Axioms/Applied sciences/Introduction/Section

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In mathematics, we encounter many structures which occur again and again. E.g., the rational numbers and the real numbers share many common properties, but they also differ in regard with certain other properties. This observation is the fundament for the axiomatic approach to mathematics. In this approach, one puts several structural properties which appear in a certain context consistently into a new concept. The goal of this is to infer in a logical way further properties from basic properties. The argumentation does then not take place on the level of specific familiar examples like the real numbers, but on the level of the properties themselves in a logical-deductive manner. The gain of this method is that one has to do the mathematical conclusions only once on the abstract level of the properties, and these conclusions hold then for all models which fulfill the basic properties. At the same time, one recognizes logical dependencies and hierarchies between the properties. Basic properties of mathematical structures are called axioms.

In the axiomatic approach, the properties (their principles and rules) are in the center. Mathematical objects which obey these rules are then examples or models for these concepts. In particular, one chooses properties which, on one hand, are easy to formulate, and which, on the other hand, allow strong conclusions. The advantages of this approach are the following items.

    • The mathematical objects are based on a set-theoretical-logical foundation, there is no need to rely on intuitive assumptions.
    • It is always clear which argumentation to establish a certain property is allowed: only the logical deduction of the property from the axioms.
    • Fundamental properties are stressed and get an important place. One develops a hierarchy between fundamental properties and deduced properties.
    • Structural similarities become visible, which are not visible from an intuitive standpoint.
    • Many statements which can be deduced from an axiomatic system do not need the complete axiomatic system, only some part of it. Therefore, it is possible to group the axioms into smaller units. If one can deduce the statement from a subset of the axioms, the statement is then true for every mathematical object which fulfils only this subset.
    • By giving "counter-examples“ one can show that certain properties do not follow from certain other properties.
    • This approach is economically efficient as it avoids the repetition of conclusions.

As disadvantages, one can consider the following items.

    • huge conceptual effort.
    • Abstract, sometimes formal and counter-intuitive procedure.
    • Seemingly "trivial properties“ need a justification if they are not explicitly listed in the axiomatic system.