Area plays an important role in modern mathematics.

It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory.

While the roots of formalised logic go back to Aristotele, the end of the 19th and early 20th centuries saw the development of modern logic and formalised mathematics.

Sets are of great importance in mathematics ; in fact, in modern formal treatments, most mathematical objects ( numbers, relations, functions, etc.

This subject constitutes a major part of modern mathematics education.

The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo – Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent.

Methods for breaking modern cryptosystems often involve solving carefully constructed problems in pure mathematics, the best-known being integer factorization.

In the early modern age, Victorian schools were reformed to teach commercially useful topics, such as modern languages and mathematics, rather than classical subjects, such as Latin and Greek.

In modern mathematics, Euclidean spaces form the prototypes for other, more complicated geometric objects.

In modern mathematics, the theory of fields ( or field theory ) plays an essential role in number theory and algebraic geometry.

Nowadays, functors are used throughout modern mathematics to relate various categories.

Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects ( for example, numbers and functions ) from all the traditional areas of mathematics ( such as algebra, analysis and topology ) in a single theory, and provides a standard set of axioms to prove or disprove them.

In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.

It inaugurated a new Western-based education system for all young people, sent thousands of students to the United States and Europe, and hired more than 3, 000 Westerners to teach modern science, mathematics, technology, and foreign languages in Japan ( O-yatoi gaikokujin ).

During the Meiji period ( 1868 – 1912 ), leaders inaugurated a new Western-based education system for all young people, sent thousands of students to the United States and Europe, and hired more than 3, 000 Westerners to teach modern science, mathematics, technology, and foreign languages in Japan ( Oyatoi gaikokujin ).

From Robertson, Madison learned mathematics, geography, and modern and ancient languages.

Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics.

A guided tour through the various branches of modern mathematics.

An online mathematics encyclopedia under construction, focusing on modern mathematics.

The development of algebraic geometry from its classical to modern forms is a particularly striking example of the way an area of mathematics can change radically in its viewpoint, without making what was correctly proved before in any way incorrect ; of course mathematical progress clarifies gaps in previous proofs, often by exposing hidden assumptions, which progress has revealed worth conceptualizing.

Intuitionistic logic substitutes constructability for abstract truth and is associated with a transition from the proof to model theory of abstract truth in modern mathematics.

This is the only case in modern history of a people of Britannic origin submitting without continued struggle to what they view as foreign domination.

While sovereignty has roots in antiquity, in its present usage it is essentially modern.

Neither is primary experience understood according to the attitude of modern empiricism in which nothing is thought to be received other than signals of sensory qualities producing their responses in the appropriate sense organs.

Shakespeare's Shylock, too, is of dubious value in the modern world.

But a modern Oedipus who is doomed because he cannot oppose his own childhood is only pathetic, and for renouncing the mystery in favor of psychological truth he gives up the claim on our sympathies.

A characteristic expression of such concern and inquiry is found in Joseph P. Lyford's Introduction To The Agreeable Autocracies, a recent paperback study of the institutions of modern democratic society.

The men who speculate on these institutions have, for the most part, come to at least one common conclusion: that many of the great enterprises and associations around which our democracy is formed are in themselves autocratic in nature, and possessed of power which can be used to frustrate the citizen who is trying to assert his individuality in the modern world ''.

As Sandburg said at the time: `` It is as ancient as the medieval European ballads brought to the Appalachian Mountains, it is as modern as skyscrapers, the Volstead Act, and the latest oil well gusher ''.

Here we may observe that at least one modern philosophy of history is built on the assumption that ideas are the primary objectives of the historian's research.

The Chicago contingent of modern critics follow Aristotle so far in this direction that it is hard to see how they can compare one poem with another for the purpose of evaluation.

Diplomatic is another area for which the dawn of the twentieth century marks the beginning of modern standards of scholarship.

This is certainly an irrational dogmatism, in which the modern mind attempts to understand the spirit of the sixteenth century on twentieth-century terms.

Rexroth is a longtime jazz buff, a name-dropper of jazz heroes, and a student of traditional as well as modern jazz.

In addition to his experiments in reading poetry to jazz, Patchen is beginning to use the figure of the modern jazz musician as a myth hero in the same way he used the figure of the private detective a decade ago.

This angry and exasperated stance which Patchen has maintained in his poetry for almost fifteen years has been successfully modulated into a kind of woe that is as effective as anger and still expresses his disapproval of the modern world.

But the firm has recognized the tight dollar and the tourist's desire to visit the `` smaller, less-traveled and relatively inexpensive countries '', and is now prepared to teach modern Greek and Portuguese through recordings.

Perhaps there is more truth than we are wont to admit in the conviction of that ornament of Tarheelia, Robert Ruark's grandfather, who was persuaded that the great curse of the modern world is `` all this gallivantin' ''.

The possibility of recall into the Army is part of the price that a modern American has to pay for the enviable heritage of liberty which he enjoys.

Under modern conditions, this is especially true of the ready reserve.

However, its modern one-story layout is designed to increase our production capacity, permit more efficient manufacturing, and substantially reduce current repair and maintenance costs.