Pythagoras believed that behind the appearance of things, there was the permanent principle of mathematics, and that the forms were based on a transcendental mathematical relation.

The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.

In mathematics, a contraction mapping, or contraction, on a metric space ( M, d ) is a function f from M to itself, with the property that there is some nonnegative real number < math > k < 1 </ math > such that for all x and y in M,

The scope of this school was not limited to theological subjects ; science, mathematics and humanities were also taught there.

In 1963, he earned a Ph. D. in mathematics ( advisor: Marshall Hall ) from the California Institute of Technology, and began to work there as associate professor and began work on The Art of Computer Programming.

However, there is no exact, universally agreed, definition of the term " discrete mathematics.

In mathematics, a directed set ( or a directed preorder or a filtered set ) is a nonempty set A together with a reflexive and transitive binary relation ≤ ( that is, a preorder ), with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤ c.

" For example: in mathematics, it is known that 2 + 2 = 4, but there is also knowing how to add two numbers and knowing a person ( e. g., oneself ), place ( e. g., one's hometown ), thing ( e. g., cars ), or activity ( e. g., addition ).

A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another ; see graph ( mathematics ) for more detailed definitions and for other variations in the types of graph that are commonly considered.

Gödel's incompleteness theorem, another celebrated result, shows that there are inherent limitations in what can be achieved with formal proofs in mathematics.

Kepler lived in an era when there was no clear distinction between astronomy and astrology, but there was a strong division between astronomy ( a branch of mathematics within the liberal arts ) and physics ( a branch of natural philosophy ).

The problem often arises in resource allocation where there are financial constraints and is studied in fields such as combinatorics, computer science, complexity theory, cryptography and applied mathematics.

During its golden age, there was a flourishing of mathematics, astronomy, literature, and art.

Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory.

While the mathematics permits calculation of many quantities that can be measured experimentally, there is a definite theoretical limit to values that can be simultaneously measured.

In mathematics, the natural numbers are the ordinary whole numbers used for counting (" there are 6 coins on the table ") and ordering (" this is the 3rd largest city in the country ").

While Greek astronomy — thanks to Alexander's conquests — probably influenced Indian learning, to the point of introducing trigonometry, it seems to be the case that Indian mathematics is otherwise an indigenous tradition ; in particular, there is no evidence that Euclid's Elements reached India before the 18th century.

In 967, Borrell II of Barcelona ( 947 – 992 ), visited the monastery, and the abbot asked the Count to take Gerbert with him so that the lad could study mathematics in Spain and acquire there some knowledge of Arabic learning.

The mathematics of the spectral behavior reveals that there are two regions of particular interest:

In mathematics a topological space is called separable if it contains a countable dense subset ; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

He initially chose engineering as his field of study since at the time he was convinced that there was nothing new to discover in mathematics.

However, for many Muslims there is no distinction ; all forms of art, the natural world, mathematics and science are all creations of God and therefore are reflections of the same thing-that is, God's will expressed through His Creation.

* Articles are usually between five and twenty pages and are complete descriptions of current original research findings, but there are considerable variations between scientific fields and journals – 80-page articles are not rare in mathematics or theoretical computer science.

This is an unsolved problem which probably has never been seriously investigated, although one frequently hears the comment that we have insufficient specialists of the kind who can compete with the Germans or Swiss, for example, in precision machinery and mathematics, or the Finns in geochemistry.

Next September, after receiving a degree from Yale's Master of Arts in Teaching Program, I will be teaching somewhere -- that much is guaranteed by the present shortage of mathematics teachers.

But because science is based on mathematics doesn't mean that a hot rodder must necessarily be a mathematician.

Like primitive numbers in mathematics, the entire axiological framework is taken to rest upon its operational worth.

In the new situation, philosophy is able to provide the social sciences with the same guidance that mathematics offers the physical sciences, a reservoir of logical relations that can be used in framing hypotheses having explanatory and predictive value.

So, too, is the mathematical competence of a college graduate who has majored in mathematics.

The principal of the school announced that -- despite the help of private tutors in Hollywood and Philadelphia -- Fabian is a 10-o'clock scholar in English and mathematics.

In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the central tendency of a collection of numbers taken as the sum of the numbers divided by the size of the collection.

The term " arithmetic mean " is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.

In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent.

The use of the soroban is still taught in Japanese primary schools as part of mathematics, primarily as an aid to faster mental calculation.

In mathematics and computer science, an algorithm ( originating from al-Khwārizmī, the famous Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī ) is a step-by-step procedure for calculations.

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.

:" A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence.

Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.

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

There is no prize awarded for mathematics, but see Abel Prize.