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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

#101

Post by nsonha »

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.

Re: Terry Tao on some desirable properties of mathematical notation

#102

Post by judofyr »

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…

> 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.

Are you saying that it was the notation alone which caused you to completely stop reading the book? I can't see that I share that experience. There are certainly times where I don't fully understand a proof or a lemma due to uncertainty about the notation, but that's rarely a _blocker_. I just skip it, continue reading and come back to it later.

I agree with the GP here: The main thing which blocks _me_ on a book is the subject itself and how the author presents it. The problem is rarely the notion by itself, but the fact that the author uses concepts which I don't know about.

EDIT: It should also be pointed out that the GP didn't say that notation was _never_ a problem, only that it's a much _smaller_ problem than "those presented with the subject itself". The big question about your experience would be if you would have been unblocked if the notation was explained in English, or if it would just uncover that it uses concepts which haven't been fully explained to you.

Re: Terry Tao on some desirable properties of mathematical notation

#103

Post by simion314 »

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.

>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.

Imagine you would write the code on paper by hand and have to repeat some long var or function name all the time . Even with IDEs I see programming languages using shorter keywords like "var","const","int" then the full word.

Because you are agains single letter symbols let me know what are your suggestions for PI, or X(in polynomials or equations) and show me a screenshot on how some real world physics equation solving would look like with camel case named symbols.

Re: Terry Tao on some desirable properties of mathematical notation

#104

Post by steventhedev »

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…

I think this is partially because it's applicable to so many different domains (cryptography, statistics, etc) that each have their own notational quirks. Avoiding collisions between notation in the application domain is more important than preserving some "consistent" probability notation (similar to the examples given with dot products). But the issue isn't even limited to notation: Chebyshev has at least 9 valid ways to spell his name (more if you include non-latin alphabets).

The biggest issue I've encountered is borrowing lecture slides from different universities/lecturers and the notation changes between slides on the same topic (even small things like using square brackets instead of parentheses).

Re: Terry Tao on some desirable properties of mathematical notation

#105

Post by diffeomorphism »

chongli wrote: post #81
Koshkin wrote: post #60

Earlier quoted context omitted.

I would argue that in fact most of mathematical notation happens to be intelligible to non-mathematicians (engineers, scientists, etc.)

I would disagree. Engineers, scientists, etc. have their own notation conventions as well. They may be aware of mathematical notation but only in specific areas of math related to their field (i.e. calculus and linear algebra for Newtonian physics). I would not expect a mechanical engineer to understand the notation used by graph theorists or algebraic geometers.

Neither would I expect this of mathematicians. If you never use graph theory there is no reason to know all the quirks of the domain-specific notation. This would be like expecting any programmer to be familiar with all programming languages.

Re: Terry Tao on some desirable properties of mathematical notation

#106

Post by diffeomorphism »

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…

> I've met way more people interesting in learning math than performing it, so we try to make notation as clear and possible.

That is like trying to learn an instrument by looking at it but never touching it. Notation like other tools is made to useful. Also if you never played an instrument you would think they are each their own little completely different world. That's totally wrong for ya.

Re: Terry Tao on some desirable properties of mathematical notation

#107

Post by Smaug123 »

logicslave wrote: post #98
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…

"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.

Re: Terry Tao on some desirable properties of mathematical notation

#108

Post by Smaug123 »

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.

Longhand matrix notation itself is quite good at this. It all but guarantees that you've specified the right number of components in the matrix, and it plays to human strengths in making it easy to check that a matrix is e.g. diagonal or upper-triangular.

I'm going to claim that string diagrams have reasonably good error-detection properties, too, again because they lay facts out in space in a way that humans are quite well optimised for.

Re: Terry Tao on some desirable properties of mathematical notation

#109

Post by Smaug123 »

nl wrote: post #52
chongli wrote: post #44

Earlier quoted context omitted.

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…

To be honest I think my (and most people on this thread) real complaint isn't with notation per-say (as in Euclidean vs Hilbert notation etc, as discussed by Tao). It's really about the lazy habits of many who work at the intersection of math and computer science, and use maths to express themselves without defining things. > The other problem with academic papers is that the authors generally don't care about reprod…

(Irrelevant nit: the phrase is "per se", meaning "by virtue of itself" in Latin.)

Re: Terry Tao on some desirable properties of mathematical notation

#110

Post by nl »

Smaug123 wrote: post #109
nl wrote: post #52

Earlier quoted context omitted.

To be honest I think my (and most people on this thread) real complaint isn't with notation per-say (as in Euclidean vs Hilbert notation etc, as discussed by Tao). It's really about the lazy habits of many who work at the intersection of math and computer science, and use maths to express themselves without defining things. > The other problem with academic papers is that the authors generally don't care about reprod…

(Irrelevant nit: the phrase is "per se", meaning "by virtue of itself" in Latin.)

Yes you are entirely right, and it's outside my edit window to fix.
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