Lexicon of Arguments

Philosophical and Scientific Issues in Dispute
 
[german]


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Theses I
Theses II

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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.
I 213
Insisting on "clean" solutions only comes up with meta mathematical needs.

Terminology/Writing:
Rule arrow: >>
Implication imp
The following applies to all: V
Rule (VP) A(y) imp B >>Vx A(x) imp B.
I 214
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.
I 217
We can "linearize" such rule systems: i.e. introduce basic rules without premises and then continue in ascending order.
I 219
The complete syntactic capture of evidence is ideal.
>Proofs, >Provability, >Syntax, >Formalization.

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