|Systems, philosophy of science: systems are compilations of rules for the formation of statements on a previously defined subject domain. Apart from the - usually recursive - rules for the combination of expressions or signs, the specification of the vocabulary or sign set of the system is also required. See also axioms, axiom systems, theories, strength of theories, expressiveness, rules, order, recursion, models, structure, system theory._____________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
System/Aspects/Provability/Waismann: One could also have proved the equivalences in Russell's system
~p equi ~p v ~p
p v q equi ~(~p v ~p ) v~(~q v ~q)
but would one have expressed with this Sheffer's discovery? Not at all! One could speak of the discovery of a new aspect.
Again the question: could one look for this aspect? No. That something can be seen in this way can only be seen when it is seen.
That one aspect is possible is only seen when it is there. You can simply underline the newly discovered, so you give a new sign.
The formulas with the underlining do something different than those without underlining, they make the new structure visible.
E.g. Suppose there is a tribe of people somewhere who owns our decimal system, and calculates exactly as we do, but infinite decimal fractions remain unknown to them. People stop the division, e.g. at the 5th digit.
1/3 = 0.333333.
The periodicity would not be noticeable to them, they would not have to think that this always goes on like this.
After the discovery of the infinite decimal fractions one "sees" the calculation differently! This is the discovery that one sees the infinite possibility of progressing into the calculation.
The emphasis on the return of the rest is the expression that he has discovered the induction. We must not forget that the division with underlining is a different type of calculation.
E.g. (5 + 3)² = 5² +2 x 5 x 3 + 3². According to one calculation this is at the same time a proof for (7 +8)²= 7² + 2 x 7 x 8 + 8², but not according to the other calculation!
We would sometimes have to underline the different digits, sometimes underline them twice.
For example, is the x times x not the same as x²? It is a new system._____________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