|Proof in logic, mathematics: finite string of symbols, which derives a statement in a system from the axioms of the system together with already proven statements._____________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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|Friedrich Waismann Suchen und Finden in der Mathematik 1938 in Kursbuch 8 Mathematik 1967
Proof/meaningful/senseless/Waismann: e.g. how is the statement "This man died five hours ago" to be proved? The application of a medical discovery is called the proof that it is so.
This discovery does not alter the meaning of the sentence "This man died five hours ago". The discovery discovered that a particular hypothesis is true.
In mathematics it is different: the mathematical proof could not be described before it was found.
"Soluble" means a structure that cannot be described without knowing it.
E.g. the method of decomposition introduces something completely new, just as the negative numbers are something entirely new.
Suppose we had only explained the multiplication, then terms such as "dividend" "divisor", etc. would have no meaning at all and the question about them would be misplaced.
E.g. if we are looking for an Easter egg in the room, the question about the concept of the Easter egg does not make any sense, because we can describe the Easter egg almost arbitrarily exactly.
For example, in the case of the threefold division of the angle, one cannot speak of possibility or impossibility as far as precise concepts are concerned.
Suppose, for example, in an arithmetic in which only the multiplication is known, one can ask "Is this number decomposed?", i.e. have we carried out a multiplication in which it emerged as a product? On the other hand one cannot ask: "Is it decomposable?"
The other question must be answered only when we go over to the next calculus of the division.
But even here it is not really meaningful to speak of a possibility, but of a rule. To say, we cannot disassemble the number, makes the wrong impression, we would try and then encounter an obstacle.
In reality, we are expanding our system and are not seeing new possibilities, but have new rules.
If one wants, one can continue to search with a circle and a ruler, which is not wrong in itself and is not forbidden by the proof.
It just does not mean anymore what it used to mean._____________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.
Einführung in das mathematische Denken Darmstadt 1996
Logik, Sprache, Philosophie Stuttgart 1976