
Towards a Better Notation for Mathematics (2010) [pdf] - stared
https://christopherolah.files.wordpress.com/2013/03/mathnotation-chrisolah.pdf
======
jhanschoo
As the author admits in his update, this document is not very interesting.
When I saw the title I was mainly interested in perhaps a formalism of
mathematical exposition and proof notation.

The problem with the author's proposals is this: he proposes replacing well-
known, well-distinguished symbols (alphabets and digits) with symbols that
reflect a certain geometric interpretation.

First, the geometric symbols are difficult to read, and a small error (either
in printing or in reading) can mean a huge change in meaning.

Second, the obvious criticism that they are too radical a departure from the
current convention for them to be widely adopted.

Most salient, however, is that they offer no benefit to the development of
mathematics. The mathematical enterprise is based on abstraction,
generalization, and symbolic manipulation. The symbols proposed reflect only
the most banal representation of the mathematical objects they discuss.

For example, he discusses representing the natural numbers with a number line.
But this is an elementary schooler's notion of natural numbers. Mathematicians
would like to discuss other representations and constructions of the natural
numbers, say, their Peano axiomatization, as Church numerals, as an algebraic
structure, as indices, as a countable set, and many more. Hence it is honestly
more convenient, and less intrusive to the discussion of novel mathematical
ideas and perspectives, to simply use a flavorless symbol of the alphabet,
rather than one that is associated with a particular geometric interpretation.

Finally, I'd like to point out that while many things have changed in
mathematics since ancient times, we still follow Aristotle in using letters to
denote mathematical objects, and Euclid's rigorous exposition and proof, and
actually use them today with far more frequency than they ever did.

~~~
colah3
Hi, I'm the author. Please keep in mind that I wrote this when I was 16.

Thanks for your comments -- I agree with many of them. The only part of this
that I still think is at all interesting is the section on quantifiers, and to
a lesser extent the octal numbers. And I don't think they're that interesting.
In general, while I do think there is interesting potential for improving
notation -- Feynman diagrams come to mind as a somewhat modern example of
notation impacting research -- I don't think this is what it looks like.

> he proposes replacing well-known, well-distinguished symbols (alphabets and
> digits) with symbols that reflect a certain geometric interpretation.

That's what most of it is, although the quantifiers are a more "grammatical"
change. I think the best defense I can level of the symbol proposals is that
the people most effected by low-level notational choices like symbols are
probably young students learning them for the first time. It seems plausible
there might be significant pedagogical gains in other symbols.

~~~
gizmo686
Math major here. I too came out of your paper thinking that the quantifiers
are the only interesting part of you paper; interesting enough, in fact, that
I may start using them myself. One criticism is that there is no way to use
them for variables that occur in more than one place. That is, you could not
use them to right "for all x in R, x<x+1". I propose a simple addition of an
optional subscript for the variable name in the quantifier.

My other criticism is that it seems less natural to read that notation. With
the standard notation, you can simply replace the symbol with English words
and read it. While the transliteration in your notation is likely possible, it
seems more complicated. In this sense, I would probably view this notation
more as a shorthand. For pedagogical purposes, I feel like separating the
qualifier from the usage of the variable is a good idea.

~~~
giomasce
It is not the only problem, unfortunately. For example, there is no way to
specify the order of quantifiers, which is very important (if you commute
existiential and universal quantifiers, in general you change the meaning of a
formula). So, for example, the definitions of pointwise and uniform continuity
would we written identically, and that is not tolerable.

Probably you could solve also this by adding another index to quantified
quantities, but at that point are you really solving the problem of
complexity? Complexity is not only measured in terms of verbosity (i.e.,
length of the expressions you produce); ability to quickly locate information
and ease of manipulation are also important, and it seems to me that this
formalism fails completely. In the "traditional" formalism you separate the
"final fact" you want to assert from its "conditions", given by the
quantifiers; usually, you want to process those pieces of information at
different times. Also, you have rules for quantifier introduction and
elimination, which seems to be a nightmare with the proposed formalism.

That said, I do encourage new proposals and experimentation with notations, as
with anything else in mathematics, even from young students. There have been
cases in maths in which some good idea for a new notation made an entire field
much easier (I am thinking for example to Einstein notation for tensor
calculus).

------
gabrielgoh
A lot of the problems with mathematical notation, I think, are solved problems
in programming languages. Mathematicians could learn a lot from programmers in
this regard

\- Variable Scope. In mathematical writing, the convention is that every
variable is a global variable - and in a paper you quicky run out of variable
names to use. There should be a way to declare the scope of a variable. For
this chapter only, x is this vector.

\- Strict Typing. You skim a paper, and you see "x + y". But what is x? a
matrix, a vector, a scalar, an operator, a set, a random variable, you have no
idea. The "+" operator is so overloaded x and y could be anything. Having to
hunt down the definitions for every variable is a time consuming and frankly
wasteful process. Conventions are built up around this (upper case letters A,B
at the beginning of the alphabet are matrices, but upper case letters at the
end X,Y are random variables) but they are mostly rater silly. There needs to
be an easy way to infer the type of a variable quickly.

\- Anonymous Objects. There is no way to declare a function without either
giving a name to it (f(x) = x^2) or at least declaring its input symbols (x
|=> x^2). But you can't go, say (x |=> x^2)(y). Using a symbol to represent a
disposable function seems a waste of a valuable global variable (see again, 1)
and is inelegant.

~~~
vlasev
> Variable scope

This goes against the very nature of what mathematical variables are like! The
whole point is that the symbols are suggestive. This kind of thing allows the
reader to quickly get up to speed with what the symbols are all about. As a
side-effect, an unconventional use of symbols is usually quite grating to the
experienced mathematician. Usually:

\- x, y, z stand for variables

\- i, j, k stand for subscripts and superscripts

\- m, n stand for integers

\- theta, phi stand for angles

\- alpha, beta stand for real numbers of some sort

\- capital letters are usually sets, transformations and such

> Strict typing

Again. Mathematics is about abstraction. Here "+" means that you can add the
two objects, whatever they are. That's all there is to it. Of course, they are
mentally typecast to the more general type. The point of this overloading is
that, again, its suggestive nature makes it easier to read and write. The
abuse is so pervasive that something like 1 + A can easily mean that, yes, 1
is the identity matrix.

> Anonymous Objects

I agree. Although it's much easier to, once and for all, declare "Let f(x) =
x^2" and use "f" instead of the anonymous function every time.

All that this kind of thing highlights is, IMHO, the fundamental difference
between being a mathematician and being a software developer (and I've said it
before if you look through my comment history):

\- Mathematicians manipulate a given symbol, often in the thousands of times,
in their heads or on paper \- Software developers read a given line of code,
often in the thousands of times, in order to understand what the code does

These differences drive, IMHO, 99% of the difference in notation.

~~~
dmurray
> As a side-effect, an unconventional use of symbols is usually quite grating
> to the experienced mathematician.

There's a great spoof paper that demonstrates this by giving each variable the
next available letter of the alphabet. Something like "Let a be a set of
points in the plane. Let b, c and d be elements of a. Let e be the angle
formed by..."

It becomes unreadable very quickly. Sadly I can't find the original now -
anyone know what I'm talking about?

------
skadamat
Some great initial ideas here, but the focus still seems to be on notation for
pencil and paper. This is the medium of the past and it's worth thinking about
how mathematical ideas should be represented in a computing medium. Some
potential ideas here:

\- [http://geometry.mrao.cam.ac.uk/](http://geometry.mrao.cam.ac.uk/) \-
[http://worrydream.com/KillMath/](http://worrydream.com/KillMath/)

------
aqsalose
All the trigonometry symbols look too much alike. (Small round things with
small scribbles of size of the ant's leg attached to them).

Likewise, base-8 numerals look fun. However, one benefit of the regular
numbers is that they look both pleasant and distinctive from each other in the
flow of the regular text. Both in print and also if written by hand (at least
if you add the extra horizontal bar to the 7).

Actually, if one were to adopt all of the proposed notation, it appears that
one might lost in the forest of slightly different combinations of arrows that
mean different things.

The quantifier notation is the only thing I actually might like. (It's also
backwards-compatible.)

------
stared
From "Surely You're Joking, Mr. Feynman!":

"While I was doing all this trigonometry, I didn't like the symbols for sine,
cosine, tangent, and so on. To me, "sin f" looked like s times i times n times
f! So I invented another symbol, like a square root sign, that was a sigma
with a long arm sticking out of it, and I put the f underneath. For the
tangent it was a tau with the top of the tau extended, and for the cosine I
made a kind of gamma, but it looked a little bit like the square root sign.
Now the inverse sine was the same sigma, but left -to-right reflected so that
it started with the horizontal line with the value underneath, and then the
sigma. That was the inverse sine, NOT sink f--that was crazy! They had that in
books! To me, sin_i meant i/sine, the reciprocal. So my symbols were better."

And some example here:
[https://tex.stackexchange.com/questions/274463/feynman-
trig-...](https://tex.stackexchange.com/questions/274463/feynman-trig-
notation-creating-custom-characters).

------
Houshalter
My favorite new math notation is the triangle of power:
[https://www.youtube.com/watch?v=sULa9Lc4pck](https://www.youtube.com/watch?v=sULa9Lc4pck)

I slightly modified it to avoid subscript and keep it closer to existing
notation for powers:
[http://i.imgur.com/hOz9MXa.jpg](http://i.imgur.com/hOz9MXa.jpg)

------
CharlesMerriam2
I think the author misses the basis of innovation of notation: mathematicians
are lazy.

A simple exercise, often used to teach children, is to start without notation.
Do exercises like these (translated from Hungarian way of saying numerals):

Write the questions the teacher speaks and give answers: "Two tens three plus
one ten six equals?" Three tens nine. "Seven tens one plus two equals?" Seven
tens three. ... Do about five of these. Children will ask if they can write
less. Do a few more with comparisons ("is greater than or equal to"),
simplifying equations ("is equivalent to"), lines of equations ("has the same
solutions as") and children understand.

These leads to fixing notation that causes errors: "6x3" can read like "6x
times 3". Poor handwriting confuses "6' with "8". Pi is usually a less
interesting number than 2*pi. Kinematics use all four corners of a letter,
e.g., "joint x, in frame r, at time t, ..."

There is space for better notation, or at least an appreciation of how it ends
up where it does.

------
danieltillett
This might be well known, but is there are good resource for knowing what the
various mathematical symbols mean? I only got as far as first year uni maths
and I find that when I look at maths papers that are interesting for a problem
I want to solve I can't make much progress on working out what all the symbols
mean.

Almost all the time once I see the code that implements the mathematics then
it is easy to understand, but making sense of all the different symbols is
very hard especially when different fields seem to use the same symbols to
meant totally different things.

~~~
giomasce
> especially when different fields seem to use the same symbols to meant
> totally different things

That is the problem: there are by far too few letters and symbols to describe
all the objects used in modern mathematics. You need to reuse symbols, and
therefore any paper must declare the symbols they use. Sometimes they are even
forced to use the same symbol for different things, so you need to have some
experience in the field to understand which is the correct reading each time.

The symbol declaration can be done explicitly (by stating the fact, usually at
the beginning or in a dedicated section; some books also have a table of
notations with references to definitions) or implicitly, when the paper is
targeted to scholars that are most probably already familiar with that
notation. The latter is probably not very friendly to newcomers, but in most
cases if you do not know what implicitly defined objects are, then you
probably cannot understand the paper anyway.

In short, there are very few "short ways" into mathematics. If you want to
understand things, you have to spend time and study them.

~~~
danieltillett
It is a huge job to bring my mathematics up to the required level :)

About the only way I have been able to short-circut the process is outsource
to someone who can turn the paper into code - once they do this the maths
becomes easy for me to understand.

------
honestoHeminway
A better Notation is a digital one, which can be zoomed by clicking upon a
component. Click on the Summation Symbol to get to the Wikipedia Article for
the Summation.
[https://en.wikipedia.org/wiki/Summation](https://en.wikipedia.org/wiki/Summation)
Click on a worked upon item to see its origin and a history of access and
usage within the formuala/proof.

TL,DR; A better Mathematic Notation is a VS for math.

------
suyash
This is a genius paper. We should think about evolving the language of
mathematics just like other languages (programming, english etc) evolve to
become more friendly and empowering. I love the idea behind it, and of course
the proposal has many points to be debated as to how and what but the core
idea is genius.

------
paulpauper
replacing cosines and sines with dials makes typesetting a nightmare. imho I
don't see this as being helpful at all.

~~~
messe
Not only that, but at a glance they're near identical.

------
throwaway7645
Interesting thought, but Kenneth Iverson got the Turing Award for APL which
was a programming language implementation for his new mathematical notation
system. His paper "Notation as a Tool of Thought" is quite interesting. You
can take a look at any APL code and see it in action. Cool stuff.

------
al2o3cr
The part on quantifiers reminds me of point-free notation in Haskell. Not in a
_good_ way, either...

------
quantum_state
The title is a bit misleading though enticing ... IMO, it is not an even a
dead horse ...

------
mbid
> Towards a Better Notation for Mathematics _for Engineers_

