notation | Christopher Olah's Blog

Posts Tagged ‘notation’

Math Notation Experiment “Paper”

March 20, 2013

A while back, I wrote a post on some unusual math notation I was playing with. I actually took it much further than that and drafted up a paper full of different ideas over the following months. At the time, I was hoping it might be publishable but then got discouraged and never did anything with it… Except show it to people in person and promise I’d put it online soon. In any event, it’s one of those things I’ve been meaning to put up for forever.

So here it is. I’m particularly proud of the quantifier stuff and the numerals.

When one considers how complicated the ideas mathematical notation must represent are, it clearly does quite a good job. Despite this, the author believes that any claim that modern mathematical notation is the best should be met with extreme scepticism.

Mathematical notation is a natural language: no one sat down and constructed it, but rather it formed gradually by people making changes that are adopted. Most changes are not adopted; whether they are depends on a variety of factors including: mathematical utility, ease of adoption (the average individual doesn’t want to spend hours learning), dissemination, and shear dumb luck. The first two of these qualities are associated to real properties in notation, forming the necessary selective pressure for evolution to occur.

Evolution is a blind watchmaker: the world around us is filled with examples of the stupidity of biological evolution (the classic example being halibut). Similarly, evolution is also a blind language and notation designer. In particular, it is held back by a strong selective force against change, since people would need to adopt it, and so evolution doesn’t effectively explore the full notation-space. This staticism means that even the most outrageous notations remain unchallenged by virtue of age.

My Math Notation

February 9, 2010

Over the last year I’ve begun to diverge quite heavily from standard Mathematical notation. My experience in CS is compelling me to document this API notation.

Purpose

While real Mathematicians have been surprisingly supportive of me inventing my own notation, I feel compelled to justify it. My thoughts can be summarised in the syllogism:

Some notations allow certain thoughts to be expressed more easily than other notations.
The ease with which thoughts can be expressed changes the thoughts that are expressed.
Therefore notation can change thought.

Given this, I wish to make my notation capable of more conveniently expressing otherwise awkward thoughts.

Notation 1: $\{set\}$

Very early on this year, I got tired of writing: “For an open set $u$ ,” and “Let $s$ be a compact set.”

I was also exposed to the notation of writing the `set (class) of sets’ as $\{set\}$ . So I decided to use this notation for expressing any set. For example: $u \in\{op en\}$ and $s\in\{compact\}$ .

More recently, I’ve been treating them as functions that returns sets to express more complicated ideas. For example, I’d replace “let $p$ be an accumulation point of the set $s$ on the topology $T$ ” with $p\in\{accumulation\}(s,T)$ .

Great for notes. Even outside math.

Notation 2: $S^\forall, ~~ S^\exists$

I recently got sick and tired of writing long chains of for alls and exists. I noticed that there were some very common patterns in there use. A typical example would be:

$(\forall a \in A)(\forall b\in B)(\exists c\in C)(a+b=c)$

We’re just saying that for any member of this set, or for some member of that set… Why can’t we just have an operator that acts on the set to do this for us?

I don’t see any reason. $S^\forall$ : Place holder for all members of $S$ . $S^\exists$ : There is at least one member of $S$ that can take this place.

Note: if $S^\forall$ or $S^\exists$ is used multiple times in one formula, they are the same variable. If you want to specify a different one:

$S^{\forall'}$ : Any member of $S$ that is not the same as $S^\forall$ .
$S^{'\forall}$ : Any member of $S$ , possibly the same as $S^\forall$ .

Note: Order of operations for evaluating these operators: Nested, left to right of first instances. If one wishes to vary from this order, denote the operators alternative position $n$ as $S^{n\forall}$ or $S^{n\exists}$ . If the standard notation is used in conjunction with this, the standard notation takes precedence in order of operations.

Example 1: Accumulation Point Defenition

$p \in \{accumulation\}(S,T) \iff \{s|p\in s\in T\}^\forall\cup S \neq 0$

Example 2: Hausdorff Space

$S \in T_2 (X,T) \iff$ $N(S^\forall)^\exists\cap N(S^{\forall'})^\exists =0$

Tags:math, notation
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Anonymous Functions in Math and CS

December 17, 2009

Consider:

$f(x) = ...$

This is how one normally declares a function in math: by describing what the value of the function applied to a variable would be. One consequence of this approach is that ir necessitates a variable name, in this case $f$ , being assigned to the function.

This may not sound significant or problematic, but consider what the parallel of this in a different class of objects, like numbers would be. Before doing anything, we’d have to assign a variable to the number! For example, we couldn’t just write $3*2$ , we’d have to write $x=2; y=3; x*y$ . This may still not seem significant. After all, it’s only functions; people only have to define them occasionally.

I’d like to mention a theory that someone (Kelly Rose, one of the leaders — read teachers — of hacklab.to‘s math reading group) mentioned to me recently, that the reason two people (Newton and Leibniz) effectively simultaneously discovered calculus after people failed to for however many thousand years was that notation had developed that allowed for the discovery. (Interesting side note: this individual suggested that category theory may be the next big change.)

If this is theory is correct (and I’m not sure how we could every be certain) then it is the best example I can think of, of something I’ve been thinking about for several weeks: the importance of notation. (I’m using the term `notation’ loosely as including something that isn’t truly a new idea but is a new way of presenting the idea.)

It seems obvious to me that if one makes an idea difficult to represent, a person will avoid using it (or make an easier way to represent it). In some ways, this is applying the Sapir-Whorf hypothesis to things other than spoken language: the medium (traditional language, visualisation, notation, etc) of thinking effects the thoughts themselves. (If you’ve been nodding along, consider the effects English must be having on you mind; you may now wish to look at Lojban.)

I believe this is true, to an extent that I am probably failing to convey. (It’s why, for example, I find it deeply troubling that $\pi$ isn’t defined as $2\pi$ : think about all the students who would have had a better understanding of trigonometry and more respect for math in general if it wasn’t for the fact that Pi Is Wrong.)

But we’ve digressed. Back to functions. We need to ask, is there a way to avoid defining a variable when using functions. In fact there is: $x \to ...$ . For example, the following to statements are identical:

$f(x)=x^2+3; f(2)$

$(x\to x^2+3)(2)$

We call $(x\to x^2+3)$ an anonymous function. Another notation for this is using lambda calculus. We represent a function with $\lambda x. ~ ...$ . For example, the earlier example would be $(\lambda x . x^2+3)$ .

This has endless usefullness in math. It can provide an ellegant way to represent the Mandelbrot set, for example:

$\left\{x\in \mathbb{C}| \lim_{n\to\infty}(z\to z^2+x)^n(x)\neq\pm\infty\right\}$

Translation for non-mathematicians: the set of complex numbers $x$ such that if we square x, add x, square that and add x, and so on infinitely, we won’t end up with infinity.

We can also use this to define functions that return functions using it. For example, we can define the differential operator, $\frac{d}{dx}$ :

$\frac{d}{dx}(f) = (x \to \lim_{h\to 0} \frac{f(x+h)-f(x)}{h})$

We can also use anonymous functions in programing. For example, python allows us to use lambda calculus in it. The following is perfectly valid:

>>> (lambda x: x*x +3)(2)
7

Now, you may have noticed that we can do some really interesting things with this. And could do even more interesting things, if we could make recursive functions.

Some programing languages support anonymous recursive functions. For example, in javascript, functions are passed a `callee’ argument that allows one to recurse on itself. Thus, we can implement an anonymous function that gives us a term in the Fibonacci sequence.

(function (x) {
  if (x <= 2) {
    return 1;
  } else {
    return arguments.callee(x - 1) + arguments.callee(x - 2);
  }
})(6) // = 8