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.
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.
Very early on this year, I got tired of writing: “For an open set ,” and “Let be a compact set.”
I was also exposed to the notation of writing the `set (class) of sets’ as . So I decided to use this notation for expressing any set. For example: and .
More recently, I’ve been treating them as functions that returns sets to express more complicated ideas. For example, I’d replace “let be an accumulation point of the set on the topology ” with .
Great for notes. Even outside math.
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:
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. : Place holder for all members of . : There is at least one member of that can take this place.
Note: if or is used multiple times in one formula, they are the same variable. If you want to specify a different one:
- : Any member of that is not the same as .
- : Any member of , possibly the same as .
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 as or . If the standard notation is used in conjunction with this, the standard notation takes precedence in order of operations.
Example 1: Accumulation Point Defenition
Example 2: Hausdorff Space