|Analogy: formal parallelism. Intends show that from a similar case, similar conclusions can be drawn._____________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. |
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Analogies/Science/Mathematics/Searching/Finding/Waismann: The mathematician who searches, proceeds with analogies.
1. He thinks of constructions of other regular figures, he is familiar with.
2. He thinks of a figure, whose sides are actually equal in length. Important: the connection with the empirical and the mathematical figure exists, but is a non-mathematical!
They are, therefore, non-mathematical aspects which are the leading stars of mathematical research. The question in mathematics does not give the investigation an objective, but only a direction.
E.g. Brouwer's question whether there is in the development of the number π a place where the digits 0123456789 follow one another. The term "development of the number π" does not help me with the question.
Suppose we gain the possibility to answer the question by finding a formula which indicates the digits of π. Thus, we trace back the question, whether this sequence exists, to another question.
Then we believe that it is still the same concept. One thinks that one sought both times in the same space, namely in the development of π.
The misconception is that a strip passes us, on the other hand the idea of the strip has led to the direction of this whole investigation.
E.g. Suppose, for example, that a law on the distribution of the prime numbers is found with the help of function theory. Then one believes that one has discovered a new property in the previous concept of the prime number.
One does not see that the term has been inserted into a new context, that is, has created a new prime number concept! The two prime number terms, however, correspond to each other as the concept of the cardinal number to that of the positive whole real number. They do not coincide, they only correspond to each other.
E.g. After the discovery of the North Pole, we have not two earths, one with and one without the North Pole, but after discovering the law of the prime number distribution, we have two types of prime numbers._____________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.
Intuitionism: An Introduction (Study in Logic & Mathematics) 1971