# Help! We have forgotten how to write math stuff

## Main Question or Discussion Point

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?

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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 :)

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

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jedishrfu
Mentor
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.

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

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.

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.

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.

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.

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
Gold Member
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
Gold Member
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 thats rather poor reason to want to abolish useful notation.

WannabeNewton
Oh. I think thats 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.

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
Gold Member
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.

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.

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

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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
Mentor
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.

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?

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