Axiom

Évariste Galois, Abductive reasoning, Abstract algebra, Al-Farabi, Al-Ghazali, Al-Kindi, Alan Turing, Alfred North Whitehead, Alfred Tarski, Algebraic topology, Alonzo Church, Ancient Greece, Angle, Angles, Antinomy, Argument, Argumentation theory, Aristotle, Arithmetic, Assignment (mathematical logic), Averroes, Avicenna, Axiom, Axiom (disambiguation), Axiom of choice, Axiom of countability, Axiom schema, Axiom scheme, Axiomatic set theory, Axiomatic system, Axiomatization, Bertrand Russell, Boethius, Boolean algebra (logic), Boolean function, Charles Sanders Peirce, Circle, Class (set theory), Cogency, Combinatorics, Commutative, Completeness, Complex analysis, Computability logic, Conditional probability, Conservative extension, Consistency, Consistent, Constructivism (mathematics), Continuum hypothesis, Corollary, Critical thinking, David Hilbert, Decidability (logic), Decision, Deductive reasoning, Deductive system, Deontic logic, Dharmakirti, Dialetheism, Differential geometry, Differential topology, Dignāga, Doxastic logic, Element (mathematics), Elliptic geometry, Encyclopædia Britannica Eleventh Edition, Epistemic logic, Ergodic theory, Euclid, Euclid's Elements, Euclid's postulates, Euclidean geometry, Explanation, Fakhr al-Din al-Razi, Fallacy, Fictionalism, Field (mathematics), Finitary relation, Finite, Finitism, First-order arithmetic, First-order language, First-order logic, Forcing (mathematics), Formal language, Formal proof, Formal semantics, Formal system, Formalism (mathematics), Formation rule, Formula (mathematical logic), Fuzzy logic, Gödel's completeness theorem, Gödel's first incompleteness theorem, Gödel's second incompleteness theorem, Galois theory, Geminus, Geometry, Georg Cantor, George Boole, Gerhard Gentzen, Giuseppe Peano, Gottfried Leibniz, Gottlob Frege, Greek language, Grothendieck universe, Group (algebra), Group (mathematics), Group theory, Henri Poincaré, Hilary Putnam, History of logic, Homology theory, Homotopy theory, Hyperbolic geometry, Ibn Hazm, Ibn Taymiyyah, Ibn al-Nafis, Implication, Independence (mathematical logic), Indian logic, Inductive reasoning, Inference, Infinite, Informal logic, Integer, Interpretation (logic), Intuitionism, Intuitionistic logic, Is logic empirical?, Isomorphism, Isuzu Axiom, Kanada, Kurt Gödel, Löwenheim-Skolem theorem, Line-line intersection, Linear logic, Linear space, List of axioms, List of basic topics in logic, List of boolean algebra topics, List of fallacies, List of logicians, List of mathematical logic topics, List of paradoxes, List of rules of inference, List of set theory topics, List of topics in logic, Logic, Logic in China, Logic in Islamic philosophy, Logic in computer science, Logical atomism, Logical connective, Logical consequence, Logical truth, Logicism, Lotfi Asker Zadeh, Mathematical logic, Mathematical proof, Mathematical theory, Mathematician, Mathematics, Measure theory, Metalogic, Modal logic, Model theory, Modus ponens, Monadic predicate calculus, Morse-Kelley set theory, Mozi, Nagarjuna, Naive set theory, Natural number, Negation, Neo-logicism, Nominalism, Non-classical logic, Non-monotonic logic, Non-standard analysis, Number theory, Nyāya Sūtras, Organon, Pāṇini, Paraconsistent logic, Paradox, Parallel postulate, Paul Cohen (mathematician), Peano arithmetic, Peano axioms, Philosopher, Philosophical logic, Philosophical realism, Philosophy of logic, Physical space, Plane geometry, PlanetMath, Platonic realism, Point set topology, Polygon, Posterior analytics, Postulate, Predicate (mathematical logic), Predicate calculus, Predicate logic, Prime Number, Probability, Proclus, Proof theory, Proposition, Propositional calculus, Propositional logic, Propositional variable, Quantification, Real analysis, Real numbers, Reason, Reasoning, Recursion theory, Relevance logic, Representation theory, Right angle, Ring (mathematics), Rudolf Carnap, Rule of inference, Rules of inference, Russell's paradox, Satisfiability, Satisfiability and validity, Saul Kripke, Science, Second-order arithmetic, Second-order logic, Self-evidence, Set (mathematics), Set theory, Shahab al-Din Suhrawardi, Soundness, Straight line, Strongly inaccessible cardinal, Successor function, Syllogism, Syntax (logic), T. L. Heath, Table of logic symbols, Tautologies, Tautology (logic), Temporal logic, Term logic, Theorem, Thoralf Skolem, Topic outline of logic, Topological space, Triangle, Truth table, Type theory, Unary function, Validity, Verbal noun, Von Neumann–Bernays–Gödel set theory, Well-formed formula, Wikisource, Wiktionary, Willard Van Orman Quine, Zermelo–Fraenkel set theory, Zermelo-Frankel axioms,

Évariste Galois (French pronunciation: [evaʁist ɡalwa]) (October 25, 1811 – May 31, 1832) was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem. His work laid the foundations for Galois theory, a major branch of abstract algebra, and the subfield of Galois connections. He was the first to use the word "group" (French: groupe) as a technical term in mathematics to represent a group of permutations. A radical Republican during the monarchy of Louis Philippe in France, he died from wounds suffered in a duel under shadowy circumstances^[1] at the age of twenty.

Abductive reasoning

Abduction is a method of logical inference introduced by Charles Sanders Peirce which comes prior to induction and deduction for which the colloquial name is to have a "hunch". Abductive reasoning starts when an inquirer considers of a set of seemingly unrelated facts, armed with an intuition that they are somehow connected. The term abduction is commonly presumed to mean the same thing as hypothesis; however, an abduction is actually the process of inference that produces a hypothesis as its end result^[1]. It is used in both philosophy and computing.

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and real or complex numbers, often now called elementary algebra. The distinction is rarely made in more recent writings.

Al-Ghazali

Abū Ḥāmid Muḥammad ibn Muḥammad al-Ghazālī (1058-1111) (Persian: ابو حامد محمد ابن محمد الغزالی), often Algazel in English, was born and died in Tus, in the Khorasan province of Persia. He was an Islamic theologian, jurist, philosopher, cosmologist, psychologist and mystic of Persian origin,^[3][4] and remains one of the most celebrated scholars in the history of Sunni Islamic thought. He is considered a pioneer of the methods of doubt and skepticism,^[5] and in one of his major works, The Incoherence of the Philosophers, he changed the course of early Islamic philosophy, shifting it away from an Islamic metaphysics influenced by ancient Greek and Hellenistic philosophy, and towards an Islamic philosophy based on cause-and-effect that was determined by God or intermediate angels, a theory now known as occasionalism.

Al-Kindi

Abū Yūsuf Yaʻqūb ibn Isḥāq al-Kindī (Arabic: أبو يوسف يعقوب إبن إسحاق الكندي‎) (c. 801–873 CE), also known to the West by the Latinized version of his name Alkindus, was an Arab Iraqi polymath:^[1] an Islamic philosopher, scientist, astrologer, astronomer, cosmologist, chemist, logician, mathematician, musician, physician, physicist, psychologist, and meteorologist.^[2] Al-Kindi was the first of the Muslim Peripatetic philosophers, and is known for his efforts to introduce Greek and Hellenistic philosophy to the Arab world,^[3] and as a pioneer in chemistry, cryptography, medicine, music theory, physics, psychology, and the philosophy of science.

Alan Turing

Alan Mathison Turing, OBE, FRS (pronounced /ˈtjʊərɪŋ/, TYOOR-ing; 23 June 1912 – 7 June 1954), was an English mathematician, logician, cryptanalyst, and computer scientist. He was influential in the development of computer science and providing a formalisation of the concept of the algorithm and computation with the Turing machine, playing a significant role in the creation of the modern computer.^[1]

Alfred North Whitehead

Alfred North Whitehead, OM (February 15, 1861 – December 30, 1947) was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education. He co-authored the epochal Principia Mathematica with Bertrand Russell.

Alfred Tarski

Alfred Tarski (January 14, 1901, Warsaw, Russian-ruled Poland – October 26, 1983, Berkeley, California) was a Polish logician and mathematician. Educated in the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death.^[1]

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism. In many situations this is too much to hope for and it is more prudent to aim for a more modest goal, classification up to homotopy equivalence.

Alonzo Church

Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem.

Ancient Greece

Ancient Greece is the civilisation belonging to the period of Greek history lasting from the Archaic period of the 8th to 6th centuries BC to 146 BC and the Roman conquest of Greece after the Battle of Corinth. At the center of this time period is Classical Greece, which flourished during the 5th to 4th centuries, at first under Athenian leadership successfully repelling the military threat of Persian invasion. The Athenian Golden Age ends with the defeat of Athens at the hands of Sparta in the Peloponnesian War in 404 BC.

Angle

In geometry and trigonometry, an angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle (Sidorov 2001). The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide with the other (see "Measuring angles", below). Where there is no possibility of confusion, the term "angle" is used interchangeably for both the geometric configuration itself and for its angular magnitude (which is simply a numerical quantity).

Angles

The Angles is a modern English word for a Germanic-speaking people who took their name from the ancestral cultural region of Angeln, a district located in Schleswig-Holstein, Germany. The Angles were one of the main groups that settled in Britain in the post-Roman period, founding several of the kingdoms of Anglo-Saxon England, and their name is the root of the name "England".

Antinomy

Antinomy (Greek αντι-, against, plus νομος, law) literally means the mutual incompatibility, real or apparent, of two laws. It is a term used in logic and epistemology.

Argument

In logic, an argument is a set of one or more meaningful declarative sentences (or "propositions") known as the premises along with another meaningful declarative sentence (or "proposition") known as the conclusion. A deductive argument asserts that the truth of the conclusion is a logical consequence of the premises; an inductive argument asserts that the truth of the conclusion is supported by the premises. Deductive arguments are valid or invalid, and sound or not sound. An argument is valid if and only if the truth of the conclusion is a logical consequence of the premises and (consequently) its corresponding conditional is a necessary truth. A sound argument is a valid argument with true premises.

Argumentation theory

Argumentation theory, or argumentation, is the interdisciplinary study of how humans should, can, and do reach conclusions through logical reasoning, that is, claims based, soundly or not, on premises. It includes the arts and sciences of civil debate, dialogue, conversation, and persuasion. It studies rules of inference, logic, and procedural rules in both artificial and real world settings.

Aristotle

Aristotle (Greek: Ἀριστοτέλης, Aristotélēs) (384 BC – 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, politics, government, ethics, biology, and zoology. Together with Plato and Socrates (Plato's teacher), Aristotle is one of the most important founding figures in Western philosophy. Aristotle's writings constitute a first at creating a comprehensive system of Western philosophy, encompassing morality and aesthetics, logic and science, politics and metaphysics.

Arithmetic

Arithmetic or arithmetics (from the Greek word αριθμός = 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. It involves the study of quantity, especially as the result of combining numbers. In common usage, it refers to the simpler properties when using the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers. Professional mathematicians sometimes use the term (higher) arithmetic^[1] when referring to more advanced results related to number theory, but this should not be confused with elementary arithmetic.