Welcome to an open book for mathematics, physics and computer science!
What is BookofProofs?
BookofProofs (BoP) is an open online book dedicated to mathematics, physics or computer science, which is completely written by the Internet community. It can be browsed (and downloaded - working on it) completely for free and for educational purposes.
It is also our policy to keep the book as a pure educational resource, which is free of advertising.
The mission of BoP is to promote the competency of thinking logically and formulating proofs as basic skills rather than something reserved for academia, to broaden the knowledge of the axiomatic method, and to drive awareness of its supreme importance for any kind of scientific progress.
Unlike article-based wikis structuring all information by categories, BoP is structured more like an open book, in which you can create parts, chapters, sections, and subsections, and develop the theories by adding theorems, proofs, definitions, examples, explanations, exercises, and solutions. The book contains only tagged entries, i.e. each entry has a unique id, which makes it referenceable independently from the structure of the book.
BoP's unique features
A unique feature of BoP is that some hyperlinks - those, which are contained in proofs and in definitions - are not only used to better navigate through the book but have a special meaning and follow a strict pattern: The pattern allows for tracking the logical predecessors and the logical successors of all theorems and definitions. This feature strictly follows the axiomatic method, enabling the reader to recognize, which theorems or definitions follow logically from which other theorems or definitions, and to track this logical chain back to the axioms, upon which a theory is based. It is also possible to follow a theory the other way round - starting with axioms, and identifying one-by-one all the theorems or definitions derived from them in a theory.
To make this pattern work, it is important that each theorem, lemma, proposition, and corollary has at least one proof and that its text body contains exactly those hyperlinks, which are necessary for the logical conclusions in the proof. The same restrictions apply for hyperlinks which have to be included in the text body of each definition. Other types of hyperlinks, i.e. those contained in the text body of the theorems themselves, in examples, in explanations, in exercises, won't do the trick, and will be ignored when tracking the logical chain. Nevertheless, you are encouraged to set also such hyperlinks to make it easier for the reader to navigate through the book.
As the same mathematical theories and concepts can be derived from different sets of axioms, only one of these possibilities could be depicted in the logical structure of BoP. In accordance with the mission statement, it is not the goal of the book to provide means of deriving mathematical theories from any possible sets of axioms but rather to focus on one valid way and on promoting and demonstrating the power of the axiomatic method as a tool and achievement of human thinking.
Another important feature of BoP is that it does not consist of separate self-contained open books, one for each discipline. This means for you as a co-author that it is not necessary (and not allowed) to reintroduce mathematical concepts already defined in the book (like e.g. sets, functions, numbers, ...), just because you need using the concepts in the branch of mathematics you are currently working on. This means for you as a reader that, throughout all disciplines of BoP, all mathematical concepts are introduced and defined only once. Thus, they remain always referenced consistently and, if needed, can quickly be re-stated/corrected at a central place. Every catalog entry has been thoroughly reviewed to ensure high quality and avoid such redundancies. Unless it is absolutely necessary (for instance in case of branch-specific conventions), we also pay particular attention to keeping a unique notation of all mathematical concepts throughout the whole book.
How to contribute?
The BoP project is a collaborative effort. With your help, this book will become more comprehensive and cover at least all main branches of modern mathematics, theoretical physics, and computer sciences. The growing library of BoP currently consists of 5672 entries, contributed since 1 May 2014 (launch of the project) by 116 author(s). Last updates were provided 1 day ago.
There is a list of people who are members of our community. While browsing the site as a member please provide feedback by leaving a comment or correcting any errors you encounter. Another important option to help our project is to post new or edit existing results, and to tell other poeple about us.
You can also support us by giving us backlinks, or like us on social media!Sign up to access to all contents or to contribute!