Economics Dictionary of ArgumentsHome![]() | |||
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Induction: Induction in logic is a type of reasoning in which we draw general conclusions from specific observations. It is the opposite of deductive reasoning, where we draw specific conclusions from general premises. See also Deduction, Grue, Generalization, Generality, Conclusions._____________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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Luitzen E. J. Brouwer on Induction - Dictionary of Arguments
Waismann I 70 Proof/Induction/Intuitionism/Brouwer/Waismann: If it is said that the proof applies to all numbers, one has to be clear that one only determines by the proof the meaning of the word "all". >Proofs, >Provability, >All/every. And this meaning is different than e.g. "All the chairs in this room are made of wood". For when I deny the last statement, this means that there is at least one that is not made of wood. If, however, I deny "A applies to all natural numbers", that means only: One of the equations in the proof of A is false, but not, there is a number for which A does not apply. >Negation, >Existential quantification, >Universal quantification, >Quantification. The general formula in mathematics and the existence statement do not belong to the same logical system. (Brouwer: the incorrectness of a statement does not mean the existence of a counterexample). >Existence, >Existence statements, >Non-existence. Now the performance of the induction becomes clear: it is not a conclusion that carries to infinity. The set a + b = b + a is not an abbreviation for infinitely many individual equations, as little as 0.333 ... is an abbreviation, and the inductive proof is not the abbreviation for infinitely many syllogisms (VsPoincaré). In fact, we begin with the formulation of the formulas a+b = b+a a+(b+c) = (a+b)+c a whole new calculus, which cannot be inferred from the calculations of arithmetic in any way. >Calculus, >Derivation, >Derivability._____________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. |
Brouwer, L. E. J. Waismann I F. Waismann Einführung in das mathematische Denken Darmstadt 1996 Waismann II F. Waismann Logik, Sprache, Philosophie Stuttgart 1976 |