to errors.

Well as a lowly physicist I take pride in not suffering from mathematical rigor mortis, abusing notation, and generally making mathematicians cringe in dismay.

 

[WannabeNewton]

I don't know how mathematicians feel about dirac notation but Einstein notation doesn't seem to be too rare amongst the mathematicians. Lee for example uses it in both his smooth and Riemannian manifolds texts.

 

[SW VandeCarr]

How about a different way to write simple arithmetic? We regularly write expressions like f(x,y).

Why not +(3,5)? With longer horizontal lists you only need to write + once.

 

[DiracPool]

Woahh woah. All this talk makes me think I should hold off on learning maths until we come up with a "sensible" notation. I'm glad I read this before I started. I knew there was something fishy about that Dirac notation..."Bra-Ket", gimme a break

 

[MarneMath]

I never really liked the usage of P(X = x). I always felt it confused new people to probability. I can't recall how many times I've had to explain the difference between the X and x -_-.

 

[jedishrfu]

Liebnitz notation vs Newton notation is an interesting case. Since CM is interested in time rates of change then Newton streamlined things using the top dot. Liebnitz saw a more general use for calculus and so used the dx notation which while less compact covered arbitrary rates of change.

I never liked the inverse function notation that looks to close to exponentiation notation.

 

[SW VandeCarr]

I never really liked the usage of P(X = x). I always felt it confused new people to probability. I can't recall how many times I've had to explain the difference between the X and x -_-.

This is the usual way to express the probability density of X at x which is taken to mean the cumulative value of a probability density function from X=0 to X=x; 0 \leq x \leq 1. How would you prefer to do it?

 

[FreeMitya]

On an aesthetic level, I find Einstein notation very beautiful but I personally don't find Dirac notation elegant in the same way. I know I always bring up Maxwell's equations but I mean cmon how can you not marvel at the beauty of these equations especially using Einstein's notation: $\nabla^{a}F_{ab} = -4\pi j_{b}, \nabla_{[a}F_{bc]} = 0$. They look even better in terms of differential forms $dF = 0, d\star F = 4\pi \star j$ but that doesn't really help my point so let's focus on the former xD.

I completely agree. In fact, the beauty of mathematical notation is one of the things that I enjoy most about the subject. Whenever I walk into my room and see my notebook open on my desk, the maths always puts a smile on my face.

 

[WannabeNewton]

I completely agree. In fact, the beauty of mathematical text is one of the things that I enjoy most about the subject. Whenever I walk into my room and see my notebook on my desk, the maths always puts a smile on my face.

Oh thank god, I thought I was doing physics some kind of disservice by liking it largely in part due to the elegant mathematics. Nice to see someone else feels the same way too. I think I have some threads of mine in the GR section with some pretty tensor calculus that reflects what I mean ; here's a recent one: https://www.physicsforums.com/showthread.php?t=688422 (ironically I call the calculations less than elegant lol).

 

[FreeMitya]

Oh thank god, I thought I was doing physics some kind of disservice by liking it largely in part due to the elegant mathematics. Nice to see someone else feels the same way too. I think I have some threads of mine in the GR section with some pretty tensor calculus that reflects what I mean ; here's a recent one: https://www.physicsforums.com/showthread.php?t=688422 (ironically I call the calculations less than elegant lol).

To address what I boldfaced, certainly not. I think promoting not only the conceptual beauty of physics but also the superficial beauty is important when trying to attract new students. How incredible is the fact that in these symbols is contained so much information about reality?

 

[WannabeNewton]

How incredible is the fact that in these symbols is contained so much information about reality?

It is quite amazing indeed, and probably one of the biggest reasons I love physics. It truly is something to marvel at. By the way, I didn't parse my paragraph above correctly. When I said "feels the same way" I meant likes physics in part due to aesthetic appeal.

 

[jedishrfu]

Micro are you trying to make a comprehensive system of mathematical symbols for all of mathematics?

If so then you should also consider making it unambiguous and programmable (follow an ll sub(n) grammar).

This is why I mentioned APL earlier as it handles n dimensional arrays with ease and broke programmers away from the for loop.

 

[HeLiXe]

something...anything different to represent the operation functions for cross and dot product. even the target symbol would be better.

also the use of rho for partial derivatives in Leibniz notation can be particularly annoying.

throwing one physics one in ...the vector symbol with L for angular momentum is one I have always found particularly annoying. Something else like the line and curved arrow used to represent the axial tilt and direction of rotation (used in astronomy) would be nice.

 

[Office_Shredder]

The only thing that needs to change is
\sin^2(x)

This needs to die in a fire

 

[bp_psy]

Whenever I see \int dx dy dz f(stuff) I know I will have a bad day.

 

[DiracPool]

Whenever I see \int dx dy dz f(stuff) I know I will have a bad day.

Geez, that's easy, even I can do triple integrals:

Btw, you forgot the other two integral signs..guess you're having a bad day

 

[micromass]

Geez, that's easy, even I can do triple integrals:

The point is that he wrote dx before the function instead of after it. He's complaing that

\int dx f(x)

is used instead of

\int f(x)dx

Btw, you forgot the other two integral signs..guess you're having a bad day

It's not really wrong not to write the triple integral sign. It's clear from context anyway.
Actually, if you've seen measure theory, then you might actually always prefer not to write the triple integral sign. For example, you would have

\int f d\mu\otimes d \nu

which is a double integral, but it is written here as an integral over a product measure. Writing

\int \int f d\mu d\nu

would give you the iterated integral. The two are not always equal, but they are in a very large class of situations (the relevant result is Fubini-Tonelli theorem).

So I think it's actually better not to write the double integral sign. Other people might disagree with me and write it. I guess it's a matter of taste.

 

[DiracPool]

The point is that he wrote dx before the function instead of after it.

Ah, yes, that might be right. I though he was simply integrating the whole thing over 1, and the (stuff) was just an afterthought, not the function. Kind of like "I hate this so and so 'stuff'"

 

[FlexGunship]

The plus sign ("+") looks too much like the letter "t". So... I don't know... make it, like, a squiggle or something.

 

[jk22]

Have you noticed that sometimes cos^2(x) is used for cos(x)^2 while the first should mean cos(cos(x)) and the second cos(x)*cos(x). I thought maybe it was because the first notation could spare a parenthesis ?

 

[I like Serena]

Have you noticed that sometimes cos^2(x) is used for cos(x)^2 while the first should mean cos(cos(x)) and the second cos(x)*cos(x). I thought maybe it was because the first notation could spare a parenthesis ?

I still wonder what $\sin^{-2}x$ means. Is it $(\arcsin x)^2$, or is it $(\csc x)^2$?I'd like to get rid of all the ambiguities in math notation.
So I also have problems with the fact

... that $\text{sinc }x$ can mean either $\frac {\sin x} x$ or $\frac {\sin(\pi x)} {\pi x}$.

... that Fourier Transforms are not properly standardized.

... that the meaning of $\theta$ and $\phi$ in spherical coordinates is not properly standardized.

 

[DiracPool]

The plus sign ("+") looks too much like the letter "t". So... I don't know... make it, like, a squiggle or something.

I think we're OK with "+,-,*, and /", let's not go overboard here. I don't want to start unlearning stuff I learned in preschool at my age!

 

[I like Serena]

Ah, yes, that might be right. I though he was simply integrating the whole thing over 1, and the (stuff) was just an afterthought, not the function. Kind of like "I hate this so and so 'stuff'"

An integration is actually a summation.
Didn't we learn in pre-school that multiplication has a higher priority than summation?
I prefer not to unlearn that.

 

[WannabeNewton]

I hate the annoying anti-symmetrization brackets used for things like wedge product and exterior derivatives. Like honestly, who ever thought $\nabla_{[e}\omega_{a_1...a_n]}$ was better than $d\omega$, not to mention it is quite cumbersome during proofs.

 

[micromass]

An integration is actually a summation.

Actually, it's the converse. A summation is integration wrt a special measure.

 

[HeLiXe]

I still wonder what $\sin^{-2}x$ means. Is it $(\arcsin x)^2$, or is it $(\csc x)^2$?

@_@ Have a heart ILS! geez. I tend to go for the latter tho.

 

[jedishrfu]

so Micro what's the motivation behind your question?

Are you designing a better math?

or writing a post-apocalyptic sci-fi novel?

 

[strangerep]

I don't know how mathematicians feel about dirac notation but Einstein notation doesn't seem to be too rare amongst the mathematicians.

I know at least one (applied) mathematician who dislikes both, and prefers to write a Dirac braket product as something like $\phi^T \psi$. Personally, I find both notations to be useful in different situations. Being multilingual is usually beneficial.

Re Einstein index notation, I also like Penrose's generalization to "abstract index notation". It looks a lot like the usual Einstein notation, but its meaning generalizes to infinite-dimensional spaces.

 

[Fredrik]

The only thing that needs to change is
\sin^2(x)

This needs to die in a fire

I don't have a problem with this at all.

Have you noticed that sometimes cos^2(x) is used for cos(x)^2 while the first should mean cos(cos(x)) and the second cos(x)*cos(x).

Why should the first one refer to a composition? If you're working with a set of functions on which both products and compositions are defined, what notation would you use for the product of f and g? Wouldn't you use $fg$? In that case, the $f^2$ notation is very natural too.

This also explains $\sin^2(x)$.

 

[yenchin]

I don't know how mathematicians feel about dirac notation but Einstein notation doesn't seem to be too rare amongst the mathematicians. Lee for example uses it in both his smooth and Riemannian manifolds texts.

I recommend reading the very nice article: "Mathematical Surprises and Dirac’s Formalism in Quantum Mechanics", F. Gieres, Rep.Prog.Phys. 63 (2000) 1893, arXiv:quant-ph/9907069v2.

"We discuss the problems and shortcomings of this formalism as well as those of the bra and ket notation introduced by Dirac in this context. In conclusion, we indicate how all of these problems can be solved or at least avoided."

"...the verdict of major mathematicians like J.Dieudonne is devastating [5]: “When one gets to the mathematical theories which are at the basis of quantum mechanics, one realizes that the attitude of certain physicists in the handling of these theories truly borders on the delirium. [...] One has to wonder what remains in the mind of a student who has absorbed this unbelievable accumulation
of nonsense, a real gibberish! It should be to believe that today’s physicists are only at
ease in the vagueness, the obscure and the contradictory."

 

[Fredrik]

Nothing in mathematics annoys me as much as the physicist's description of a tensor as "something that transforms as (blah-blah-blah)". I don't think I even want to talk about it. I get angry just thinking about it. So I'll just mention some mildly irritating things.

I don't like it when people write something like f(x) and refer to it as a "function". It's not. f is the function. f(x) is an element of its codomain. So f(x) is usually a number.

"Find the derivative of x sin ax." The correct answer is: "The derivative of a number is not defined, you idiot". But I doubt that you will get the maximum number of points if you write this on an exam.

I also don't like when people write $\frac{d}{dx}f$ or $\frac{d}{dt}f$ for the derivative of a function f. Either use a notation like Df, that doesn't have an irrelevant variable symbol in the part that tells you to take the derivative of something, or write $\frac{d}{dx}f(x)$.

The latter notation is perfectly fine, because
$$\frac{d}{dx}(x\sin ax)$$ is read as "the value at x of the derivative of the function $t\mapsto t\sin at$". Of course when I explain that to someone, they always ask what t is. **Facepalm**

I also don't like that people don't use the simple notation $(AB)_{ij}=\sum_{k=1}^n A_{ik}B_{kj}$ in the definition of matrix multiplication. Most physics students who see this don't even recognize this as the definition.

I don't like the term "functions of many variables". It makes sense when we talk informally about how a statement like $x+y+z$ makes z "a function of x and y", but it's inappropriate when we use the actual definition of "function". A function $f:\mathbb R^3\to\mathbb R$ isn't a function "of many variables". It takes one element of the domain as input, and that element can be represented by one variable.

 

[Fredrik]

When one gets to the mathematical theories which are at the basis of quantum mechanics, one realizes that the attitude of certain physicists in the handling of these theories truly borders on the delirium. [...] One has to wonder what remains in the mind of a student who has absorbed this unbelievable accumulation of nonsense, a real gibberish!

I love that quote.