Philosophy 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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Henri Poincaré on Induction - Dictionary of Arguments
Waismann I 70 Induction/Brouwer/Poincaré/Waismann: the power of induction: it is not a conclusion that carries to infinity. The sentence a + b = b + a is not an abbreviation for infinitely many individual equations, as well as 0.333 ... is not an abbreviation, and the inductive proof is not the abbreviation for infinitely many syllogisms (VsPoincaré). In fact, with the formulation of the formulas we begin 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, >Infinity, >Abbreviations, >Equations. But: Principle/Induction/Calculus/Definition/Poincaré/Waismann: ... this is the correct thing in Poincaré's assertion: the principle of induction cannot be proved logically. >Proofs, >Provability. VsPoincaré: But he does not represent, as he thought, a synthetic judgment a priori; it is not a truth at all, but a determination: If the formula f(x) applies for x = 1, and f(c + 1) follows from f(c), let us say that "the formula f(x) is proved for all natural numbers". --- A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967 46 Induction/PoincaréVsHilbert: in some of his demonstrations, the principle of induction is used and he asserts that this principle is the expression of an extra-logical view of the human mind. Poincaré concludes that the geometry cannot be derived in a purely logical manner from a group of postulates. >Geometry, >Postulates, >Derivation, >Derivability. 46 Induction is continually applied in mathematics, inter alia also in Euclid's proof of the infinity of the prime numbers. >Euclid, >Primes. Induction principle/Poincaré: it cannot be a law of logic, for it is quite possible to construct a mathematics in which the principle of induction is denied. Hilbert, too, does not postulate it among his postulates, so he also seems to be of the opinion that it is not a pure postulate._____________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. |
Waismann I F. Waismann Einführung in das mathematische Denken Darmstadt 1996 Waismann II F. Waismann Logik, Sprache, Philosophie Stuttgart 1976 |
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Ed. Martin Schulz, access date 2024-04-19