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

Re: Terry Tao on some desirable properties of mathematical notation

#71

Post by tmpz22 » Wed, Jul 22, 2020, 3:09 AM UTC

wtallis wrote: post #58

tmpz22 wrote: post #16

Earlier quoted context omitted.

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 rea…

In many university mathematics curriculums, discrete math is the course used to transition students from the computation-oriented mindset (instilled in them by high school and early undergraduate courses up through calculus) to the more abstract and proof-oriented mindset. Thus, teaching you to use that "really really advanced level math notation" is often one of the primary goals of the course, even if it seems like…

We were never taught the notations, he would just pull out dozens of glyphs from his vast experience in mathematics - not the standard notations you might find in a text book. I won't deny that I could've researched these topics deeply in my own time enough to keep up with the graduate student, but I myself had no interest in being a graduate math student just to pass the course.

Re: Terry Tao on some desirable properties of mathematical notation

#72

Post by foobar_ » Wed, Jul 22, 2020, 3:10 AM UTC

wheresmycraisin wrote: post #67

foobar_ wrote: post #65

Earlier quoted context omitted.

I'm guessing there was a physical intuition behind the theorem, if you can simulate it you will probably do something better than the proof. Now it's your turn to tell me why 1 + 1 = 2.

Do you realize that you sound like you have a second-grade math education with your example? Also, do you realize simulation is not feasible for more than a few dimensions, if that?

I guess you missed the sarcasm. If you want me to s-p-e-l-l it out - maths is convention.

Re: Terry Tao on some desirable properties of mathematical notation

#73

Post by [deleted] » Wed, Jul 22, 2020, 3:16 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…

[deleted]

Re: Terry Tao on some desirable properties of mathematical notation

#74

Post by ramblenode » Wed, Jul 22, 2020, 3:17 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 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.

Re: Terry Tao on some desirable properties of mathematical notation

#75

Post by dhash » Wed, Jul 22, 2020, 3:18 AM UTC

moonchild wrote: post #31

Another interesting notation is iverson notation. See Notation as a Tool of Thought [1]. Here's the inner product (note that this is actually general inner product): c ≡ u +.× v 1. https://www.jsoftware.com/papers/tot.htm

Iverson notation originated from APL, which was itself born out of the horror that Iverson saw when presented with "standard mathematical notation". Its use of strange, otherwise cryptic notation was influenced by two main themes: it was originally designed on a blackboard, so strange glyphs were normal, and a desire to become unmoored from "standard mathematical notation" in order to strengthen principle 1. of OP - Unambiguity. Casting off historic baggage and canonicalizing mathematical notation under the principles of OP was APL's prime goal, and it does a damn good job of it.

I wish they taught it to math majors.

APL's a wonderful rabbit hole to fall down, and J was my eso-lang of choice last year.

Re: Terry Tao on some desirable properties of mathematical notation

#76

Post by lmkg » Wed, Jul 22, 2020, 3:20 AM UTC

btrettel wrote: post #61

Terry Tao mentions that notation can help with error detection. Anyone here aware of some good examples? One that I like is that in Einstein notation you can't have 3 of the same index, e.g., u_i u_i u_i is invalid.

From the example notations that he gives, all three Einstein notations as well as the Penrose notation make the indices explicit in a way where a mismatch or misalignment will stand out.

Another good example is the Liebnitz notation for derivatives. Proper application of the chain rule visually resembles how fractions cancel: dy/dz dz/dx = dy/dx. It's very easy for the eye to follow and make sure that the cancellation is valid. Newton's notation doesn't make that as easy.

Re: Terry Tao on some desirable properties of mathematical notation

#77

Post by dragonwriter » Wed, Jul 22, 2020, 3:21 AM UTC

vii wrote: post #69

Enumerating what we want from notation helps us understand how far we are from the ideal. The whimsical introduction of Notation to talk about notation makes it practical. Given a domain in mathematics, adding notation (e.g. modulo arithmetic) can make complex notions pretty to express and quick to prove. I used to really enjoy this and tried to redefine notation for each exposition. It's shorter and prettier, but ju…

> domain specific languages, while much more efficient are unpopular

JSX seems pretty popular, and when XML was popular similar XML embeddings were, as well. Templating languages are popular. Heck, the relative popularity of “general purpose” programming languages is not consistent across domains, with domain fit being a factor even for general purpose languages.

Re: Terry Tao on some desirable properties of mathematical notation

#78

Post by lmm » Wed, Jul 22, 2020, 3:26 AM UTC

dragonwriter wrote: post #77

vii wrote: post #69

Enumerating what we want from notation helps us understand how far we are from the ideal. The whimsical introduction of Notation to talk about notation makes it practical. Given a domain in mathematics, adding notation (e.g. modulo arithmetic) can make complex notions pretty to express and quick to prove. I used to really enjoy this and tried to redefine notation for each exposition. It's shorter and prettier, but ju…

> domain specific languages, while much more efficient are unpopular JSX seems pretty popular, and when XML was popular similar XML embeddings were, as well. Templating languages are popular. Heck, the relative popularity of “general purpose” programming languages is not consistent across domains, with domain fit being a factor even for general purpose languages.

> JSX seems pretty popular, and when XML was popular similar XML embeddings were, as well. Templating languages are popular.

I don't think any of these things are popular; in my experience people using them mostly hate them, or at best grudgingly accept that they're the least-bad option.

Re: Terry Tao on some desirable properties of mathematical notation

#79

Post by rytill » Wed, Jul 22, 2020, 3:26 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.)

Not true at all, there are several times I've attempted to read through a textbook only to be stopped by notation because something was introduced prior to being referenced, or notation is overloaded with multiple meanings.

I consistently have run into "perceived or real" confusing mathematical notation as an impediment to learning in a way that programming languages have never, ever caused me.

Does no one else feel this way? I can't be alone, and like the others responding to you have said, your claim does not seem substantiated.

Re: Terry Tao on some desirable properties of mathematical notation

#80

Post by dhosek » Wed, Jul 22, 2020, 3:33 AM UTC

Koshkin wrote: post #49

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 nothin…

I remember, as a callow college freshman waiting in the hallway of the math department to be able to go into a classroom reading an article which talked about mathematical writing and the first thing it said was to prefer English text over mathematical symbols in numerous cases (e.g., writing "For all $x$ in the Reals" over $\forall x\in\mathbb{R}$). As someone who was highly skilled in TeX at a time when such skills were still fairly rare (many colleges, if they even had TeX available on their time share systems, were still using the am fonts instead of the cm fonts as the latter had only been introduced two years earlier), I remember being upset that my backslash skills were thus denigrated.