## Philosophy Dictionary of ArgumentsHome | |||

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Numbers: whether numbers are objects or concepts, has been controversial in the philosophical discussion for millennia. The most widely accepted definition today is given by G. Frege (G. Frege, Grundlagen der Arithmetik 1987, p. 79ff). Frege-inspired notions represent numbers as classes of classes, or as second-level terms, or as that with one measure the size of sets. Up until today, there is an ambiguity between concept and object in the discussion of numbers. See also counting, sets, measurements, mathematics, abstract objects, mathematical entities, theoretical entities, number, platonism._____________ 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. | |||

Author | Concept | Summary/Quotes | Sources |
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W.V.O. Quine on Numbers - Dictionary of Arguments I 219 Not all abstract objects are properties: numbers, classes, functions, geometric shapes, ideas, possibilities - give up or re-trace abstract objects - can be distinguished by the faithful use of "-ness" from concrete objects. - - - II 26 Numbers: quantification is objectification, numeric name - diagonals: are irrational, scope: is transcendental. Measure: measuring scale: is a multidigit general term, puts physical objects in relation to pure numbers - counting: measuring a class. >Measuring/Quine. II 28 Numbers/Ontology: Numbers are merely a "facon de parler". - Higher classes needed to replace numbers - otherwise there would only be physical objects. - - - IX 54 Numbers/Frege/Quine: like predecessors (ancestor): Definition Predecessor/Frege: the common elements of all classes for which the initial condition was fulfilled: "y ε z" and the seclusion condition: which resulted in "a" "z 0 ε z]}. - Problem: the successor relation could also lead to things that are not numbers. Numbers/Quine: we will mainly use them as a measure of multiplicities (that is how Frege had defined them). - a has x elements"- the scheme goes back to Frege: a has 0 elements ↔ a = Λ. - s has S°x elements ↔ Ey(y ε a n _{y} has x elements. IX 59 Numbers/Zermelo: (1908) ^{(1)} takes Λ as 0, then {x} as S°x for each x. (i.e. "{x}" always one more than x! - {x} successor of x! - As numbers we then receive Λ, {Λ}, {{Λ}}.. etc.IX 59ff Numbers/Von Neumann (1923) ^{(2)} regards every natural number as the class of the previous numbers: 0 becomes Λ again, - but successor S°x does not become {x}, but x U {x}. (Combined with) - 1: as in Zermelo: equal {Λ} - but 2: {0,1} or {∧,{∧}}. - 3: {0,1,2} or {Λ,{Λ},{Λ,{Λ,{Λ}}}.For von Neuman this says that a has x elements, that a ~ x. (number, equipotent) - that’s just the "a ~ {y: y < x}" from chapter 11, because for von Neumann is x = {y: y ‹ x}. Decimal numbers/dimidial numbers/decimal system/Quine: Example en gros: comes from "large quantities" = 1 dozen times 1 dozen. Score: = 20. Decimal system: Example 365 = 3 x 10² + 6 x 10 ^{1} + 5 x 10^{0}.1. Zermelo, E. (1908). Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen, 65, 261-281. http://dx.doi.org/10.1007/BF01449999 2. Neumann, John von. (1923) Zur Einführung der transfiniten Zahlen; in: Acta Scientiarum Mathematicarum (Szeged); Band: 1; Nummer: 4; Seite(n): 199-208; - - - XIII 41 Exponent/high numbers/high zero/high 0: why is n ^{0} = 1 and not 0? Because we want that n ^{m + n} is always n^{m x n}. Example m = 0: then n^{1} is = n^{0}, or n = n^{0} x ^{n}; therefore n^{0} must be = 1. Decimal system: the positions correspond to a built-in abacus. Comma/Decimal Point: was inspired by negative exponents: Example 3.65 = 3 x 100 + 6 10-1 + 5 x 10-2. Counting/Division: had little to do with each other before this breakthrough. Because division happened on the basis of division by 2, while at the same time already in the decimal system was counted. Real numbers: some are finite, e.g. ½ = 0.5. Decimal Numbers: their correspondence with real numbers is not perfect: each finite decimal number is equivalent to an infinite: Example 5 to .4999... Solution: the correspondence can simply be made perfect by forgetting the ".5" and sticking to the ",4.999". Infinite/infinite extension/decimal number/Quine: Example a six-digit decimal number like 4.237251 is the fraction (ratio) 4,237,251 1 million Infinite decimal number: is then approximated as a limit value by the series of fractions, which of ever longer fractions, is represented by sections of this decimal number. Limit value: can be here again a fraction Example .333..., or .1428428... or irrational e.g. in the case of 3,14159 ((s) N.B.: here for the first time a number before the decimal point, because the concrete number is π). XIII 42 Infinite decimal numbers/Quine: we must not regard them as expressions! This is because real numbers that exceed any means of expression are ((s) temporarily) written as infinite decimal numbers. ((s) So one (necessarily finitely written decimal number) can correspond to several real numbers). Decimal system/Quine: each number >= 2 could function instead of the 10 as basis of a number system. The larger the base, the more compact the notation of the multiplication table. Dual System/binary/"dimidial"/binary numbers/binary system/Quine: from "0" and "1", i.e. numbers are divided by halves (partes dimidiae): Example 365 = 2 ^{8} + 2^{2} + 2^{5} + 2^{3} + 2^{2} + 2^{0}.N.B.: Law: every positive integer is a sum of distinct multiples of 2. This is only possible with 2 as a base, no other number! I.e. at 365 the 10² does not occur once, but three times. Decimal Comma/binary: in binary notation: the places on the right are then negative powers of 2. Example ,0001 is a 16th. Real numbers/binary notation: nice consequence: if we consider the series of real numbers between 0 and 1 (without the 0), we have a 1:1 correspondence between these real numbers and the infinite classes of positive integers. Solution: each binary represented real number is identified with a binary extension which is infinite in the sense that there is no last "1". XIII 43 Integers: the corresponding class of integers is then that of the integers that count the places where the "1" occurs. For example, suppose the binary representation of the real number in question begins with "001011001": the corresponding class of integers will then begin with 3,5,6 and 9. Because "1" occurs at the third, fifth, sixth and ninth digit of the binary expansion. N.B.: the class thus determined is therefore infinite! Because there is no last occurrence of "1" in the binary expansion. And vice versa: Real numbers: every infinite class of positive integers defines a real number by specifying all places where "1" occurs instead of "0". _____________ 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. |
Quine I W.V.O. Quine Word and Object, Cambridge/MA 1960 German Edition: Wort und Gegenstand Stuttgart 1980 Quine II W.V.O. Quine Theories and Things, Cambridge/MA 1986 German Edition: Theorien und Dinge Frankfurt 1985 Quine III W.V.O. Quine Methods of Logic, 4th edition Cambridge/MA 1982 German Edition: Grundzüge der Logik Frankfurt 1978 Quine V W.V.O. Quine The Roots of Reference, La Salle/Illinois 1974 German Edition: Die Wurzeln der Referenz Frankfurt 1989 Quine VI W.V.O. Quine Pursuit of Truth, Cambridge/MA 1992 German Edition: Unterwegs zur Wahrheit Paderborn 1995 Quine VII W.V.O. Quine From a logical point of view Cambridge, Mass. 1953 Quine VII (a) W. V. A. Quine On what there is InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (b) W. V. A. Quine Two dogmas of empiricism InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (c) W. V. A. Quine The problem of meaning in linguistics InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (d) W. V. A. Quine Identity, ostension and hypostasis InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (e) W. V. A. Quine New foundations for mathematical logic InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (f) W. V. A. Quine Logic and the reification of universals InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (g) W. V. A. Quine Notes on the theory of reference InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (h) W. V. A. Quine Reference and modality InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VII (i) W. V. A. Quine Meaning and existential inference InFrom a Logical Point of View, , Cambridge, MA 1953 Quine VIII W.V.O. Quine Designation and Existence, in: The Journal of Philosophy 36 (1939) German Edition: Bezeichnung und Referenz InZur Philosophie der idealen Sprache, J. Sinnreich (Hg), München 1982 Quine IX W.V.O. Quine Set Theory and its Logic, Cambridge/MA 1963 German Edition: Mengenlehre und ihre Logik Wiesbaden 1967 Quine X W.V.O. Quine The Philosophy of Logic, Cambridge/MA 1970, 1986 German Edition: Philosophie der Logik Bamberg 2005 Quine XII W.V.O. Quine Ontological Relativity and Other Essays, New York 1969 German Edition: Ontologische Relativität Frankfurt 2003 Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |

Ed. Martin Schulz, access date 2024-04-18