Despite these facts, most mathematicians accept the axiom of choice as a valid principle for proving new results in mathematics.

One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up.

This list could be expanded to include most fields of mathematics, including measure theory, ergodic theory, probability, representation theory, and differential geometry.

The most general setting in which these words have meaning is an abstract branch of mathematics called category theory.

Overall, his contributions are considered the most important in advancing chemistry to the level reached in physics and mathematics during the 18th century.

It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.

On the Infinite was Hilbert ’ s most important paper on the foundations of mathematics, serving as the heart of Hilbert's program to secure the foundation of transfinite numbers by basing them on finite methods.

Arithmetic or arithmetics ( from the Greek word ἀριθμός, arithmos " number ") is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations.

The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics ( for example Venn diagrams and symbolic reasoning about their Boolean algebra ), and the everyday usage of set theory concepts in most contemporary mathematics.

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

Categories now appear in most branches of mathematics, some areas of theoretical computer science where they correspond to types, and mathematical physics where they can be used to describe vector spaces.

Sometimes referred to as the Princeps mathematicorum ( Latin, " the Prince of Mathematicians " or " the foremost of mathematicians ") and " greatest mathematician since antiquity ", Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians.

The originality of Descartes ' thinking, therefore, is not so much in expressing the cogito — a feat accomplished by other predecessors, as we shall see — but on using the cogito as demonstrating the most fundamental epistemological principle, that science and mathematics are justified by relying on clarity, distinctiveness, and self-evidence.

This program is still recognizable in the most popular philosophy of mathematics, where it is usually called formalism.

As he began to understand physics and how physicists were using mathematics, he developed a coherent mathematical theory for what he found, most importantly in the area of integral equations.

The Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics.

His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics ( especially geometry ) from the time of its publication until the late 19th or early 20th century.

It is, properly speaking, the most universal mathematics of all.

The physicist Richard Feynman called Euler's formula " our jewel " and " one of the most remarkable, almost astounding, formulas in all of mathematics.

In 1828, he attempted the entrance examination for the École Polytechnique, the most prestigious institution for mathematics in France at the time, without the usual preparation in mathematics, and failed for lack of explanations on the oral examination.

Galois ' most significant contribution to mathematics by far is his development of Galois theory.

He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses.

This time Lewis had his own rifle in his hands, and he threw some answering fire back at the mysterious far-off shot, then spent most of the day searching out the area.

The natural world then, plus poetry and some kinds of art, receives from the most ordinary of Persians a great deal of attention.

Yet often fear persists because, even with the most rigid ritual, one is never quite free from the uneasy feeling that one might make some mistake or that in every previous execution one had been unaware of the really decisive act.

And the life they lead is undisciplined and for the most part unproductive, even though they make a fetish of devoting themselves to some creative pursuit -- writing, painting, music.

This sentence would have most of the characteristics of a question, but it has some of the characteristics of a statement because the questioner has conveyed the fact that he has no faith in his own timepiece or the one attached to his car.

Anyone who tried to remedy some of the most glaring defects in our form of democracy was denounced as a traitorous red whose real purpose was the destruction of our government.

After a year in a studio on Sheridan Square, having married an American girl who was a native of Virginia, Helion moved to a village in the Blue Ridge mountains, where he produced some of the most imposing of his abstract canvases.

nor is there need to add that among them are some of the most highly individualized and most successful of his characters.

and, `` I do think that families are the most beautiful things in all the world '', burst out Jo some five hundred pages later in that popular story of the March family, which had first appeared when Henrietta was eight ; ;

Whether you experienced the passion of desire I have, of course, no way of knowing, nor indeed have I wished with even the most fleeting fragment of a wish to know, for the fact that one constitutes by one's mere existence so to speak the proof of some sort of passion makes any speculation upon this part of one's parents' experience more immodest, more scandalizing, more deeply unwelcome than an obscenity from a stranger.

This understanding, of course, may in its turn take many forms and some of these -- especially those most interesting to the student of comparative literature -- are essentially historical.

The confused rambling of guerrilla warfare, such as most of Garibaldi's campaigns were, was brought to life by Trevelyan's pen in some of the best passages in the books.

There is no explanation of terms nor a qualification that most such revolts have been dealt with by force -- only a bald dogmatism that they must, because of some undefined compulsion, be so repelled.

Moreover, he believed that most professional historians lacked some of the essential instruments for a proper study of history.

The new column by Maurice Stans regarding business scandals, is fair and accurate in most respects and his solution to the problem has some merit.

The scientists, in and out of government, do not agree on some of the most vital points, at least publicly.

But he painted some of the boldest and most original pictures of his time, and even after nearly half a century, the tense, tormented world he put on canvas has lost none of its fascination.

This is one of the most constructive suggestions made in this critical field in years, and I certainly hope it sparks some action.

At the time Alex arrived he was engaged in some sort of intimate communication with the hen, who had settled herself on the nest most peacefully after the occurrences of the morning.

`` Chickens have short memories '', the doctor remarked, `` that's why they are better company than most people I know '', and he went on to break some important news to Alex.

While this may well be true in general, I believe it is also important to keep in mind that some recent developments suggest that over the next year or so military electronics may be one of the most strongly growing areas in an economy which is not expanding rapidly in other directions.

In his gym the professor has some of the most `` knocked out '' equipment since Vic Tanny.

It is true that most architectural schools have five year courses, some even have six or more.

An examination of some forty catalogs of schools offering courses in interior design, for the most part schools accredited by membership in the National Association of Schools of Art, and a further `` on the spot '' inspection of a number of schools, show their courses adhere pretty closely to the recommendations.