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The Axiomatic Approach on BookofProofs

Modern mathematics uses the so-called axiomatic method - in each discipline, there is a small number of facts, called axioms, which are considered as absolutely certain and which do not have to be proven. Beginning from these axioms (sometimes complemented by definitions), one proves theorems using logical reasoning. Based on the proven theorems, one proves new theorems, and so forth. The axiomatic method is - like a snow ball - very powerful!

Criticism

The axiomatic method is very powerful as it allows to construct a complex theory from simple and (sometimes) easy to understand basic facts, called axioms. It has been very successful since about 300 B.C.E, when the Greek mathematician Euklid used it for the first (historically attested) time in his famous Elements. The axiomatic method has continued to be very successful over centuries and it still works very well today.

However, there are some substantial theoretical limits, known as the Gödel's Incompleteness Theorems. They basically state that (i) if we have a consistent theory derived from an axiom system \(\Sigma\), there will always be statements formulated in this theory, which cannot be proved or disproved using this theory; and (ii) it is impossible by means of this theory to prove its consistency (i.e. lack of contradictions). In other words, it is for us theoretically impossible (and for this reason it will never be possible in BookOfProofs) to establish a complete axiom system (i.e. one, which is sufficient to prove or disprove the truth of any statement formulated in the theory), nor it will ever be possible for us to ensure that there are no contradictions in the theory derived from this axiom system.

Another difficulty is that it is impossible to create an axiom system without implicitly using other axioms. For instance, even the above-mentioned Gödel's Incompleteness Theorems, proving the limits to mathematics, imply that each statement has to be either true or false. However, we know that this is only an idealization of our real-life experience (just think about statements like "I love you" or "It is 10 o'clock"). This idealization is known as the law of excluded middle). In addition, there are other logical systems (e.g. the fuzzy logic), which successfully do without the law of excluded middle and are still consistent.

Despite these theoretical (and philosophical?) difficulties, BookOfProofs consequently uses the axiomatic method as it is the best approach known up to date to establish generally excepted mathematical theories.


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