Welcome to BookofProofs
BookofProofs (BoP) is an open book dedicated to mathematics, physics, and computer science. Its goal is to broaden the public knowledge of the axiomatic method, without which the development of these topics over the centuries wouldn't have been possible.
The ability to formulate mathematical proofs using the axiomatic method should be taught as a basic skill like reading or writing.
This motto is what BoP is all about.
BoP was launched in February 2014 and started as a collaborative effort. Although there have been 140+ registered users between 2014 and 2018, only three (3) contributions were posted by other users than me during this time. I was forced to realize that it is very hard to mobilize any kind of activity among the community and that it takes me too much time to maintain both, the contents and the functionality of the site. Therefore, after 4 years I decided to remove the functionality enabling co-authoring (collaboration, community, user access, etc.) from my site. Now, I invest my free time only to the contents. If you are still interested in co-authoring or want to provide any kind of feedback, please contact me.
The Axiomatic Method
You can learn the axiomatic method immediately. Just follow these steps:
- Assert the truth of one or more statements (and call them axioms or postulates). Add the postulates to a list (and call that list theory).
- Given all statements in your theory, logically derive new statements which are true (and call them propositions or theorems).
- Add the newly derived theorems to your theory.
- Continue with step 2.
The method is very powerful. It works like a snowball and allows constructing complex theories from easy to understand basic axioms.
A Simple Example ("Little Bird Theory"):
- Axiom 1: "All ravens are black". Axiom 2: "Every bird is a raven". In its first iteration, our "Little Bird Theory" consists of the axioms 1 and 2.
- Statement 1: All birds are black. Proof: Let a thing be a bird. According to Axiom 2, the thing is a raven. According to Axiom 1, the thing must be black.
- After this proof, our "Little Bird Theory" became bigger. Now, it consists of the axioms 1 and 2, including the statement 1.
- We could now continue with step 2:
- Statement 2: White things are not birds. Proof: Let a thing be white. According to Statement 1, the thing cannot be a bird.
- After this proof, our "Little Bird Theory" became even bigger. Now, it consists of the axioms 1 and 2, including the statements 1 and 2.
The "Little Bird Theory" is nonsense, since it disagrees with our daily experience. But apart from some technical details, which are unimportant here (eg. we haven't defined how exactly we derive new theorems using logical steps), there is nothing to complain about this theory.
The success of the mathematics, physics and computer sciences based on the axiomatic method lies in the clever choice of axioms at the very beginning of a new theory. The better the choice of axioms, the more likely the axiomatic method will produce a theory which has better applications in the real world or makes better predictions about the real world. For instance, the axiom "The speed of light is constant whether the ray be emitted by a stationary or by a moving body" allowed Albert Einstein at the beginning of the 20th century to develop his Theory of Relativity. Now, we harvest this theory by applications like the GPS system, which wouldn't work, if the axiom was wrong.
Publications available on BoP
All publications listed below can be downloaded for free. Enjoy! Please note that how you can use each publication is restricted by the license referenced or stated in each publication.
Publications based on the axiomatic method
There is a lot there to discover but most parts are still under construction...
- The foundation of the axiomatic method and metalanguage of mathematics - Logic
- Set Theory - a foundation of mathematics
- Number systems and arithmetics - a clarification of 'numbers'
- Algebra - dealing with structured sets and their properties
- Calculus - a branch studying the change rates of processes
- Topology - a study of the location properties of space
- Geometry, please also note the classic Euclid's Elements
- Combinatorics - a systematic study of counting methods
- Probability Theory and Statistics - a discipline dealing with random processes
- Number Theory - the study about divisibility and prime numbers
- Graph Theory - deals with theoretical and practical problems which can be modelled by edges and nodes.
- Knot Theory - a systematic study of knots and links
- Game Theory - a search for winning strategies in games
- Theoretical Physics - a mathematical description of the physical world
- Theoretical Computer Science - a foundation of computing