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Terry Tao on some desirable properties of mathematical notation

mathoverflow.net

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Re: Terry Tao on some desirable properties of mathematical notation

#111

Post by datastoat »

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 if you’re referring to the entire distribution. Or Pr_X(x|Y=y) for a conditional density. But this notation still has problems when you’re working with hairier conditional distributions, or with distributions that are neither discrete nor continuous.

(Source: used to be a mathematical probabilistic, now working in ML.)

Re: Terry Tao on some desirable properties of mathematical notation

#112

Post by diffeomorphism »

zamfi wrote: post #19
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.

> 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.

Go back a few centuries and you get long form Latin instead. Or go back much further and you will find the quadratic formula described as taking the number that the second term is increased by, diminishing it by half and then.... . Basically an essay just to say -q/2.

Sure, you could write

-CoefficientInFrontOfXToTheOne/2

But then each row of your calculation will need line breaks and will be hard to understand. It will take forever to write and it will take forever to read. No thank you.

Re: Terry Tao on some desirable properties of mathematical notation

#113

Post by Smaug123 »

foobar_ wrote: post #72
wheresmycraisin wrote: post #67

Earlier quoted context omitted.

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.

(I missed the sarcasm, and honestly I still can't see it after you've pointed it out - and I'm British, supposedly a native expert!)

Re: Terry Tao on some desirable properties of mathematical notation

#114

Post by moonchild »

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

Note: depending on context, you might want to replace ≡ with ← or ←→.

Re: Terry Tao on some desirable properties of mathematical notation

#115

Post by rocqua »

zamfi wrote: post #90
ianhorn wrote: post #36

Earlier quoted context omitted.

If your job consisted of calculating with sequences of changes of programs, with no copy paste available, you'd probably feel differently. For example, a fairly roundabout derivation of a change of base for logarithms (pretending we forget log(a^x) = x log(a) for arbitrary base): We're trying to derive that a^x = b^(x * log_b(a)). Or in verbose descriptive terms, oldBase `exponentiate` oldExponent = newBase `exponent…

Every domain — every language — has its basic jargon, and algebra is no exception. I’m not suggesting that mathematicians simply replace every symbol with a word that hints at meaning; that’s too literal an interpretation of what I wrote. “exponentiate” is no more descriptive than “^”. But there are alternative ways of describing that derivation that are not as symbol-manipulation heavy; you would certainly not commu…

I think most people who wanted to communicate this proof in person would say 'let's find a whiteboard, or do you have some paper'? And then proceed to write down the first version of the proof.

Because it is better communicated in notation than in words.

Re: Terry Tao on some desirable properties of mathematical notation

#116

Post by quietbritishjim »

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.

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 equation is that the objects (speed, distance, time) are visually very distinct from the operations and relationships. In the second form, there's a bit of a word soup so you need to "manually" parse the sentence rather than letting your eyes (really the visual cortex) do that bit of the processing for you.

Re: Terry Tao on some desirable properties of mathematical notation

#117

Post by tsimionescu »

rytill wrote: post #79
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…

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 simple fact of not knowing the names of the operators, which makes it impossible to read the formulas cursively in you mind (for example, when you encounter 'a + b' you can read it in your mind as 'a plus b', but for something like 'a b', even if given `a b = 2*a+b`, reading the text becomes a mental chore).

Re: Terry Tao on some desirable properties of mathematical notation

#118

Post by prox »

nsonha wrote: post #101
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…

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.

Re: Terry Tao on some desirable properties of mathematical notation

#119

Post by auggierose »

quietbritishjim wrote: post #116
Smaug123 wrote: post #107

Earlier quoted context omitted.

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.

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.

Re: Terry Tao on some desirable properties of mathematical notation

#120

Post by carlob »

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'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.

[0] http://reference.wolfram.com/language/Notation/guide/Notatio...

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