|Method: a method is a procedure agreed on by participants of a discussion or research project. In the case of violations of a method, the comparability of the results is in particular questioned, since these no longer come from a set with uniformly defined properties of the elements._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
|Friedrich Waismann Suchen und Finden in der Mathematik 1938 in Kursbuch 8 Mathematik 1967
Method/Searching/Finding/Waismann: Can you describe exactly what you are looking for? For example, a student is looking for the construction of a regular pentagon. Caution: the design does not depend on accuracy, because also an inaccurately designed construction can be correct!
From this it can already be seen that the construction has no necessary connection with the concept of a figure, which has five equal sides when measuring.
In reality, we have quite different concepts of a pentagon, which roughly correspond to each other like the physical geometry to the mathematical one. We can speak of a measured and a constructed pentagon. (The constructed is, of course, not an ideal one, which stands beside the measured, but a concept, which is determined by the method of construction.)
E.g. Let us search for the root of 436. Let us assume that I randomly extract numbers, then multiply them by themselves, and compare with 436.
How is it shown that I was looking for it? This was not to be seen in my calculating operations.
How can I recognize my search? Probably only because I have a feeling of disappointment at the end.
Let us suppose that I calculate root 436 according to the usual calculation method, then I will not come into the situation of speaking about my feelings.
For the process is what is called the search of the root here, while the other is not a search for this root. One and the other can only be sought in a very different sense.
Finding/Mathematics/Waismann: If a description is complete, then the object is found. This is not the case in the life situation.
The activity does not yet contain the objective. Some will not call it a search. For the search in mathematics it is characteristic that one cannot describe the searched beforehand or it can only be decribed seemingly.
Looking along a way and looking for the way are two completely different things.
E.g. Multiplication tasks in numerical arithmetic: there are infinitely many questions and answers here, but they all belong to one and the same schema.
Another system is the elementary trigonometry. If I know their rules, I can control the sentence:
sin = tgx times cos x, but not the sentence:
sin x = x x³/3! + x5/5 !.
The latter sentence cannot be understood from elementary trigonometry. The sine of the elementary trigonometry and the sine of the analysis are quite different concepts.
E.g. It is called a task to obtain the root from three, or to divide an angle into three parts. (compare also: tasks of integrating with those of differentiation).
The former is considered easier, but it is forgotten that they are tasks in a completely different sense. The difference, however, is not a psychological one.
The great mathematical problems, if they are solvable, are solved by the construction of new calculi. They are therefore suggestions for the construction of such calculi.
One first searches the room, that is, the calculus, the conceptual system in which his question gets a clear meaning._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. The note [Author1]Vs[Author2] or [Author]Vs[term] is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
Einführung in das mathematische Denken Darmstadt 1996
Logik, Sprache, Philosophie Stuttgart 1976