|Syntheticity, philosophy: something is synthetic that comes from a composition of previously separate entities and has new qualities in the composition. In the philosophical discussion the analytic is opposed to the synthetic the one which satisfies stricter criteria, by not introducing qualities which were not previously found in the object of investigation. See also analyticity/syntheticity, analytical, analysis, emergence._____________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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|Waismann I 70
Induction/Brouwer/Poincaré/Waismann: the power of induction: 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 well as 0.333 ... is not 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. But:
Principle/Induction/Calculus/Definition/Poincaré/Waismann: ... this is the correct thing in Poincaré's assertion that the principle of induction cannot be proved logically. 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"._____________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.
Einführung in das mathematische Denken Darmstadt 1996
Logik, Sprache, Philosophie Stuttgart 1976