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Terry Tao on some desirable properties of mathematical notation

mathoverflow.net

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Re: Terry Tao on some desirable properties of mathematical notation

#21

Post by ianhorn »

JadeNB wrote: post #3

I found this post a shame. (The post itself, not putting it here; I love seeing math posts on HN, and automatically upvote. Bringing hackers and mathematicians together is highly worthwhile for both.) Usually Tao's posts are so insightful, and crystallise some idea so perfectly that it feels like I was just on the cusp of discovering it myself—a rare talent, and hard to cultivate since it goes against the ego. In thi…

I found the idea of (ab)using mathematical notation to talk about mathematical notation to be entertaining and got the sense that he was kinda just having fun for himself. I also got the impression that he was potentially nerd sniped. The rest of the post beside the `Notation` function was definitely a good informative read.

Re: Terry Tao on some desirable properties of mathematical notation

#22

Post by riazrizvi »

Unambiguity as an adjective is slippery. Mathematical notation must be concise, because a key purpose is to provide understanding, which it achieves by focused abstraction. So when you search for notation to model some real world system, you leave things out, as such it leaves room for interpretation when remapping back to the real world, ie there is ambiguity. I think this #1 item should really be termed Consistency, because above all, notation must not contradict itself.

Re: Terry Tao on some desirable properties of mathematical notation

#23

Post by Koshkin »

zamfi wrote: post #19
jstanley wrote: post #12

Earlier quoted context omitted.

> Math notation sucks because none of it maps to things non-mathematicians know. This doesn't mean maths notation sucks any more than vim's user interface sucks because it doesn't make sense to non-vim-users. Mathematical notation presumably mostly makes perfect sense to the kind of people who deal with mathematical notation all day long.

> Mathematical notation presumably mostly makes perfect sense to the kind of people who deal with mathematical notation all day long. Maybe the overuse of opaque names leads to self-selection of who becomes a mathematician? Single-letter non-descriptive variable and functions names would “make sense” to programmers who use it all day long too — but that alone doesn’t make it a good idea.

Mathematicians will never agree to write dblEulerConstant instead of e.

Re: Terry Tao on some desirable properties of mathematical notation

#24

Post by h-jones »

Koshkin wrote: post #18

Difficulties, if any, perceived or real, arising in connection with notation, are usually incomparably smaller than those presented with the subject itself. (Personally, I only wish mathematical notation were better integrated with software in general and programming languages in particular.)

I think differential geometry may present the closest exception to this, not only is the notation often incredibly dense and subtle (i.e. spacing between indices when raising and lowering) but often everyone seems to have their personal favorite take on any given notation.

Re: Terry Tao on some desirable properties of mathematical notation

#25

Post by quantdev »

zamfi wrote: post #19
jstanley wrote: post #12

Earlier quoted context omitted.

> Math notation sucks because none of it maps to things non-mathematicians know. This doesn't mean maths notation sucks any more than vim's user interface sucks because it doesn't make sense to non-vim-users. Mathematical notation presumably mostly makes perfect sense to the kind of people who deal with mathematical notation all day long.

> Mathematical notation presumably mostly makes perfect sense to the kind of people who deal with mathematical notation all day long. Maybe the overuse of opaque names leads to self-selection of who becomes a mathematician? Single-letter non-descriptive variable and functions names would “make sense” to programmers who use it all day long too — but that alone doesn’t make it a good idea.

I'm both a programmer and a mathematician. Mathematics is written in English (or another human language) plus added symbols. The symbols are what we refer to as the notation. It's not comparable to programming language. Here the words are equal to the symbols.

Also, no better notation could help you understand most modern mathematics. There's no better choice of symbols or names that will help you understand Galois theory. You simply need to understand high-school algebra, then group theory, then Galois theory.

Re: Terry Tao on some desirable properties of mathematical notation

#26

Post by canjobear »

zamfi wrote: post #19
jstanley wrote: post #12

Earlier quoted context omitted.

> Math notation sucks because none of it maps to things non-mathematicians know. This doesn't mean maths notation sucks any more than vim's user interface sucks because it doesn't make sense to non-vim-users. Mathematical notation presumably mostly makes perfect sense to the kind of people who deal with mathematical notation all day long.

> Mathematical notation presumably mostly makes perfect sense to the kind of people who deal with mathematical notation all day long. Maybe the overuse of opaque names leads to self-selection of who becomes a mathematician? Single-letter non-descriptive variable and functions names would “make sense” to programmers who use it all day long too — but that alone doesn’t make it a good idea.

Opaque names have the advantage that they don't trigger any potentially misleading associations and they emphasize the abstract nature of what is being discussed. A variable named x could be anything, even something that the mathematician/programmer didn't anticipate.

Re: Terry Tao on some desirable properties of mathematical notation

#27

Post by pubby »

jstanley wrote: post #12
pubby wrote: post #9

Imagine I give you a list of words and ask you to remember them. 5 minutes later, I ask you to give me those words in reverse order. Not too hard, right? Now imagine if those words I gave you were in Vietnamese, or some language you don't speak. Suddenly the task becomes much more confusing. You aren't remembering a small handful of objects and ideas, but instead trying to juggle the individual syllables in your head…

> Math notation sucks because none of it maps to things non-mathematicians know. This doesn't mean maths notation sucks any more than vim's user interface sucks because it doesn't make sense to non-vim-users. Mathematical notation presumably mostly makes perfect sense to the kind of people who deal with mathematical notation all day long.

The question is if we should optimize for pencil efficiency or education. To me, it's obvious: I've met way more people interesting in learning math than performing it, so we try to make notation as clear and possible.

Vim, like math notation, is optimized for efficiency. That's great for power users, but it's not what you would use if you wanted to teach someone how to word process.

(BTW, try imaging Vim where each file mandates its own special keybindings. That's math notation for ya.)

Re: Terry Tao on some desirable properties of mathematical notation

#28

Post by chongli »

pubby wrote: post #9

Imagine I give you a list of words and ask you to remember them. 5 minutes later, I ask you to give me those words in reverse order. Not too hard, right? Now imagine if those words I gave you were in Vietnamese, or some language you don't speak. Suddenly the task becomes much more confusing. You aren't remembering a small handful of objects and ideas, but instead trying to juggle the individual syllables in your head…

Now imagine if those words I gave you were in Vietnamese, or some language you don't speak.

Math notation sucks because none of it maps to things non-mathematicians know.

I'm struggling to see how someone could conclude the latter statement from the former. Why is it reasonable for Vietnamese to be unintelligible to non-Vietnamese speakers, yet unreasonable for mathematics to be unintelligible to non-mathematicians?

Re: Terry Tao on some desirable properties of mathematical notation

#29

Post by mxcrossb »

zamfi wrote: post #17
pubby wrote: post #9

Imagine I give you a list of words and ask you to remember them. 5 minutes later, I ask you to give me those words in reverse order. Not too hard, right? Now imagine if those words I gave you were in Vietnamese, or some language you don't speak. Suddenly the task becomes much more confusing. You aren't remembering a small handful of objects and ideas, but instead trying to juggle the individual syllables in your head…

In addition, the same notation is often overloaded in new domains, where the association with the prior domain is metaphorical, or even simply allegorical. As a novice programmer, it took years to get out of the habit of using non-descriptive single-letter variable and function names. It’s a terrible habit, and makes code incomprehensible — and programmers know it. Math, as taught to kids, is full of single-letter va…

Are you sure it’s not the opposite? Plenty of students can solve a set of equations. But when you start asking them about trains speeding in the night...

Re: Terry Tao on some desirable properties of mathematical notation

#30

Post by chongli »

pubby wrote: post #27
jstanley wrote: post #12

Earlier quoted context omitted.

> Math notation sucks because none of it maps to things non-mathematicians know. This doesn't mean maths notation sucks any more than vim's user interface sucks because it doesn't make sense to non-vim-users. Mathematical notation presumably mostly makes perfect sense to the kind of people who deal with mathematical notation all day long.

The question is if we should optimize for pencil efficiency or education. To me, it's obvious: I've met way more people interesting in learning math than performing it, so we try to make notation as clear and possible. Vim, like math notation, is optimized for efficiency. That's great for power users, but it's not what you would use if you wanted to teach someone how to word process. (BTW, try imaging Vim where each…

Vim, like math notation, is optimized for efficiency. That's great for power users, but it's not what you would use if you wanted to teach someone how to word process.

Vim is a text editor, not a word processor. Additionally, Vim is a tool designed for power users. Mathematical notation is also meant for power users. When we teach mathematics, we introduce the notation gradually, so students have time to pick it up. This is a process which takes decades from Kindergarten through PhD. Just as a Vim user would scoff at being forced to write in Notepad, a mathematics PhD would scoff at being forced to express their ideas in Kindergarten-level mathematical notation.

try imaging Vim where each file mandates its own special keybindings. That's math notation for ya.

That's how Emacs works, and how Vim works when you install filetype-specific plugins.

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