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Comment by chilie | Hacker News Reader

These are just issues that someone unfamiliar with a field would face. None of them are problems for those of us in the field.

First, the expectation thing. He's using a special case, E(X), and complaining that the more general case doesn't follow the general case. It's like saying "Well the plural of mouse is mice but the plural of house isn't hice!". The general definition of expectation (for a discrete probability) is

E(f(x)) = sum f(x_i)* p(x_i)

If you start with this general definition, both E(X) and E(X^2) are perfectly natural. The author's error of starting with the special case in no way implies an issue with notation.

And how is the fact that Wikipedia is inconsistent between E(X) and E[X] in any way mathematical notation's fault? If you read a novel that starts using ' for quotes and switches to ", that's an issue with the novel (assuming its not stylistic) and not an issue with the typography in general.

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> These are just issues that someone unfamiliar with a field would face. None of them are problems for those of us in the field.

True, but it raises a "barrier to entry" (on purpose, or by mistake) because it is almost impossible to enter the field without a supervisor/colleges that provide the "semi-supervision" needed to learn the notation.

I can understand how those in the field think that's a good thing, but for the rest of humanity it probably isn't... Look e.g. what has happened in academic operating system research: innovation has moved from Berkeley and Bell Labs to the Linux kernel mailing list. Are academic OS researchers better off because of it? Probably not. Is the world better off? You bet!

> It's like saying "Well the plural of mouse is mice but the plural of house isn't hice!"

Yes, that's exactly what he is saying:

> Math is a language that is about as consistent as English, and that's on a good day.

Except that math is not a language. (Otherwise one could call chemistry a language, too.)

It might be in Florida. http://www.npr.org/sections/ed/2016/03/01/468695376/french-s...

Unfamiliar is a weasel word. It's a No-True-Scottsman. And I don't mean, that it is entirely wrong, I am saying you are unjustly putting a limit to whom you deem worthy for the field. There is no need for imprecision other than speed breaking things while you go.

> "Well the plural of mouse is mice but the plural of house isn't hice!"

Are irregular word forms necessary or essential? Probably, but I doubt you could explain why. Hence you are not qualified to ridicule anyone. It's a perfectly valid complaint, IMHO, but probably only loosely related to the math example, which I can't be bothered to follow at the moment.

I just saw that I misunderstood. I don't mean to say "hice" would be good. Mouses would be good.

I have taken ten years of math in school and then calculus, discrete math, linear algebra. That's all useless if I want to follow math in a simple research paper because they're using notation common in THAT field and it has nothing to do with the notation in another math field.

And it's all V hat superscript pi subscript h. It's not like code, where I get descriptive variable names. And you thought pi meant pi? No, it means policy in THIS context.

Reading a "simple" research paper requires a different approach compared to reading a newspaper or a blog. I hope the following explanation helps.

The notation actually helps to keep things simple. I think of it as a kind of metalinguistic programming [1] where a notation is introduced which then makes the important parts easier to understand.

I am not mathematically inclined but I have to read papers containing maths quite a lot of the time. I tend to read them 3 times.

The first time, I tend to skip the equations altogether and just get a feeling for the paper - what is it about, is it useful for me to read?

The second time I have a pen and a highlighter where I actually label the mathematical symbols with arrows and words (using the textual descriptions). I also highlight important sentences. I think of this stage as trying to make the paper as clear as possible for later reading.

In the third stage I am trying to understand the paper as a whole - something it seems you try to do on the first read, I am familiar with the frustration because this is what I used to do.

I quite enjoy reading papers now and I have more respect for the notation.

[1] https://en.wikipedia.org/wiki/Metalinguistic_abstraction

> The second time I have a pen and a highlighter where I actually label the mathematical symbols with arrows and words (using the textual descriptions).

Gee, it's almost as if we should be writing the words out instead of writing one letter variable names.

Funny how programmers found this to be good practice and mathematicians still write with a notation that's purposefully unreadable.

I feel the equations are nicer when they are small and so I have sympathy for people using short variables names.

One thing I would like is to see is more use of labels as described here:

https://tex.stackexchange.com/questions/36863/adding-labels-...

> Gee, it's almost as if we should be writing the words out instead of writing one letter variable names

That's how things were way back in the day, and it was terrible.

I've taken years of English Studies in school, then German, Greek, Russian. That's all useless if I want to read a French book, because they are using a language common in THAT country, and it has nothing to do with the language in another country.

That's not a direct equivalent, but it is close; somewhat equivalent but distinct notations arise in Math and Physics because authors weren't working together (much like natural languages). But otherwise it is "turtles all the way down" - it can only be "simple" or "needlessly complicated" if you assume something about the reader's knowledge.

All of

    integrate(f(x)dx,dx)
    sum f(x_i) for i=1..n
    f(x_1) + f(x_2) + .... + f(x_3)

Are essentially equivalent to a mathematician, often with a preference for the first, whereas someone unfamiliar may claim that only the 3rd is clear and the others are unreadable. A friend of mine was in a classroom where the lecturer started with the first form, when to the 2nd and 3rd over class objections, and finally switching to something like "our function result at the first data point, added to our function result at the second data point, ....". This was an OR class for students pursuing an MBA.

This sounds so much like something the greybeards used to say in the 80s: "If it was hard to write, it should be hard to understand"

Basically, it's not the fault of our systems; it's the user's fault. Once he learns the arcane incantations, he'll understand why our way is the better way.

Computer UX has finally progressed beyond this arrogance. Why not math?

The reason is that difficulty of mathematics lies not in notation - it is in ideas and techniques. There is no doubt that a good modern UX complete with graphics, animation, and audio would facilitate understanding of the ideas, but, just as in programming, the need for textual notation cannot be overcome. Category Theory is an interesting example: while a lot of reasoning in it is done by "diagram chasing", if you look at a book on this theory, you will find more textual proofs and formulas than diagrams. Even more strikingly, same is true for topology, differential geometry etc.

So why are years of studying math insufficient for one to be "familiar" with it?

Years of studying what, though? Calculus, most of linear algebra, discrete math are all calculation based. Math papers are proof based. If you spend years studying proof based math (analysis, algebra, topology, and so on) then you'll be familiar enough with proof based math to understand it.

It's like saying "why isn't spending years studying spelling not sufficient to understand Ulysses?"

Ok, then why are we spending years learning "spelling"?

Probably because the goal was different from read Ulysses.

And I doubt most subject areas have the goal of reading proofs.

I'm sorry but I don't see how this is an answer to my question.

We spend years learning "spelling", because the mathematical equivalent (arithmetic, up through calculus, linear algebra, differential equations, discrete math, etc) is incredibly valuable on its own—and that that value doesn't really require familiarity with the proof-based foundations underpinning it. Learning real analysis is valuable if you're going to be a mathematician, but as an engineer it won't help you get more out of calculus (at least, not compared to the cost of learning it).

I believe you have adopted your opinion of what should go into the first set of brackets of yours.

An engineer is expected to allot too much time to math to turn out unfamiliar with it. I think we should spend less effort on practicing cleaning fish and more on learning to forage for fish.

> I believe you have adopted your opinion of what should go into the first set of brackets of yours.

I'm not really sure what you mean by that.

Certainly I'm not of the opinion that proof-based math is useless; I studied math in grad school, and stayed as far away from anything applied as I could. But I don't think it offers much value to engineers or scientists. Let mathematicians establish the foundations, and let scientists and engineers build on top of that. They're different skill sets, with different goals.

> "Well the plural of mouse is mice but the plural of house isn't hice!"

That's the strangest defense. English notation is awful.

"These are just issues that someone unfamiliar with a field would face. None of them are problems for those of us in the field."

How is this argument different from defending code with bad variable naming by stating that it doesn't cause issues to anyone familiar with the code base?

In the other words, E(...) has a hidden lambda there, and the fully consistent usage would be E(X -> X^2) (instead of E(X^2)) and so on. The covariance thing would be E((X,Y) -> X*Y) with an implied domain being a Cartesian product of that of X and Y. Of course we humans can easily infer the domains, and writing explicit domains every time is not efficient.

Well, formally a random variable is already a function which assigns values to elements of the probability space (outcomes). The expected value is just another name for an integral over this space. When the probability space is discrete, "the integral over a probability space" is just another name for a weighted sum. The domain is always the same: it's the probability space.

The real bad notation (which is employed here, actually) is f² meaning (x ↦ f(x)²) while at the same time f⁻¹ means the inverse function of f instead of (x ↦ f(x)⁻¹).

I understood math better once I used it for coding.

Linear algebra and matrices, for example, by creating a simple 3D simulation and later studying a circuit simulation.

Same with physics and other sciences, which are typically taught more like math calculation and/or memorization classes.

It would be nice to see more math documented and taught as code.

>It's like saying "Well the plural of mouse is mice but the plural of house isn't hice!".

Just because English sucks and doesn't make any sense doesn't help the case for mathematical notation.

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