The aim of this section is to clarify and emphasize the logical structure of a mathematical text. As a rule, a mathematical text is divided into definitions, sentences and proofs. The reader often encounters the following types of structures:
Definition: New concepts are being introduced (defined). In a more informal interpretation new concepts can also be introduced directly in the text (as often done in these notes). The purpose of the definition is to unambiguously anchor (define) concepts. The author of the definition agrees with the reader on what it is the meaning of a term. This is very important. Without clearly defined terms, there is a danger that two people would not be able to agree, because everyone is talking about something else but both use the same name for it.
Theorem: An important statement that deserves a numerical designation in the text, or even to be named after its authors.
Proof: A structure containing evidence of a previous statement, for example a theorem, but also a lemma, corollary (see in the following). Since it is typically longer than the statement formulation, its ending is usually marked with an ending symbol.1 In BIE-ZMA we usually use the Halmos symbol of tombstone $\square$. The reader can also often find the abbreviation Q.E.D. coming from the Latin quod erat demonstrandum („as it was to be proven“). The reader can find more about proofs in section 2.3.
You might also see:
Lemma 2: An auxiliary claim that does not have a wider application in itself3, but it is used in the proof of one of the immediately following theorems.
Corollary: Claims very straightforward from previous theorems, or reformulation of previous theorems into another context. Typically with a very simple proof (practically just a straightforward utilization - i.e. application - of previous theorems).
At this point, I would like to take a short note about a frequent student's „mistake“. It is often the case to encounter the sentence „to define a theorem“ This points to a fundamental misunderstanding by the users of this senseless words. They probably misunderstand the word „define“ with „verbatim copy“. „To define a theorem“ is not possible in principle. You can define a term and then give a certain statement about that term, that is, the theorem. But here you have to prove, verify that the theorem is true. Fortunately, claims in mathematics cannot be defined.
Readers might be more familiar to notations using XML language. The structure of a mathematical text can then be seen as follows:
<definition>
...
</definition>
<theorem>
...
</theorem>
<proof>
...
</proof>
Apparently, presenting the reader with the text in this way would be typographically crazy. However, it should be noted that the source LaTeX code of this document uses this approach.
Of course, most mathematical texts are not composed only of the above described structures. Additional comments, examples or diagrams are often given for the reader's convenience, explaining further context regarding the discussed topic.
This structured approach of writing can be found not only in mathematics, but also in other technical and professional literature. For example, in the IT field, let us mention the documentary genre, or specification of standards, where a strong emphasis is placed on the logical structure of the text.