2.3 What is a proof?

The word proof raises irrational resistance in many students. In this chapter, we will try to whitewash its reputation. A proof is nothing more than a logical argument ensuring the validity of a claim. It is an answer to the inquiring question „why?“ In this chapter, we will try to outline the meaning of this term in a broader context and we will show some simple standard proofs.

Students at our faculty often come up with the notion that there is no need for proofs, it is only necessary to know the statements of theorems. However, this is a very short-sighted approach, especially for the following reasons.

  • As already mentioned, a proof is nothing but a logical argument. It is based on assumptions and conclusions are reached by logical steps. Therefore, learning a proof improves not only the knowledge of the studied objects, but also the argumentative and expressive skills. It develops the capability of unambiguously describing and expressing ideas.

  • The proof reveals to the student why the claim is true. It is then easier to remember its statement (e.g. its assumptions). Without studying the proof, the student loses understanding of the context and resorts to learning sentences by heart (which is not enriching for him4 nor manageable).

  • Most of the proofs, especially the so-called constructive ones, give direct guidance (algorithm) to solve problems.

  • No superior authority (teacher, professor, guru) other than logics decides, on the correctness of the proof, and thus the truth of the proven claim. Once proven, it stays proven forever. Such an absoluteness of mathematics is beautiful.

An old concise comparison says: studying mathematics without proofs is like playing football without a ball. In short, mathematics without proofs does not make much sense!