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Continuum: The continuum in mathematics is a compact, connected, metric space. It is a mathematical concept that captures the idea of a continuous, unbroken whole. The real numbers, for example, are a continuum. See also Real numbers, Continuum hypothesis, Compactness.
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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

W.V.O. Quine on Continuum - Dictionary of Arguments

XIII 45
Def Discrete/Discreteness/Quine: an order of numbers or other objects is discrete if each object has an immediate predecessor or successor or both. For example, the integers are discrete. One the other hand:
Def dense: fractions are dense and not discrete.
Real numbers: are more than dense: they are continuous.
Discrete/continuous: here we compare them as opposites.
Discreteness: we need it to learn to count by distinguishing the objects.
Irrational Numbers/Cantor: Theorem: most will always escape us.
Number Theory: deals with integers.
Real numbers: are used by the sciences. The contrast of both is illuminated by a pair of theorems:
Goedel's Theorem: no proof procedure can encompass all truths of elementary number theory.
Tarski's theorem: truth in the parallel theory of real numbers can be checked routinely, e.g. by a computer.
N.B.: both systems are identical in their notation! The difference lies in the different interpretation of the variables (or their domains, one time the positive integers with the 0, the other time the positive real numbers with the 0).
XIII 47
This leads to a difference in the truth of the formulas!
a) real numbers: here the true formulas are a set that can be handled
b) elementary number theory: not here.
Continuity/Discreteness/Language/Quine: the interaction of the two terms is not limited to mathematics, it also exists in language: phonemes impose discreteness on the phonetic continuum.
Discreteness/Quine: also allows yellowed or damaged manuscripts to be returned to a fresh state. The discreteness of the alphabet helps that the small deviations (e.g. yellowing) add up to larger ones.
Continuum/Continuity: images, on the other hand, are a continuous medium: i.e. there are no standards for repairing a damaged copy or correcting a poor copy.
Technology: here discreteness is often combined with continuity. Example clock: it should give the impression of moving continuously.
XIII 48
Film/Quine: Continuity here is due to the weakness of our perception. Similar to the clock or our thinking about atoms over the millennia.
Planck time/Quine: here we have the next approach of nature to continuity.


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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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Ed. Martin Schulz, access date 2024-02-27
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