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 this case, though: I'm a professional mathematician, and as prone as anyone in my discipline to use mathematical language to describe not strictly mathematical things, but the pseudo-mathematisation here ("Notation^{-1}(C)", for example) seems more like wit than clarity. Not that there's anything wrong with wit, but in this case it seems to me that it's at the expense of, rather than a pleasant addition to, the central point.
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?
(EDIT: I was scared about making this post, since there's rightfully a lot of respect and appreciation for Tao—and I hope it's clear that I concur on both counts—and I wasn't sure how my reticence on his post would go over; but I'm super glad I asked. Thanks so much to everyone downthread; these are wonderful responses and I feel that it benefited me a lot to read them.)
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.
I'm a programmer by trade, so my knowledge of mathematics as a whole is limited by what I happened to learn doing my undergrad in CS, so a lot of the mathematic-specific examples in this post were lost on me.
What I found particularly insightful here is applying this mathematical notation to programming languages and their syntaxes. His notation described as
```
Notation:{well-formed expressions}→{abstract objects in 𝑋}
```
isn't too far off from what most programming languages implement at some level.
As a result of this, what properties do programming languages share with mathematical notations, and why are some languages deemed more "expressive" than others? How much does the "expressiveness" of a certain language in a domain lead to better understanding of the abstractions beneath them?
To answer your question; I don't think this post explicitly increased my understanding of notations (especially not in the context of mathematics), but rather led me to ponder the importance of them in communicating & extending abstract concepts effectively across domains.
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?
I am a physicist. Abusing mathematical notation is sort of what pays the bills for us, so... :D Seriously, though, why should meta-mathematics, not use math like notation. The post can be rewritten without that semi-formal notation and not change much, but I don't mind it all. If it is wit, I appreciate the enjoyment.
I am bookmarking the post as I found it valuable for next time I invent notation.
Honestly, it doesn't bother me at all. I'm used to slogans like "syntax and semantics are adjoint functors" or "meaning is a functor from syntax to semantics" from category theory. In the opening definition, for example, I'm reading the LHS as a source category whose objects are abstract syntax trees and whose arrows are substitutions between said trees, and the RHS as any target category of interest. We're giving meanings to notation by sending the notation along a functor, and syntactic/formal manipulations of the notation correspond to manipulations of the objects which they "mean"/represent.
Framed this way, the properties listed are (1) (part of) functorality, (2) surjectiveness, (3/4) smallness, (5) renormalizability (!!), (6) smoothness/continuity of some sort, and (8) the ability to have natural transformations applied. Only (7) is culturally dependent. (1) and (8) are free for all functors!
To address your question on the head (I am not a mathematician), this post gave me some properties of good notation which I hadn't considered before. On one hand, yeah, it's kind of obvious that notation should somehow be able to surjectively reach all possible objects of interest, while still being small enough to definitely not reach everything. On the other hand, renormalization is kind of a dense topic, so it's surprising to see it arise here.
> 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?
No, but I did learn about new dialects and conceptualize mathematical notations better. I think for someone new to languages/notations the first set of detailed bullet points are useful. For more experienced practitioners I think it sums up to "use the right tool for the job".
What I found very interesting was the latter set of bullet points where he presents a sort of Rosetta Code (http://rosettacode.org/wiki/Rosetta_Code) of math. I am familiar with thousands of computer languages but not a mathematician and not familiar with what I'm sure are hundreds (thousands/more?) of Mathematical notations, so the few examples he lists here have me intrigued.
I think the notation is intentionally pseudo notation on purpose.
A large part of math is knowing notation and definition. When I started studying higher levels of Mathematics a big part of me that was keeping my understanding was figuring out the notation and the definitions.
Graph theory is so notorious for this that I can remember a joke that everyones notation for graph theory is slightly different.
As an engineer, not a mathematician, I'm glad that good mathematicians care about good notation. The relatively-elementary maths that I learnt didn't always have good notation, the tradition seems suited to chalk and pen i.e. complex glyphs are easy, but perhaps because with hand writing the size and position of elements can be ambiguous, too much notation is overloaded and re-used. Even simple stuff like an exponent of -1 meaning inverse function, it's hardly unusual for it to be mixed up with numerical exponents.
One bugbear is that mathematical writing leaks into engineering science. I wouldn't ask professional mathematicians to start caring about units or change their style while communicating amongst themselves, but in my view textbooks ought to define notation before they use it, and clearly define the units used in all expressions.
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.
I'm not sure I agree about the fact that notation is a pseudo-mathematisation.
For example in Mathematica there is a (mostly deprecated) package called Notation`[0] that does just this kind of stuff. I have to admit that it's not really used in production code anymore as MakeBoxes and MakeExpression are more fine-grained and robust.
Thus said I have to admit that the transformation between 2-D boxes and M-expression is not as foundational as what Tao is talking about, however the whole field of designing programming languages is deep down an exercise in defining notation, the transformation mentioned above just make this a bit more explicit.
> I'm not sure I agree about the fact that notation is a pseudo-mathematisation.
I definitely don't think that notation is pseudo-mathematisation; good notation is inordinately powerful in enabling good mathematics (and bad notation can make even simple mathematics hard). What I meant to describe as pseudo-mathematisation was the discussion of notation in what seemed to me in a (to me) unnecessarily formally mathematical way.
> 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...?
> 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...?
I definitely think that he's earned the right to contribute to the mathematical community in whatever way he sees fit, and, especially if it benefits other people, which it clearly does, then my opinion of it shouldn't and doesn't matter a hill of beans. I just usually feel that he combines those missions—not that he has to, but that he does—and felt that he didn't here; but, again, that is, if at all, very much my problem and not at all his.
I wouldn't be able to precisely break down his notation (and I'm guessing it's not intended that way?) but just following his thought process has improved my intuition by seeing some of these ideas I had toyed with before reframed in a more mature way. At the very least, a lot of interesting analogies to consider.
I think the post would be more interesting if it came with examples of existing notations that do particularly well and particularly bad on the listed criteria.
ex-theoretical physicist here. I read Terry's post and I totally saw your point about "Notation^{-1}(C)" being more wit than clarity. Of course, I also share your respect and appreciation for Tao.
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 this case, though: I'm a professional mathematician, and as prone as anyone in my discipline to use mathematical language to describe not strictly mathematical things, but the pseudo-mathematisation here ("Notation^{-1}(C)", for example) seems more like wit than clarity. Not that there's anything wrong with wit, but in this case it seems to me that it's at the expense of, rather than a pleasant addition to, the central point.
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?
(EDIT: I was scared about making this post, since there's rightfully a lot of respect and appreciation for Tao—and I hope it's clear that I concur on both counts—and I wasn't sure how my reticence on his post would go over; but I'm super glad I asked. Thanks so much to everyone downthread; these are wonderful responses and I feel that it benefited me a lot to read them.)