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Mathematical Notation Is Awful

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Re: Mathematical Notation Is Awful

#81

Post by ivansavz »

zimbatm wrote: post #69

If I could fix one things in Maths, it would be to introduce an explicit import statement. Right now it's very hard to work back what the symbols mean in a specific context unless you're familiar with the field. from url/to/geometric-algebra.pdf import X; I don't mind the overloading too much and it would always be possible to alias symbols in case two or more fields are user together.

Very cool idea!

I think this would work both at the logical level (e.g., the concept of azimutal angle in a xyz-coordinate system) and as a stylesheet level (theta for physcisits, and phi for mathematicians).

Such annotations would really help with browsing math content—you can see what prerequisites concepts are used in a paper from its import statements, without the need to read the whole thing. Also, you could browse from the other end (reverse-imports / uses), looking for docs that make use of a given concept.

Re: Mathematical Notation Is Awful

#82

Post by Koshkin »

julian37 wrote: post #30
chilie wrote: post #13

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…

> 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.)

Re: Mathematical Notation Is Awful

#83

Post by wckronholm »

Koshkin wrote: post #82
julian37 wrote: post #30

Earlier quoted context omitted.

> 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...

Re: Mathematical Notation Is Awful

#84

Post by posterboy »

posterboy wrote: post #35
chilie wrote: post #13

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…

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 ex…

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

Re: Mathematical Notation Is Awful

#85

Post by Koshkin »

kstenerud wrote: post #79
chilie wrote: post #13

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…

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.

Re: Mathematical Notation Is Awful

#86

Post by paxcoder »

mcphage wrote: post #71
paxcoder wrote: post #31

Earlier quoted context omitted.

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"?

Re: Mathematical Notation Is Awful

#87

Post by nshm »

As a developer I'm not quite happy with a single-letter names. Of course it's ok for minor local things to be named 'x' but for more important values and functions there is no problem these days to have a readable names just like in software. So you could actually read of a paper without guessing and searching those epsilons, lambdas and cappas and cryptic symbols. Use expectation(x) instead of e(x), use mean(y) instead of \hat y and so on.

Re: Mathematical Notation Is Awful

#88

Post by stdbrouw »

nshm wrote: post #87

As a developer I'm not quite happy with a single-letter names. Of course it's ok for minor local things to be named 'x' but for more important values and functions there is no problem these days to have a readable names just like in software. So you could actually read of a paper without guessing and searching those epsilons, lambdas and cappas and cryptic symbols. Use expectation(x) instead of e(x), use mean(y) inst…

It really depends. Within a given field certain concepts are so common that you really do want a way to express them in the most terse possible way, just as programmers use i for index, err for error and n for counts. This allows you to put more stuff in a smaller space which means you don't have to jump back and forth to understand something -- sort of like how it's nice to put different parts of an app into different files and even to split them up into separate libraries, but go too far and the flow of the application becomes opaque.

Have you ever had a chance to read some of those really old mathematical proofs that didn't use any mathematical symbols at all? They're a nightmare to try and understand. There's a tradeoff. No disagreement, though, that mathematicians tend not to be very great at finding the sweet spot where the trade-off balances :-)

(Also, the sample mean of y would be `\bar y`, hats are for estimators in general.)

Re: Mathematical Notation Is Awful

#89

Post by Smaug123 »

wolfgke wrote: post #72
pwdisswordfish wrote: post #58

Earlier quoted context omitted.

> Now that's a fine thing in a computer language -- and there are philosophers who say mathematical truth is no more than computation. But mathematical notation is for humans to talk about things that have meanings that humans can understand. Then how come an unfamiliar piece of source code is easier to follow than a mathematics paper using unfamiliar notation? You can look up the definition of each function and lear…

> A mathematical formalism that could be interpreted in a purely mechanical way would be a huge improvement. Look at Homotopy Type Theory: > https://homotopytypetheory.org/

I feel obliged to point out that HoTT is… really rather abstract. I'm just starting to learn about it now in my spare time, and I hold a master's degree in mathematics specialising in logic.

Re: Mathematical Notation Is Awful

#90

Post by jokoon »

nkozyra wrote: post #62
jokoon wrote: post #52

Funny, I said the same about Andrew Ng's ML course. Ultimately, if what you're teaching is going to end up in software, why use math at all? Use code or pseudo code. I don't think it's bad to just give the working algorithm without having to prove the math. Really how many students will end up being computer scientists anyway, and research and write about new methods of doing AI, and do the actual math? So few. I gue…

Understanding underlying math is what allows people to create improved algorithms. If it's just "implement scikit" then you barely even need a developer. Understanding why things work is still important and why software companies still routinely test on algorithm design.

If you're a beginner at ML, the first course don't really need to be so thorough on math, especially for simple things like regression.

I'm not saying math is dispensable, I'm saying that it could be used much less because code allows it in many situation.

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