idealmathematics
Main
Aspects of this topic are discussed in the following places at Britannica.
Assorted References
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rings in number theory
( in algebra, modern: Rings in number theory )
...where the coefficients a1, …, an are integers.) Their work introduced the important concept of an ideal in such rings, so called because it could be represented by “ideal elements” outside the ring concerned. In the late 19th century the German mathematician David Hilbert used ideals...
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work of Kummer
( in Kummer, Ernst Eduard )
German mathematician whose introduction of ideal numbers, which are defined as a special subgroup of a ring, extended the fundamental theorem of arithmetic (unique factorization of every integer into a product of primes) to complex number fields.
use in
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Dedekind’s number theory ( in Dedekind, Richard
)
...is, the idea of number—Dedekind published Über die Theorie der ganzen algebraischen Zahlen (1879; “On the Theory of Algebraic Whole Numbers”). There he proposed the “ideal” as a collection of numbers that may be separated out of a larger collection, composed of algebraic integers that satisfy polynomial equations with ordinary integers as...
in algebra: Ideals )Finally, Dedekind introduced the concept of an ideal. A main methodological trait of Dedekind’s innovative approach to algebra was to translate ordinary arithmetic properties into properties of sets of numbers. In this case, he focused on the set I of multiples of any given integer and pointed out two of its main properties: If n and m are two numbers in I, then...
in mathematics: The theory of numbers )In Germany Richard Dedekind patiently created a new approach, in which each new number (called an ideal) was defined by means of a suitable set of algebraic integers in such a way that it was the common divisor of the set of algebraic integers used to define it. Dedekind’s work was slow to gain approval, yet it illustrates several of the most profound features of modern mathematics. It was...
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democracy democracy
As noted above, Aristotle found it useful to classify actually existing governments in terms of three “ideal constitutions.” For essentially the same reasons, the notion of an “ideal democracy” also can be useful for identifying and understanding the democratic characteristics of actually existing governments, be they of city-states, nation-states, or larger...
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rings in number theory algebra, modern
...where the coefficients a1, …, an are integers.) Their work introduced the important concept of an ideal in such rings, so called because it could be represented by “ideal elements” outside the ring concerned. In the late 19th century the German mathematician David Hilbert used ideals...
-
work of Kummer Kummer, Ernst Eduard
German mathematician whose introduction of ideal numbers, which are defined as a special subgroup of a ring, extended the fundamental theorem of arithmetic (unique factorization of every integer into...
use in
-
Dedekind’s number theory ( in Dedekind, Richard
)
...is, the idea of number—Dedekind published Über die Theorie der ganzen algebraischen Zahlen (1879; “On the Theory of Algebraic Whole Numbers”). There he proposed the “ideal” as a collection of numbers that may be separated out of a larger collection, composed of algebraic integers that satisfy polynomial equations with ordinary integers as...
in algebra: Ideals )Finally, Dedekind introduced the concept of an ideal. A main methodological trait of Dedekind’s innovative approach to algebra was to translate ordinary arithmetic properties into properties of sets of numbers. In this case, he focused on the set I of multiples of any given integer and pointed out two of its main properties: If n and m are two numbers in I, then...
in mathematics: The theory of numbers )In Germany Richard Dedekind patiently created a new approach, in which each new number (called an ideal) was defined by means of a suitable set of algebraic integers in such a way that it was the common divisor of the set of algebraic integers used to define it. Dedekind’s work was slow to gain approval, yet it illustrates several of the most profound features of modern mathematics. It was...
in analytic philosophy, a language that is precise, free of ambiguity, and clear in structure, on the model of symbolic logic, as contrasted with ordinary language, which is vague, misleading, and sometimes contradictory. In the Tractatus Logico-Philosophicus (1922), the Viennese-born philosopher Ludwig Wittgenstein viewed the role of language as providing a “picture of reality.” Truth was seen as making logical propositions that correspond to reality. An ideal language was thus seen as the necessary criterion for determining the meaning, or meaninglessness, of statements about the world.
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early modern logic logic, history of
...Aristotelian logic (broadly conceived to include scholastic logic) is best explained by the changing and quite new goals that logic took on in the modern era. One such goal was the development of an ideal logical language that naturally expressed ideal thought and was more precise than natural languages. Another goal was to develop methods of thinking and discovery that would accelerate or...
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language Analysis Positivism
...Oxford philosophers, of the Cambridge Analyst John Wisdom, and others; and (2) the ideology, essentially that of Carnap, usually designated as logical reconstruction, which builds up an artificial language. In the procedures of “ordinary language” Analysis, an attempt is made to trace the ways in which a person commonly expresses himself. In this manner, many of the traditional...
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Logical Positivism semantics
...epistemology of science. To such “logical” Positivists as the German-born philosopher Rudolf Carnap, for instance, the symbolism of modern logic represented the grammar (syntax) of an “ideal” language. Because the Logical Positivists were, at...
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theories of solutions liquid
...solution is ideal and γ1 = γ2 = 1. In the general case, neither HE nor SE is zero, but two types of semi-ideal solutions can be designated: in the first, SE is zero but HE is not; this is called a regular solution. In the second,...
a common mental construct in the social sciences derived from observable reality although not conforming to it in detail because of deliberate simplification and exaggeration. It is not ideal in the sense that it is excellent, nor is it an average; it is, rather, a constructed ideal used to approximate reality by selecting and accentuating certain elements.
The concept of the ideal type was developed by German sociologist Max Weber, who used it as an analytic tool for his historical studies. Some writers confine the use of ideal types to general phenomena that recur in different times and places (e.g., bureaucracy), although Weber also used them for historically unique occurrences (e.g., his famous Protestant ethic).
Problems in using the ideal type include its tendency to focus attention on extreme, or polar, phenomena while overlooking the connections between them, and the difficulty of showing how the types and their elements fit into a conception of a total social system.
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