In traditional logic Logic is the study of arguments. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident In epistemology , a self-evident proposition is one that is known to be true by understanding its meaning without proof, or subject to necessary decision Decision making can be regarded as the mental processes resulting in the selection of a course of action among several alternatives. Every decision making process produces a final choice. The output can be an action or an opinion of choice. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.

In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Unlike theorems In mathematics, a theorem is a statement which has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproved proposition that is believed to, simply because they are starting points; there is nothing else from which they logically follow (otherwise they would be classified as theorems).

Logical axioms are usually statements that are taken to be universally true (e.g., A and B implies A), while non-logical axioms (e.g., a + b = b + a) are actually defining properties for the domain of a specific mathematical theory (such as arithmetic Arithmetic or arithmetics 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). When used in the latter sense, "axiom," "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.

Outside logic and mathematics, the term "axiom" is used loosely for any established principle of some field.

Etymology

The word "axiom" comes from the Greek Greek , an independent branch of the Indo-European family of languages, is the language of the Greeks. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. In its ancient form, it is the language of classical ancient Greek literature and the New Testament of word ἀξίωμα (axioma), a verbal noun A verbal noun is a noun formed directly as an inflexion of a verb or a verb stem, sharing at least in part its constructions. This term is applied especially to gerunds, and sometimes also to infinitives and supines from the verb ἀξιόειν (axioein), meaning "to deem worthy", but also "to require", which in turn comes from ἄξιος (axios), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the ancient Greek Ancient Greece is the civilization 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 BC, at first under Athenian philosophers Philosophy is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language. It is distinguished from other ways of addressing fundamental questions by its critical, generally systematic approach and its reliance on rational argument. The word "philosophy" comes from the an axiom was a claim which could be seen to be true without any need for proof.

The root meaning of the word 'postulate' is to 'demand'; for instance, Euclid Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry." He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics demands of us that we agree that the some things can be done, e.g. any two points can be joined by a straight line, etc.^[1]

Ancient geometers maintained some distinction between axioms and postulates. While commenting Euclid's books Proclus Proclus Lycaeus , called "The Successor" or "Diadochos" (Greek Πρόκλος ὁ Διάδοχος Próklos ho Diádokhos), was a Greek Neoplatonist philosopher, one of the last major Classical philosophers (see Damascius). He set forth one of the most elaborate and fully developed systems of Neoplatonism. He stands near the remarks that "Geminus Geminus of Rhodes , was a Greek astronomer and mathematician, who flourished in the 1st century BC. An astronomy work of his, the Introduction to the Phenomena, still survives; it was intended as an introductory astronomy book for students. He also wrote a work on mathematics, of which only fragments quoted by later authors survive held that this [4th] Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property".^[2] Boethius Anicius Manlius Severinus Boëthius, commonly called Boethius was a Christian philosopher of the early 6th century. He was born in Rome to an ancient and important family which included emperors Petronius Maximus and Olybrius and many consuls. His father, Flavius Manlius Boethius, was consul in 487 after Odoacer deposed the last Western Roman translated 'postulate' as petitio and called the axioms notiones communes but in later manuscripts this usage was not always strictly kept.

Historical development

Early Greeks

The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (syllogisms, rules of inference), was developed by the ancient Greeks, and has become the core principle of modern mathematics^{[citation needed]}. Tautologies In logic, a tautology is a formula which is true in every possible interpretation. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense today excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions (theorems In mathematics, a theorem is a statement which has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other, if we are talking about mathematics) must be proven with the aid of these basic assumptions. However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms axiom and postulate hold a slightly different meaning for the present day mathematician, than they did for Aristotle Aristotle (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 and Euclid Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry." He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics.

The ancient Greeks considered geometry Geometry "Earth-measuring" is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by as just one of several sciences Science is, in its broadest sense, any systematic knowledge that is capable of resulting in a correct prediction or reliable outcome. In this sense, science may refer to a highly skilled technique, technology, or practice^{[citation needed]}, and held the theorems of geometry on par with scientific facts. As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge. Aristotle's posterior analytics The Posterior Analytics is a text from Aristotle's Organon that deals with demonstration, definition, and scientific knowledge. The demonstration is distinguished as a syllogism productive of scientific knowledge, while the definition marked as the statement of a thing's nature, ... a statement of the meaning of the name, or of an equivalent is a definitive exposition of the classical view.

An “axiom”, in classical terminology, referred to a self-evident assumption common to many branches of science. A good example would be the assertion that

When an equal amount is taken from equals, an equal amount results.

At the foundation of the various sciences lay certain additional hypotheses which were accepted without proof. Such a hypothesis was termed a postulate. While the axioms were common to many sciences, the postulates of each particular science were different. Their validity had to be established by means of real-world experience. Indeed, Aristotle warns that the content of a science cannot be successfully communicated, if the learner is in doubt about the truth of the postulates.

The classical approach is well illustrated by Euclid's Elements Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the, where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of "common notions" (very basic, self-evident assertions).

Postulates

1. It is possible to draw a straight line In Euclidean geometry, a line is a straight curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height. Lines are an idealization of such objects and have no width or height at all and are usually considered to be infinitely long. Lines are a fundamental concept in some from any point to any other point.
2. It is possible to extend a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment is either an edge if they are adjacent vertices, or continuously in a straight line.
3. It is possible to describe a circle A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the centre . The common distance of the points of a circle from its centre is called its radius with any center and any radius.
4. It is true that all right angles In geometry and trigonometry, a right angle is an angle which bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. As a rotation, a right angle corresponds to a quarter turn are equal to one another.
5. ("Parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that:") It is true that, if a straight line falling on two straight lines make the interior angles In geometry a polygon is traditionally a plane figure that is bounded by a closed path or circuit, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain). These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices or corners. An n-gon is a polygon with n sides on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or a line. Distinguishing these cases, and finding the intersection point have use, for example, in computer graphics, motion planning, and collision detection on that side on which are the 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 less than the two right angles.

Common notions

1. Things which are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.

Modern development

A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, propositions In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true propositions. The series of formulas which is constructed, theorems) and definitions^{[citation needed]}. This abstraction, one might even say formalization, makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts.

Structuralist mathematics goes further, and develops theories and axioms (e.g. field theory In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic, group theory In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. While these are familiar from, topology Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology, vector spaces) without any particular application in mind. The distinction between an “axiom” and a “postulate” disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid's fifth postulate we get theories that have meaning in wider contexts, hyperbolic geometry In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line l and point P not on l, there is exactly one line through P that does not intersect l; for example. We must simply be prepared to use labels like “line” and “parallel” with greater flexibility. The development of hyperbolic geometry taught mathematicians that postulates should be regarded as purely formal statements, and not as facts based on experience.

When mathematicians employ the axioms of a field In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic, the intentions are even more abstract. The propositions of field theory do not concern any one particular application; the mathematician now works in complete abstraction. There are many examples of fields; field theory gives correct knowledge about them all.

It is not correct to say that the axioms of field theory are “propositions that are regarded as true without proof.” Rather, the field axioms are a set of constraints. If any given system of addition and multiplication satisfies these constraints, then one is in a position to instantly know a great deal of extra information about this system.

Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and logic Logic is the study of arguments. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of logic itself can be regarded as a branch of mathematics. Frege Friedrich Ludwig Gottlob Frege was a German mathematician who became a logician and philosopher. He was one of the founders of modern logic, and made major contributions to the foundations of mathematics. As a philosopher, he is generally considered to be the father of analytic philosophy, for his writings on the philosophy of language and, Russell Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, socialist, pacifist, and social critic. He spent most of his life in England; he was born in Wales where he also died, aged 97, Poincaré Jules Henri Poincaré (French pronunciation: [ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe]) was a French mathematician, theoretical physicist, and a philosopher of science. Poincaré is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime, Hilbert David Hilbert /ˈdaːfɪt ˈhɪlbʌt/ was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of, and Gödel Kurt Gödel (German pronunciation: [kʊʁt ˈɡøːdl̩] ; April 28, 1906, Brno, Moravia – January 14, 1978, Princeton, New Jersey, USA) was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, are some of the key figures in this development.

In the modern understanding, a set of axioms is any collection In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context. In work on ZF set theory, the notion of class is informal, whereas other set theories, such as NBG set of formally stated assertions from which other formally stated assertions follow by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be consistent In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model; this is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term; it should be impossible to derive a contradiction from the axiom. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom.

It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, could be derived from a consistent collection of basic axioms. An early success of the formalist program was Hilbert's formalization of Euclidean geometry Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, whose Elements is the earliest known systematic discussion of geometry. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. Although many of Euclid's results had been, and the related demonstration of the consistency of those axioms.

In a wider context, there was an attempt to base all of mathematics on Cantor's Georg Ferdinand Ludwig Philipp Cantor was a mathematician, best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the set theory Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. Here the emergence of Russell's paradox In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory of Richard Dedekind and Frege leads to a contradiction. The very same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other, and similar antinomies of naive set theory Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most contemporary mathematics raised the possibility that any such system could turn out to be inconsistent.

The formalist project suffered a decisive setback, when in 1931 Gödel showed that it is possible, for any sufficiently large set of axioms (Peano's axioms, for example) to construct a statement whose truth is independent of that set of axioms. As a corollary, Gödel proved that the consistency of a theory like Peano arithmetic is an unprovable assertion within the scope of that theory.

It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of natural numbers, an infinite but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern Zermelo–Fraenkel axioms for set theory. The axiom of choice, a key hypothesis of this theory, remains a very controversial assumption. Furthermore, using techniques of forcing (Cohen) one can show that the continuum hypothesis (Cantor) is independent of the Zermelo–Franekel axioms. Thus, even this very general set of axioms cannot be regarded as the definitive foundation for mathematics.

Mathematical logic

In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical axioms and non-logical axioms (somewhat similar to the ancient distinction between "axioms" and "postulates" respectively)

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