Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Terry Tao on some desirable properties of mathematical notation

mathoverflow.net

Post reply on HN
Print view View original

121–130 of 229 posts

Re: Terry Tao on some desirable properties of mathematical notation

#121

Post by quietbritishjim »

auggierose wrote: post #119
quietbritishjim wrote: post #116

Earlier quoted context omitted.

Do you really think > speed = distance / time is less clear than > speed is the ratio of distance over time ? To me the first equation is genuinely easier to read for the purposes of understanding, not just for formal manipulation. For larger equations the difference is only more stark, not less. I have a maths PhD so your comment about symbols being for novices doesn't apply. I suppose the beauty of the first equati…

The problem is when it is not quite clear what the symbolic notation stands for. With division, that tends to be less of a problem.

* I find it quite unusual in practice for genuinely new symbolic notation to be used by an author. Maybe that just reflects the fields I read about most (information theory, Bayesian modelling, harmonic analysis).

* Usually you don't come across a journal article or even blog post with a single isolated equation. So any new or unusual notation can be explained once and reused many times.

* Even if you did have an isolated equation with unusual notation, I still think it's more clear to define the notation and then show the equation rather than spelling it out it words! (I'm sure you could find terrible counterexamples!) The visual benefit seems so great to me that it would be worth splitting it into two parts. A bit like a diagram or a graph can make things clearer even if it needs a bit of explanation.

Re: Terry Tao on some desirable properties of mathematical notation

#122

Post by Darkstryder »

Steal this idea: a Shazam of mathematical notation. In an app you would draw (or take a picture) of a mathematical symbol you don’t recognize and get a link to the appropriate Wikipedia page.

My biggest pet peeve with mathematical symbols is the difficulty of looking them up when you don’t know them already. If I’m reading a text on a topic I'm unfamiliar with, I can at least google the keywords I don't know. This is difficult with symbols.

Re: Terry Tao on some desirable properties of mathematical notation

#123

Post by jpm48 »

Darkstryder wrote: post #122

Steal this idea: a Shazam of mathematical notation. In an app you would draw (or take a picture) of a mathematical symbol you don’t recognize and get a link to the appropriate Wikipedia page. My biggest pet peeve with mathematical symbols is the difficulty of looking them up when you don’t know them already. If I’m reading a text on a topic I'm unfamiliar with, I can at least google the keywords I don't know. This is…

you could try something like https://mathpix.com/ and then look up what the LaTeX symbol is.

Re: Terry Tao on some desirable properties of mathematical notation

#124

Post by OscarCunningham »

tsimionescu wrote: post #117
rytill wrote: post #79

Earlier quoted context omitted.

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…

There are some cases where this is true (the most infamous being Roman numerals, where the notation makes even simple arithmetic difficult - XI * VI = LXVI???), but in general notation is just something you learn once and mostly remember, as long as you understand the concepts. However, I fully agree that reading a text that uses notation you are not familiar with without introducing it is almost futile. Even the sim…

Incidentally, the example of roman numeral multiplication you chose is actually fairly easy. Just break it up as (XI×V) + (XI×I). Then since X×V=L and I×V=V (Romans would have these memorised just like we memorise multiplication tables for all the decimal digits) so the first multiplication is LV and the second is XI, which add to give LXVI by simply interleaving the letters.

Re: Terry Tao on some desirable properties of mathematical notation

#125

Post by cdu1 »

Darkstryder wrote: post #122

Steal this idea: a Shazam of mathematical notation. In an app you would draw (or take a picture) of a mathematical symbol you don’t recognize and get a link to the appropriate Wikipedia page. My biggest pet peeve with mathematical symbols is the difficulty of looking them up when you don’t know them already. If I’m reading a text on a topic I'm unfamiliar with, I can at least google the keywords I don't know. This is…

Great idea. I'm a fan of this app/site for finding the Latex command for a particular symbol:

https://detexify.kirelabs.org/classify.html.

Then just need to look up the Latex on Wikipedia:

https://en.wikipedia.org/wiki/List_of_mathematical_symbols

Re: Terry Tao on some desirable properties of mathematical notation

#126

Post by jvvw »

Smaug123 wrote: post #107
logicslave wrote: post #98

Earlier quoted context omitted.

"does in fact nothing to help the reader better understand what is being said" For a mathematician it is the opposite, they just wish it would be written with symbols so that they could know precisely what the book is trying to say

I don't think this is true in general. It may be true for a novice, who needs all the available help to keep them rigorous (but even then, there is definitely room for reading-to-build-intuition), but symbols definitely slow you down while you translate them.

I've got a PhD in maths and can translate symbols into concepts in my brain much quicker than words into concepts in general. Also writing symbolically forces a rigour on the writer. I've been trying to read some semi-mathematical stuff written by scientists but non-mathematicians recently and it is painful trying to figure out what they really mean!

Re: Terry Tao on some desirable properties of mathematical notation

#127

Post by foerbert »

prox wrote: post #118
nsonha wrote: post #101

Earlier quoted context omitted.

Compressing does help it. To trained eyes a couple of lines is easier to read and less ambiguous than one or 2 pages of English explanation, which is hard to follow for both trained and untrained eyes if you're gonna encouter a bunch of them while reading a proof.

To me it seems language can also be interpreted as more ambiguous, where as well defined symbols and notation are far more rigorous in definition and use.

Except you have instances of misuse of those well-defined symbols and notations. Without any accompanying language it can be very difficult to not only determine that there is an error, but then also figure out how to fix it.

And other times in some fields there is a lack of well-defined symbols and notations. I don't recall the specifics, but in game theory there is some notation that is ambiguous. There are two methods or forms of representation, and the field is split on which version the notation stands for. It's also frequently not mentioned explicitly in papers, so you have to dig around a bit to see if you can find a usage that makes the meaning clear.

I don't know why we can't just have both.

Re: Terry Tao on some desirable properties of mathematical notation

#128

Post by jjgreen »

foobar_ wrote: post #65
wheresmycraisin wrote: post #62

Earlier quoted context omitted.

Ok, then convince me. Write 'software' of, say, the proof of the the dominated convergence theorem or something else reasonably advanced and let's compare it to the proof in conventional math notation.

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.

See Principia Mathematica, A. N. Whitehead and B. Russell, Proposition 110.643

Re: Terry Tao on some desirable properties of mathematical notation

#129

Post by quietbritishjim »

datastoat wrote: post #111
dopu wrote: post #91

Is it just me, or does probability theory in general have fairly terrible notation? Ambiguity between random variables and their distributions because of them simply being distinguished by being upper-case or lower-case, writing likelihood functions alternatively with an L() or p(), and using p() (with different arguments) to refer to different probability distributions. Perhaps I'm just having such a difficult time…

Probability notation used in ML and engineering has this problem, of overloading p(). Probability notation as used by probabilists in maths departments is completely different: it’s more explicit, and sometimes more clunky. There’s a hybrid notation that I prefer, for example “Pr_X(x)” for the density function of random variable X at point x; you drop X if the random variable is clear from the context, and you drop x…

I used to hate the way Bayesian ML people used p(...), until I realised that strictly speaking for a conditional variable we ought to be writing: p_X|Y=y(x). The variable is X|Y=y so all that ought to be in the subscript.

It's definitely worthwhile everyone using the full notation at least once so they can get a feel for what's really going on. I've spoken to Bayesian ML professionals who are especially unconfortable with that because it conditions on a zero-probability event (if Y is continuous)... of course p(x|y) does too, they just weren't thinking about it before! And (as I think you're getting at) the appreviated p(x|y) simply throws away information e.g. there's no way to represent the identity p_Y(x)=p_X(x) without adding back some sort of subscript.

But on the other hand p(x|y) is obviously much visually cleaner. If you're writing out a more complex identity and the abbreviated notation isn't ambiguous then it generally communicates the idea much more clearly because there's so much less visual noise.

Re: Terry Tao on some desirable properties of mathematical notation

#130

Post by rini17 »

xelxebar wrote: post #53
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…

The frustration is inarguably real, but... Programming language syntax sucks because none of it maps to things that non-programmers know. Every time a new token is introduced, whether it be an ascii symbol or a word it's one more mapping your brain has to create to remember it. And on top of this you have to remember how these actually execute too. Yes I said execute - most programming concepts have so many implement…

Except it's much easier to look up definition and examples for these tokens, without going so deep into the rabbit-hole.
Post reply on HN