postulate
As postulate (of Latin postulatus , Demanded, Requested, claimed in court or Alleged) one is principle for a discussion, a theory or a formal system called, of any new terms introduced, but not from the given definitions derived can be. A postulate is considered an axiom if other theorems of the system or everyday experience can be derived from it, the validity of which is already known or has been decided. The validity of a postulate can be attacked, contested and refuted at the level of the metatheory , e.g. B. if another sentence is found in its place, which has at least the same justification force.
mathematics
In mathematics , unproven or unprovable statements that should be assumed to be true in conclusions or systems of proof are also called postulates. Axioms are also understood as purely logical principles of a system, whereas postulates are understood as principles that do not only contain logical symbols.
The use of postulates comes from Euclidean geometry , in which a distinction is made between definitions, postulates and principles. Euclid's text speaks of aitēmata (postulates) and koinai ennoiai (axioms, literally common terms , Latin communes animi conceptiones ). The recognition of a thesis or theorem was not required, but that a certain construction is possible, e.g. B. that any two points can be connected with exactly one straight line, or that a circle can be drawn around every center point with every radius. Today, in mathematical practice, a clear distinction is no longer made between requirement and principle, i.e. postulate and axiom.
Proclus distinguishes aitēmata as confirming by a proof (similar to the hypotheseis of Aristotle) from axiomata as not requiring any proof. He also assigns postulates to geometry and axioms to all sciences dealing with quantities and spatial expansion. Archimedes understands axiomata also definitions and calls the postulates lambanomena .
In older logicism an attempt was made to achieve a general foundation without axioms, only on the basis of logical definitions. It was assumed that the definitions and axioms relate to necessarily valid facts as their extension , and that their role as fundamental propositions behind which one cannot go back is essential. In formalism, however, axiomatizations were understood as arbitrary determinations of formal systems, which differ from one another through internal and external quality criteria (e.g. decidability and completeness (logic) , expressibility of known mathematical propositions). For formalism, the simplest mathematical terms are implicitly defined by the axioms set up. While logicism eliminates postulates, postulates and axioms coincide in formalism.
philosophy
The Aristotelian philosophy of science distinguishes between
- Principle as necessary by itself: axiom understandable for everyone
- Principle as a requirement ( hypothesis ): understandable for the learner in the relevant science
- Principle as a postulate ( aitēma ): not understandable for the learner in the relevant science or contrary to his opinion; including principally provable sentences that are currently accepted or used without proof.
In Immanuel Kant's terminology, "postulate" is a "practical, immediately certain sentence or principle that determines a possible action, which is assumed to be immediately certain of the way it is carried out." (Immanuel Kant: AA IX, 112– Logic lecture) He distinguishes mathematical postulates from postulates of practical reason: The postulates of practical reason are a subjectively necessary assumption for moral action, the mathematical postulates are for Kant objectively necessary and true propositions, which do not follow from concepts, but on the idea of mathematical objects are recognized a priori as constructs of the imagination (Immanuel Kant: AA V, 11 - Critique of Practical Reason , cf. also Immanuel Kant: AA III, 198 - Critique of Pure Reason , A 234 / B 286)
In epistemology and philosophy of science, the term “postulate” is sometimes used more generally in the sense of a normative requirement .
Moritz Schlick advocated the thesis: “Postulates in the sense of the old philosophy do not exist” - namely as “a rule that we must adhere to under all circumstances”. Rather, “postulate” should designate an empirically expedient instruction for the formation of statements.
physics
In today's physics, the terms “postulate” and “axiom” are used interchangeably. Since physical theories can be axiomatized in different ways, a certain physical statement can have the status of an axiom in one formulation of the theory, but the status of a theorem in another, equivalent formulation. For example, the classical point mechanics can optionally be formulated on the basis of Newton's laws , the Lagrange formalism or the Hamilton-Jacobi formalism . In the former case, for. B. Newton's 3rd law has the status of a postulate or an axiom, in the other two cases it is a theorem .
Web links
Individual evidence
- ^ Entry postulatus in: Charlton T. Lewis, Charles Short, A Latin Dictionary .
- ↑ Cf. for example Anton Hügli , Poul Lübcke: Philosophielexikon , Kröner, Stuttgart 1991, sv "Postulat".
- ↑ Cf. In primum Euclidis Elementorum librum commentarii, ed. G. Friedlein, Teubner, Leipzig 1873, digitized version, pages 181-183.
- ↑ Analytica posteriora 76b 23-34.
- ↑ Immanuel Kant, Collected Writings. Ed .: Vol. 1-22 Prussian Academy of Sciences, Vol. 23 German Academy of Sciences in Berlin, from Vol. 24 Academy of Sciences in Göttingen, Berlin 1900ff., AA . IX, 112 - Logic Lecture
- ↑ Immanuel Kant, Collected Writings. Ed .: Vol. 1-22 Prussian Academy of Sciences, Vol. 23 German Academy of Sciences in Berlin, from Vol. 24 Academy of Sciences in Göttingen, Berlin 1900ff., AA . V, 11– Critique of Practical Reason
- ↑ Immanuel Kant, Collected Writings. Ed .: Vol. 1-22 Prussian Academy of Sciences, Vol. 23 German Academy of Sciences in Berlin, from Vol. 24 Academy of Sciences in Göttingen, Berlin 1900ff., AA . III, 198– Critique of Pure Reason , A 234 / B 286
- ↑ Moritz Schlick: The causality in contemporary physics , in: Die Naturwissenschaften 19 (1931), 145-62, here 155; also in: J. Friedl / H. Rutte (eds.): Die Wiener Zeit : Essays, Articles, Reviews 1926-1936, Springer, Vienna 2008, pp. 231–292, here 269.