axioms

Although cohomology is fundamental to modern algebraic topology, its importance was not seen for some 40 years after the development of homology. The concept of dual cell structure, which Henri Poincaré used in his proof of his Poincaré duality theorem, contained the germ of the idea of cohomology, but this was not seen until later. There were various precursors to cohomology. In the mid-1920s, J.W. Alexander and Lefschetz founded the intersection theory of cycles on manifolds. On an n-dimensional manifold M, a ...

See also:

Cohomology, Cohomology - History, Cohomology - Cohomology theories, Cohomology - Eilenberg-Steenrod theories, Cohomology - Extraordinary cohomology theories, Cohomology - Other cohomology theories

Read more here: » Cohomology: Encyclopedia II - Cohomology - History

Let φ be a well-formed formula in which w,x,y, … occur freely, but not y, z, or u. Let ψ be a well-formed formula in which w, y, v … occur freely, but not x, z, or u. ...

See also:

Tarski's axioms, Tarski's axioms - The axioms, Tarski's axioms - Non-Euclidean geometry

Read more here: » Tarski's axioms: Encyclopedia II - Tarski's axioms - The axioms

It is not sufficiently recognized that Peirce’s career was that of a scientist, not a philosopher; and that during his lifetime he was known and valued chiefly as a scientist, only secondly as a logician, and scarcely at all as a philosopher. Even his work in philosophy and logic will not be understood until this fact becomes a standing premise of Peircian studies. (Max Fisch, in Moore and Robin 1964: 486). Upon this first, and in one sense sole, rule of reason, that in order to learn you must desire to learn, and in so desiring not ...

See also:

Charles Peirce, Charles Peirce - Life, Charles Peirce - Reception, Charles Peirce - Works, Charles Peirce - Peirce's philosophy, Charles Peirce - Pragmatism, Charles Peirce - Scholastic realism, Charles Peirce - Formal perspective, Charles Peirce - Dynamics of representation, Charles Peirce - Normative sciences, Charles Peirce - Parallels with Leibniz, Charles Peirce - Bibliography, Charles Peirce - Primary literature, Charles Peirce - Secondary literature

Read more here: » Charles Peirce: Encyclopedia II - Charles Peirce - Peirce's philosophy

Naïve Empiricism: Our ideas and theories need to be tested against reality and not be affected by preconceived notions. Constructive Empiricism: According to this view of science coined by Bas C. van Fraassen (The Scientific Image, 1980), we should only ask that theories accurately describe observable parts of the world. Theories that meet these requirements are considered "empirically adequate". If a theory becomes well established, it should be "accepted". What that means is the theory is believed to be empirically accurate, used to solve further problems, and u ...

See also:

Empiricism, Empiricism - Empiricism and Science, Empiricism - Empiricism in history, Empiricism - Classical Empiricism, Empiricism - Modern Empiricism, Empiricism - Radical Empiricism, Empiricism - Moderate Empiricism, Empiricism - Other forms, Empiricism - Criticisms, Empiricism - Kuhn's The Structure of Scientific Revolutions, Empiricism - Constructivism, Empiricism - Quantum mechanics

Read more here: » Empiricism: Encyclopedia II - Empiricism - Other forms

The political activities of Sheikh Taqiuddin an-Nabhani started very early. Before establishing Hizb ut-Tahrir he had no organised political activity, save for the period in his teens and twenties he had spent with the famous mujahid Sheikh Izz_ad-Din_al-Qassam, whom he helped lay down plans for the well known revolutionary upheavals against British colonial rule, and against plans to set up the state of Israel for jews. He also mixed with the Muslim_Brotherhood, and exchanged views with Sayyid Qutb. Many of his early al-Azhar colleagues lat ...

See also:

Taqiuddin al-Nabhani, Taqiuddin al-Nabhani - Background, Taqiuddin al-Nabhani - Education, Taqiuddin al-Nabhani - Career, Taqiuddin al-Nabhani - Thought, Taqiuddin al-Nabhani - Philosophy & Theology, Taqiuddin al-Nabhani - Politics, Taqiuddin al-Nabhani - Encyclopedia Quote, Taqiuddin al-Nabhani - Hizb ut-Tahrir, Taqiuddin al-Nabhani - Death, Taqiuddin al-Nabhani - Books

Read more here: » Taqiuddin al-Nabhani: Encyclopedia II - Taqiuddin al-Nabhani - Politics

Constant functor: A very boring functor C → D is one which maps every object of C to a fixed object X in D and every morphism in C to the identity morphism on X. Such a functor is called a constant or selection functor. Power sets: The power set functor P : Set → Set maps each set to its power set and each function to the map which sends to its image . One can also consider the contravariant power set functor which sends f to the map which sends USee also:

Functor, Functor - Definition, Functor - Covariance and contravariance, Functor - Examples, Functor - Properties, Functor - Relation to other categorical concepts

Read more here: » Functor: Encyclopedia II - Functor - Examples

The study of structure starting with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by everyday numbers. Long standing questions about ruler-and-compass constructions were finally settled by Galois ...

See also:

Areas of mathematics, Areas of mathematics - Foundations / general, Areas of mathematics - Algebra, Areas of mathematics - Analysis, Areas of mathematics - Geometry, Areas of mathematics - Applied mathematics, Areas of mathematics - Probability and statistics, Areas of mathematics - Computational sciences, Areas of mathematics - Physical sciences, Areas of mathematics - Non-physical sciences

Read more here: » Areas of mathematics: Encyclopedia II - Areas of mathematics - Algebra

Primitive recursive functions take natural numbers or tuples of natural numbers as arguments and produce a natural number. A function which takes n arguments is called n-ary. The basic primitive recursive functions are given by these axioms: The constant function 0 is primitive recursive. The successor function S, which takes one argument and returns the succeeding number as given by the Peano postulates, is primitive recursive. The projection functions Pin, which take n arguments and return ...

See also:

Primitive recursive function, Primitive recursive function - Definition, Primitive recursive function - Examples, Primitive recursive function - Addition, Primitive recursive function - Subtraction, Primitive recursive function - Limitations, Primitive recursive function - Bibliography

Read more here: » Primitive recursive function: Encyclopedia II - Primitive recursive function - Definition

Three of those papers (on Brownian motion, the photoelectric effect, and special relativity) deserved Nobel Prizes according to some physicists. Only the paper on the photoelectric effect would win one. What makes these papers remarkable is that, in each case, Einstein boldly took an idea from theoretical physics to its logical consequences and managed to explain experimental results that had baffled scientists for decades. See also:

Annus Mirabilis Papers, Annus Mirabilis Papers - Papers, Annus Mirabilis Papers - Background, Annus Mirabilis Papers - Photoelectric effect, Annus Mirabilis Papers - Brownian motion, Annus Mirabilis Papers - Special relativity, Annus Mirabilis Papers - Matter and energy equivalence, Annus Mirabilis Papers - Commemoration

Read more here: » Annus Mirabilis Papers: Encyclopedia II - Annus Mirabilis Papers - Papers

This is a complex question and, for simplicity of exposition, we will here consider only vacuous truth as concerns logical implication, i.e., the case when S has the form P ⇒ Q, and P is false. This case strikes many people as odd, and it's not immediately obvious whether all such statements are true, all such statements are false, some are true while others are false, or what. ...

See also:

Vacuous truth, Vacuous truth - Examples, Vacuous truth - Scope of the concept, Vacuous truth - Arguments of the semantic truth of vacuously true logical statements, Vacuous truth - Arguments that at least some vacuously true statements are true, Vacuous truth - Arguments for taking all vacuously true statements to be true, Vacuous truth - Arguments that only some vacuously true statements are true, Vacuous truth - Summary, Vacuous truth - Difficulties with the use of vacuous truth, Vacuous truth - Vacuous truths in mathematics

Read more here: » Vacuous truth: Encyclopedia II - Vacuous truth - Arguments of the semantic truth of vacuously true logical statements

A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. By a theorem of Peter Freyd, this definition is equivalent to the following "piecemeal" definition: A category is preadditive if it is enriched over the monoidal category Ab of abelian groups. This means that all hom-sets are abelian groups and the composition of morphisms is bilinear. A preadditive category is ...

See also:

Abelian category, Abelian category - Definitions, Abelian category - Examples, Abelian category - Elementary properties, Abelian category - Related concepts, Abelian category - History

Read more here: » Abelian category: Encyclopedia II - Abelian category - Definitions

In scientific usage, a theory does not mean an unsubstantiated guess or hunch, as it does in other contexts. Neither is a scientific theory a fact. Scientific theories are never proven to be true, but can be disproven. All scientific understanding takes the form of hypotheses, theories, or laws. Theories are typically ways of explaining why things happen, often, but not always after their occurrence is no longer in scientific dispute. In referring to the "theory of global warming" for example, the worldwide ...

See also:

Theory, Theory - Etymology, Theory - Science, Theory - Models, Theory - Types of theories, Theory - Further explanation of a scientific theory, Theory - Characteristics, Theory - Mathematics, Theory - Other fields, Theory - List of famous theories, Theory - Reference

Read more here: » Theory: Encyclopedia II - Theory - Science

The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g., addition, subtraction, mul ...

See also:

Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions

Read more here: » Mathematics: Encyclopedia II - Mathematics - History

No Answers in Genesis is a site maintained by members of Australian Skeptics led by retired civil servant John Stear for the purpose of rebutting AiG. Some creationists use a statement from the scientist and atheist Richard Dawkins that evolution makes it possible to be an intellectually fulfilled atheist to argue that Stear's atheism adds a bias to his arguments. He has responded (to similar arguments): The fact is that what I believe personally has no bearing on my management of a web site that is dedicated to, and is obv ...

See also:

Answers in Genesis, Answers in Genesis - History, Answers in Genesis - Tax-exempt status, Answers in Genesis - The Creation Museum, Answers in Genesis - Facts and figures, Answers in Genesis - Teachings and beliefs, Answers in Genesis - Methodology, Answers in Genesis - Mission, Answers in Genesis - Apologetic method, Answers in Genesis - AiG's views on cosmology and astronomy, Answers in Genesis - AiG's views on moral and social issues, Answers in Genesis - Life issues, Answers in Genesis - Homosexuality, Answers in Genesis - Evolution and race, Answers in Genesis - Culture and media, Answers in Genesis - Criticism, Answers in Genesis - Is Answers in Genesis a scientific organization?, Answers in Genesis - Criticisms of specific claims made by Answers in Genesis, Answers in Genesis - Controversy over Interview with Richard Dawkins, Answers in Genesis - Answers in Genesis and September 11 the Indian Ocean tsunami and hurricane Katrina disasters, Answers in Genesis - Do Answers in Genesis and mainstream scientists agree about defining evolution?

Read more here: » Answers in Genesis: Encyclopedia II - Answers in Genesis - Criticism

What precedes this point is intended as an informal introduction to the axioms of the primary arithmetic and primary algebra, and hopefully provides the reader with an intuitive sense of their motivation and rationale. The next order of business is to give the exact forms of the axioms that are used in the following more formal development, devolving from Peirce's various systems of logical graphs via Spencer-Brown's Laws of Form (LOF). In formal proofs, a variation of the annotation scheme from LOF will be used to mark each st ...

See also:

Logical graph, Logical graph - Abstract point of view, Logical graph - In lieu of a beginning, Logical graph - Duality logical and topological, Logical graph - Computational representation, Logical graph - Quick tour of the neighborhood, Logical graph - Primary arithmetic as semiotic system, Logical graph - Primary algebra as pattern calculus, Logical graph - Formal development, Logical graph - Axioms, Logical graph - Frequently used theorems, Logical graph - Exemplary proofs

Read more here: » Logical graph: Encyclopedia II - Logical graph - Formal development

Every Boolean algebra (A, , ) gives rise to a ring (A, +, *) by defining a + b = (a ¬b) (b ¬a) (this operation is called "symmetric difference" in the case of sets and XOR in the case of logic) and a * b = a b. The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the 1 of the Boolean algebra. This ring has the property that a * a = a for all ...

See also:

Boolean algebra, Boolean algebra - Formal definition, Boolean algebra - Examples, Boolean algebra - Order theoretic properties, Boolean algebra - Principle of duality, Boolean algebra - Other notation, Boolean algebra - Homomorphisms and isomorphisms, Boolean algebra - Boolean rings, ideals and filters, Boolean algebra - Representing Boolean algebras, Boolean algebra - Axiomatics for Boolean algebras

Read more here: » Boolean algebra: Encyclopedia II - Boolean algebra - Boolean rings, ideals and filters

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that ins ...

See also:

Mathematics, Mathematics - History, Mathematics - Inspiration, pure and applied mathematics, and aesthetics, Mathematics - Notation, language, and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Common misconceptions

Read more here: » Mathematics: Encyclopedia II - Mathematics - Inspiration, pure and applied mathematics, and aesthetics

Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict grammar (under the influence o ...

See also:

Mathematics, Mathematics - History, Mathematics - Inspiration, pure and applied mathematics, and aesthetics, Mathematics - Notation, language, and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Common misconceptions

Read more here: » Mathematics: Encyclopedia II - Mathematics - Notation, language, and rigor

Main articles: category, morphism A category C consists of a class ob(C) of objects: a class hom(C) of morphisms. Each morphism f has a unique source object a and target object b. We write f: a → b, and we say "f is a morphism from a to b". We write hom(a, b) [or Hom(a, b), or homC(a, b)] to denote the hom-class of all morphisms from < ...

See also:

Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories, objects, and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions, limits, and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

Read more here: » Category theory: Encyclopedia II - Category theory - Categories, objects, and morphisms

AiG has controversially remarked that September 11 is an "encouraging reminder" that "God allows tragedies to awaken us", and a reminder that "this nation needs God" [60]. They have also claimed that "It is likely that the perpetrators were ones fuelled in part by an intense hatred of the Christianity they still associate with America." [61] AiG has published an article [62] claiming that the Indian Ocean Earthquake and resulting tsunami (in which approximately 275,000 people died) must be attributed to the action of God, and not just ...

See also:

Answers in Genesis, Answers in Genesis - History, Answers in Genesis - Tax-exempt status, Answers in Genesis - The Creation Museum, Answers in Genesis - Facts and figures, Answers in Genesis - Teachings and beliefs, Answers in Genesis - Methodology, Answers in Genesis - Mission, Answers in Genesis - Apologetic method, Answers in Genesis - AiG's views on cosmology and astronomy, Answers in Genesis - AiG's views on moral and social issues, Answers in Genesis - Life issues, Answers in Genesis - Homosexuality, Answers in Genesis - Evolution and race, Answers in Genesis - Culture and media, Answers in Genesis - Criticism, Answers in Genesis - Is Answers in Genesis a scientific organization?, Answers in Genesis - Criticisms of specific claims made by Answers in Genesis, Answers in Genesis - Controversy over Interview with Richard Dawkins, Answers in Genesis - Answers in Genesis and September 11, the Indian Ocean tsunami and hurricane Katrina disasters, Answers in Genesis - Do Answers in Genesis and mainstream scientists agree about defining evolution?

Read more here: » Answers in Genesis: Encyclopedia II - Answers in Genesis - Answers in Genesis and September 11, the Indian Ocean tsunami and hurricane Katrina disasters

Main articles: universal property, limit (category theory) Using the language of category theory, many areas of mathematical study can be cast into appropriate categories, such as the categories of all sets, groups, topologies, and so on. These categories surely have some objects that are "special" in a certain way, such as the empty set or the product of two topologies. Yet, in the definition of a category, objects are considered to be atomic; i.e. we do not know, whether an object A is a set, a topology, or any other a ...

See also:

Category theory, Category theory - Background, Category theory - Historical notes, Category theory - Categories, objects, and morphisms, Category theory - Some properties of morphisms, Category theory - Functors, Category theory - Natural transformations and isomorphisms, Category theory - Universal constructions, limits, and colimits, Category theory - Equivalent categories, Category theory - Further concepts and results, Category theory - Higher-dimensional categories

Read more here: » Category theory: Encyclopedia II - Category theory - Universal constructions, limits, and colimits

There are some other conditions on topological spaces that are sometimes classified with the separation axioms, but these don't fit in with the usual separation axioms as completely. Other than their definitions, they won't be discussed here. X is semiregular if the regular open sets form a base for the open sets of X. Any regular space must also be semiregular. X is fully normal if every open cover has an open star refinement. Every fully normal space must also be both normal regular and paracompact. In fact, fully normal spaces actually have more to do ...

See also:

Separation axiom, Separation axiom - Separated sets and topologically distinguishable points, Separation axiom - Definitions of the axioms, Separation axiom - Relationships between the axioms, Separation axiom - Other separation axioms, Separation axiom - Sources

Read more here: » Separation axiom: Encyclopedia II - Separation axiom - Other separation axioms