Talk:Famous Theorems of Mathematics

From Wikibooks, the open-content textbooks collection

Jump to: navigation, search

Contents

[edit] Book Intentions

What is the intention of this book? Is this book going to only focus on definitions and examples of each proof, or will there also be detailed explanations for each proof aimed at helping to teach and learn each proof or will a separate math book be needed to understand the proofs and to put them into context? --darklama 13:30, 13 September 2007 (UTC)

The book focuses on proofs so certainly there will be detailed proofs. Examples and definitions should be included to supplement the proof.
The two objectives of the book in my view are:
  • Complement other wikibooks by providing them with a place where they can put the tedious proofs of their theorems while they themselves concentrate on the explanations part. (I am assuming that such a complementary book is allowed in wikibooks as a book with a glossary of math terms is allowed.)
  • Focus on the beauty of mathematical proofs as such. For example a result may have a geometric proof, an algebraic proof, a rigorous proof from the foundation etc. All such proofs can be included. The pythagoras theorem has many proofs. Trigonometric identities have geometric proofs. The Axiom of Choice has many equivalent formulations.
Please help contribute to the book. Cheers--Shahab 05:22, 6 October 2007 (UTC)
Sure books can have glossaries, but a glossary is secondary and not the main part of a book, nor is it a separate book. Thats actually why I asked what the intention is. So if its suppose to let other books concentrate on the explanation parts, then am I right in saying that this book would not be explaining the proofs at all? You first say that there will be detailed proofs in answer to my question which was asking about "detailed explanations for each proof", but go on to explain the objectives which seems to suggest instead that it will be relying on other math books to provide explanations for the proofs.
Even reference books I've read have detailed explanations, usually assuming some prior knowledge of the subject, but still explaining some things as to refresh/remind people with professionals/advanced knowledge of the subject, and put whats being referenced into context. If this book to be complimentary, sounds like it should still be having explanations, but directed at people with advanced knowledge, leaving the details needed to help beginners to other math books. Perhaps that is already whats intended?
I can't unfortunately help contribute to this book, because I don't know any proofs or at least I don't think I do. What proofs are covered isn't an issue. I'm just checking up to be sure its not intended to be a dictionary of math proofs. I'm not trying to scare you, discourage you or anything, just want to help insure you've got the right project for your work in the form you want it. --darklama 13:47, 6 October 2007 (UTC)


What about creating the entire proof tree from the axioms, definitions and lemmas? That is a dream I have thought of for a long time. Paxinum 13:43, 7 October 2007 (UTC)

I too wanted to have some kind of proof tree in the book (see my sandbox). That would be a good idea. Anyway in response to Darklama, the proofs will be certainly explained. The explanations in other books will be of the concepts. An example is this page. Cheers--Shahab 10:49, 8 October 2007 (UTC)
Great! Is there anyone that has an idea on how one organizes proofs/lemmas, definitions and axioms? Maybe creating a template for proofs, like "This proof uses proof(s) A,B,C", and try to motivate every (non-trivial) step by linking to the proof/axiom used? Paxinum 09:02, 9 October 2007 (UTC)


How about this:
Full Proof
This result uses the following proof:

Similar templates can be made for axioms and definitions. I got this from the Measure Theory book. Cheers--Shahab 10:50, 9 October 2007 (UTC)

That looks great, I was thinking something like this User:Paxinum/Proof styles. Paxinum 15:11, 9 October 2007 (UTC)
Nice. I would suggest the phrase This result uses the following: instead of Depends on. Can you make the template? Cheers--Shahab 11:12, 10 October 2007 (UTC)
I put it under Resources on the front page. This is just an example. Paxinum 20:29, 11 October 2007 (UTC)

[edit] Suggestion for editors

All proofs available on the wikipedia site and on other wikibooks can be copied here. Formatting can be done later. Right now we need to add more content to the book. Also this link can be used to watch changes related to the entire book.--Shahab 10:57, 18 October 2007 (UTC)

[edit] Structure

This book needs a consistent and suitable structure for its purpose.

At the moment, each page (on the whole) relates to an entire branch of mathematics, with an assortment of theorems and a proof or two of each. This is going to be no use as the number of theorems and proofs increases. We need to break down the pages into subcategories and theorems. We should also avoid vague headings like "Theorem 1" as is currently on Proofs in Number Theory.

Before this was started, I drafted some pages for a proofs book, intending to put it up when I've got a bit of content ready. The structure I had in mind was basically to categorise the theorems hierarchically and have a page for each theorem, which has a subpage for each proof.

What do we think? I think it would be good to work towards a 'one page, one theorem' principle.

If nobody minds, I'll make a start on improving the structure.

While I'm at it, how should Proofs in Geometry be subcategorised? Sections that should be included include:

  • Plane Euclidean geometry
  • Solid Euclidean geometry
  • Projective geometry
  • Elliptic geometry
  • Hyperbolic geometry

but I'm not sure what the best hierarchy would be. Some theorems would cross the boundaries between these sections - how we should put these in is also something to think about. -- Smjg 18:10, 3 November 2007 (UTC)

Definitely a hierarchical structure is needed in the book. What I feel is that we should adopt 2 approaches side by side in developing the book. One approach is to frame the sequence of the proofs by building different pages for each topics like in matrix theory proofs and while simultaneously also populating the book with proofs just dumped under the pages of the fields in mathematics. After sufficient proofs have been added then they can be categorised further. The one page- one proof principle is good but should be applied for lengthy and/or interesting proofs (such as Pythagoras, FTA etc). Cheers--Shahab 16:01, 4 November 2007 (UTC)
So if I understand correctly, you're proposing that the content and the structure should be developed independently of each other, with theorems/proofs dumped under the higher-level pages pending splitting into subpages. All well; moreover, I think we should concentrate on categorising theorems rather than categorising proofs. Once we've got the theorems categorised, we can look at whether it's worth subcategorising proofs of the same theorem, e.g. geometrical vs. algebraic proofs of Pythagoras' theorem.
One thing I don't get: What's with the use of "but" in your last sentence? Did you by any chance miss out a word to the effect of "only"?
Also, I think we should get rid of the phrase "Proofs in" from chapter titles. It's redundant, inconsistently applied and making some page titles excessively long. -- Smjg 17:14, 4 November 2007 (UTC)
I agree that categorising by theorems is correct and that the word proof from the chapter title should be removed. One thing that we have to keep in mind is that to maintain accessibility the theorem/proofs should not lie more then 2 levels deep in the hierarchy. What we can do is to create lists in the 1 level deep subject page eg the Algebra page for the various branches of Algebra (eg linear algebra, abstract algebra etc). Each list should contain its subdivisions and a sublist of links to various pages of proofs. For example:

Abstract Algebra

  • Group Theory (Basic proofs, Isomorphism Theorems, Sylow Theorems, Fundamental Theorem of finitely generated abelian groups etc)
  • Ring Theory(...)

All this should be on the Algebra page only and so only 1 level deep. The 2nd level should contain the theorems/proofs. This way accessibility will be easy. Cheers--Shahab 09:36, 5 November 2007 (UTC)

Let me think ... so
I suppose this would work. Actually, there's still one thing for which I think we should use the third level: if so many proofs of the same theorem have been contributed that it becomes desirable to split them across pages. I don't think this would compromise ease of navigation, partly because it would be done only occasionally, and partly because people are likely to want to find the theorem before choosing from the various (groups of) proofs of it to look at. -- Smjg 12:13, 5 November 2007 (UTC)
One point to keep in mind is that the link titled Basic Proofs should actually point to Basic Proofs (Group Theory) in the example above. A similar title should be used for names which are associated with many theorems eg Cauchy's Theorem (Group Theory), Cauchy's Theorem (Complex Analysis) etc. --Shahab 15:20, 5 November 2007 (UTC)
I'm not sure whether "Basic Proofs (Group Theory)" is a good title, or if it should be "Basic Group Theory" instead. But it's a suitable notation for distinguishing theorems with the same name. Moreover, what shall we do with the titles of the as-yet-uncreated "Definitions in" pages? I personally think incorporating definitions into the same first-level subpages as the proofs would work or, if not, making them The Book of Mathematical Proofs/Algebra/Definitions and similar. -- Smjg 23:28, 6 November 2007 (UTC)
What about instead of having duplicate pages with multiple theorems, there was one page for each theorem classification? Each page could include an explanation/definition of the classification and a list of links to associated theorems/proofs. So for example there could be a "Group Theory" page and a "Complex Analysis" page both with a link to a page called "Cauchy's Theorem". --darklama 23:52, 6 November 2007 (UTC)
Not sure if that would help. In fact it would then be confusing to link theorems with multiple people names. For example Peano-Cauchy theorem would then point to the page on Cauchy's theorem. Also it would really be better to keep a page related to one field only. Our ultimate aim is to build a whole hierarchy of proofs. In response to Smjg I think definitions can be incorporated inside the proofs. In fact many proofs provide us with the definitions. For example the Ramsey number is a number which exists due to Ramsey's theorem & so it should be logically defined just after Ramsey's theorem is proved.--Shahab 17:35, 7 November 2007 (UTC)
I'm now going to make a start at implementing these ideas.... -- Smjg 00:05, 9 November 2007 (UTC)

I have built a list or a TOC for the Number Theory page. Please take a look at the format and give me your opinion. Also while deciding on the subtopics to include in the list I found that

1. Unless the theorem is sufficiently big to merit a page of its own it should be kept within a page containing lots of other theorems related to the same topic. For example the prime numbers page can contain the fundamental theorem of arithmetic. However this should not preclude the fundamental theorem to have a seperate mention on the list of subtopics so that it remains accessible to casual browsers.

2.It will be a generally good idea to list the topics rather then the theorems because a majority of the theorems don't have names but are extremely useful.--Shahab 10:54, 11 November 2007 (UTC)

The structure in itself seems OK, but I'm not sure about the choice of subsubtopics. Maybe we should try developing the chapter and see how it goes. And are we going to have headings in sentence case or title case? You can't seem to make up your mind even within the course of a single editing operation. -- Smjg 16:09, 11 November 2007 (UTC)

I've just had a go at writing a contents page for The Book of Mathematical Proofs/Geometry. Please tell me what you think. -- Smjg 12:59, 22 November 2007 (UTC)

Nice, although I'd prefer small case in the subtopics. I'll soon start work on the number theory page. Lets develop the chapters and see what we learn from that. One more thing: The standard on the wikipedia site is that in a section heading only the first letter of the first word, letters in acronyms, and the first letter of proper nouns are capitalized; all other letters are in lower case (Funding of UNESCO projects, not Funding of UNESCO Projects). Do you think that we should implement that policy here?--Shahab 15:33, 22 November 2007 (UTC)
While sentence-case titles are a standard on Wikipedia, titles on Wikibooks are a mixture. This applies to both page titles and section headings within a page. I don't think there's an official Wikibooks policy, but the consensus seem to be 'each book for itself', while it should be kept consistent within any one book. At the moment and for this case, I'm not sure which convention I prefer.... -- Smjg 16:35, 22 November 2007 (UTC)

[edit] Which form of English?

Should we standardise (or standardize?) on British or American spelling/terminology? It's going to affect quite a bit (centre/center, trapezium/trapezoid JTNAF) and possibly even some page titles. We should certainly address such dialectal differences as trapezium/trapezoid - the question at the moment is which forms should be in general use throughout the book. -- Smjg 13:16, 24 November 2007 (UTC)

[edit] Book Intentions 2

Originally I started the book with the aim of creating a collection of all possible mathematical definitions, theorems and proofs. Increasingly I started feeling that this amounts to just creating an encyclopaedia of mathematics, a function well served elsewhere. The book has too gigantic a scope if built with this intention and is doomed to fail. The only alternative I feel would be to change the original objectives of the book, and instead to create a repository of the beautiful/good proofs in mathematics, without any regard to the various fields they belong to. For example, the Pythagoras Theorem, Euler's Theorem, Fermats theorem etc can qualify as beautiful proofs. The community can decide by consensus on the talk page whether any proof is worthy of being included or not. Any comments?--Shahab (talk) 17:49, 2 August 2008 (UTC)

I propose changing the name of the book to Famous theorems of mathematics in accordance with the discussion here and for the same reasons as stated above.--Shahab (talk) 06:01, 3 August 2008 (UTC)
Retrieved from "http://en.wikibooks.org/wiki/Talk:Famous_Theorems_of_Mathematics"