Philosophy Dictionary of ArgumentsHome
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| Number theory: Number theory is a branch of mathematics that studies the integers and related objects including prime numbers, divisibility, congruences, modular arithmetic, diophantine equations, analytic number theory, algebraic number theory, geometric number theory. See also Numbers, Geometry, Equations._____________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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W.V.O. Quine on Number Theory - Dictionary of Arguments
IX 81 Elementary Number Theory/Quine: this is the theory that can only be expressed with the terms "zero, successor, sum, power, product, identity" and with the help of connections from propositional logic and quantification using natural numbers. One can omit the first four of these points or the first two and the fifth. But the more detailed list is convenient, because the classical axiom system fits directly to it. Quine: our quantifiable variables allow other objects than numbers. However, we will now tacitly introduce a limitation to "x ε N". Elementary Number Theory/Quine: less than/equal to: superfluous here. "Ez(x + z = y)" - x ε N > Λ + x = x. - x,y ε N >{x} + y = {x+y}. IX 239 Relative Strength/Proof Theory/Theory/Provability/Quine: Goedel, incompleteness theorem (1931)(1). Since number theory can be developed in set theory, this means that the class of all theorems IX 239 (in reality, all the Goedel numbers of theorems) of an existing set theory can be defined in that same set theory, and different things can be proved about it in it. >Set Theory/Quine. Incompleteness Theorem: as a consequence, however, Goedel showed that set theory (if it is free of contradiction) cannot prove one thing through the class of its own theorems, namely that it is consistent, i.e., for example, that "0 = 1" does not lie within it. If the consistency of one set theory can be proved in another, then the latter is the stronger (unless both are contradictory). Zermelo's system is stronger than type theory. >Type theory, >Strength of theories, >Set theory, >Provability. 1.Kurt Gödel: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. In: Monatshefte für Mathematik und Physik. 38, 1931, S. 173–198, doi:10.1007/BF01700692 - - - II 178 Elementary number theory is the modest part of mathematics that deals with the addition and multiplication of integers. It does not matter if some true statements will remain unprovable. This is the core of Goedel's theorem. He has shown how one can form a sentence with any given proof procedure purely in the poor notation of elementary number theory, which can be proved then and only then if it is wrong. But wait! The sentence cannot be proved and still be wrong. So it is true, but not provable. Quine: we used to believe that mathematical truth consists in provability. Now we see that this view is untenable to mathematics as a whole. II 179 Goedel's incompleteness theorem (the techniques applied there) has proved useful in other fields: Recursive number theory, or recursion theory for short. Or hierarchy theory. >Goedel/Quine. III 311 Elementary Number Theory/Quine: does not even have a complete proof procedure. Proof: reductio ad absurdum: suppose we had it with which to prove every true sentence in the spelling of the elementary number theory, III 312 then there would also be a complete refutation procedure: to refute a sentence one would prove its negation. But then we could combine the proof and refutation procedure of page III 247 to a decision procedure. V 165 Substitutional Quantification/Referential Quantification/Numbers/Quine: Dilemma: the substitutional quantification does not help elementary number theory to any ontological thrift, for either the numbers run out or there are infinitely many number signs. If the explanatory speech of an infinite number sign itself is to be understood again in the sense of insertion, we face a problem at least as serious as that of numbers - if it is to be understood in the sense of referential quantification, then one could also be satisfied from the outset uncritically with object quantification via numbers. >Quantification/Quine. V 166 Truth conditions: if one now assumes substitutional quantification, one can actually explain the truth conditions for them by numbers by speaking only of number signs and their insertion. Problem: if numerals are to serve their purpose, they must be as abstract as numbers. Expressions, of which there should be an infinite number, could be identified by their Goedel numbers. No other approach leads to a noticeable reduction in abstraction. Substitutional quantification: forces to renounce the law that every number has a successor. A number would be the last, but the substitutional quantification theorist would not know which one. It would depend on actual inscriptions in the present and future. (Quine/Goodman 1947). This would be similar to Esenin Volpin's theory of producible numbers: one would have an unknown finite bound. V 191 QuineVsSubstitutional Quantification: the expressions to be used are abstract entities as are the numbers themselves. V 192 NominalismVsVs: one could reduce the ontology of real numbers or set theory to that of elementary number theory by establishing truth conditions for substitutional quantification on the basis of Goedel numbers. >Goedel Numbers/Quine. QuineVs: this is not nominalistic, but Pythagorean. It is not about the high estimation of the concrete and disgust for the abstract, but about the acceptance of natural numbers and the rejection of most transcendent numbers. As Kronecker says: "The natural numbers were created by God, the others are human work". QuineVs: but even that is not possible, we saw above that the subsitutional quantification over classes is basically not compatible with the object quantification over objects. V 193 VsVs: one could also understand the quantification of objects in this way. QuineVs: that wasn't possible because there aren't enough names. You could teach space-time coordination, but that doesn't explain language learning. X 79 Validity/Sentence/Quantity/Schema/Quine: if quantities and sentences fall apart in this way, there should be a difference between these two definitions of validity about schema (with sentences) and models (with sentences). But it follows from the Löwenheim theorem that the two definitions of validity (using sentences or sets) do not fall apart as long as the object language is not too weak in expression. Condition: the object language must be able to express (contain) the elementary number theory. Object Language: In such a language, a scheme that remains true in all insertions of propositions is also fulfilled by all models and vice versa. >Object Language/Quine The requirement of elementary number theory is rather weak. Def Elementary Number Theory/Quine: speaks about positive integers by means of addition, multiplication, identity, truth functions and quantification. Standard Grammar/Quine: the standard grammar would express the functors of addition, multiplication, like identity, by suitable predicates. X 83 Elementary Number Theory/Quine: is similar to the theory of finite n-tuples and effectively equivalent to a certain part of set theory, but only to the theory of finite sets. XI 94 Translation Indeterminacy/Quine/Harman/Lauener: ("Words and Objections"): e.g. translation of number theory into the language of set theory by Zermelo or von Neumann: both versions translate true or false sentences of number theory into true or false sentences of set theory. Only the truth values of sentences like e.g. "The number two has exactly one element", which had no sense before translation, differ from each other in both systems. (XI 179: it is true in von Neumann's and false in Zermelo's system, in number theory it is meaningless). XI 94 Since they both serve all purposes of number theory in the same way, it is not possible to mark one of them as a correct translation._____________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 In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (b) W. V. A. Quine Two dogmas of empiricism In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (c) W. V. A. Quine The problem of meaning in linguistics In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (d) W. V. A. Quine Identity, ostension and hypostasis In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (e) W. V. A. Quine New foundations for mathematical logic In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (f) W. V. A. Quine Logic and the reification of universals In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (g) W. V. A. Quine Notes on the theory of reference In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (h) W. V. A. Quine Reference and modality In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (i) W. V. A. Quine Meaning and existential inference In From 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 In Zur 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 |
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> Counter arguments against Quine
> Counter arguments in relation to Number Theory
Ed. Martin Schulz, access date 2024-04-16