:: wikimiki.org ::

Infinite

Infinite

Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. The word infinity comes from Latin : "Infinito", unending. In theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. In philosophy, infinity can be attributed to space and time, as for instance in Kant's first antinomy. In both theology and philosophy, infinity is explored in articles such as the Ultimate, the Absolute, God, and Zeno's paradoxes. In mathematics, infinity is relevant to or the subject matter of articles such as mathematical limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals, Russell's paradox, hyperreal numbers, projective geometry, extended real numbers and the Absolute Infinite. By some, infinity is considered to be not a number but a concept of increase beyond bounds. In popular culture, we have Buzz Lightyear's rallying cry, "To infinity — and beyond!", which may also be viewed as the rallying cry of set theorists considering large cardinals. For a discussion about infinity and the physical universe, see Universe.

History

Ancient view of infinity

The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800 BC - 800 BC) which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. Jaina mathematicians were the first to conceive of different orders of infinity, including one they called unenumerable (innumerable).[http://www.infinityfoundation.com/mandala/t_es/t_es_agraw_jaina.htm] [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html] The concept of different orders of infinity would remain unknown in Europe until the late 19th century. In Europe, the traditional view derives from Aristotle: :"... it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence the infinite is potential, never actual; the number of parts that can be taken always surpasses any assigned number." [Physics 207b8] This is often called potential infinity; however there are two ideas mixed up with this. One is that it is always possible to find a number of things that surpasses any given number, even if there are not actually such things. The other is that we may quantify over infinite sets without restriction. For example, ∀n∈Z(∃m∈Z[m>n∧P(m)]), which reads, "for any integer n, there exists an integer m > n such that P(m)". The second view is found in a clearer form by medieval writers such as William of Ockham: :"Sed omne continuum est actualiter existens. Igitur quaelibet pars sua est vere existens in rerum natura. Sed partes continui sunt infinitae quia non tot quin plures, igitur partes infinitae sunt actualiter existentes." (But every continuum is actually existent. Therefore any of its parts is really existent in nature. But the parts of the continuum are infinite because there are not so many that there are not more, and therefore the infinite parts are actually existent.) The parts are actually there, in some sense. However, on this view, no infinite magnitude can have a number, for whatever number we can imagine, there is always a larger one: "There are not so many (in number) that there are no more". Aquinas also argued against the idea that infinity could be in any sense complete, or a totality.

Views from the Renaissance to modern times

Galileo (during his long house arrest in Siena after his condemnation by the Inquisition) was the first to notice that we can place an infinite set into one-to-one correspondence with one of its proper subsets (any part of the set, that is not the whole). For example, we can match up the "set" of even numbers with the natural numbers as follows: :1, 2, 3, 4, ... :2, 4, 6, 8, ... It appeared, by this reasoning, as though a set which is naturally smaller than the set of which it is a part (since it does not contain all the members of that set) is in some sense the same size. He thought this was one of the difficulties which arise when we try, "with our finite minds", to comprehend the infinite. :"So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal", "greater", and "less", are not applicable to infinite, but only to finite, quantities." [On two New Sciences, 1638] The idea that size can be measured by one-to-one correspondence is today known as Hume's principle, although Hume, like Galileo, believed the principle could not be applied to infinite sets. Locke, in common with most of the empiricist philosophers, also believed that we can have no proper idea of the infinite. They believed all our ideas were derived from sense data or "impressions", and since all sensory impressions are inherently finite, so too are our thoughts and ideas. Our idea of infinity is merely negative or privative. :"Whatever positive ideas we have in our minds of any space, duration, or number, let them be never so great, they are still finite; but when we suppose an inexhaustible remainder, from which we remove all bounds, and wherein we allow the mind an endless progression of thought, without ever completing the idea, there we have our idea of infinity ... yet when we would frame in our minds the idea of an infinite space or duration, that idea is very obscure and confused, because it is made up of two parts very different, if not inconsistent. For let a man frame in his mind an idea of any space or number, as great as he will, it is plain the mind rests and terminates in that idea; which is contrary to the idea of infinity, which consists in a supposed endless progression." (Essay, II. xvii. 7., author's emphasis) Famously, the ultra-empiricist Hobbes tried to defend the idea of a potential infinity in the light of the discovery by Evangelista Torricelli, of a figure (Gabriel's horn) whose surface area is infinite, but whose volume is finite. Not reported, this motivation of Hobbes came too late as curves having infinite length yet bounding finite areas were known much before. Such seeming paradoxes are resolved by taking any finite figure and stretching its content infinitely in one direction; the magnitude of its content is unchanged as its divisions drop off geometrically but the magnitude of its bounds increases to infinity by necessity. Potentiality lies in the definitions of this operation, as well-defined and interconsistent mathematical axioms. A potential infinity is allowed by letting an infinitely-large quantity be cancelled out by an infinitely-small quantity.

Modern philosophical views

Modern discussion of the infinite is now regarded as part of set theory and mathematics, and generally avoided by philosophers. An exception was Wittgenstein, who made an impassioned attack upon axiomatic set theory, and upon the idea of the actual infinite, during his "middle period". (see also Logic of antinomies :"Does the relation m = 2n correlate the class of all numbers with one of its subclasses? No. It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of classes, of which one is correlated with the other, but which are never related as class and subclass. Neither is this infinite process itself in some sense or other such a pair of classes ... In the superstition that m = 2n correlates a class with its subclass, we merely have yet another case of ambiguous grammar." (Philosophical Remarks § 141, cf Philosophical Grammar p. 465) Unlike the traditional empiricists, he thought that the infinite was in some way given to sense experience. :"... I can see in space the possibility of any finite experience ... we recognise [the] essential infinity of space in its smallest part." "[Time] is infinite in the same sense as the three-dimensional space of sight and movement is infinite, even if in fact I can only see as far as the walls of my room." :"... what is infinite about endlessness is only the endlessness itself."

Infinity symbol

The precise origins of the infinity symbol

\infty

are unclear. One possibility is suggested by the name it is sometimes called — the lemniscate, from the Latin lemniscus, meaning "ribbon". One can imagine walking forever along a simple loop formed from a ribbon. A popular explanation is that the infinity symbol is derived from the shape of a Möbius strip. Again, one can imagine walking along its surface forever. This possible explanation is probably incorrect, however, since the symbol had been in use to represent infinity for over two hundred years before August Ferdinand Möbius and Johann Benedict Listing discovered the Möbius strip in 1858. John Wallis is usually credited with introducing

\infty

as a symbol for infinity in 1655 in his De sectionibus conicus. One conjecture about why he chose this symbol is that he derived it from a Roman numeral for 1000 that was in turn derived from the Etruscan numeral for 1000, which looked somewhat like CIƆ and was sometimes used to mean "many". Another conjecture is that he derived it from the Greek letter ω (omega), the last letter in the Greek alphabet. The infinity symbol is represented in Unicode by the character ∞ (∞).

Mathematical infinity

Infinity in real analysis

In real analysis, the symbol

\infty

, called "infinity", denotes an unbounded limit.

x \rightarrow \infty

means that x grows beyond any assigned value, and

x \rightarrow -\infty

means x is eventually less than any assigned value. Points labeled

\infty

and

-\infty

can be added to the real numbers as a topological space, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat

\infty

and

-\infty

as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also introduces a line at infinity in plane geometry, and so forth for higher dimensions. Infinity is often used not only to define a limit but as if it were a value in the extended real numbers in real analysis; if f(t) ≥ 0 then
-

\int_^ \, f(t) dt \  = \infty

means that f(t) does not bound a finite area from 0 to 1
-

\int_^ \, f(t) dt \  = \infty

means that the area under f(t) is not finite
-

\int_^ \, f(t) dt \  = 1

means that the area under f(t) approaches 1

Infinity in complex analysis

As in real analysis, in complex analysis the symbol

\infty

, called "infinity", denotes an unbounded limit.

x \rightarrow \infty

means that the magnitude

|x|

of x grows beyond any assigned value. A point labeled

\infty

can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is still a one-dimensional complex manifold and called the extended complex plane or the Riemann sphere. In this context is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of

\infty

at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations.

Arithmetic properties of infinity

Infinity is not a real number but may be considered part of the extended real number line, in which arithmetic operations involving infinity may be performed.

Infinity with itself

\infty + \infty = \infty \cdot \infty = (-\infty) \cdot (-\infty) = \infty

(-\infty) + (-\infty) = \infty \cdot (-\infty) = (-\infty) \cdot \infty = (-\infty)

Operations involving infinity and real numbers

-\infty < x < \infty

x + \infty = \infty

and

x + (-\infty) = (-\infty)

x - \infty = -\infty

x - (-\infty) = \infty

= 0

and

= 0

# If

0 then x \cdot \infty = \infty and x \cdot (-\infty) = (-\infty) .
# If -\infty then x \cdot \infty = -\infty and x \cdot (-\infty) = \infty .

Undefined operations

0 \cdot \infty

and

0 \cdot (-\infty)

\infty + (-\infty)

and

(-\infty) + \infty

^0

1^

Notice that

[ = 0] \not\equiv [0 \cdot \infty = x]

. This is because zero times infinity is undefined.

Infinity in set theory

A different type of "infinity" are the ordinal and cardinal infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null (

\aleph_0

), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work by Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of collections, or sets. Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. An infinite set can simply be defined as one having the same size as at least one of its "proper" parts; this notion of infinity is called Dedekind infinite. Cantor defined two kinds of infinite numbers, the ordinal numbers and the cardinal numbers. Ordinal numbers may be identified with well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one to one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes. Our intuition gained from finite sets breaks down when dealing with infinite sets. One example of this is Hilbert's paradox of the Grand Hotel.

Mathematics without infinity

Leopold Kronecker rejected the notion of infinity and began a school of thought in the philosophy of mathematics called finitism, which led to the philosophical and mathematical school of mathematical constructivism.

Use of infinity in common speech

In common parlance, infinity is often used in a hyperbolic sense. For example, "The movie was infinitely boring, but we had to wait forever to get tickets." In video games, "infinite lives" and "infinite ammo" usually mean a never-ending supply of lives and ammunition. An infinite loop in computer programming is a conditional loop construction whose condition always evaluates to true. As long as there is no external interaction (such as switching the computer off, or the heat death of the universe), the loop will continue to run for all time. In practice however, most programming loops considered as infinite will halt by exceeding the (finite) number range of one of its variables. See halting problem. These terms describe things that are only theoretically infinite; it is impossible to play a video game for an infinite period of time or keep a computer running for an infinite period of time. The number Infinity plus 1 is also used sometimes in common speech.

Physical infinity

In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. counting). It is therefore assumed by physicists that no measurable quantity could have an infinite value, for instance by taking an infinite value in an extended real number system (see also: hyperreal number), or by requiring the counting of an infinite number of events. It is for example presumed impossible for any body to have infinite mass or infinite energy. There exists the concept of infinite entities (such as an infinite plane wave) but there are no means to generate such things. Likewise, perpetual motion machines theoretically generate infinite energy by attaining 100% efficiency or greater, and emulate every conceivable open system; the impossible problem follows of knowing that the output is actually infinite when the source or mechanism exceeds any known and understood system. This point of view does not mean that infinity cannot be used in physics. For convenience sake, calculations, equations, theories and approximations often use infinite series, unbounded functions, etc., and may involve infinite quantities. Physicists however require that the end result be physically meaningful. In quantum field theory infinities arise which need to be interpreted in such a way as to lead to a physically meaningful result, a process called renormalization.

Infinity in cosmology

An intriguing question is whether actual infinity exists in our physical universe: Are there infinitely many stars? Does the universe have infinite volume? Does space "go on forever"? This is an important open question of cosmology. Note that the question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By walking/sailing/driving straight long enough, you'll return to the exact spot you started from. The universe, at least in principle, might have a similar topology; if you fly your space ship straight ahead long enough, perhaps you would eventually revisit your starting point. If the universe is indeed ever expanding as science suggests then you could never get back to your starting point even on an infinite time scale.

Three types of infinities

Besides the mathematical infinity and the physical infinity, there could also be a philosophical infinity. There are scientists who hold that all three really exist and there are scientists who hold that none of the three exist. And in between there are the various possibilities. Rudy Rucker, in his book Infinity and the Mind -- the science and philosophy of the mind (1982), has worked out a model list of representatives of each of the eight possible standpoints. The footnote on p.335 of his book suggests the consideration of the following names: Abraham Robinson, Plato, Thomas Aquinas, L.E.J. Brouwer, David Hilbert, Bertrand Russell, Kurt Gödel and Georg Cantor.

Infinity in science fiction

The Hitchhiker's Guide to the Galaxy contains the following definition of infinity: :"Bigger than the biggest thing ever and then some, much bigger than that, in fact really amazingly immense, a totally stunning size, real 'Wow, that's big!' time. Infinity is just so big that by comparison, bigness itself looks really titchy. Gigantic multiplied by colossal multiplied by staggeringly huge is the sort of concept we are trying to get across here." Another quote from The Hitchhiker's Guide to the Galaxy states: "Infinity itself looks flat and uninteresting. Looking up into the night sky is looking into infinity -- distance is incomprehensible and therefore meaningless." Rudy Rucker's novel White Light describes a mathematician who leaves his body and travels to a kind of afterworld that includes a mountain whose Absolute Infinite height matches that of the class of all ordinals. Georg Cantor makes an appearance as a character, and the hero finds a physical correlate for Cantor's Continuum Problem.

References

-
- #

External links

- [http://www.earlham.edu/~peters/writing/infapp.htm A Crash Course in the Mathematics of Infinite Sets], by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1-59. The stand-alone appendix to Infinite Reflections, below. A concise introduction to Cantor's mathematics of infinite sets.
- [http://www.earlham.edu/~peters/writing/infinity.htm Infinite Reflections], by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 (1998) 1-59.
- [http://pespmc1.vub.ac.be/INFINITY.html Infinity, Principia Cybernetica]
- [http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html Hotel Infinity]
- [http://samvak.tripod.com/infinite.html The concepts of finiteness and infinity in philosophy]
- [http://uk.geocities.com/frege@btinternet.com/cantor/Phil-Infinity.htm Source page on medieval and modern writing on Infinity]

Note

Large cardinals are quantitative infinities defining the number of things in a collection, which are so large that they cannot be proven to exist in the ordinary mathematics of Zermelo-Fraenkel plus Choice (ZFC). Category:Mathematics Category:Philosophical terminology Category:Philosophy of mathematics Category:Theology ko:무한 ja:無限 simple:Infinity

Theology

Theology is reasoned discourse concerning God (Greek θεος, theos, "God", + λογος, logos, "word" or "reason"). It can also refer to the study of other religious topics. A theologian is a person learned in theology. religious topics

History of the term

The word "Theology" is derived from Hellenistic Greek, but its meaning has changed significantly through its use in the European Christian thought of the Middle ages and Enlightenment The term theologia is used in Classical Greek literature, with the meaning "discourse on the gods or cosmology" (see Lidell and Scott's Greek-English Lexicon for references). Since the authority of Hellenistic city states was partly based on religious observance, those who first sought to ask difficult questions about the gods were often viewed as heretics, or in the language of the day "atheists". Socrates is famous for having been condemned to death for teaching youths atheism (though in fact he had not). Plato, his pupil, wrote several discourses on the gods, though his doctrine of forms and emanations would be more significant for later Theology. Aristotle divided theoretical philosophy into mathematice, phusike and theologike, with the latter corresponding roughly to metaphysics, which for Aristotle included discussion of the nature of the divine. The term has since been appropriated by a number of Eastern and Western religious traditions. Drawing on Greek sources, the Latin writer Varro influentially distinguished three forms of such discourse: mythical (concerning the myths of the Greek gods), rational (philosophical analysis of the gods and of cosmology) and civil (concerning the rites and duties of public religious observance). Christian writers, working within the Hellenistic mould, began to use the term to describe their studies. It appears once in some biblical manuscripts, in the heading to the book of Revelation: apokalupsis ioannou tou theologou, "the revelation of John the theologos". There, however, we are probably dealing with a slightly different sense of the root logos, to mean not "rational discourse" but "word" or "message": ho theologos here is probably meant to tell us that the author of Revelation has presented God's revealed messages – words of God, logoi tou theou – not that he was a "theologian" in the modern English sense of the word. Other Christian writers used the term with several different ranges of meaning. # Some Latin authors, such as Tertullian and Augustine followed Varro's threefold usage, described above. # In patristic Greek sources, theologia could refer narrowly to the discussion of the nature and attributes of God. # In other patristic Greek sources, theologia could also refer narrowly to the discussion of the attribution of divine nature to Jesus. (It is in this sense that Gregory Nazianzus was nicknamed "the theologian": he was a staunch defender of the divinity of Christ.) # In medieval Greek and Latin sources, theologia (in the sense of "an account or record of the ways of God") could refer simply to the Bible. # In scholastic Latin sources, the term came to denote the rational study of the doctrines of the Christian religion, or (more precisely) the academic discipline which investigated the coherence and implications of the language and claims of the Bible and of the theological tradition (the latter often as represented in Peter Lombard's Sentences, a book of extracts from the Church Fathers). It is the last of these senses which lies behind most modern uses (though the second is also found in some academic and ecclesiastical contexts), and while the term "theology" can refer to any discussion of the nature of God or the gods, or indeed the discussion of any religious topic, it is also regularly used to denote the academic study (in Universities, seminaries and elsewhere) of the doctrines of Christianity, or of any other religion, or of the relationships and contrasts between various different religions, although the latter is a field more usually termed "comparative religion."

A brief history of "Theologies"

::Main article: History of theology Classical Greek theology (c.700 BC to 323 BC). Various forms of systematic and philosophical reflection on Ancient Greek religion and Greek mythology arose in the classical period - from Hesiod's attempts to organise the diverse materials of mythology into a unified Theogony to the more properly philosophical analysis reportedly carried out by Socrates. Plato's Timaeus and Aristotle's Metaphysics Book Lambda are two of the most influential writings of Classical Greek theology. Hellenistic theology (323 BC to 529 AD). Philosophical reflection on the gods, on religion, and on the origins and governance of the Universe, flourished in the Hellenistic period amongst both Greek- and Latin-speaking thinkers. Amongst the very diverse movements of Hellenistic philosophy in which theological reflection could be found were Cynicism, Stoicism, Epicureanism, Middle Platonism, and Neoplatonism. Influential texts include Cleanthes' Hymn to Zeus, Cicero's de Natura Deorum, Lucretius' de Rerum Natura, Epictetus' Enchiridion, and Plotinus' Enneads. Hellenistic theology, which could be deemed to last until the suppression of the Athenian Academy in 529 by Justinian I, overlaps with early Jewish and early Christian theology (see below), and several strands of thought important particularly to early Christian thought arise within Hellenistic circles: attempts to explain the apparent caprice of the gods, Atheism, the development of monotheism, the idea of God as first cause or form of the Good, the dualism of spirit and matter in humanity, and redemption (the release of the spirit from its material prison to a higher spiritual world) through knowledge. See also Greek mythology - Hellenistic rationalism and Ancient Greek religion - Theology Early Jewish theology (to c.200 AD). Two strands of Jewish theology develop in this period. On the one hand, there are those oral traditions of Rabbinic exegesis (Midrash) and legal discussion (Mishnah) that eventually began to be written down towards the end of the 2nd Century AD. Important figures include Gamliel I, Yohanan ben Zakkai, Gamliel II, Rabbi Akiva, and Rabbi Judah haNasi. On the other hand, there is the attempt to accommodate traditional Jewish exegesis of the Jewish Scriptures and tradition with Greek philosophy - a strand of thought of which Philo is the best known proponent. The destruction of the Jerusalem temple in 70 AD and the dispersion of many Jews from Israel had a profound effect on Jewish Theology. Early Christian theology, coming partly from Hellenistic Judaism, therefore had no trouble in expressing itself in the Greek language (i.e. the New Testament). Whilst the conception of a canon of sacred books was inherited from Judaism, their interpretation soon came to be heavily influenced by Greek allegorical methods (e.g. Origen). Origen" during the long decline of the Roman Empire]] Patristic Theology (c. 100 – 500 AD) is so called because certain men (Fathers or "Patroi") concerned themselves with determining the degree to which the Christian faith could be accommodated to Hellenistic thought. Irenaeus of Lyons wrote to combat those who made Christianity into Gnostic Theology. Justin Martyr sought to use Hellenistic philosophy and Natural Theology to justify Christianity to the Romans. Later Theologians especially sought to show how three divine persons could be one in substance (the Trinity, see Council of Nicea) and how Jesus (a man of material flesh, see Council of Chalcedon) could at also be divine. These statements though held to be philosophically illogical were nevertheless held to be true, human reason being incapable of understanding them. This was an important development that would define the Theology of the Middle Ages in Islam as well as Christianity. Important theologians were Athanasius, Gregory of Nazanzius, Gregory of Nyssa, Origen, Ambrose, Augustine and Jerome. The fall of the Roman empire affected Theology in two main ways; Firstly monasticism became more popular and ascetic, and mystical theology therefore became more prevalent. Secondly, the increasing influence of the Bishop of Rome (The Pope) in theological doctrine and cultural differences between the two remnants of the Roman empire caused the doctrine of apostolic succession to be more important. The two sides finally split in 1054. The collapse of the Roman Empire meant that most Theology occurred in Monasteries with few of the resources of classical scholarship available. Over time many local variations in Theology developed and the traditions of pre-Christian religions were sometimes included in Theology as well as practice. Likewise, in the East, (Greece and the Levant) Theology became increasingly influenced by speculative neo-Platonism. The epistle of Dionysius the Areopagite was a popular guide with such ideas. Many monks came to emphasize the idea of the inherent evil of the world. Islam established itself in this atmosphere and began also to practice Theology. Although Islam is often considered to lack a "Theology" as in Christianity there were many attempts to frame Islamic ideas within Greek thought, especially during the early abbassids and the reign of the caliph al-mamun. However, this movement, Mu’tazilism, became discredited through the Abassids attempts to use it to enforce religious unity, and the popular and orthodox considered Hellenistic thought to be unhelpful and error. Theology would continue to be practiced, but was usually done so by an elite of intellectuals whose ideas would seldom be made public. These included Al-Kindi, Al-Farabi, Averroes, Avicenna and Al-Ghazali. High Medieval theology in Western Europe combined the Theology inherited from Dark-age monasticism with new learning from classical Hellenistic documents from the Islamic world. Thomas Aquinas, Anselm, John Duns Scotus and Peter Abelard were among the most important Theologians of this period. Peter Abelard]] The Renaissance yielded scholars the ability to read the scriptures in their original languages and this in part stimulated the Reformation, a Theological movement that based its "Protests" on a new understanding of the Bible. Most important were Martin Luther, John Calvin, Zwingli, Melancthon, Martin Bucer and the Anabaptists. Their Theology was developed by successors such as Theodore Beza, the English Puritans and Francis Turretin. The Catholic counter-reformation spearheaded by the Jesuits under Ignatius Loyola took their Theology from the decisions of the Council of Trent. The overall result of the Reformation was therefore to highlight distinctions of belief that had previously co-existed uneasily. The fall of Constantinople in the east, 1453, led to a significant shift of gravity to the rising state of Russia, the "Third Rome". The Renaissance would also stimulate a program of reforms by patriarchs of prayer books. A movement called the "Old believers" consequently resulted and influenced Russian Orthodox Theology in the direction of conservatism and Erastianism. After the Reformation protestant groups continued to splinter, leading to a range of new Theologies. The "Enthusiasts" were so named because of their emotional zeal. These included the Methodists, the Quakers and Baptists. Another group sought to reconcile Christian faith with "Modern" ideas, sometimes causing them to reject beliefs they considered to be illogical, including the Nicene creed and Chalcedonian creed. these included Unitarians and Universalists. The Nineteenth Century saw the rise of biblical criticism, new knowledge of religious diversity in other continents and above all the growth of science. This led many church men to espouse a form of Deism. This, along with concepts such as the brotherhood of man and a rejection of miracles led to what is called "Classic Liberalism". Immensely influential in its day, classic liberalism suffered badly as a result of the two world wars and fell prey to the criticisms of postmodernism.postmodernism Theologian]] Postmodern theology seeks to respond to the challenges of post modern and deconstructionist thought, and has included the death of God movement, Process Theology, Feminist theology and Queer Theology and most importantly Neo-orthodox Theology. Karl Barth, Rudolf Bultmann and Reinhold Niebuhr were Neo-Orthodoxies main representatives. In particular Barth labeled his Theology "Dialectical Theology", a reference to existentialism. The predominance of Classic Liberalism resulted in many reactionary movements amongst conservative believers. Evangelical theology, Pentecostal or Renewal theology and Fundamentalist theology, often combined with Dispensationalism, all moved from the fringe into the academy. Marxism stimulated the significant rise of Liberation Theology which can be interpreted as a challenge to Academic Theology that fails to challenge the establishment and help the poor. From the late nineteenth century to the early twentieth many groups established themselves that derived many of their beliefs from protestant evangelical groups but significantly differed in doctrine. These include the Jehovah's Witnesses, the Latter Day Saints and many so called "cults". Many of these groups use the protestant version of the bible and typically interpret it in a fundamentalist fashion, adding, however, special prophecy or scriptures, and typically denying the trinity and the full deity of Jesus Christ. Ecumenical Theology sought to discover a common consensus on theological matters that could bring the many Christian denominations together. As a movement it was successful in helping to provide a basis for the establishment of the World council of churches and for some reconciliation between more established denominations. But ecumenical theology was nearly always the concern of liberal theologians, often protestant ones. The movement for ecumenism was opposed especially by fundamentalists and viewed as flawed by many neo-orthodox theologians. The pattern of challenge from a changing world, liberal response from official representatives and orthodox backlash from conservatives is found also in the history of Islam and Judaism. Reform Judaism represents a liberal interpretation as against Orthodox Judaism, and moderate or Liberal Islam continues to be theologically distinct from Islamic Fundamentalism, notably its Wahabi and Deobandi Schools. As other religions came to be studied in Western post Christian academies the term Theology was applied to them, though, as noted below, this may be a serious misnomer!

Theology and religions other than Christianity

In academic theological circles, there is some debate as to whether theology is an activity peculiar to the Christian religion. If so we should distinguish Christian Theology from others. It is seen by some to be a term only appropriate to the study of a deity (a theos) within a presupposed belief in the ability to speak and reason about the subject (in logia) - and so to be less appropriate in religious contexts which are organized differently (i.e. religions without a deity, or which deny that such subjects can be studied logically). reason For example, some academic courses on Buddhism which are dedicated to the rational investigation of a Buddhist understanding of the world prefer the designation Buddhist philosophy to the term Buddhist theology, since Buddhism lacks the same conception of a theos. The same might be said of Hinduism which has many devas (deities). See for example, Vaishnava Theology, Advaita Vedanta and Hinduism#Nature of God. Moreover, the application of the term Theology to religions similar to Christianity can be misleading. in Islam, theological discussion which parallels Christian theological discussion has been a minor and even slightly disreputable activity, named "Kalam"; the Islamic analogue of Christian theological discussion would more properly be the investigation and elaboration of Islamic law, or "Fiqh". In Judaism the historical absence of political authority has meant that most theological reflection has happened within the context of the Jewish community and synagogue, rather than within specialised academic institutions. Nevertheless Jewish Theology has been historically very active and highly significant for Christian and Islamic Theology. Once again, the Jewish analogue of Christian theological discussion would more properly be Rabbinical discussion of Jewish law and Jewish Biblical commentaries.

Theology and the Academy

Theology has a significantly problematic relationship to Academia that is not shared by any other subject. Most universities founded before the modern era grew out of the church schools and monastic institutions of Western Europe during the High Middle Ages (e.g. University of Bologna, Paris University and Oxford University). They were founded to train young men to serve the church in Theology and Law (often Church or Canon Law). At such Universities Theological study was incomplete with Theological practice, including preaching, prayer and the Mass. Ancient Universities still maintain some of these links (e.g. having Chapels and Chaplains) and are more likely to teach Theology than other institutions. During the High Middle Ages theology was therefore the main subject at universities, being named "The Queen of the Sciences" alongside the Trivium and Quadrivium that young men were expected to study. This meant that the other subjects (including Philosophy) existed primarily to help with theological thought. With the Enlightenment universities began to change, teaching a wide range of subjects, especially in Germany, and from a Humanistic perspective. Theology was no longer the principle subject and Universities existed for many purposes, not only to train Clergy for established churches. Theology thus became unusual as the only subject to maintain a confessional basis in otherwise secular establishments. As a result theology is often distinguished from many other established Academic disciplines that cover the same subject area. Those who contend it is different claim it is distinguished by its viewpoint (it is studied from within a faith, rather than from without) and its practical involvement (theology cannot be truly studied or understood without a practical faith). Many of the early Church Fathers described the theologian as a person who "truly prays.". Non-religious theologians often disagree with these viewpoints, arguing that the term theology covers the study of religion or peoples' beliefs about God, rather than God himself. They also argue that human reason alone is sufficient to understand such subjects and that prayer and worship are not necessary. Nevertheless theology should be distinguished from the following disciplines; Comparative religion/Religious studies Philosophy of Religion The History of Religions Psychology of Religion Sociology of Religion All of these approach religion with humanistic presuppositions and assume a uniformity in religious faith and experience, unlike most theology.

Theological studies in different institutions

In Europe, the traditional places for the study of theology have been universities and seminaries. Typically the protestant state churches have trained their ministers in universities while the Catholic church has used seminaries. However, the secularization of European states has closed down the theological faculties in many countries while the Catholic church has increased the academical level of its priests by founding a number of pontifical universities. However, at least Finland and Sweden have state universities with faculties of theology training Lutheran priests as well as teachers and scholars of religion. As study of theology in these countries includes a strong (Christian) humanist content, graduates of theology who do not wish to embark on clerical career may find work also in marketing, business or administration, although this is frowned upon by many. In the United States, study of theology does not enjoy state endorsement due to the nature of the constitution of United States. Theological studies (often called Biblical studies) take place in a large number of universities, the academic level of which may vary considerably. The academic freedom of thought in many of these institutions may not reach the level of the faculties of theology in European state universities. Theologians ending up with view deemed "heretical" by the denomination upholding the institution may find themselves out of work.

Divisions of theology

Theology can be divided up in any number of ways. Many of these divisions have originated in the study of the Christian religion, although some have been adapted and extended to apply to other religions, or to the study of multiple religions. The most established distinctions are Systematic Theology, Biblical Studies/Biblical Theology, Historical Theology and Pastoral Theology. Theology can also be divided up into : Academic subdisciplines;
- Biblical Theology - focused on the investigation and interpretation of a religions' scriptures, especially noting different emphases (theologies) within different biblical books.
- Biblical Studies - focused on the interpretation and exegesis of the Bible.
- Comparative religion - focused on the comparison of common themes among different religious traditions
- Historical Theology - focused on the intellectual history of the religion
- Moral Theology - explores the moral and ethical dimensions of the religious life
- Patrology - studies the teaching of Church Fathers.
- Practical Theology - dedicated to the practical application of theological insights. Generally includes the subdisciplines of pastoral theology, homiletics, and Christian education, among others.
- Systematic theology (doctrinal theology, dogmatic theology or philosophical theology) - focused on the attempt to arrange and interpret the ideas current in the religion. Topic (or by 'Loci');
- Angelology (less common than it used to be) - angels, the unseen world
- Bibliology (a less common term than most of the others) - the Bible, the nature and means of its inspiration, etc.; hermeneutics is the study of proper biblical interpretation (exegesis).
- Christology (normally only in Christianity) - Jesus Christ, the nature of Christ, the relationship between the divine and human in Christ
- Covenant theology, an interpretive grid that understands God's plans in the Old and New Testaments as being a result of God's covenant with his chosen people. This movement is an alternative to Dispensationalism.
- Demonology (much less common than it used to be) - Satan, demons, evil spirits
- Dispensational Theology - an interpretative grid that views God's relationship with the created order as passing through successive "dispensations", in each of which the covenants of the previous one(s) may no longer be valid.
- Ecclesiology - the church
- Eschatology - literally, the study of 'last things' or 'ultimate things'. Covers subjects such as death and the afterlife, the end of history, the end of the world, the last judgment, the nature of hope and progress, etc.
- Gaudiya Vaishnava Theology - the Vaishnava Theology which emphasizes the devotee's relationship to the "Divine Couple," Radha and Krishna, and looks to Caitanya Mahaprabhu as an avatar embodying both Radha and Krishna.
- Harmatiology (often considered under 'soteriology') - sin
- Krishnology - the discourse concerning the Hindu deity Krishna within the context of Vaishnava Theology.
- Missiology (often a subsection of ecclesiology) - missions, evangelism, etc.
- Radhavallabha Theology is the Vaishnava Theology of Harivamsa Gosvami, who started the Radhavallabha sect. His theology emphasizes devotion to Radharani. This sect also has a famous temple in Vrindavan of the same name.
- Soteriology - the nature and means of salvation
- Theodicy - Attempts at reconciling the existence of all the evil and suffering in the world with the nature and power of the God or gods of the religion
- Theological anthropology - nature of human being, formerly known as the Doctrine of Man.
- Theology Proper - God or the divine: attributes, nature, and relation to the world. Often includes discussion of Creation and providence. See the nature of God in Western theology.
- Pneumatology - the Holy Spirit or divine Spirit; sometimes also 'geist' as in Hegelianism and other philosophico-theological systems;
- Vaishnava Theology is the theological discourse concerning the Hindu deity Vishnu and/or one of His avatar. Modes;
- Apophatic theology (or negative theology; sometimes contrasted with "cataphatic theology") - the discussion of what God is not, or the investigation of how language about God breaks down
- dialectical theology
- Natural theology - the discussion of those aspects of theology that can be investigated without the help of revelation, scriptures or tradition (sometimes contrasted with "positive theology") - the discussion of those aspects of theology Movements;
- Black theology
- Ecumenical theology
- Evangelical theology
- Feminist theology
- Holocaust theology(In response to the horrors of the Holocaustespecially in relation to Theodicy,
- Liberal theology
- Liberation theology
- Neo-Orthodoxy
- Paleo-Orthodoxy
- Postliberal theology or Narrative theology
- Postmodern theology
- Queer Theology
- Revisionist theology
- Transcendental Theology

Quotes

- "Theology is the effort to explain the unknowable in terms of the not worth knowing." - H.L. Mencken
- "An authentic theology will not allow man to be obsessed with himself." - Thomas F. Torrance in Reality and Scientific Theology
- "Theology announces not just what the Bible says but what it means." - J. Kenneth Grider in A Wesleyan-Holiness Theology (Kansas City: Beacon Hill, 1994), p. 19.

External links

- [http://catholicapologeticsofamerica.blogspot.com Catholic Apologetics of America] (Roman Catholic)
- [http://swami-center.org/en/text/Theology.html General Theology — the Science about God] (New Age)
- [http://www.monergism.com/systematic.html Monergism: Systematic Theology] (conservative Calvinist)
- [http://www.geocities.com/dbusnipe/subjective_truth/theological.htm Theological Links] (Humor)
- [http://www.theopedia.com Theopedia] (conservative Calvinist)
- [http://www.theowiki.com/index.php/Main_Page TheoWiki] (InterFaith)
- [http://wesley.nnu.edu/ Wesley Center for Applied Theology] (Wesleyan/Holiness)
- [http://gbgm-umc.org/umhistory/wesley/ The Wesleys and their Times] (Wesleyan/Methodist)
- ja:神学 simple:Theology

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.

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

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.

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

Latin

Latin is an ancient Indo-European language originally spoken in the region around Rome called Latium. It gained great importance as the formal language of the Roman Empire. All Romance languages, those being most notably Spanish, French, Portuguese, Italian, and Romanian, are descended from Latin, and many words based on Latin are found in other modern languages such as

Infinite

History

Ancient view of infinity

Views from the Renaissance to modern times

Modern philosophical views

Infinity symbol

Mathematical infinity

Infinity in real analysis

Infinity in complex analysis

Arithmetic properties of infinity

Infinity with itself

Operations involving infinity and real numbers

Undefined operations

Infinity in set theory

Mathematics without infinity

Use of infinity in common speech

Physical infinity

Infinity in cosmology

Three types of infinities

Infinity in science fiction

See also

References

External links

Note

Theology

History of the term

A brief history of "Theologies"

Theology and religions other than Christianity

Theology and the Academy

Theological studies in different institutions

Divisions of theology

Quotes

See also

External links

Mathematics

History

Inspiration, pure and applied mathematics, and aesthetics

Notation, language, and rigor

Is mathematics a science?

Overview of fields of mathematics

Major themes in mathematics

Quantity

Structure

Space

Change

Foundations and methods

Discrete mathematics

Applied mathematics

Important theorems

Important conjectures

History and the world of mathematicians

Mathematics and other fields

Common misconceptions

See also

Bibliography

External links

Latin