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Axiomatic Reasoning

Axiomatic reasoning

In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system; usually though the effort towards complete formalisation brings diminishing returns in certainty, and a lack of readability for humans. Therefore discussion of axiomatic systems is normally only semi-formal. A formal theory typically means an axiomatic system, for example formulated within model theory. A formal proof is a complete rendition of a proof within a formal system.

Properties

An axiomatic system is said to be consistent if it lacks contradiction, i.e. the ability to derive both a statement and its negation from the system's axioms. In an axiomatic system, an axiom is called independent if it is not a theorem that can be derived from other axioms in the system. A system will be called independent if each of its underlying axioms is independent. Although independence is not a necessary requirement for a system, consistency is. An axiomatic system will be called complete if every statement is either derivable or its negation is derivable.

Models

A mathematical model for an axiomatic system is a well-defined set, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system. The existence of a concrete model
- proves the consistency of a system. Models can also be used to show the independence of an axiom in the system. By constructing a valid model for a subsystem without a specific axiom, we show that the omitted axiom is independent if its correctness does not necessarily follow from the subsystem. Two models are said to be isomorphic if a one-to-one correspondence can be found between their elements, in a manner that preserves their relationship. An axiomatic system for which every model is isomorphic to another is called categorial, and the property of categoriality ensures the completeness of a system.
- A model is called concrete if the meanings assigned are objects and relations from the real world, as opposed to an abstract model which is based on other axiomatic systems. The first axiomatic system was Euclidean geometry.

Axiomatic method

The axiomatic method is often discussed as if it were a unitary approach, or uniform procedure. With the example of Euclid to appeal to, it was indeed treated that way for many centuries: up until the beginning of the nineteenth century it was generally assumed, in European mathematics and philosophy (for example in Spinoza's work) that the heritage of Greek mathematics represented the highest standard of intellectual finish (development more geometrico, in the style of the geometers). This traditional approach, in which axioms were supposed to be self-evident and so indisputable, was swept away during the course of the nineteenth century, by the development of Non-Euclidean geometry, the foundations of real analysis, Cantor's set theory and Frege's work on foundations, and Hilbert's 'new' use of axiomatic method as a research tool. For example, group theory was first put on an axiomatic basis towards the end of that century. Once the axioms were clarified (that inverse elements should be required, for example), the subject could proceed autonomously, without reference to the transformation group origins of those studies. Therefore, there are at least three 'modes' of axiomatic method current in mathematics, and in the fields it influences. In caricature, possible attitudes are: #Accept my axioms and you must accept their consequences; #I reject one of your axioms and accept extra models; #My set of axioms defines a research programme. The first case is the classic deductive method. The second goes by the slogan be wise, generalise; it may go along with the assumption that concepts can or should be expressed at some intrinsic 'natural level of generality'. The third was very prominent in the mathematics of the twentieth century, in particular in subjects based around homological algebra. It is easy to see that the axiomatic method has limitations outside mathematics. For example, in political philosophy axioms that lead to unacceptable conclusions are likely to be rejected wholesale; so that no one really assents to version 1 above.

See also


- Axiomatization
- Model theory
- Gödel's incompleteness theorem Category:Mathematical logic Category:Systems

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]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

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

Mathematical theory

Theory has a number of distinct meanings in different fields of knowledge, depending on the context and their methodologies.

Etymology

The word ‘theory’ derives from the Greek ‘theorein’, which means ‘to look at’. According to some sources, it was used frequently in terms of ‘looking at’ a theatre stage, which may explain why sometimes the word ‘theory’ is used as something provisional or not completely resembling real. The term ‘theoria’ (a noun) was already used by the scholars of ancient Greeks.

Science

In scientific usage, a theory does not mean an unsubstantiated guess or hunch, as it does in other contexts. Neither is a scientific theory a fact. Scientific theories are never proven to be true, but can be disproven. All scientific understanding takes the form of hypotheses, theories, or laws. Theories are typically ways of explaining why things happen, often, but not always after the fact that they happen is no longer in scientific dispute. In referring to the "theory of global warming" for example, the worldwide temperatures have been measured and seem to be increasing. The "theory of global warming" refers instead to scientific work that attempts to explain how and why this could be happening. In various sciences, a theory is a logically self-consistent model or framework for describing the behavior of a certain natural or social phenomenon, thus either originating from or supported by experimental evidence (see scientific method). In this sense, a theory is a systematic and formalized expression of all previous observations made that is predictive, logical, testable, and has never been falsified. In physics, the term theory is generally used for a mathematical framework derived from a small set of basic principles, capable of producing experimental predictions for a given category of physical systems. A good example is electromagnetic theory, which encompasses the results that can be derived from Maxwell's equations. This theory is usually taken to be synonymous with classical electromagnetism. The term theoretical is used in science to describe a result that is predicted by theory but has not yet been observed. For example, until recently, black holes were considered theoretical. It is not uncommon in the history of physics for theory to produce predictions that are later confirmed by experiment; failed predictions, however, also occur, and sometimes work to falsify a theory. Conversely, at any time in the study of physics there can also be confirmed experimental results that are not yet explained by theory. For a given body of theory to be considered part of established scientific knowledge, it is usually necessary for it to characterize a critical experiment, namely an experimental result not predicted by any existing established theory. Unfortunately, usage of the term theory is muddled by scientists in such examples as string theory and various theories of everything, which are more correctly characterized at present as a bundle of competing hypotheses or a protoscience. A hypothesis, however, is still vastly more reliable than a conjecture, which is at best an untested guess consistent with selected data and often simply a belief based on non-repeatable experiments, anecdotes, popular opinion, "wisdom of the ancients," commercial motivation, or mysticism. Even worse, theory has almost the opposite meaning in common use than its definition in the sciences, and this change can be seen in modern dictionaries which now list theory as a "guess or hunch" in preference to the former scientific definition that used to be the dominant one. In everyday language, a theory is (Morrison, 2005, p. 39): :...a hunch that a detective comes up with in a murder mystery. It is one of several competing ideas, none of them proved. Fringe theories and conspiracy theories are crazy ideas that are out of the mainstream. New medicines or changes in the tax laws may be good in theory but don't work in practice. Among some scientists, theorists are thought to lack solid grounding in the facts... Even scientists tend to use the now common definition in everyday speech and writing, being more careful in published material. Yet a California Academy of Sciences exhibit on fossils included this line: "Scientists have a number of theories about why ammonites develop spines on their shells" (emphasis added; from Morrison, 2005).

Models

Humans construct theories in order to explain, predict and master phenomena (e.g. inanimate things, events, or the behaviour of animals). In many instances, this is seen to be the construction of models of reality. A theory makes generalizations about observations and consists of an interrelated, coherent set of ideas and models. According to Stephen Hawking in A Brief History of Time, "a theory is a good theory if it satisfies two requirements: It must accurately describe a large class of observations on the basis of a model that contains only a few arbitrary elements, and it must make definite predictions about the results of future observations." He goes on to state, "any physical theory is always provisional, in the sense that it is only a hypothesis; you can never prove it. No matter how many times the results of experiments agree with some theory, you can never be sure that the next time the result will not contradict the theory. On the other hand, you can disprove a theory by finding even a single observation that disagrees with the predictions of the theory." This is borne out by what Isaac Asimov said in "Understanding Physics". He spoke of theories as "arguments" where one deduces a "scheme" or model. Arguments or theories always begin with Hawking's "arbitrary elements" which are here described as "assumptions". An assumption according to Asimov is "something accepted without proof, and it is incorrect to speak of an assumption as either true or false, since there is no way of proving it to be either. (If there were, it would no longer be an assumption.) It is better to consider assumptions as either useful or useless, depending on whether deductions made from them corresponded to reality. .. On the other hand, it seems obvious that assumptions are the weak points in any argument, as they have to be accepted on faith in a philosophy of science that prides itself on its rationalsim. Since we must start somewhere, we must have assumptions, but at least let us have as few assumptions as possible." (See Ockham's razor) An example of using assumptions to formulate a theory is when Albert Einstein put forth his Special Theory of Relativity. He took two phenomena that had been observed i.e. that the "addition of velocities" is valid (Galilean transformation) and that light did not appear to have an "addition of velocities" (Michelson-Morley experiment). He assumed that both of these were correct and formulated his theory based on these assumptions by simply altering the Galilean transformation to accommodate the lack of addition of velocities with regard to the speed of light. Therefore, the model created in his theory is based on the assumption that light maintains a constant velocity (or more precisely the speed of light is a constant). An example of how theories are models can be seen from theories on the planetary system. The Greeks formulated theories that were recorded by the astronomer Ptolemy. In Ptolemy's planetary model, the earth was at the center, the planets and the sun made circular orbits around the earth, and the stars were on a sphere outside of the orbits of the planet and the earth. Retrograde motion of the planets was explained by smaller circular orbits of individual planets. This could actually be built into a literal model and illustrated as a model. Mathematical calculations could be made for the prediction of where the planets would be to a great degree of accuracy, so that this model of the planetary system survived over 1500 years until the time of Copernicus. So one can see how a theory is a model of reality that explains certain scientific facts yet may not be a true picture of reality and another more accurate theory can later replace the previous model.

Types of theories

There are two uses of the word theory; a supposition which is not backed by observation is known as a conjecture, and if backed by observation it is a hypothesis. Most theory evolves from hypotheses, but the reverse is not true: many hypotheses turn out to be false and so do not evolve into theory. A theory is different from a theorem. The former is a model of physical events and cannot be proved from basic axioms. The latter is a statement of mathematical fact which logically follows from a set of axioms. A theory is also different from a physical law in that the former is a model of reality whereas the latter is a statement of what has been observed. Theories can become accepted if they are able to make correct predictions and avoid incorrect ones. Theories which are simpler, and more mathematically elegant, tend to be accepted over theories which are complex. Theories are more likely to be accepted if they connect a wide range of phenomena. The process of accepting theories, or of extending existing theory, is part of the scientific method.

Further explanation of a scientific theory

As noted above, in common usage a theory is defined as little more than a guess or a hypothesis. But in science and generally in academic usage, a theory is much more than that. A theory is an established paradigm that explains all or much of the data we have and offers valid predictions that can be tested. In science, a theory is not considered fact or infallible, because we can never assume we know all there is to know. Instead, theories remain standing until they are disproved, at which point they are thrown out altogether or modified to fit the additional data. Theories start out with empirical observations such as "sometimes water turns into ice." At some point, there is a need or curiosity to find out why this is, which leads to a theoretical/scientific phase. In scientific theories, this then leads to research, in combination with auxiliary and other hypotheses (see scientific method), which may then eventually lead to a theory. Some scientific theories (such as the theory of gravity) are so widely accepted that they are often seen as laws. This, however, rests on a mistaken assumption of what theories and laws are. Theories and laws are not rungs in a ladder of truth, but different sets of data. A law is a general statement based on observations. A canonical example of a disproved theory is the geocentric model of the universe proposed by Ptolemy. Evidence, in the form of Galileo's observation of the phases of Venus in 1610, was produced which was completely incompatible with the predictions set forth by the theory. This falsification, though, did not necessarily mean that only one alternative theory was necessarily the "correct" replacement — both the Copernican system and the Tychonian system predicted the phases of Venus.

Characteristics

In science, a body of descriptions of knowledge is usually only called a theory once it has a firm empirical basis, i.e., it # is consistent with pre-existing theory to the extent that the pre-existing theory was experimentally verified, though it will often show pre-existing theory to be wrong in an exact sense, # is supported by many strands of evidence rather than a single foundation, ensuring that it probably is a good approximation if not totally correct, # makes predictions that might someday be used to disprove the theory, # is tentative, correctable and dynamic, in allowing for changes to be made as new data is discovered, rather than asserting certainty, and # is the most parsimonious explanation, sparing in proposed entities or explanations, commonly referred to as passing Occam's Razor. This is true of such established theories as special and general relativity, quantum mechanics, plate tectonics, evolution, etc. Theories considered scientific meet at least most, but ideally all, of the above criteria. The fewer which are matched, the less scientific it is; those that meet only several or none at all, cannot be said to be scientific in any meaningful sense of the word. Karl Popper described the characteristics of a scientific theory as: 1. It is easy to obtain confirmations, or verifications, for nearly every theory — if we look for confirmations. 2. Confirmations should count only if they are the result of risky predictions; that is to say, if, unenlightened by the theory in question, we should have expected an event which was incompatible with the theory — an event which would have refuted the theory. 3. Every "good" scientific theory is a prohibition: it forbids certain things to happen. The more a theory forbids, the better it is. 4. A theory which is not refutable by any conceivable event is non-scientific. Irrefutability is not a virtue of a theory (as people often think) but a vice. 5. Every genuine test of a theory is an attempt to falsify it, or to refute it. Testability is falsifiability; but there are degrees of testability: some theories are more testable, more exposed to refutation, than others; they take, as it were, greater risks. 6. Confirming evidence should not count except when it is the result of a genuine test of the theory; and this means that it can be presented as a serious but unsuccessful attempt to falsify the theory. (I now speak in such cases of "corroborating evidence.") 7. Some genuinely testable theories, when found to be false, are still upheld by their admirers — for example by introducing ad hoc some auxiliary assumption, or by reinterpreting the theory ad hoc in such a way that it escapes refutation. Such a procedure is always possible, but it rescues the theory from refutation only at the price of destroying, or at least lowering, its scientific status. (I later described such a rescuing operation as a "conventionalist twist" or a "conventionalist stratagem.") One can sum up all this by saying that the criterion of the scientific status of a theory is its falsifiability, or refutability, or testability."--end quote

Mathematics

In mathematics, the word theory is used informally to refer to certain distinct bodies of knowledge about mathematics. This knowledge consists of axioms, definitions, theorems and computational techniques, all related in some way by tradition or practice. Examples include group theory, set theory, Lebesgue integration theory and field theory. The term "theory" also has a formal usage in mathematics, particularly in mathematical logic and model theory. A theory in this sense is a set of statements closed under certain rules of inference. A typical theory will present certain axioms and rules, corresponding to a useful or interesting abstraction, and then derive non-obvious theorems from those axioms. The resulting theorems often provide solutions to real-world problems which correspond to the original abstraction. Obvious examples include arithmetic (abstracting the concept of number), geometry (the concept of space), and probability (the concept of randomness). However, Gödel's incompleteness theorem shows that no consistent theory capable of defining the concept of natural numbers can derive all true statements about those numbers. This sets a fundamental limit to the applicability of any mathematical system.

Other fields

Theories exist not only in the so-called hard sciences; but in all fields of academic study, from philosophy to music to literature. In the humanities, theory is often used as an abbreviation for critical theory or literary theory, referring to continental philosophy's aesthetics or its attempts to understand the structure of society and to conceptualize alternatives. In philosophy, theoreticism refers to the overuse of theory.

List of famous theories


- Mathematics: Axiomatic set theory - Chaos theory - Graph theory - Number theory - Probability theory
- Statistics : Extreme value theory
- Physics: Theory of relativity - Special relativity - General relativity - Quantum field theory - Acoustic theory - Antenna theory
- Planetary science: Giant impact theory
- Biology: Evolution by natural selection - Cell theory
- Chemistry: Atomic theory - Kinetic theory of gases
- Geology: Continental drift - Plate tectonics
- Climatology: Global warming
- Humanities: Critical theory
- Sociology: Social Theory - Critical social theory - Value theory
- Philosophy: Speculative reason
- Literature: Literary theory
- Music: Music theory
- Computer science: Algorithmic information theory - Computation theory
- Games: Rational choice theory - Game theory
- Other: Obsolete scientific theories - Phlogiston theory

See also


- Scientific method

Reference


- Morrison, David. 2005. "Only a theory? Framing the evolution/creation issue". Skeptical Inquirer, 29 (6): 37-41.
- Karl Popper, Conjectures and Refutations, London: Routledge and Keagan Paul, 1963, pp. 33-39; from Theodore Schick, ed., Readings in the Philosophy of Science, Mountain View, CA: Mayfield Publishing Company, 2000, pp. 9-13. Theories Category:Scientific method Category:Mathematical terminology Category:Philosophy of science ja:理論

Formal system

In logic, mathematics, and computer science, a formal system is a formal grammar used for modelling purposes. Formalization is the act of creating a formal system, in an attempt to capture the essential features of a real-world or conceptual system in formal language. In mathematics, formal proofs are the product of formal systems, consisting of axioms and rules of deduction. Theorems are then recognised as the possible 'last lines' of formal proofs. The point of view that this picture encompasses mathematics has been called formalist. The term has been used pejoratively. On the other hand, David Hilbert founded metamathematics as a discipline designed for discussing formal systems; it is not assumed that the metalanguage in which proofs are studied is itself less informal than the usual habits of mathematicians suggest. To contrast with the metalanguage, the language described by a formal grammar is often called an object language (i.e., the object of discussion - this distinction may have been introduced by Carnap). It has become common to speak of a formalism, more-or-less synonymously with a formal system within standard mathematics invented for a particular purpose. This may not be much more than a notation, such as Dirac's bra-ket notation. Mathematical formal systems consist of the following: # A finite set of symbols which can be used for constructing formulae. # A grammar, i.e. a way of constructing well-formed formulae out of the symbols, such that it is possible to find a decision procedure for deciding whether a formula is a well-formed formula (wff) or not. # A set of axioms or axiom schemata: each axiom has to be a wff. # A set of inference rules. # A set of theorems. This set includes all the axioms, plus all wffs which can be derived from previously-derived theorems by means of rules of inference. Unlike the grammar for wffs, there is no guarantee that there will be a decision procedure for deciding whether a given wff is a theorem or not. Category:Formal languages Category:Mathematical logic Category:Formal methods Category:Systems

Proof

The word proof can mean:
- originally, a test assessing the validity or quality of something.
- a rigorous, compelling argument, including:
  - a logical argument or a mathematical proof (see also proof theory).
  - a legal proof.
  - a large accumulation of scientific evidence.
- as an adjective, to mean firm or successful in resisting.
  - Rabbit-Proof Fence (film), film based on a book.
- a photograph, print or page layout presented as an example for proofreading or approval.
- alcoholic proof, a measure of how much ethanol is in an alcoholic beverage.
- proof coinage, a coin made as an example of a particular strike.
- Proof (play), a play by David Auburn.
- Proof (1991 film), an Australian film by Jocelyn Moorhouse starring Hugo Weaving and Russell Crowe.
- Proof (2005 film), a film by John Madden.
- Proof (rapper), a rapper and member of the group D12.
- Yeast (baking) yeast or leavening agent, with sugar and water before making activator, beer, bread, cider, dough, or wine.
- The testing process for determining the integrity of a firearm. See Birmingham Gun Barrel Proof House. ja:証明 simple:Proof



Isomorphism

:This article describes mathematical isomorphism. For the sociological term, see isomorphism (sociology). In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich. Douglas Hofstadter provides an informal definition: :The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where "corresponding" means that the two parts play similar roles in their respective structures. (Gödel, Escher, Bach, p. 49) Formally, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e. structure-preserving mappings. If there exists an isomorphism between two structures, we call the two structures isomorphic. Isomorphic structures are "the same" at some level of abstraction; ignoring the specific identities of the elements in the underlying sets, and focusing just on the structures themselves, the two structures are identical. Here are some everyday examples of isomorphic structures.
- A solid cube made of wood and a solid cube made of lead are both solid cubes; although their matter differs, their geometric structures are isomorphic.
- A standard deck of 52 playing cards with green backs and a standard deck of 52 playing cards with brown backs; although the colours on the backs of each deck differ, the decks are structurally isomorphic — if we wish to play cards, it doesn't matter which deck we choose to use.
- The Clock Tower in London (that contains Big Ben) and a wristwatch; although the clocks vary greatly in size, their mechanisms of reckoning time are isomorphic.
- A six-sided die and a bag from which a number 1 through 6 is chosen; although the method of obtaining a number is different, their random number generating abilities are isomorphic. This is an example of functional isomorphism, without the presumption of geometric isomorphism. For example, if one object consists of a set X with an ordering ≤ and the other object consists of a set Y with an ordering \sqsubseteq then an isomorphism from X to Y is a bijective function f : X → Y such that : f(u) \sqsubseteq f(v) iff u ≤ v. Such an isomorphism is called an order isomorphism. Or, if on these sets, the unknown binary operations \star and \Diamond are defined, respectively, then an isomorphism from X to Y is a bijective function f : X → Y such that : f(u) \Diamond f(v) = f(u \star v) for all u, v in X. When the objects in question are groups, such an isomorphism is called a group isomorphism. Similarly, if the objects are fields, it is called a field isomorphism. In universal algebra, one can give a general definition of isomorphism that covers these and many other cases. The definition of isomorphism given in category theory is even more general. In graph theory, an isomorphism between two graphs G and H is a bijective map f from the vertices of G to the vertices of H that preserves the "edge structure" in the sense that there is an edge from vertex u to vertex v in G iff there is an edge from f(u) to f(v) in H. In linear algebra, an isomorphism can also be defined as a linear map between two vector spaces that is one-to-one and onto.

See also


- automorphism
- homomorphism
- epimorphism
- isomorphism class
- monomorphism
- morphism Category:Abstract algebra Category:Algebra Category:Category theory



Spinoza

Benedictus de Spinoza (November 24, 1632 – February 21, 1677), was named Baruch Spinoza by his synagogue elders and known as Bento de Espinosa or Bento d'Espiñoza in his native Amsterdam. Along with René Descartes and Gottfried Leibniz, he was one of the great rationalists of 17th-century philosophy. He is considered the founder of modern Biblical criticism. His magnum opus was the Ethics.

Life

Born to a Sephardic family among the Portuguese Jews of Amsterdam, he gained fame for his positions of pantheism and neutral monism, as well as the fact that his Ethics was written in the form of postulates and definitions, as though it were a geometry treatise. In the summer of 1656, he was excommunicated because of apostasy from the Jewish community for his claims that God is the mechanism of nature and the universe, having no personality, and that the Bible is a metaphorical and allegorical work used to teach the nature of God, both of which were based on a form of Cartesianism (see René Descartes). Following his excommunication, he adopted the first name Benedictus (the Latin equivalent of his given name, either Baruch or Bento). The terms of his excommunication were quite severe; excerpts from the text may be found at [http://www.mnstate.edu/mouch/spinoza/excomm.html]. After his excommunication, he lived and worked for a while in the school of Franciscus van den Enden, who taught him Latin and perhaps also introduced him to modern philosophy. In this period Spinoza also became acquainted with several Collegiants, members of a non-dogmatic and interdenominational sect with tendencies towards Rationalism. By the beginning of the 1660s Spinoza's name became more widely known and he was visited by Henry Oldenburg, with whom he would maintain a correspondence for the rest of his life. Spinoza's first publication was on the Principles of Cartesian Philosophy, but already in that work he introduced some of his own ideas. In 1665 he notified Oldenburg that he had started to work on a new book, the Theologico-Political Treatise, published in 1670. Since the public reactions to the anonymously published Theologico-Political Treatise turned unfavourable to his brand of [http://www.utm.edu/research/iep/d/descarte.htm] Cartesianism, he abstained from publishing more of his works. The Ethics and all other works, apart from the Principles of Cartesian Philosophy and the Theologico-Political Treatise, were published after his death in the Opera postuma edited by his friends. Some of the major figures whom Spinoza met include Henry Oldenburg and Leibniz.

Philosophy

Known as both the "greatest Jew" and the "greatest Atheist", Spinoza contended that God and Nature were two names for the same reality, namely the single substance (meaning "to stand beneath" rather than "matter") that underlies the universe and of which all lesser "entities" are actually modes or modifications. The argument for this single substance runs something as follows: :1. Substance exists and cannot be dependent on anything else for its existence. :2. No two substances can share an attribute. ::Proof: If they share an attribute, they would be identical. Therefore they can only be individuated by their modes. But then they would depend on their modes for their identity. This would have the substance being dependent on its mode, in violation of premise 1. Therefore, two substances cannot share the same attribute. :3. A substance can only be caused by something similar to itself (something that shares its attribute). :4. Substance cannot be caused. ::Proof: Something can only be caused by something which is similar to itself, in other words something that shares its attribute. But according to premise 2, no two substances can share an attribute. Therefore substance cannot be caused. :5. Substance is infinite. ::Proof: If substance were not infinite, it would be finite and limited by something. But to be limited by something is to be dependent on it. However, substance cannot be dependent on anything else (premise 1), therefore substance is infinite. :Conclusion: There can only be one substance. ::Proof: If there were two infinite substances, they would limit each other. But this would act as a restraint, and they would be dependent on each other. But they cannot be dependent on each other (premise 1), therefore there cannot be two substances. Spinoza contended that "Deus sive Natura" ("God or Nature") was a being of infinitely many attributes, of which extension and thought were two. His account of the nature of reality, then, seems to treat the physical and mental worlds as two different, parallel "subworlds" that neither overlap nor interact. This formulation is a historically significant panpsychist solution to the mind-body problem known as neutral monism. Spinoza was a thoroughgoing determinist who held that absolutely everything that happens occurs through the operation of necessity. For him, even human behaviour is fully determined, freedom being our capacity to know we are determined and to understand why we act as we do. So freedom is not the possibility to say "no" to what happens to us but the possibility to say "yes" and fully understand why things should necessarily happen that way. By forming more "adequate" ideas about what we do and our emotions or affections, we become the adequate cause of our effects (internal or external), which entails an increase in activity (versus passivity). This means that we become both more free and more like God, as Spinoza argues in the Scholium to Prop. 49, Part II. Spinoza's philosophy has much in common with Stoicism inasmuch as both philosophies sought to fulfil a therapeutic role by instructing people how to attain happiness (or eudaimonia, for the Stoics). However, Spinoza differed sharply from the Stoics in one important respect: he utterly rejected their contention that reason could defeat emotion. On the contrary, he contended, an emotion can be displaced or overcome only by a stronger emotion. For him, the crucial distinction was between active and passive emotions, the former being those that are rationally understood and the latter those that are not. He also held that knowledge of true causes of passive emotion can transform it to an active emotion, thus anticipating one of the key ideas of Sigmund Freud's psychoanalysis. Some of Spinoza's philosophical positions are:
- God is the natural world and has no personality.
- The natural world made itself.
- There is no real difference between good and evil.
- Everything must necessarily happen the way that it does. Therefore, there is no free will.
- Everything done by humans and other animals is excellent and divine.
- All rights are derived from the State.
- Animals can be used in any way by people for the benefit of the human race.

Spinoza's Definitions

From G. H. R. Parkinson's "Benedict de Spinoza - The Ethics and On the Improvement of the Understanding"; ISBN: 0460873474; p. 260. :"Spinoza's definitions are of the kind now commonly called 'stipulative'; that is, they tell the reader how Spinoza proposes to use certain words. Spinoza is not concerned (as a Dictionary is concerned) to describe the standard uses of words. His purpose, as he observes in the Ethics (E3: Def. XX. Expl.- p. 130) is to explain, not the meaning of words , but the nature of things. One may compare what is done by scientists, when they introduce new technical terms, or give old words a new sense, with a view to explaining what it is that interests them. For Spinoza's views about definition, cf. On the Improvement of the Understanding:[95-98]; p. 253.

Modern relevance

Albert Einstein said that Spinoza was the philosopher who had most influenced his worldview (Weltanschauung). Spinoza equated God (infinite substance) with Nature, and Einstein, too, believed in an impersonal deity. His desire to understand Nature through physics can be seen as contemplation of God. Arne Næss, the father of the deep ecology movement, acknowledged drawing much inspiration from the works of Spinoza. In the late twentieth century, there was a great increase in philosophical interest in Spinoza in Europe, particularly from a left-wing or Marxist perspective. Notable philosophers Gilles Deleuze, Antonio Negri and Étienne Balibar have each written books on Spinoza. Stuart Hampshire wrote a major English language study of Spinoza, though H H Joachim's work is equally valuable. Spinoza's portrait featured prominently on the older series of the 1000 Guilder banknote, which was legal tender in the Netherlands until the euro was introduced in 2002. The highest and most prestigious scientific prize of the Netherlands is named the Spinozapremie (Spinoza reward).

Major Works


- Ethics
- Tractatus Theologico-Politicus

Quotes

Mind and body are one and the same individual which is conceived now under the attribute of thought, and now under the attribute of extension.
-Ethics II prop. 7 I have laboured carefully, not to mock, lament, or execrate human actions, but to understand them.
-Spinoza's A Political Treatise; ISBN: 0486202496; p. 288.

Bibliography

By Spinoza


- Short Treatise on God, Man and His Well-Being
- On the Improvement of the Understanding (1662) (Project Gutenberg Entry: [http://gutenberg.net/etext/1016])
- Principles of Cartesian Philosophy (1663)
- Tractatus Theologico-Politicus (A Theologico-Political Treatise) (1670) (Project Gutenberg Entry: [http://gutenberg.net/etext/989 Part 1], [http://gutenberg.net/etext/990 Part 2], [http://gutenberg.net/etext/991 Part 3], [http://gutenberg.net/etext/992 Part 4])
- Ethica Ordine Geometrico Demonstrata (The Ethics) (1677) (Project Gutenberg Entry: [http://gutenberg.net/etext/3800])
- Hebrew Grammar (1677)

About Spinoza


- Steven Nadler, Spinoza: A Life, Cambridge U. Press, 1999, ISBN 0-521-55210-9
- Margaret Gullan-Whur, "`Within Reason: A Life of Spinoza". Jonathan Cape, 1998, ISBN 0-224-05046-X

See also


- Spinozism
- Liberalism
- Contributions to liberal theory
- Voorburg

External links


- [http://rwmeijer.ws/spinoza/ The Ethics] - Split-screen Latin/English or Latin/French
- [http://www.earlymoderntexts.com/ The Ethics]A READABLE version with all the content still there.
- [http://cf.uba.uva.nl/en/digilib/philosophy/spinheng.html Vereniging Het Spinozahuis]
- [http://www.spinoza.net The Spinoza Net]
- [http://bdsweb.tripod.com Spinoza and Spinozism] - BDSweb
- [http://www.philosophyarchive.com/text.php?era=1600-1699&author=Spinoza&text=A%20Theologico-Political%20Treatise A Theologico-Political Treatise ] -English Translation
- [http://hyperspinoza.caute.lautre.net HyperSpinoza]
- [http://www.iep.utm.edu/s/spinoza.htm Internet Encyclopedia of Philosophy - Spinoza]
- [http://atheisme.free.fr/Biographies/Spinoza_e.htm Biography of Spinoza]
- Stanford Encyclopedia of Philosophy:
  - [http://plato.stanford.edu/entries/spinoza/ Spinoza]
  - [http://plato.stanford.edu/entries/spinoza-psychological/ Spinoza's Psychological Theory]
- Spinoza, Baruch Spinoza, Baruch Spinoza, Baruch Spinoza, Baruch Spinoza, Baruch Spinoza, Baruch Spinoza, Baruch ko:스피노자 ja:バールーフ・デ・スピノザ nb:Baruch de Spinoza th:บารุค สปิโนซา



Cantor

Cantor may refer to:
- a hazzan, in Judaism is the English a professional singer who leads Jewish prayer services
- Cantus, where Cantor is the title of a member of a student society who is the main singer
- Cantor in the Roman Catholic Church is the title of the liturgical minister responsible for leading liturgical singing during the Mass. The primary role of the cantor is to intone the Responsorial Psalm during the Mass. The cantor is designated a "liturgical minister" along with lectors, acolytes, and Extraordinary Ministers of the Eucharist. Other responsibilities include leading congregational hymns, offering the Intercessions (Prayers of the Faithful), and intoning the various liturgical responses (Gloria, Sanctus, Great Amen, Agnus Dei). The cantor also performs this function in other Christian denominations. People:
- Andrés Cantor, a Spanish-language soccer announcer
- Eddie C