Terry Tao on some desirable properties of mathematical notation • Hacker News Board

Re: Terry Tao on some desirable properties of mathematical notation

#11

Post by chrispeel » Wed, Jul 22, 2020, 12:35 AM UTC

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'd like to hear especially from anyone who isn't a professional mathematician: did you feel that this post improved your understanding of the purpose and function of good notation?

Yes

I skipped past the long list of properties and went straight for his long list of ways to write the inner product of two vectors and his illustrations of why you want to use different notation in different cases.

Here's a question for you, do you think Tao's primary mission is to be a "professional mathematician" or is he just enjoying the math, enjoying teaching, enjoying notation...?

Re: Terry Tao on some desirable properties of mathematical notation

#12

Post by jstanley » Wed, Jul 22, 2020, 12:35 AM UTC

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.

Re: Terry Tao on some desirable properties of mathematical notation

#13

Post by somethingsome » Wed, Jul 22, 2020, 12:36 AM UTC

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 find it useful as he needed it for the following sentence to make sense :

A good notation should make this map Notation (and its inverse) as close to a (natural) isomorphism as possible.

Having said he want an isomorphism for Notation sums up all the following points he made explicitly (and more), so we could argue that we can keep the Notation and the isomorphism part but remove all the following point 1..9+

I think that in this way the post reach more people :) some needs only to read the first sentences, some the properties and finally some the example.

Isn't it a nice way to understand what he is trying to say in as much different ways as possible?

Re: Terry Tao on some desirable properties of mathematical notation

#14

Post by [deleted] » Wed, Jul 22, 2020, 12:37 AM UTC

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…

[deleted]

Re: Terry Tao on some desirable properties of mathematical notation

#15

Post by emmanueloga_ » Wed, Jul 22, 2020, 12:40 AM UTC

The discussion of mathematical notation reminds me of the talk by Guy Steele "It's Time for a New Old Language", discussed previously in HN [1]. That talk was focused on the Math notation that is used in computer science papers, but I feel a similar analysis could be expanded to other areas of Math.

1: https://news.ycombinator.com/item?id=15473199

Re: Terry Tao on some desirable properties of mathematical notation

#16

Post by tmpz22 » Wed, Jul 22, 2020, 12:43 AM UTC

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…

I dropped out of college at 19 and attempted to return at 21. My first math course back was discrete math and my teacher was a grad student who very clearly had no interest in teaching and was only there in order to subsidize his "real work". Keep in mind this is a large public university charging $40k/year. Going into his office hours was like going to another country, because his only method to explain math was really really advanced level math notation. He became visible frustrated that I didn't pick up on his notation. I wish I could say that was my only teacher who has no business teaching.

I dropped out again and haven't looked back. That system wasn't for me and didn't care about my success or its ability to transfer knowledge.

Re: Terry Tao on some desirable properties of mathematical notation

#17

Post by zamfi » Wed, Jul 22, 2020, 12:44 AM UTC

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 variable names (or worse, as you point out, using non-Roman letters and other symbols) that are devoid of any hints to the uninitiated of what their meaning might be.

I wonder what it would be like if math were taught with descriptive names — starting with the basics, for functions and variables, and used those names consistently.

Would it be more verbose? Absolutely — but I’m pretty sure it would also be a huge kindness to learners.

Re: Terry Tao on some desirable properties of mathematical notation

#18

Post by Koshkin » Wed, Jul 22, 2020, 12:45 AM UTC

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.)

Re: Terry Tao on some desirable properties of mathematical notation

#19

Post by zamfi » Wed, Jul 22, 2020, 12:46 AM UTC

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.

> 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.

Re: Terry Tao on some desirable properties of mathematical notation

#20

Post by mmhsieh » Wed, Jul 22, 2020, 12:49 AM UTC

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 thought it worked. It was his way of demonstrating one of his own requirements:

(Preservation of quality, I) Every "natural" concept in 𝑋 X should be easily expressible using the notation.

So, if he is going to commit to his idea of the notation-to concept-isomorphism, it would have been weirder if he did NOT introduce the notion of the inverse map.