## My Math Notation

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:

1. Some notations allow certain thoughts to be expressed more easily than other notations.
2. The ease with which thoughts can be expressed changes the thoughts that are expressed.
3. 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 $S \in T_2 (X,T) \iff$ $N(S^\forall)^\exists\cap N(S^{\forall'})^\exists =0$

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### 5 Responses to “My Math Notation”

1. Nikolay Degtyarev Says:

Hi. I don’t think that second notation is good one. For me it is harder to read then ordinary one.

• christopherolah Says:

Hey Nikolay!

It’s interesting that you dislike the second notation. One contributing factor may be the that the low quality images of equations the LaTeX is translated into make superscripts awkward to read…. Then again, this notation probably isn’t for everyone.

One really nice thing about this notation is that it reduces the number of variables you are working with. Makes things more compact, too.

2. Sniffnoy Says:

The second notation has serious problems, I think, primarily in that it encourages confusion of levels/types. I had to look at your first example quite a few times before I realized that that set isn’t getting intersected with S, rather sets *from* that set are getting intersected with S. Similarly to realize that the second one really was talking about Hausdorffness and not, say, normality. The fact is, there are enough things in math where type confusion is likely to occur *even when* it’s written perfectly clearly; this notation introduces it when it’s not a problem, and when it is a problem would make things truly awful. There are some other problems, too, but this is the major one.

• christopherolah Says:

This is definitely a valid criticism. Everyone once and a while I revisit the topic of notation and I’ve basically come to view my superscript notation not as a replacement but as an alternative that is sometimes better.

In defense of this notation though, I think that the standard notation of going (∀x..) is a big tricky for outsiders to follow because keeping track of x when you haven’t internalized a convention is hard. For example, I now know that in topology u and v are open sets, F and G closed ones, calligraphic letters collections of sets, and so on… But when I started it made it extremely difficult to follow proofs.

In any case, I’ve been working through the back of _Counterexamples In Topology_ in this notation when I have free time to “stress test” it. The results have been somewhat interesting in terms of how I’ve rephrased ideas or created new topology specific notations.

I’ve also experimented in alternative notations for other areas of math… That is sorely due for a post.

3. Math Notation Experiment “Paper” | Christopher Olah's Blog Says:

[…] while back, I wrote a post on some unusual math notation I was playing with. I actually took it much further than that and […]