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

Re: Terry Tao on some desirable properties of mathematical notation

#41

Post by JadeNB » Wed, Jul 22, 2020, 1:34 AM UTC

riazrizvi wrote: post #22

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…

> I think this #1 item should really be termed Consistency, because above all, notation must not contradict itself.

This is a good goal, but I'm not sure it's the primary goal; the phrase 'abuse of notation' exists precisely to describe its breakage, with even the best mathematicians and expositors engaging in it, and I think insisting on no abuse of notation leads us rapidly to the style of impenetrable Principia-style logic, or of modern formal proofs—both of which have their place (at least the latter …), but neither of which should govern all mathematical discourse.

As with all writing, I think that part of being a good mathematical writer is knowing the rules so that you can figure out when to break them unintentionally, rather than stumbling into it accidentally.

Re: Terry Tao on some desirable properties of mathematical notation

#42

Post by JadeNB » Wed, Jul 22, 2020, 1:37 AM UTC

quantdev wrote: post #25

zamfi wrote: post #19

Earlier quoted context omitted.

> 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 t…

> 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.

This probably isn't completely true—as will be clear to anyone, even a subject expert, who tries to go back and read the original papers. At least part of this is due to fads in notation—in my field, I can easily read the papers of people who do similar work to mine, and struggle with the papers of those who don't, even when they're talking about exactly the same thing—but some of it must be due to genuine improvement.

Re: Terry Tao on some desirable properties of mathematical notation

#43

Post by JadeNB » Wed, Jul 22, 2020, 1:38 AM UTC

Koshkin wrote: post #23

zamfi wrote: post #19

Earlier quoted context omitted.

> 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 .

And, issues about the length of the name aside, we shouldn't! `e` isn't a double or any other standard numeric type. It is an infinite-precision number.

Re: Terry Tao on some desirable properties of mathematical notation

#44

Post by chongli » Wed, Jul 22, 2020, 1:40 AM UTC

nl wrote: post #35

chongli wrote: post #30

Earlier quoted context omitted.

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.…

I'm a power user (been programming for 30 years) and I refuse to use vim keybindings. I have too many things to remember, and there are plenty of options that give me equal power but don't demand that investment from me. Math isn't like that - I do more math than the average person and I hate having to decipher the annotation. The number of times I've tried to work out if this sigma is the same as the sigma in this o…

Mathematics is more general and applicable than Vim, I'll give you that. I wasn't the one who brought in the Vim analogy, however.

It seems like your complaints are with academic papers, not with mathematical notation specifically. This is a problem that is universal to academic papers. Try reading a critical theory paper, for example, and you'll find it's extremely dense with critical theory jargon the authors don't bother explaining at all.

The problem with academic papers is that they aren't intended for a general audience. The authors of papers are often operating in subfields that are so small that they've actually met most of the other people who will be reading their papers. In that situation, conventions and jargon arise naturally among colleagues. Variable names form a part of these conventions such that, in a more common example, sigma will mean standard deviation among statisticians but mean singular value among linear algebraists.

The other problem with academic papers is that the authors generally don't care about reproducibility, consistency, clarity, pedagogy, or even intelligibility. They're optimizing for quantity of papers published, not quality. As long as their expert peer reviewers understand and give the green light for publication, that's good enough.

Re: Terry Tao on some desirable properties of mathematical notation

#45

Post by joppy » Wed, Jul 22, 2020, 1:40 AM UTC

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…

You say "I've met way more people interesting in learning math than performing it", but I have found it downright impossible to learn mathematics without performing it. And I daresay this is true of most people -- a well-known feeling amongst mathematics graduate students is to read a chapter of a text, think we have understood it, then turn to the very first exercise and get completely stumped and have to backtrack.

A lot of mathematics is optimised for pencil efficiency for a reason, and it's not at odds with learning mathematics.

Re: Terry Tao on some desirable properties of mathematical notation

#46

Post by Kednicma » Wed, Jul 22, 2020, 1:47 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…

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.

Re: Terry Tao on some desirable properties of mathematical notation

#47

Post by riazrizvi » Wed, Jul 22, 2020, 1:50 AM UTC

JadeNB wrote: post #41

riazrizvi wrote: post #22

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…

> I think this #1 item should really be termed Consistency, because above all, notation must not contradict itself. This is a good goal, but I'm not sure it's the primary goal; the phrase 'abuse of notation' exists precisely to describe its breakage, with even the best mathematicians and expositors engaging in it, and I think insisting on no abuse of notation leads us rapidly to the style of impenetrable Principia-st…

Great point. It is bad to nitpick consistency when you are in the initial stages of developing a model outline, and looking to capture the most important points. What's the right term for this notational quality? Precedence?

Re: Terry Tao on some desirable properties of mathematical notation

#48

Post by dilap » Wed, Jul 22, 2020, 1:52 AM UTC

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.)

> with the subject itself

that's the thing though, you can never grapple with the subject itself, only representations thereof. this is a mix of feelings/images/movements inside your head & mechanical manipulations of the notation; furthermore, the notation itself influences our internal model/feeling of the subject.

Re: Terry Tao on some desirable properties of mathematical notation

#49

Post by Koshkin » Wed, Jul 22, 2020, 2:09 AM UTC

Mathematical notation is great at facilitating formal manipulations. This is its critical feature, and without it we would get stuck at the level of ancient mathematics. This is the reason it was invented a few hundred years ago in the first place. That said, I find that notation is often abused in texts as a mere substitute for the normal human language which, while allowing to compress the text, does in fact nothing to help the reader better understand what is being said but rather looks like a crazy mess of characters and other marks in a multitude of fonts, styles and sizes the only purpose of which seems to be to cause an eye strain.

Re: Terry Tao on some desirable properties of mathematical notation

#50

Post by foobar_ » Wed, Jul 22, 2020, 2:18 AM UTC

No one uses mathematical notation for practical purposes. This is just like the medival music notation which is neither practical nor what modern composers use, which is more visual in nature. Infact modernism is a rejection of medievalism.

I think in the future programming will force all mathematicians to code or give out simulations. Most mathematical notation was intended to be throwaway by the original authors, thats why there are so many notations. Trying to find relevance in them is a pointless exercise. Much like 80x20, tabs vs spaces ... most of the original intent is lost and what survives is guff meant for ceremonious purposes.