By Venema G.

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Traditionally the end of a proof was indicated with the abbreviation QED, which stands for the Latin phrase quod erat demonstrandum (which was to be demonstrated). In this book we mark the end of our proofs with the symbol . Like each individual proof, the overall structure of the collection of theorems and proofs in an axiomatic system should be logical and sequential. Within any given proof, it is legitimate to appeal only to the axioms, a theorem that has been previously stated and proved, a deﬁnition that has been stated earlier, or to an earlier step in the same proof.

Another function of deﬁnitions is to identify and highlight key structures and concepts. Axioms The second part of an axiomatic system is a list of axioms. 1 The axioms are statements that are accepted without proof. They are where the subject begins. Everything else in the system should be logically deduced from them. All relevant assumptions are to be stated in the axioms and the only properties of the undeﬁned terms that may be used in the subsequent development of the subject are those that are explicitly spelled out in the axioms.

Once things have been clariﬁed in this way, the kind of logical reasoning that characterizes mathematics becomes an incredibly powerful and effective tool. The history of mathematics is full of examples of surprising practical applications of mathematical ideas that were originally discovered and developed by people who were completely unaware of the eventual applications. 4 1. It is said that Hilbert once illustrated his contention that the undeﬁned terms in a geometry should not have any inherent meaning by claiming that it should be possible to replace point by beer mug and line by table in the statements of the axioms.