BranchesHistoryHelpLogin
Welcome guest
You're not logged in.
245 users online, thereof 0 logged in

Thank you for your message! I'll respond as soon as possible.

BookOfProofs

BookofProofs

About

About

BoP is an open book dedicated to mathematics, physics, and computer science. Its goal is to broaden the public knowledge of the axiomatic method.

Motto

Motto

The ability to formulate mathematical proofs using the axiomatic method should be taught as a basic skill like reading or writing.

This is what BoP is all about!

Project

Project

BoP was launched in February 2014 and started as a collaborative effort, but it was too hard to mobilize the community. For the time being, BoP remains a single-author project.

Feedback

Feedback

If you are still interested in co-authoring or want to provide any kind of feedback, please

contact me

What is the "Axiomatic Method"?

The axiomatic method is one of the greatest inventions of mankind, without which the development of mathematics, physics and technology over the centuries wouldn't have been possible.

  1. 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).
  2. Given all statements in your theory, logically derive new statements which are true (and call them propositions or theorems).
  3. Add the newly derived theorems to your theory.
  4. 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"):

  1. 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.
  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.
  3. After this proof, our "Little Bird Theory" became bigger. Now, it consists of the axioms 1 and 2, including the statement 1.
  4. 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.

Selected Publications

All publications listed below can be downloaded for free. Enjoy! Some of them are still work-in-progress. Usage or reproduction may be restricted by the license referenced or stated in each publication.

Logic
Image by ElisaRiva from Pixabay
Set Theory
Image by Clker-Free-Vector-Images from Pixabay
Number Systems and Arithmetics
Image by Dean Norris from Pixabay
Algebra
Image by OpenClipart-Vectors from Pixabay
Calculus
Image by OpenClipart-Vectors from Pixabay
Topology
Image by Chrom72 from Pixabay
Geometry
Image by gingertea from Pixabay
Euclid's Elements
Image by maxek from Pixabay
Combinatorics
Image by Clker-Free-Vector-Images from Pixabay
Probability Theory and Statistics
Image by OpenClipart-Vectors from Pixabay
Number Theory
Image by Clker-Free-Vector-Images from Pixabay
Graph Theory
Image by Gerd Altmann from Pixabay
Knot Theory
Image by Clker-Free-Vector-Images from Pixabay
Game Theory
Image by Peggy und Marco Lachmann-Anke from Pixabay
Theoretical Physics
Image by OpenClipart-Vectors from Pixabay
Theoretical Computer Science
Image by Gerd Altmann from Pixabay
History of Mathematics
Image by Momentmal from Pixabay
Dudeney's Amusements in Mathematics
Image by Karen Arnold from Pixabay