# Philosophy Dictionary of Arguments

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Imaginary numbers: An imaginary number is a number that is multiplied by the imaginary unit i, which is defined by its property i^2 = -1. The square of an imaginary number bi is -b^2. For example, 5i is an imaginary number, and its square is -25. Imaginary numbers are used in mathematics to solve equations that have no real solutions. They are also used in physics and engineering to describe phenomena such as electromagnetism and quantum mechanics. See also Quantum Mechanics.
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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 Imaginary Numbers - Dictionary of Arguments

XIII 30
Imaginary Numbers/Quine: are actually of the same type as real numbers, they were only introduced later.
They were only used to be able to draw roots from negative numbers.
Equation: always has n solutions if the highest exponent is n.
Real numbers: are the positive numbers and the 0.
XIII 30
Negative real numbers/Quine: in order to get them in the first place, we first need a new kind of proportions (ratios) together with irrational numbers.
Solution: we use excellent real numbers (positive and negative) to distinguish them from (positive) real numbers.
Notation: excellent (signed, designated) real numbers: are notated as ordered pairs (gP) '0,x' and 'x, 0'.
Ordered pairs/gP/Order/Quine: an artificial way to construct an ordered pair is for example {{x,y},x}... Here x is element of both elements. ((s) Thus, the order is determined). Then we can easily get y out as well.
Imaginary unit: notation i: = √-1.
Def imaginary number: is any product yi, where y is a signed real number.
Def complex number: is any sum x + yi, where x and y are signed real numbers (called positive or negative signed). Because of the "indigestibility" of i, the sum is not commutative. I.e. the sum cannot be broken up differently. Example 5 = 3 + 2 = 4 + 1.
This is the reason why complex numbers are often used to represent points of a plane.
XIII 31
Complex Number/Tradition: previously (in the 19th century) they were assumed to be ordered pairs of two designated real numbers.
Proportions/Ratio/Rational Numbers/Quine: have two senses.
Positive Integers: have three senses.
Complex numbers: the same thing happens here. Example a) √2, as originally constructed, b) the positive real number + √2, c) the complex number √2, thus √2 + 0i, thus <√2,0>.
Real number: can always be represented as a complex number with the imaginary part = 0.
N.B.: now the rational numbers have four senses and the positive integers have five senses! But that does not matter in practice. Also not as philosophical constructions. In "set theory and its logic" I have almost completely eliminated these doublings.
Complex numbers with the imaginary part 0 become marked real numbers and these become unmarked normal real numbers etc.
Numbers/Quine: (set theory and its logic): at the end all these numbers (complex, imaginary, real, rational) become natural numbers. Only the latter are doubled, only once, from the natural number n to the rational number 1/n.
>Numbers/Quine
, >Number Theory/Quine.

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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:

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

Ed. Martin Schulz, access date 2024-03-01