User:Breakmode

Some thoughts

1. Axiom Sets
   • Right now, there are no axioms listed in the book. Before we can provide any proofs, we have to have axioms somewhere that we can refer to.
   • The current table of contents is vague about where the axioms are going to go. I assume that we aren't forcing ourselves to define everything from within set theory? That we're going to start the other sections off with higher-level axioms?
   • When looking at any given theorem, the reader should be able to tell what axioms it follows from. For an example of this not happening, look at the Geometry page, which tentatively puts four different axiom sets all on the same chapter page/level. The page The Book of Mathematical Proofs/Geometry/Parallel Lines is placed in the box for Euclidean Geometry, but the page name does not reflect that. In addition, the other geometries will want pages about their parallel lines as well, causing naming conflicts.
   • As the axioms you start with determine what theorems you can prove, they are our most basic means of categorization. I suggest that the first level of the book's hierarchy, what we give links to in the table of contents, be groupings by Axiom Set instead of by Field of Mathematics, as is currently the case.
     • An alternate method would drop the Axiom Sets down a level: Geometry/Euclidean Geometry/Parallel Lines. I don't like this method as much because it makes the names longer, it doesn't provide any information that isn't redundant, and boundaries between fields can be hard to describe while the axioms must be exactly stated.
     • The table of contents' links to each Axiom Set can still be organized by branch: /TableOfContentsDemo
2. Page Naming
   • I'd like to suggest we do number our theorems, and that each theorem, no matter how small, is given its own page named with its number. (Axiom Set/Theorem 1, Axiom Set/Theorem 2, Axiom Set/Axiom 1)
   • Proposed page layout: The_Book_of_Mathematical_Proofs/Template:Theorem
   • Once a theorem is written in the book, it is not just going to be used by the readers; it is going to be used in the proof of further theorems. We need to be able to easily refer to and link to any theorem (and its proof). The "this result depends on" box isn't a bad idea, but it can't be used in-line in the course of a proof. As well, it could be bothersome to have to always refer to nameless theorems by their complete statement. Within an Axiom Set, numbered theorems could be linked to as [T12]. If necessary, they can be linked to from other Axiom Sets as [AxiomSetName/T12]. Of course, any text, including a theorem's statement as part of a proof, could still be turned into a link to the numbered page.
   • Under the current subtopic page setup, each subtopic page is forced to hold a lot of information for each of the many theorems that will be on it. Individual pages for each theorem give definite spaces for all this information, and prevent the subtopic pages from becoming unwieldy to navigate.
     • Name. Some theorems are known by more than one name, some theorems have no proper name. The 'Known As' section of the theorem page layout can hold any number alternate names in its bulleted list.
     • Proofs. Some theorems can be proven in different ways. With each theorem on its own page, it can have as much space as it needs for alternate proofs.
     • About and See Also. I see the About section as something written in paragraph form, not limited to a list of facts presented as trivia.
   • Numbered theorems on individual pages encourage us to refer to the theorems we have already included, because there is a definite page to link to, and to include the theorems we want to refer to, because they are needed for the proof of another theorem we want to include, which cannot be entered out of order as it would mess up the numbering. (Compare this to a proof entered on a subtopic page as a block of text, where there is no guarantee that the support it uses has been included in the book yet.) Discussion of what theorems are needed before a certain theorem can be added could occur either on the Axiom Set's discussion page or perhaps on a separate "Requesting Proof" page.
   • That actual textbooks number their theorems for internal reference shows that this kind of thing can work, and won't be difficult for readers, especially with the subtopic pages still existing in some form to provide access by subject. I'd also like to see an Index section providing access by name, and the Axiom Set pages providing access by direct statement. More details on those in the next section.

Book Layout

• The Book of Mathematical Proofs
  • Axiom Set
    • Axiom 1
    • Axiom 2
    • Definition 1
    • Definition 2
    • Theorem 1
    • Theorem 2
    • Topic Page
    • Topic Page
  • Axiom Set
  • Axiom Set
  • Index

Each set of axioms has its own chapter at The Book of Mathematical Proofs/Axiom Set. That page will use the following layout: User:Breakmode/AxiomSetPageLayout. There are 4 types of pages that will make up the Axiom Set chapter: Axiom pages, Definition pages, Theorem pages, and Topic pages. Each page in the chapter is linked to under the matching heading on The Book of Mathematical Proofs/Axiom Set.

• Axiom and Definition pages use the following layout:The_Book_of_Mathematical_Proofs/Template:Axiom
• Theorem pages uses the following layout: The_Book_of_Mathematical_Proofs/Template:Theorem
• Axioms, Definitions, and Theorems will be numbered for reference within the book. The page names will use this numbering as well; Axiom 1, Definition 3, Theorem 12, etc. As seen on the sample layout pages, there is a "Known As" section where all alternate names for the theorem/axiom/definition will be listed.
• The Topic pages are similar in function to the Branch/Subtopic pages currently in use. They do not have to prove the theorems, as that is done on the theorem page. Their purpose is to organize and present the theorems.
• The Index provides access to pages by name (alphabetically), like those included in the "Known As" sections.

Axiom Set Page Layout

User:Breakmode/AxiomSetPageLayout

Getting Started

1. Choose an Axiom Set
   • Create the axiom set page; The Book of Mathematical Proof/Axiom Set
   • Put a link to that page in the table of contents
2. Determine the axioms
   • List the statements of the axioms in the appropriate section of Axiom Set. If you are unsure whether these are the proper axioms, in that they are generally referred to by the name "Axiom Set Name", you don't have to create the individual pages for them yet.
   • The Axiom Set discussion page can be used to discuss if these are the axioms the reader will expect to be referred to by the name Axiom Set. Citations from different sources showing the ways in which others state these axioms are encouraged.
   • Once the list and ordering of the axioms is decided, create one page Axiom Set/Axiom # for each of them. At the very least, put the statement of this axiom as used on the Axiom Set page under the axiom page's Statement section.
3. What follows directly?
   • To get activity started in the section:
   • Add any theorems that can be proven immediately. First, add the theorem's statement to the end of the numbered list in the Theorems section of the Axiom Set page. Then, create the page Axiom Set/Theorem # to match its position in the list.
   • Use the Axiom Set Name discussion page if you need help writing the proof before you update the list and create the page.
4. What are the theorems readers will expect to see?
   • To give us something to work towards, list on the discussion page:
   • Anything with a name, anything that people refer to a lot, anything other textbooks seem to think is important
   • Discuss the outline of a proof for it. What needs proven to prove this?
   • When the time comes, give it its own page.

Public Domain Math

• MetaMath
• The Elements of Euclid