„All pure mathematics — Arithmetic, Analysis, and Geometry — is built up by combinations of the primitive ideas of logic, and its propositions are deduced from the general axioms of logic, such as the syllogism and the other rules of inference.“ (p. 116) #Russell #PureMathematics #Logic
Archiv der Kategorie: Logik
Hofstadter: Incompleteness Theorem
Zitat
„All consistent axiomatic formulations of number theory include undecidable propositions. […] If consistency is the minimal condition under which symbols acquire passive meanings, then its complementary notion, completeness, is the maximal confirmation of those passive meanings. Where consistency is the property that ‚Everything produced by the system is true‘, completeness is the other way round: ‚Every true statement is produced by the system‘. […] Gödel’s Incompleteness Theorem says that any system which is ’sufficiently powerful‘ is, by virtue of its power, incomplete, in the sense that there are well-formed strings which express true statements of number theory, but which are not theorems. (There are truths belonging to number theory which are not provable within the system.)“ (p. 17, 100f [pass.]) #Hofstadter #Gödel #Escher #Bach #IncompletenessTheorem #NumberTheory
Whitehead: Logical
Zitat
„The term ‘logical’ has its ordinary meaning, including ‘logical’ consistency, or lack of contradiction, the definition of constructs in logical terms, the exemplification of general logical notions in specific instances, and the principles of inference“. (p. 71) #Whitehead #logical #LackOfContradiction
Whitehead: Coherence
Zitat
„‚Coherence,’ as here employed, means that the fundamental ideas, in terms of which the scheme is developed, presuppose each other so that in isolation they are meaningless.“ (p. 71) #Whitehead #coherence
Peirce: Deduction, Induction, and Retroduction
Zitat
„Concerning the question of the nature of the logical validity possessed by Deduction, Induction, and Retroduction, which is still an arena of controversy, I shall confine myself to stating the opinions which I am prepared to defend by positive proofs.“ (p. 22) #Peirce #deduction #induction #retroduction