Ethics cannot be based on the authoritative certainty given by mathematics and logic, or prescribed directly from the empirical findings of science.

It has also given rise to a new theory of the philosophy of mathematics, and many theories of artificial intelligence, persuasion and coercion.

In mathematics, given a set and an equivalence relation on, the equivalence class of an element in is the subset of all elements in which are equivalent to.

In mathematics, the four color theorem, or the four color map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.

In mathematics, more specifically algebraic topology, the fundamental group ( defined by Henri Poincaré in his article Analysis Situs, published in 1895 ) is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

In mathematics, given two groups ( G, *) and ( H, ·), a group homomorphism from ( G, *) to ( H, ·) is a function h: G → H such that for all u and v in G it holds that

* Homology ( mathematics ), a procedure to associate a sequence of abelian groups or modules with a given mathematical object

In mathematics, given an infinite sequence of numbers

We may represent any given proposition with a letter which we call a propositional constant, analogous to representing a number by a letter in mathematics, for instance,.

Thus, contrary to the first impression its name might convey, and as realized in specific approaches and disciplines ( e. g. Fuzzy Sets and Systems ), intuitionist mathematics is more rigorous than conventionally founded mathematics, where, ironically, the foundational elements which Intuitionism attempts to construct / refute / refound are taken as intuitively given.

Rømer was given every opportunity to learn mathematics and astronomy using Tycho Brahe's astronomical observations, as Bartholin had been given the task of preparing them for publication.

They also objected to students being refused a high school diploma if they could not perform 36 separate mathematics skills, despite being given good grades in class.

The concept has been given an axiomatic mathematical derivation in probability theory, which is used widely in such areas of study as mathematics, statistics, finance, gambling, science, artificial intelligence / machine learning and philosophy to, for example, draw inferences about the expected frequency of events.

In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ( which are thought of as bijective functions from the set M to itself ); the relationship is often written as ( G, M ).

In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order.

As discussed below, the definition given above turned out to be inadequate for formal mathematics ; instead, the notion of a " set " is taken as an undefined primitive in axiomatic set theory, and its properties are defined by the Zermelo – Fraenkel axioms.

It is common in mathematics to choose a number of hypotheses that are assumed to be true within a given theory, and then declare that the theory consists of all theorems provable using those hypotheses as assumptions.

The inaugural speech was given by influential professor of theology Gisbertus Voetius, and Bernardus Schotanus ( professor of law and mathematics ) was the university's first rector magnificus.

Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given objects.

Edward Bromhead, who knew John Boole through the Institution, helped George Boole with mathematics books ; and he was given the calculus text of Sylvestre François Lacroix by Rev.

The applied tools of the mathematics disciplines of Celestial mechanics or its subfield Orbital mechanics ( both predict orbital paths and positions ) about a center of gravity are used to generate an ephemeris ( plural: ephemerides ; from the Greek word ephemeros = daily ) which is a table of values that gives the positions of astronomical objects in the sky at a given time or times, or a formula to calculate such given the proper time offset from the epoch.

Ulam's colleague at Colorado, Jan Mycielski, has given his perspective on Ulam's contributions to mathematics.

In mathematics, a countable set is a set with the same cardinality ( number of elements ) as some subset of the set of natural numbers.

* In mathematics, a certain kind of subset of a partially ordered set.

** Filter ( mathematics ), a special subset of a partially ordered set

* Interval ( mathematics ), a range of numbers ( formally, a type of subset of an ordered set )

In mathematics, a filter is a special subset of a partially ordered set.

In algebra ( which is a branch of mathematics ), a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers.

In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is " contained " inside B, that is, all elements of A are also elements of B.

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.

In mathematics, a well-order relation ( or well-ordering ) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.

In mathematics, the infimum ( plural infima ) of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound ( also abbreviated as glb or GLB ) is also commonly used.

* Core ( functional analysis ), in mathematics, a subset of the domain of a closable operator

In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are " near " S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set ( P, ≤) is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound.

In mathematics, a complete measure ( or, more precisely, a complete measure space ) is a measure space in which every subset of every null set is measurable ( having measure zero ).

In mathematics, subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.

* Closure ( mathematics ), the smallest object that both includes the object as a subset and possesses some given property

The notions of a " decidable subset " and " recursively enumerable subset " are basic ones for classical mathematics and classical logic.

In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension, is a schema of axioms in Zermelo – Fraenkel set theory.

In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods.

In mathematics, logic and computer science, a formal language is called recursively enumerable ( also recognizable, partially decidable or Turing-acceptable ) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i. e., if there exists a Turing machine which will enumerate all valid strings of the language.

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U → R ( where U is an open subset of R < sup > n </ sup >) which satisfies Laplace's equation, i. e.