Mathematical reasoning/Introduction/Section

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In a mathematical argumentation, one tries to establish the truth of a claim by only using accepted principles. Typically, there is an audience which one would like to convince about the claim. Argumentation exists in many different contexts, in sciences, in politics, in personal relationships. There exist principles and patterns of argumentation which are typical for each context. In the political context, one invokes in general widely accepted principles like human rights, the constitution, the people and what they want, in order to enforce political decisions. The experience shows that the arguments used there are not good enough to convince everybody, and that also the interests of specific groups are represented.

Also in a mathematical argumentation one tries to establish the truth of a claim (that a computation procedure is correct, that a model is appropriate). The tools used, the strictness of the argumentation, do also depend here on the audience, the previous knowledge and motivation, the relation (bond, trust) between the persons involved (like teacher and student).

A mathematical argumentation on the scientific level shows certain standards of argumentation. A scientific argumentation exhibits the following items.

    • The strong presence of technical terms, which have to be defined and have to be used according to their definitions.
    • The existence of quite few basic principles.[1]
    • The use of logic to conclude new knowledge.
    • The free usage of knowledge, which has been already established in the sciences.
    • The free accessibility and verifiability of the results.[2]
    • The claim that the validity of the knowledge is independent of subjective wishes and values,[3]
    that it is timeless and independent of culture.[4]

In a mathematical argumentation, these items are in particular patent.[5] A strong hint for this is that there exists even a specific term for a mathematical argumentation: proof. A proven mathematical claim is called a theorem.

    • All mathematical concepts are defined precisely using only concepts which have already been defined. The definitions are built in such a way that every reasonable mathematical object either satisfies this concept or not, independent of whether we are always able to decide this.[6]
    • Mathematics is constructed
    (since about 130 years) on sets. It is organized in an axiomatic-logical way, but motivated by phenomena in the real world.
    • Logic is the main tool of mathematics. There exists
    (in principle) a complete list of allowed conclusions in propositional logic and in predicate logic.[7]
    • Proven mathematical statements
    (theorems) are used again and again.[8] This means that for a systematic account (like a lecture or a book) of an area of mathematics, the foundational material comes first and on this foundation more complex statements are built. If a theorem already proven is used somewhere, then we do not reflect about this theorem again, we only check whether in the current situation all conditions are fulfilled which the theorem requires.
    • Mathematics is published in journals and books, it is taught in lectures, it is available on the World Wide Web and in libraries.[9]
    • Mathematics is developed in a worldwide community.[10]
  1. Here runs the border between science and philosophy.
  2. This is a big difference to esoterism, where "knowledge“ is passed on to the disciples only under specific conditions (secrecy, dignity).
  3. That does not mean at all that gaining knowledge and making discoveries is without feelings. To the contrary, doing science is fun.
  4. Though the generation of knowledge is heavily dependent on age and culture.
  5. However, in mathematics an important point of the natural sciences is missing, observation, experience, experiments. Therefore, mathematics if often not considered as a natural science. But to consider mathematics as belonging to the humanities is also difficult, hence it is often considered to be a structural science.
  6. In particular, definitions by example of the form "something like“ are not allowed.
  7. This is the subject of mathematical logic.
  8. They are also not patentable.
  9. This might be different for security-relevant cryptological research.
  10. Though the major part is still in the industrialized countries, but other countries are catching up quickly.