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Axiom

Axiom

In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. Not all epistemologists agree that any axioms, understood in that sense, exist. In mathematics, an axiom is not necessarily a self-evident truth but rather, a formal logical expression used in a deduction to yield further results. Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms.

Etymology

The word axiom comes from the Greek word αξιωμα (axioma), which means that which is deemed worthy or fit or that which is considered self-evident. The word comes from αξιοειν (axioein), meaning to deem worthy, which in turn comes from αξιος (axios), meaning worthy. Among the ancient Greek philosophers an axiom was a claim which could be seen to be true without any need for proof.

Mathematics

In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical axioms and non-logical axioms.

Logical axioms

These are certain formulas in a language that are universally valid, that is, formulas that are satisfied by every structure under every variable assignment function. More colloquially, these are statements that are true in any possible universe, under any possible interpretation and with any assignment of values. Usually one takes as logical axioms some minimal set of tautologies that is sufficient for proving all tautologies in the language.

Examples

In the propositional calculus it is common to take as logical axioms all formulas of the following forms, where \phi, \psi, and \chi can be any formulas of the language: #\phi \to (\psi \to \phi) #(\phi \to (\psi \to \chi)) \to ((\phi \to \psi) \to (\phi \to \chi)) #(\lnot \phi \to \lnot \psi) \to (\psi \to \phi) Each of these patterns is an axiom schema, a rule for generating an infinite number of axioms. For example, if A, B, and C are propositional variables, then A \to (B \to A) and (A \to \lnot B) \to (C \to (A \to \lnot B)) are both instances of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and modus ponens, one can prove all tautologies of the propositional calculus. It can also be shown that no pair of these schemata is sufficient for proving all tautologies with modus ponens. These axiom schemata are also used in the predicate calculus, but additional logical axioms are needed.
Example. Let \mathfrak\, be a first-order language. For each variable x\,, the formula
x = x
is universally valid.
This means that for any variable symbol x\,, the formula x = x\, can be regarded as an axiom. Also, in this example, for this not to fall into vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by x = x\, (or, for all what matters, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol =\, has to be enforced, and mathematical logic does indeed do that. Another, more interesting example, is that which provides us with what is known as universal instantiation:
Example. Given a formula \phi\, in a first-order language \mathfrak\,, a variable x\, and a term t\, that is substitutable for x\, in \phi\,, the formula
\forall x. \phi \to \phi^x_t
is universally valid.
Informally speaking, this example allows us to state that if we know that a certain property P\, holds for every x\, and that if t\, stands for a particular object in our structure, then we should be able to claim P(t)\,. Again, we are claiming that the formula \forall x. \phi \to \phi^x_t is valid, that is, we must be able to give a "proof" of this fact, or more properly speaking, a metaproof. Actually, these examples are metatheorems of our theory of mathematical logic since we are dealing with the very concept of proof itself. Aside from this, we can also have existential generalization:
Axiom scheme. Given a formula \phi\, in a first-order language \mathfrak\,, a variable x\, and a term t\, that is substitutable for x\, in \phi\,, the formula
\phi^x_t \to \exists x. \phi
is universally valid.

Non-logical axioms

Non-logical axioms are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as groups). Thus non-logical axioms, unlike logical axioms, are not tautologies. Another name for a non-logical axiom is postulate. Almost every modern mathematical theory starts from a given set of non-logical axioms, and it was thought that in principle every theory could be axiomatized in this way and formalized down to the bare language of logical formulas. This turned out to be impossible and proved to be quite a story (see below). Non-logical axioms are often simply referred to as axioms in mathematical discourse. This does not mean that it is claimed that they are true in some absolute sense. For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing (the more general) group theory, and we can even take its negation as an axiom for the study of non-commutative groups. Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define a deductive system.

Examples

This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth). A rigorous treatment of any of these topics begins with a specification of these axioms. Basic theories, such as arithmetic, real analysis (sometimes referred to as the theory of functions of one real variable), linear algebra, and complex analysis (a.k.a. complex variables), are often introduced non-axiomatically in mostly technical studies, but any rigorous course in these subjects always begins by presenting its axioms. Geometries such as Euclidean geometry, projective geometry, symplectic geometry. Interestingly one of the results of the fifth Euclidean axiom being a non-logical axiom is that the three angles of a triangle do not by definition add to 180°. Only under the umbrella of Euclidean geometry is this always true. The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory. The development of abstract algebra brought with itself group theory, rings and fields, Galois theory. This list could be expanded to include most fields of mathematics, including axiomatic set theory, measure theory, ergodic theory, probability, representation theory, and differential geometry.
Arithmetic
The Peano axioms are the most widely used axiomatization of arithmetic. They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem. We have a language \mathfrak_ = \\, where 0\, is a constant symbol and S\, is a unary function and the following axioms: # \forall x. \lnot (Sx = 0) # \forall x. \forall y. (Sx = Sy \to x = y) # ((\phi(0) \land \forall x.\,(\phi(x) \to \phi(Sx))) \to \forall x.\phi(x) for any \mathfrak_\, formula \phi\, with one free variable. The standard structure is \mathfrak = \langle\N, 0, S\rangle\, where \N\, is the set of natural numbers, S\, is the successor function and 0\, is naturally interpreted as the number 0.
Euclidean geometry
Probably the oldest, and most famous, list of axioms are the 4 + 1 Euclid's postulates of plane geometry. This set of axioms turns out to be incomplete, and many more postulates are necessary to rigorously characterize his geometry (Hilbert used 23). The axioms are referred to as "4 + 1" because for nearly two millennia the fifth (parallel) postulate ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. Indeed, one can assume that no parallels through a point outside a line exist, that exactly one exists, or that infinitely many exist. These choices give us alternative forms of geometry in which the interior angles of a triangle add up to less than, exactly, or more than a straight line respectively and are known as elliptic, Euclidean, and hyperbolic geometries.
Real analysis
The object of study is the real numbers. The real numbers are uniquely picked out (up to isomorphism) by the properties of a complete ordered field. However, expressing these properties as axioms requires use of second-order logic. The Löwenheim-Skolem theorems tell us that if we restrict ourselves to first-order logic, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger. Some of the latter are studied in non-standard analysis.

Role in mathematical logic

Deductive systems and completeness

A deductive system consists of a set \Lambda\, of logical axioms, a set \Sigma\, of non-logical axioms, and a set \\, of rules of inference. A desirable property of a deductive system is that it be complete. A system is said to be complete if, for all formulas \phi,
if \Sigma \models \phi then \Sigma \vdash \phi
that is, for any statement that is a logical consequence of \Sigma there actually exists a deduction of the statement from \Sigma\,. This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation". Gödel's completeness theorem establishes the completeness of a certain commonly-used type of deductive system. Note that "completeness" has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms \Sigma\, of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement \phi\, such that neither \phi\, nor \lnot\phi\, can be proved from the given set of axioms. There is thus, on the one hand, the notion of completeness of a deductive system and on the other hand that of completeness of a set of non-logical axioms. The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.

Further discussion

Early mathematicians regarded axiomatic geometry as a model of physical space, and obviously there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic. Galois showed just before his untimely death that these efforts were largely wasted, but that the grand parallels between axiomatic systems could be put to good use, as he algebraically solved many classical geometrical problems. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details and modern algebra was born. In the modern view we may take as axioms any set of formulas we like, as long as they are not known to be inconsistent.

See also


- Axiomatic system
- Peano axioms
- Axiom of choice
- Axiom of countability
- Axiomatic set theory
- Parallel postulate
- Continuum hypothesis
- Axiomatization
- List of axioms

External links


- [http://us.metamath.org/mpegif/mmset.html#axioms Metamath axioms page] Category:AlgebraCategory:Logic ko:공리 ja:公理



Self-evidence

In epistemology, a self-evident proposition is one that can be understood only by one who knows that it is true. A self-evident proposition is one that can be known to be true without proof (but only by understanding what it says). Some epistemologists deny that any proposition can be self-evident. My belief that I am conscious is considered by many to be self-evident; your belief that I am conscious is not. Some examples of metaphysical propositions said to be self-evident include "A finite whole is greater than any of its parts" and "It is impossible for the same thing to be and not be at the same time in the same manner." Some examples of moral propositions said to be self-evident as cited by Alexander Hamilton in the Federalist #31 include "The means ought to be proportioned to the end", "Every power ought to be commensurate with its object" and "There ought to be no limitation of a power destined to effect a purpose which is itself incapable of limitation." In informal or colloquial speech, "self-evident" often merely means "obvious." Certain forms of argument from self-evidence are considered fallacious or abusive in debate. An example is the assertion that since an opponent disagrees with a (claimed self-evident) proposition, that he must have misunderstood it. Compare with: the concepts of primitive notion and axiom in mathematics. It is sometimes said that a self-evident proposition is one whose denial is self-contradictory. It is also sometimes said that an analytic proposition is one whose denial is self-contradictory. But these two uses of the term self-contradictory mean entirely different things. A self-evident proposition cannot be denied without knowing that one contradicts oneself (provided one actually understands the proposition). An analytic proposition cannot be denied without a contradiction, but one may fail to know that there is a contradiction because it may be a contradiction that can be found only by a long and abstruse line of logical or mathematical reasoning. Most analytic propositions are very far from self-evident. Similarly, a self-evident proposition need not be analytic: my knowledge that I am conscious is self-evident but not analytic. That being said, an analytic proposition, however long a chain of reasoning it takes to establish it, ultimately contains a tautology, and is thus only a verbal truth--a truth established through the verbal equivalence of a single meaning. For those who admit the existence of abstract concepts, the class of non-analytic self-evident truths can be regarded as truths of the understanding--truths revealing connections between the meanings of ideas. One of the most famous examples of claims to the self-evidence of a truth is found in the Declaration of Independence. The proposition that "all men are created equal" is not necessarily self-evident in a philosophically respectable sense, and the propositions which follow surely are not. However many would agree that the proposition "we ought to treat subjects known to be equal in a certain sense equally in regard to that sense" is self-evident. On the other hand the propositions described can be (as Thomas Jefferson proposed) "held" to be self-evident as the basis for practical, even revolutionary, behaviors. Category:Epistemology

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics 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, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

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 inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Quantity

This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. : :Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Structure

:Pinning down ideas of size, symmetry, and mathematical structure. : :Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory

Space

:A more visual approach to mathematics. : :Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

See also list of mathematics history topics :History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

See also


- Mathematical game
- Mathematical problem
- Mathematical puzzle
- Puzzle

Bibliography


- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links


- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
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Greek language

Greek (Greek Ελληνικά, IPA – "Hellenic") is an Indo-European language with a documented history of 3,500 years. Today, it is spoken by 15 million people in Greece, Cyprus, the former Yugoslavia, particularly The Former Yugoslav Republic of Macedonia, Bulgaria, Albania and Turkey. There are also many Greek emigrant communities around the world, such as those in Melbourne, Australia which is the third-largest Greek-populated city in the world, after Athens and Thessaloniki. Greek has been written in the Greek alphabet, the first true alphabet, since the 9th century B.C. and before that, in Linear B and the Cypriot syllabaries. Greek literature has a long and rich tradition.

History

This article does not cover the reconstructed history of Greek prior to the use of writing. For more information, see main article on Proto-Greek language. Greek has been spoken in the Balkan Peninsula since the 2nd millennium BC. The earliest evidence of this is found in the Linear B tablets dating from 1500 BC. The later Greek alphabet (q.v.) is unrelated to Linear B, and was derived from the Phoenician alphabet (abjad); with minor modifications, it is still used today. Greek is conventionally divided into the following periods:
- Mycenean Greek: the language of the Mycenean civilisation. It is recorded in the Linear B script on tablets dating from the 16th century BC onwards.
- Classical Greek (also known as Ancient Greek): In its various dialects was the language of the Archaic and Classical periods of Greek civilisation. It was widely known throughout the Roman empire. Classical Greek fell into disuse in western Europe in the Middle Ages, but remained known in the Byzantine world, and was reintroduced to the rest of Europe with the Fall of Constantinople and Greek migration to Italy.
- Hellenistic Greek (also known as Koine Greek): The fusion of various ancient Greek dialects with Attic (the dialect of Athens) resulted in the creation of the first common Greek dialect, which gradually turned into one of the world's first international languages. Koine Greek can be initially traced within the armies and conquered territories of Alexander the Great, but after the Hellenistic colonisation of the known world, it was spoken from Egypt to the fringes of India. After the Roman conquest of Greece, an unofficial diglossy of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. Through Koine Greek it is also traced the origin of Christianity, as the Apostles used it to preach in Greece and the Greek-speaking world. It is also known as the Alexandrian dialect, Post-Classical Greek or even New Testament Greek (after its most famous work of literature).
- Medieval Greek: The continuation of Hellenistic Greek during medieval Greek history as the official and vernacular (if not the literary nor the ecclesiastic) language of the Byzantine Empire, and continued to be used until, and after the fall of that Empire in the 15th century. Also known as Byzantine Greek.
- Modern Greek: Stemming independently from Koine Greek, Modern Greek usages can be traced in the late Byzantine period (as early as 11th century). Two main forms of the language have been in use since the end of the medieval Greek period: Dhimotikí (Δημοτική), the Demotic (vernacular) language, and Katharévousa (Καθαρεύουσα), an imitation of classical Greek, which was used for literary, juridic, and scientific purposes during the 19th and early 20th centuries. Demotic Greek is now the official language of the modern Greek state, and the most widely spoken by Greeks today. It has been claimed that an "educated" speaker of the modern language can understand an ancient text, but this is surely as much a function of education as of the similarity of the languages. Still, Koinē , the version of Greek used to write the New Testament and the Septuagint, is relatively easy to understand for modern speakers. Greek words have been widely borrowed into the European languages: astronomy, democracy, philosophy, thespian, etc. Moreover, Greek words and word elements continue to be productive as a basis for coinages: anthropology, photography, isomer, biomechanics etc. and form, with Latin words, the foundation of international scientific and technical vocabulary. See English words of Greek origin, and List of Greek words with English derivatives.

Classification

Greek is an independent branch of the Indo-European language family. The ancient languages which were probably most closely related to it, Ancient Macedonian language (which may be regarded as a dialect of Greek) and Phrygian, are not well enough documented to permit detailed comparison. Among living languages, Armenian seems to be the most closely related to it.

Geographic distribution

Modern Greek is spoken by about 15 million people mainly in Greece and Cyprus. There are also Greek-speaking populations in Georgia, Ukraine, Egypt, Turkey, Albania, Former Yugoslav Republic of Macedonia and Southern Italy. The language is spoken also in many other countries where Greeks have settled, including Armenia, Australia, Austria, Belgium, Bulgaria, Canada, Denmark, France, Germany, Netherlands, Sweden, United Kingdom, and the United States.

Official status

Greek is the official language of Greece where it is spoken by about 99.5% of the population. It is also, alongside Turkish, the official language of Cyprus. Due to the membership of Greece and Cyprus, Greek is one of the 20 official languages of the European Union.

Phonology

This section generally describes the post-Classic phonology of the Greek language. :All phonetic transcriptions in this section use the International Phonetic Alphabet

Vowel sounds

Greek has 5 vowel sounds, all phonemic:

Ancient Greece

Ancient Greece is the term used to describe the Greek-speaking world in ancient times. It refers not only to the geographical peninsula of modern Greece, but also to areas of Hellenic culture that were settled in ancient times by Greeks: Cyprus, the Aegean coast of Turkey (then known as Ionia), Sicily and southern Italy (known as Magna Graecia), and the scattered Greek settlements on the coasts of what are now Albania, Bulgaria, Egypt, Libya, southern France, southern Spain, Catalonia, Georgia, Romania, and Ukraine. There are no fixed or universally agreed upon dates for the beginning or the end of the Ancient Greek period. In common usage it refers to all Greek history before the Roman Empire, but historians use the term more precisely. Some writers include the periods of the Greek-speaking Mycenaean civilization that collapsed about 1100 BC, though most would argue that the influential Minoan was so different from later Greek cultures that it should be classed separately. In the modern Greek school-books, "ancient times" is a period of about 1000 years (from the catastrophe of Mycenae until the conquest of the country by the Romans) that is divided in four periods, based on styles of art as much as culture and politics. The historical line starts with Greek Dark Ages (1100–800 BC). In this period artists use geometrical schemes such as squares, circles, lines to decorate amphoras and other pottery. The archaic period (800–500 BC) represents those years when the artists made larger free-standing sculptures in stiff, hieratic poses with the dreamlike "archaic smile". In the classical years (500–323 BC) artists perfected the style that since has been taken as exemplary: "classical", such as the (Parthenon). In the Hellenistic years that followed the conquests of Alexander (323–146 BC), also known as Alexandrian, aspects of Hellenic civilization expanded to Egypt and Bactria. Traditionally, the Ancient Greek period was taken to begin with the date of the first Olympic Games in 776 BC, but many historians now extend the term back to about 1000 BC. The traditional date for the end of the Ancient Greek period is the death of Alexander the Great in 323 BC (The following period is classed Hellenistic) or the integration of Greece into the Roman Republic in 146 BC. These dates are historians' conventions and some writers treat the Ancient Greek civilization as a continuum running until the advent of Christianity in the third century AD. Ancient Greece is considered by most historians to be the foundational culture of Western Civilization. Greek culture was a powerful influence in the Roman Empire, which carried a version of it to many parts of Europe. Ancient Greek civilization has been immensely influential on the language, politics, educational systems, philosophy, art and architecture of the modern world, particularly during the Renaissance in Western Europe and again during various neo-Classical revivals in 18th and 19th century Europe and The Americas.

Origins

The Americas The Greeks are believed to have migrated southward into the Greek peninsula in several waves beginning in the late 3rd millennium BC, the last being the Dorian invasion. The period from 1600 BC to about 1100 BC is described in History of Mycenaean Greece known for the reign of King Agamemnon and the wars against Troy as narrated in the epics of Homer. The period from 1100 BC to the 8th century BC is a "dark age" from which no primary texts survive, and only scant archaeological evidence remains. Secondary and tertiary texts such as Herodotus' Histories, Pausanias' Description of Greece, Diodorus' Bibliotheca and Jerome's Chronicon, contain brief chronologies and king lists for this period. The history of Ancient Greece is often taken to end with the reign of Alexander the Great, who died in 323 BC. Subsequent events are described in Hellenistic Greece. Any history of Ancient Greece requires a cautionary note on sources. Those Greek historians and political writers whose works have survived, notably Herodotus, Thucydides, Xenophon, Demosthenes, Plato and Aristotle, were mostly either Athenian or pro-Athenian. That is why we know far more about the history and politics of Athens than of any other city, and why we know almost nothing about some cities' histories. These writers, furthermore, concentrate almost wholly on political, military and diplomatic history, and ignore economic and social history. All histories of Ancient Greece have to contend with these limits in their sources.

The rise of Hellas

In the 8th century BC Greece began to emerge from the Dark Ages which followed the fall of the Mycenaean civilization. Literacy had been lost and the Mycenaean script forgotten, but the Greeks adapted the Phoenician alphabet to Greek and from about 800 BC written records begin to appear. Greece was divided into many small self-governing communities, a pattern dictated by Greek geography, where every island, valley and plain is cut off from its neighbors by the sea or mountain ranges. 800 BC. It was the greatest architectural statement of 5th century BC Greece]] As Greece recovered economically, its population grew beyond the capacity of its limited arable land, and from about 750 BC the Greeks began 250 years of expansion, settling colonies in all directions. To the east, the Aegean coast of Asia Minor was colonized first, followed by Cyprus and the coasts of Thrace, the Sea of Marmara and south coast of the Black Sea. Eventually Greek colonization reached as far north-east as present day Ukraine. To the west the coasts of Albania, Sicily and southern Italy were settled, followed by the south coast of France, Corsica, and even northeastern Spain. Greek colonies were also founded in Egypt and Libya. Modern Syracuse, Naples, Marseille and Istanbul had their beginnings as the Greek colonies Syracusa, Neapolis, Massilia and Byzantium. By the 6th century BC Hellas had become a cultural and linguistic area much larger than the geographical area of Greece. Greek colonies were not politically controlled by their founding cities, although they often retained religious and commercial links with them. The Greeks both at home and abroad organised themselves into independent communities, and the city (polis) became the basic unit of Greek government. First Crete, then in short order the other Greek city-states, adopted the formal practice of pederasty. From its ritual roots in Indo-European prehistory, the practice was elevated to prominence, influencing pedagogy, warfare and social life, and becoming a central feature of Hellenic culture for the next thousand years.

Social and political conflict

The Greek cities were originally monarchies, although many of them were very small and the term "King" (basileus) for their rulers is misleadingly grand. In a country always short of farmland, power rested with a small class of landowners, who formed a warrior aristocracy fighting frequent petty inter-city wars over land and rapidly ousting the monarchy. About this time the rise of a mercantile class (shown by the introduction of coinage in about 680 BC) introduced class conflict into the larger cities. From 650 BC onwards, the aristocracies had to fight not to be overthrown and replaced by populist leaders called tyrants (tyrranoi), a word which did not necessarily have the modern meaning of oppressive dictators. By the 6th century BC several cities had emerged as dominant in Greek affairs: Athens, Sparta, Corinth, and Thebes. Each of them had brought the surrounding rural areas and smaller towns under their control, and Athens and Corinth had become major maritime and mercantile powers as well. Athens and Sparta developed a rivalry that dominated Greek politics for generations. In Sparta, the landed aristocracy retained their power, and the constitution of Lycurgus (about 650 BC) entrenched their power and gave Sparta a permanent militarist regime under a dual monarchy. Sparta dominated the other cities of the Peloponnese, with the sole exceptions of Argus and Achaia. In Athens, by contrast, the monarchy was abolished in 683 BC, and reforms of Solon established a moderate system of aristocratic government. The aristocrats were followed by the tyranny of Pisistratus and his sons, who made the city a great naval and commercial power. When the Pisistratids were overthrown, Cleisthenes established the world's first democracy (500 BC), with power being held by an assembly of all the male citizens. But it must be remembered that only a minority of the male inhabitants were citizens, excluding slaves, freedmen and non-Athenians.

The Persian Wars

In Ionia (the modern Aegean coast of Turkey) the Greek cities, which included great centres such as Miletus and Halicarnassus, were unable to maintain their independence and came under the rule of the Persian Empire in the mid 6th century BC. In 499 BC the Greeks rose in the Ionian Revolt, and Athens and some other Greek cities went to their aid. In 490 BC the Persian Great King, Darius I, having suppressed the Ionian cities, sent a fleet to punish the Greeks. The Persians landed in Attica, but were defeated at the Battle of Marathon by a Greek army led by the Athenian general Miltiades. The burial mound of the Athenian dead can still be seen at Marathon. Ten years later Darius's successor, Xerxes I, sent a much more powerful force by land. After being delayed by the Spartan King Leonidas I at Thermopylae, Xerxes advanced into Attica, where he captured and burned Athens. But the Athenians had evacuated the city by sea, and under Themistocles they defeated the Persian fleet at the Battle of Salamis. A year later, the Greeks, under the Spartan Pausanius, defeated the Persian army at Plataea. The Athenian fleet then turned to chasing the Persians out of the Aegean Sea, and in 478 BC they captured Byzantium. In the course of doing so Athens enrolled all the island states and some mainland allies into an alliance, called the Delian League because its treasury was kept on the sacred island of Delos. The Spartans, although they had taken part in the war, withdrew into isolation after it, allowing Athens to establish unchallenged naval and commercial power.

The dominance of Athens

Delos The Persian Wars ushered in a century of Athenian dominance of Greek affairs. Athens was the unchallenged master of the sea, and also the leading commercial power, although Corinth remained a serious rival. The leading statesman of this time was Pericles, who used the tribute paid by the members of the Delian League to build the Parthenon and other great monuments of classical Athens. By the mid 5th century the League had become an Athenian Empire, symbolised by the transfer of the League's treasury from Delos to the Parthenon in 454 BC. The wealth of Athens attracted talented people from all over Greece, and also created a wealthy leisured class who became patrons of the arts. The Athenian state also sponsored learning and the arts, particularly architecture. Athens became the centre of Greek literature, philosophy (see Greek philosophy) and the arts (see Greek theatre). Some of the greatest names of Western cultural and intellectual history lived in Athens during this period: the dramatists Aeschylus, Aristophanes, Euripides, and Sophocles, the philosophers Aristotle, Plato, and Socrates, the historians Herodotus, Thucydides, and Xenophon, the poet Simonides and the sculptor Pheidias. The city became, in Pericles's words, "the school of Hellas." The other Greek states at first accepted Athenian leadership in the continuing war against the Persians, but after the fall of the conservative politician Cimon in 461 BC, Athens became an increasingly open imperialist power. After the Greek victory at the Battle of the Eurymedon in 466 BC, the Persians were no longer a threat, and some states, such as Naxos, tried to secede from the League, but were forced to submit. The new Athenian leaders, Pericles and Ephialtes, let relations between Athens and Sparta deteriorate, and in 458 BC war broke out. After some years of inconclusive war a 30-year peace was signed between the Delian League and the Peloponnesian League (Sparta and her allies). This coincided with the last battle between the Greeks and the Persians, a sea battle off Salamis in Cyprus, followed by the Peace of Callias (450 BC) between the Greeks and Persians.

The Peloponnesian War

450 BC In 431 BC war broke out again between Athens and Sparta and its allies. The proximate cause was a dispute between Corinth and one of its colonies, Corcyra (modern-day Corfu), in which Athens intervened. The obviate cause was the growing resentment of Sparta and its allies at the dominance of Athens over Greek affairs. The war lasted 27 years, partly because Athens (a naval power) and Sparta (a land-based military power) found it difficult to come to grips with each other. Sparta's initial strategy was to invade Attica, but the Athenians were able to retreat behind their walls. An outbreak of plague in the city during the siege caused heavy losses, including Pericles. At the same time the Athenian fleet landed troops in the Peloponnese, winning battles at Naupactus (429 BC) and Pylos (425 BC). But these tactics could bring neither side a decisive victory. After several years of inconclusive campaigning, the moderate Athenian leader Nicias concluded the Peace of Nicias (421 BC). In 418 BC, however, hostility between Sparta and the Athenian ally Argos led to a resumption of fighting. At Mantinea Sparta defeated the combined armies of Athens and her allies. The resumption of fighting brought the war party, led by Alcibiades, back to power in Athens. In 415 BC Alcibiades persuaded the Athenian Assembly to launch a major expedition against Syracuse, a Peloponnesian ally in Sicily. Though Nicias was a skeptic about the Sicilian Expedition he was appointed along Alcibiades to lead the expedition. Due to accusations against him, Alcibiades fled to Sparta where he persuaded Sparta to send aid to Syracuse. As a result, the expedition was a complete disaster and the whole expeditionary force was lost. Nicias was executed by his captors. Sparta had now built a fleet to challenge Athenian naval supremacy, and had found a brilliant military leader in Lysander, who seized the strategic initiative by occupying the Hellespont, the source of Athens' grain imports. Threatened with starvation, Athens sent its last remaining fleet to confront Lysander, who decisively defeated them at Aegospotami (405 BC). The loss of her fleet threatened Athens with bankruptcy. In 404 BC Athens sued for peace, and Sparta dictated a predictably stern settlement: Athens lost her city walls, her fleet, and all of her overseas possessions. The anti-democratic party took power in Athens with Spartan support.

Spartan and Theban dominance

The end of the Peloponnesian War left Sparta the master of Greece, but the narrow outlook of the Spartan warrior elite did not suit them to this role. Within a few years the democratic party regained power in Athens and other cities. In 395 BC the Spartan rulers removed Lysander from office, and Sparta lost her naval supremacy. Athens, Argos, Thebes, and Corinth, the latter two formerly Spartan allies, challenged Spartan dominance in the Corinthian War, which ended inconclusively in 387 BC. That same year Sparta shocked Greek opinion by concluding the Treaty of Antalcidas with Persia by which they surrendered the Greek cities of Ionia and Cyprus, thus reversing a hundred years of Greek victories against Persia. Sparta then tried to further weaken the power of Thebes, which led to a war in which Thebes allied herself with the old enemy, Athens. The Theban generals Epaminondas and Pelopidas won a decisive victory at Leuctra (371 BC). The result of this battle was the end of Spartan supremacy and the establishment of Theban dominance, but Athens also recovered much of her former power. The supremacy of Thebes was short-lived. With the death of Epaminondas at Mantinea (362 BC) the city lost its greatest leader, and his successors blundered into an unsuccessful ten-year war with Phocis. In 346 BC the Thebans appealed to Philip II of Macedon to help them against the Phocians, thus drawing Macedon into Greek affairs for the first time.

The rise of Macedon

The Kingdom of Macedon was formed in the 7th century BC out of northern Greek tribes. They played little part in Greek politics before the beginning of the 4th century, but Philip was an ambitious man who had been educated in Thebes and wanted to play a larger role. In particular, he wanted to be accepted as the new leader of Greece in recovering the freedom of the Greek cities of Asia from Persian rule. By seizing the Greek cities of Amphipolis, Methone and Potidaea, he gained control of the gold and silver mines of Macedonia. This gave him the resources to realize his ambitions. Philip established Macedonian dominance over Thessaly (352 BC) and Thrace, and by 348 BC he controlled everything north of Thermopylae. He used his great wealth to bribe Greek politicians and create a "Macedonian party" in every Greek city. His intervention in the war between Thebes and Phocis brought him recognition as a Greek leader, and gave him his opportunity to become a power in Greek affairs. But despite his sincere admiration for Athens, the Athenian leader Demosthenes, in a series of famous speeches (philippics) roused the Greek cities to resist his advance. In 339 BC Thebes, Athens, Sparta and other Greek states formed an alliance to resist Philip and expel him from the Greek cities he had occupied in the north. But Philip struck first, advancing into Greece and defeating the Greek cities at Chaeronea in 338 BC. This traditionally marks the end of the era of the Greek city-state as an independent political unit, although in fact Athens and other cities survived as independent states until Roman times. Philip tried to win over Athens by flattery and gifts, but did not really succeed. He organised the cities into the League of Corinth, and announced that he would lead an invasion of Persia to liberate the Greek cities and avenge the Persian invasions of the previous century. But before he could do so he was assassinated (336 BC).

The conquests of Alexander

Philip was succeeded by his 20-year-old son Alexander, who immediately set out to carry out his father's plans. He travelled to Corinth where the assembled Greek cities recognised him as leader of the Greeks, then set off north to assemble his forces. The army with which he invaded the Persian Empire was basically Macedonian, but many idealists from the Greek cities also enlisted. But while Alexander was campaigning in Thrace, he heard that the Greek cities had rebelled. He swept south again, captured Thebes, and razed the city to the ground as a warning to the Greek cities that his power could no longer be resisted. In 334 BC Alexander crossed into Asia, and defeated the Persians at the river Granicus. This gave him control of the Ionian coast, and he made a triumphal procession through the liberated Greek cities. After settling affairs in Anatolia, he advanced south through Cilicia into Syria, where he defeated Darius III at Issus (333 BC). He then advanced through Phoenicia to Egypt, which he captured with little resistance, the Egyptians welcoming him as a liberator from Persian oppression. Darius was now ready to make peace and Alexander could have returned home in triumph, but he was determined to conquer Persia and make himself the ruler of the world. He advanced north-east through Syria and Mesopotamia, and defeated Darius again at Gaugamela (331 BC). Darius fled and was killed by his own followers, and Alexander found himself the master of the Persian Empire, occupying Susa and Persepolis without resistance. Persepolis (as an eagle) being offered wine by Ganymede. A child Eros is in the foreground.]] Meanwhile the Greek cities were making renewed efforts to escape from Macedonian control. At Megalopolis in 331 BC, Alexander's regent Antipater defeated the Spartans, who had refused to join the Corinthian League or recognise Macedonian supremacy. Alexander pressed on, advancing through what are now Afghanistan and Pakistan to the Indus river valley, and by 326 BC he had reached Punjab. He might well have advanced down the Ganges to Bengal had not his army, convinced they were at the end of the world, refused to go any further. Alexander reluctantly turned back, and died of a fever in Babylon in 323 BC. Alexander's empire broke up soon after his death, but his conquests permanently changed the Greek world. Thousands of Greeks travelled with him or after him to settle in the new Greek cities he had founded as he advanced, the most important being Alexandria in Egypt. Greek-speaking kingdoms in Egypt, Syria, Iran and Bactria were established. The Hellenistic age had begun.

See also


- Ancient Olympic Games
- Architecture of Ancient Greece
- Art in Ancient Greece
- Eleusinian Mysteries
- Fiction set in Ancient Greece
- Greek literature
- Greek mathematics
- Greek mythology
- Greek philosophy
- Greek theatre
- History of Athens
- History of the Greek language
- Homosexuality in the militaries of ancient Greece
- List of ancient Greeks
- List of ancient Greek cities
- Timeline of Ancient Greece ko:고대 그리스 ja:古代ギリシア th:กรีซโบราณ

Mathematical logic

Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. Although the layperson may think that mathematical logic is the logic of mathematics, the truth is rather that it more closely resembles the mathematics of logic. It comprises those parts of logic that can be modelled mathematically. Earlier appellations were symbolic logic (as opposed to philosophical logic), and metamathematics, which is now restricted as a term to some aspects of proof theory.

History

Mathematical logic was the name given by Giuseppe Peano to what is also known as symbolic logic. In essentials, it is still the logic of Aristotle, but from the point of view of notation it is written as a branch of abstract algebra. Attempts to treat the operations of formal logic in a symbolic or algebraic way were made by some of the more philosophical mathematicians, such as Leibniz and Lambert; but their labors remained little known and isolated. It was George Boole and then Augustus De Morgan, in the middle of the nineteenth century, who presented a systematic mathematical (of course non-quantitative) way of regarding logic. The traditional, Aristotelian doctrine of logic was reformed and completed; and out of it developed an adequate instrument for investigating the fundamental concepts of mathematics. It would be misleading to say that the foundational controversies that were alive in the period 1900-1925 have all been settled; but philosophy of mathematics was greatly clarified by the 'new' logic. While the traditional development of logic (see list of topics in logic) put heavy emphasis on forms of arguments, the attitude of current mathematical logic might be summed up as the combinatorial study of content. This covers both the syntactic (for example, sending a string from a formal language to a compiler program to write it as sequence of machine instructions), and the semantic (constructing specific models or whole sets of them, in model theory). Some landmark publications were the Begriffsschrift by Gottlob Frege, Studies in Logic by Charles Peirce, Principia Mathematica by Bertrand Russell and Alfred North Whitehead, and On Formally Undecidable Propositions of Principia Mathematica and Related Systems by Kurt Godel.

Topics in mathematical logic

The main areas of mathematical logic include model theory, proof theory and recursion theory (often now referred to as computability theory). Axiomatic set theory is sometimes considered too. There are many overlaps with computer science, since many early pioneers in computer science, such as Alan Turing, were mathematicians and logicians. The study of programming language semantics derives from model theory, as does program verification, in particular model checking. The Curry-Howard isomorphism between proofs and programs relates to proof theory; intuitionistic logic and linear logic are significant here. Calculi such as the lambda calculus and combinatory logic are nowadays studied mainly as idealized programming languages. Computer science also contributes to logic by developing techniques for the automatic checking or even finding of proofs, such as automated theorem proving and logic programming.

Some fundamental results

Some important results are:
- The set of valid first-order formulas is recursively enumerable. This follows from Gödel's completeness theorem (which establishes the equivalence of validity and provability), because the set of proofs for first-order logic formulas is recursively enumerable ("semi-decidable"). Therefore, there is a procedure that behaves as follows: Given a first-order formula as its input, the procedure eventually halts if the formula is valid, and runs forever otherwise. Some first-order theorem provers have this completeness property.
- The set of valid first-order formulas is not recursive, i.e., there is no algorithm for checking for universal validity. This follows from Gödel's incompleteness theorem.
- The set of all universally valid second-order formulas is not even recursively enumerable. This is also a consequence of Gödel's incompleteness theorem.
- The Löwenheim-Skolem theorem.
- Cut-elimination in sequent calculus.
- The independence of the continuum hypothesis, proved by Paul Cohen in 1963.

Technical reference

First-order languages and structures

Definition. A first-order language \mathfrak\, is a collection of distinct typographical symbols classified as follows: # The equality symbol =\,; the connectives \lor\,, \lnot\,; the universal quantifier \forall\, and the parentheses (\,, )\,. # A countable set of variable symbols \_^\infty\,. # A set of constant symbols \_\,. # A set of function symbols \_\,. # A set of relation symbols \_\,.
Thus, in order to specify a language, it is often sufficient to spec