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The author or concept searched is found in the following 4 entries.
| Disputed term/author/ism | Author |
Entry |
Reference |
|---|---|---|---|
| Calculus | 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. 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. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Formal Language | Thiel | Thiel I 80 Formal Language of Logic/Lorenzen/Thiel: Paul Lorenzen: "protological approach": rule system for the production of figures linearly composed of 0 and + . "A" represents such figures as schematic letter. Rules: (1) > + (2) A > A 0 (3) A > + A +. According to this "calculation" e.g. the figure ++00+ can be produced: (1), 2 times (2), then (3). I 80/81 Each figure that can be made must either have a 0 on the right or a + on the left. Test figure 0++ therefore does not work. We introduced the additional rule: (4) A > 0 A + it would be producible. On the other hand, the following rule would not allow new figures: (5) A > + + A. This is called "redundancy" (in meta mathematics "admissibility"). Such control systems can also be described as "operational logic". I 83 They can be used to introduce punctors (I 82 Example v) Proto logic is therefore still ahead of logic. >Junctions, >Logic, >Introduction, >Calculus, >Formalization, >Systems. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Proof Theory | Hilbert | Berka I 384 Proof Theory/Hilbert: first, the concepts and propositions of the theory to be examined are represented by a formal system, and treated without reference to their meaning only formally. I 385 Proof Theory: this (subsequent) investigation is dependent on the logical meaning of its concepts and conclusions. Thus formal theory is compared with a meaningful meta theory (proof theory)(1). Berka I 395 Proof Theory/Hilbert: basic thought, thesis: everything that makes up existing mathematics is strictly formalized, so that the actual mathematics becomes a set of formulas. New: the logical signs "follow" (>) and "not". Final scheme: S S › T T Where each time the premises, i.e. (S and S > T) are either an axiom, or are created by inserting an axiom or coincide with the final formula. Definition provable/Hilbert: a formula is provable if it is either an axiom or an axiom by insertion from it, or if it is the final formula of a proof. >Proofs, >Provability. Meta-Mathematics/proof theory/Hilbert: meta mathematics is now added to the actual mathematics: in contrast to the purely formal conclusions of the actual mathematics, the substantive conclusion is applied here. However, only to prove the consistency of axioms. >Axioms, >Axiom systems, >Axioms/Hilbert. In this meta-mathematics, the proofs of the actual mathematics are operated upon, and these themselves form the subject of the substantive investigation. >Meta-Mathematics. Thus the development of the mathematical totality of knowledge takes place in two ways: A) by obtaining new provable formulas from the axioms by formal concluding and B) by adding new axioms together with proof of the consistency by substantive concluding. >Consistency, >Material implication. Berka I 395 Truth/absolute truth/Hilbert: axioms and provable propositions are images of the thoughts which make up the method of the previous mathematics, but they are not themselves the absolute truths. >Truth/Hilbert. Def absolute truth/Hilbert: absolute truths are the insights provided by my proof theory with regard to the provability and consistency of the formula systems. Through this program, the truth of the axioms is already shown for our theory of proof(2). 1. K. Schütte: Beweistheorie, Berlin/Göttingen/Heidelberg 1960, p. 2f. 2. D. Hilbert: Die logischen Grundalgen der Mathematik, in: Mathematische Annalen 88 (1923), p. 151-165. |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Theories | Thiel | Thiel I 73 Theory/Mathematics/Thiel: The term "meta mathematics" had already appeared in a different meaning in the 19th century, reformulated by Hilbert. Hilbert had proved that in Euclid not all of the properties used in the geometric propositions are really developed from the basic properties recorded in the axioms. So it was incomplete. >Axioms, >Axioms/Hilbert. After Cantor's work at the end of the 19th century, it looked as if one could actually find a complete axiom system. Admittedly, no meta mathematics would have been necessary for this. >Completeness, >Incompleteness. I 75 Meta mathematics makes a difference between the proof that a statement A cannot be refuted (the proof that its opposite is not justifiable) and a "positive" justification of A. The first is a rebuttal of ~A thus a proof of ~~A, the second a proof of A. I 76 New: in meta mathematics the existential statements are interpreted more strictly. Anyone who now claims the existence of evidence must also indicate a verifiable way of constructing such evidence. Def "effective" or "constructive" assertion of existence. >Proofs, >Provability, >Syntax. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
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