|Calculus: a calculus is a system of symbols for objects (which are not further specified) as well as rules for the formation of expressions by the composition of these symbols. There are other rules for transforming composite expressions into other expressions. As long as no specified objects are accepted for the individual symbols, the calculus is not interpreted, otherwise interpreted._____________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. |
Christian Thiel on Calculus - Dictionary of Arguments
Thiel I 20/21
Calculus/Ontology/Mathematics/Thiel: Calculus Theory: It is part of the mathematician's activity both to proceed according to the rules of the calculus and to reflect on them. The boundary between mathematics and meta mathematics is questionable. The demarcation serves only certain purposes, it is sometimes obstructive: e.g. nine-probe: a number is divisible by 9, if its cross sum is divisible by 9.
Thiel I 211
Calculus/Thiel: Example: The constructive arithmetics with the calculus N and the construction equivalence of counting signs provides an operative model of the axioms. Mathematicians do not do this in practice or in books. Practice is not complete.
Insisting on "clean" solutions only comes up with meta mathematical needs.
Rule arrow: >>
The following applies to all: V
Rule (VP) A(y) imp B >>Vx A(x) imp B.
Everyday language translation: the rule (VP) states that we may pass from a valid implication formula A(y) imp B, in which "y" occurs as a free variable, to one in which the statement form "A(y)" is quantified by an existential quantifier.
Clarification: "y" must not occur freely in the conclusion of the rule and "x" must be free for yx, i.e. not within the sphere of influence of an already existing quantifier with the index "x".
However, this applies only to evidence practice. Evidence theoretical considerations require further precision. The object of the formalization can be differentiated to such an extent that we have to speak of a new object.
Thiel I 216
A "fully formalized" calculation for arithmetics in Lorenzen consists of 75 rules, including those with 7 premises.
We can "linearize" such rule systems: i.e. introduce basic rules without premises and then continue in ascending order.
The complete syntactic capture of evidence is ideal._____________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. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
Philosophie und Mathematik Darmstadt 1995