House Rules and Conventions

Last updated Nov 1, 2022

• Graph exclude

I’ve been working on this for a while and feel like I have to come up with the same conventions everytime I write because I’m not sure what the prior convention was. So, I’m starting a page where I can keep track of conventions and rules as I write.

TODO - fill this in as I go

• What role does a definition fill?

Anything that needs to be proved is a Statement. It is similar to a “Proposition”.

1. A Statement should specify the mathematical result in a mathematically precise statement.
   1. Exceptions to the above rule are when the statement is a simple connection or extension of other Statements and Definitions.
2. The Statement takes H1 (top heading) and Proof, Remark, or Properties takes H2 (first subheading).

A Counterexample is a note that provides some example to show a statement is false.

1. Use a Statement instead of a Definition if we are defining a concept with something that needs to be proven.
   • See Discrete Topology for an example.

1. A Proof should express the result as a sequence of logical steps. It should be self-contained, and reference other necessary results and concepts through links.
   1. An exception is if the proof is a simple connection of other concepts.
2. If a proof complete and mathematically precise, it should end with a $\blacksquare$. If a proof is sketched, but not necessarily precise, it should end with a $\square$. If the proof is incomplete, it should be tagged with a TODO.
3. All $\blacksquare$ or $\square$ ending a proof should occur after the period ending the last sentence.
4. Upon completion of a checkpoint (e.g. proving the Non-Degeneracy property of an Inner Product), the sentence should end with a $\checkmark$. The $\checkmark$ should occur before the period ending the sentence.
   1. some exceptions are occasionally necessary.
5. Upon reaching a contradiction, a $\unicode{x21af}$ symbol should be used. Like the $\checkmark$, it should occur before the period ending the sentence. If not referenced in the proof, Proof by Contradiction should be referenced in Other Outlinks.

1. The Properties section of a note is meant to capture the most important properties, not necessarily all of them. The more exhaustive list can be found in the “backlinks” on the rightside panel.
2. If a property needs to be referenced more than a few times, it should get its own note.

When I want to use a result I don’t have yet, I can reference a note that doesn’t exist. Obsidian will suggest this note in autocomplete. It is best practice to always complete a result whenever I create it, but that is not always feasible given limited time.

1. If the note has been started but is incomplete, I should add a TODO, along with some checkboxes about what needs to get done.

1. If I reference a more specific object, I don’t need to also reference the more general object (e.g. “Open set” instead of “Open Set”, or Continuous Function instead of continuous Function).
2. It’s okay to not reference a concept if I’ve already referenced it somewhere else. However, there is not much harm in referencing it multiple times when it is used.

1. When done writing an article, go through and link any concepts used that are not already linked in the note.
2. It’s okay if something is in the outlinks and is linked in the note. The goal of the Other Outlinks section is to create backlinks into those concepts used, as well as allow users to navigate to those concepts if they don’t understand something.
3. Other Outlinks always takes H1 (top heading).
4. Some primitive concepts like Set don’t need to be referenced on every page. The same goes for basic set operations (Set Union, Set Intersection, Complement, Subset Relation).

1. Whenever a I denote another object $A$ with some symbol $a$, I should write $a := A$ or $A =: a$. The “$:$” always goes on the side of the new symbol being defined.

1. If outside a bullet or enumerated list, all display math should go on its own line. Within a bullet or enumerated list, displaymath should be writting inline due so I can get the desired nesting.
2. If a sentence ends with displaymath, then the period should go inside the displaymath. This ensures the period is not on a different line.

1. The TODO page should not have any content. All TODOs should be written on the relevant page along with any of the following
   1. some directions about what needs to be done.
   2. A reference to an incomplete or unstarted note
   3. Some notes describing any thinking or progress I’ve made to resolving the TODO.
2. Each TODO should have associated checkboxes.
   • This is a checkbox.