Welcome to toywiki


toywiki is a collection of maths notes and an experiment on open research.

A collection of maths notes

toywiki is a collection of notes, mostly on mathematics. I wrote these notes to help myself solidify my mathematical understanding, to explore mathematics, as well as to share research and knowledge.

Style-wise, toywiki notes are meant to give a modular and intuitive presentation. I believe mathematical knowledge is highly modular and incremental, and that mathematical writing is not much dissimilar to computer programming. For example here is a table of some analogues between mathematical writing and programming.

Mathematical writing Computer programming
Theorems, lemmas, propositions, corollaries Functions and procedures
Lemmas Helper functions
Invoking a theorem Calling a function
Examples Test cases
Papers, notes, books Libraries and modules
Citing a paper, book etc. Importing a library
Declaring a variable Declaring a variable
Variables have scopes (e.g. dummy variables) Variables have scopes (e.g. local variables)
On toywiki, I try to abide by the modularity to write as few repeated statements as possible and improve the clarity and simplicity of the demonstration. It is really like a programming project, except the computer can not help debug and find errors (see below).

Despite the inspiration from computer programming, the notes are written for human rather than computers, so formality is secondary to clarity and simplicity. For example, I do not attempt to formalise mathematics completely.

I hope you find something useful here.

Contents of toywiki are under CC BY-SA 4.0 License.

Oh, why is it called toywiki? Because I think toy examples are the best tool to study mathematics :)

An experiment on open research

As an experiment on open mathematical research, toywiki is also an open-source project. It is synchronised with a Github repository, and you are welcome to clone and make your own version, or contribute to it by submitting issues or pull requests.

You can help improve the existing notes. Since the notes are informal and haven't been peer-reviewed yet, I expect there to be mistakes and typos on accuracy or attributions. If you find something fishy or outright wrong, please open an issue or submit a pull request on Github.

You can also help solve open problems. Many problems that appear in the notes or issues are marked as "Open", and they are potential research problems to work on in this project.

In general a problem may be in one of three status in a universal sense:

  1. Open. No one on Earth has a complete solution to it.
  2. Folklore. It is believed to have been solved by some, but not complete solution has been published.
  3. Solved. A complete solution has been published.

But these universal statuses are not useful, because in everyday life, a problem can only be either open or solved from a person's perspective, and unless one is omniscient, a problem open to them may fall into any one of the three categories above. They may not be aware of the solution, or that they have read the solution but there are gaps between their mathematical background and that assumed by the solution. For example, a usual problem posted on stack exchange is open to the poster until they understand a solution.

It is open problems in this sense we work on in toywiki. They can be posted on the issue page for discussion. If it is open due to ignorance about the literature, then a pointer to the solution should resolve the issue. Otherwise once they are solved, it is high time to record the solution in the wiki.

Feel free to ask questions. You are also welcome to open an issue if you have a question about any of the notes. You can also reach me by email (see the footer of this page) if you have any comments about the project itself.

About credits. Just like every other open source projects on Github, credits can be completely traced in commits and issues.

See CONTRIBUTION.md on github for detailed contribution guidelines.

Let's do some cool maths!