Economics Dictionary of ArgumentsHome
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| Mathematics: Mathematics is the study of patterns and relationships. It encompasses arithmetic, algebra, calculus, geometry, trigonometry, topology, and others. Mathematics is used in science, engineering, business, and finance. See also Physics, Logic._____________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 Mathematics - Dictionary of Arguments
Thiel I 242/243 Brouwer/Thiel: For Brouwer, all laws of formal logic are only extrapolations from ratios of finite sets. Some fail in infinite wholes. >Sets, >Set theory, >Infinity, >Finiteness. Following Jacques Herbrand there are the following criteria for the procedure of meta-mathematics (Hilbert himself has no catalog) of driteria: 1. Operate only with a finite number of objects and functions. In particular, each rule of forming expressions and each conclusion rule may contain only a finite number of premises. >Premises, >J. Herbrand. 2. The value of each function used for each argument must be unambiguously calculabe. >Unambiguity, >Functions, >Calculability. 3. Never must the set of all objects belonging to an infinite set be considered. Accordingly, the definition of a mathematical object must not be Definition > impredicative, in the sense that in the defining condition a set containing this object ("later") as an element occurs. >Impredicativity, >Predicativity. 4. The existence of an object is to be asserted only by demonstrating the same or a constructive procedure. >Existence Assertion. 5. Any assertion of a statement about "all x" of a domain must be accompanied by an instruction, how a statement can be proved for an arbitrarily presented xo from the domain A(xo). >Universal statement. I 242 Definition finite: Prohibition of the (carefree) dealing with infinite wholes. >cf. >Finitism. Hilbert accepted the new starting situation provoked by Brouwer. There have been prominent examples of errors in the history which have occured through false transfers from finite to infinite wholes. >Finiteness._____________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. T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
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