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Natural Sciences

Natural Sciences

] Natural science is the study of the physical, nonhuman aspects of the Earth and the universe around us. Natural sciences generally attempt to explain the workings of the world via natural processes rather than divine processes. The term natural science is also used to identify science as a discipline following the scientific method, in contrast to natural philosophy, or in contrast with social sciences, which use the same scientific method applied to different subjects. Natural sciences form the basis for the applied sciences. Together, the natural and applied sciences are distinguished from the social sciences on the one hand, and from the humanities, theology and the arts on the other. Mathematics is not itself a natural science, but provides many tools and frameworks used within the natural sciences. Alongside this traditional usage, more recently the words "natural sciences" are sometimes used in a way more closely matching their everyday meaning, stemming from natural history. In this sense "natural sciences" can be an alternative phrase for biological sciences, involved in biological processes, or perhaps also the earth sciences, as might distinguished from the physical sciences (more directly involved in the study of physical and chemical laws underlying the universe). See :Category:Science for articles about the individual Natural sciences

Natural sciences

- Astronomy, the study of the stars, the cosmos, etc.
- Biology, the study of life.
- Ecology, the study of the interrelationships of life.
- Chemistry, the study of chemical reactions.
- Earth science, the study of earth and specialties including:
- Geology
- Science-based or Physical Geography
- Soil science
- Physics, the study of physical laws.

External links

- [http://www.cam.ac.uk/cambuniv/natscitripos/ Natural Sciences at Cambridge University]
- [http://hrst.mit.edu/ The History of Recent Science and Technology]
- [http://www.scibooks.org/ Reviews of Books About Natural Science] This site contains over 50 previously published reviews of books about natural science, plus selected essays on timely topics in natural science. Category:Science Category:Nature ko:자연과학 ja:自然科学 th:วิทยาศาสตร์ธรรมชาติ

Universe

The terms known Universe, observable Universe, or visible Universe are often used to describe the part of the Universe that we can see or otherwise observe. Those who believe it is impossible to observe the whole continuum may use our Universe, referring only to that knowable by human beings in particular.

Expansion and age, and the Big Bang theory

The most important result of cosmology, that the Universe is expanding, is derived from redshift observations and quantified by Hubble's law. Extrapolating this expansion back in time, one approaches a gravitational singularity, a rather abstract mathematical concept, which may or may not correspond to reality. This gives rise to the Big Bang theory, the dominant model in cosmology today. The age of the Universe was estimated to be about 13.7 billion (13.7 × 10⁹) years, with a margin of error of about 1 percent (± 200 million years), according to NASA's Wilkinson Microwave Anisotropy Probe (WMAP). However, this is based on the assumption that the underlying model used for data analysis is correct. Other methods of estimating the age of the Universe give different ages. A fundamental aspect of the Big Bang can be seen today in the observation that the farther away from us galaxies are, the faster they move away from us. It can also be seen in the cosmic microwave background radiation which is the much-attenuated radiation that originated soon after the Big Bang. This background radiation is remarkably uniform in all directions, which cosmologists have attempted to explain by an initial period of rapid inflation following the Big Bang.

Size of Universe and observable Universe

There is disagreement over whether the Universe is finite or infinite in spatial extent and volume. However, the observable Universe, consisting of all locations that could have affected us since the Big Bang given the finite speed of light, is certainly finite. The edge of the cosmic light horizon is 13.7 billion light years (4.19 gpc) distant. The present distance (comoving distance) to the edge of the observable Universe is larger, since the Universe has been expanding; it is estimated to be about 78 billion light years (7.8 × 10¹⁰ light years, or 7.4 × 10²³ km). This would make the comoving volume, of the known Universe, equal to 1.9 × 10³³ cubic light years (assuming this region is perfectly spherical). The observable Universe contains about 7 × 10²² stars, organized in about 100 billion galaxies, which themselves form clusters and superclusters. The number of galaxies may be even larger, based on the Hubble Deep Field observed with the Hubble Space Telescope. The Hubble Space Telescope discovered galaxies such as Abell 1835 IR1916, which are over 13 billion light years from Earth. Both popular and professional research articles in cosmology often use the term "Universe" when they really mean "observable Universe". This is because unobservable physical phenomena are scientifically irrelevant; that is, they cannot affect any events that we can perceive. See also Causality (physics). We live in the centre of the Universe that we observe, in apparent contradiction to the Copernican principle which says that the Universe is more or less uniform and it has no distinguished centre. This is simply because light does not travel infinitely fast, and we make observations of the past. As we look further and further away, we see things from epochs (times) closer and closer to the limit of time, which equals zero, according to the Big bang model. Since light travels at the same speed in any direction towards us, it is reasonable to suggest that we live at the center of our observable Universe.

Shape of the Universe

An important open question of cosmology is the shape of the Universe. Mathematically, which 3-manifold represents best the spatial part of the Universe? Firstly, whether the Universe is spatially flat, i.e. whether the rules of Euclidean geometry are valid on the largest scales, is unknown. Currently, most cosmologists believe that the observable Universe is very nearly spatially flat, with local wrinkles where massive objects distort spacetime, just as a lake is (nearly) flat. This opinion was strengthened by the latest data from WMAP, looking at "acoustic oscillations" in the cosmic microwave background radiation temperature variations. Secondly, whether the Universe is multiply connected, is unknown. The Universe has no spatial boundary according to the standard Big Bang model, but nevertheless may be spatially finite (compact). This can be understood using a two-dimensional analogy: the surface of a sphere has no edge, but nonetheless has a finite area. It is a two-dimensional surface with constant curvature in a third dimension. The 3-sphere is a three-dimensional equivalent in which all three dimensions are constantly curved in a fourth. If the Universe is indeed spatially finite, as described, then traveling in a "straight" line, in any given direction, would theoretically cause one to eventually arrive back at the starting point. Strictly speaking, we should call the stars and galaxies "views" of stars and galaxies, since it is possible that the Universe is multiply-connected and sufficiently small (and of an appropriate, perhaps complex, shape) that we can see once or several times around it in various, and perhaps all, directions. (Think of a house of mirrors.) If so, the actual number of physically distinct stars and galaxies would be smaller than currently accounted. Although this possibility has not been ruled out, the results of the latest cosmic microwave background research make this appear very unlikely.

Fate of the Universe

Depending on the average density of matter and energy in the Universe, it will either keep on expanding forever or it will be gravitationally slowed down and will eventually collapse back on itself in a "big crunch". Currently the evidence suggests not only that there is insufficient mass/energy to cause a recollapse, but that the expansion of the Universe seems to be accelerating and will accelerate for the whole of eternity (see accelerating Universe). Other ideas of the fate of our Universe include the Big Rip, the Big Freeze, and Heat Death of the Universe theory. For a more detailed discussion of other theories, see the ultimate fate of the Universe.

Multiverse

There is some speculation that multiple universes exist in a higher-level multiverse (also known as a megaverse), our Universe being one of those universes (lower case). For example, matter that falls into a black hole in our Universe could emerge as a "Big Bang," starting another universe. However, all such ideas are currently untestable and cannot be regarded as anything more than speculation. The concept of parallel universes are understood only when related to string theory.

Other terms

Different words have been used throughout history to denote "all of space", including the equivalents in various languages of "heavens", "cosmos" and "world". Although words like world and its equivalents in other languages now almost always refer to the planet Earth, they previously referred to everything that exists—see Copernicus, for example—and still sometimes do (as in "the whole wide world"). Some languages use the word for "world" as part of the word for "outer space", e.g. in the German word "Weltall".

References

- Albert Einstein (1952). Relativity: The Special and the General Theory (Fifteenth Edition), ISBN 0-517-88441-0

External links

- [http://www.anzwers.org/free/universe/index.html Richard Powell: An Atlas of the Universe] - a series of images at various scales, with explanations.
- [http://www.shekpvar.net/~dennis/Elib/Astronomicon/Astronomicon/Cosmos/cosmos.html Cosmos - an Illustrated Dimensional Journey from microcosmos to macrocosmos]
- [http://universe-review.ca/ A Review of the Universe - Structures, Evolutions, Observations, and Theories]
- [http://www.space.com/scienceastronomy/age_universe_030103.html Age of the Universe at Space.Com]
- [http://slate.msn.com/id/2087206/ My So-Called Universe] by Jim Holt, on various arguments for and against an infinite Universe and parallel universes
- [http://www.hep.upenn.edu/~max/multiverse1.html Parallel Universes] by Max Tegmark
- [http://www.astro.princeton.edu/~mjuric/universe/ Logarithmic Maps of the Universe]
- [http://setiathome.ssl.berkeley.edu/ Seti@Home - the Search for Extraterrestrial Intelligence]
- [http://www.exploreuniverse.com/ic Universe - Space Information Centre ] by Exploreuniverse.com
- [http://hypertextbook.com/facts/1999/TopazMurray.shtml Number of Galaxies in the Universe] Category:Environments ko:우주 ms:Alam Semesta ja:宇宙 simple:Universe

Category:Academic disciplines

Disciplines, studied in Academia. See List of academic disciplines. Category:Education Disciplines

Natural philosophy

Natural philosophy is a term applied to the objective study of nature and the physical universe before the development of modern science. In other words, all forms of science historically developed out of philosophy or more specifically natural philosophy. At older universities, long-established Chairs of Natural Philosophy are nowadays occupied mainly by professors of physics. In fact, our notions of science and scientists date only to the 19th century. Before then, the word "science" simply meant knowledge and the label of scientist did not exist. Natural philosophy was the term whose usage preceded our current term science in the sense that prior to the replacement of the term "natural philosophy" with the term science, the term science was used exclusively (and comparatively rarely) as a synonym for knowledge or study and when the subject of that knowledge or study was 'the workings of nature', then the term "natural philosophy" would be used. Robert Boyle wrote what is considered to be a seminal work on the distinction between nature and metaphysics called A Free Enquiry into the Vulgarly Received Notion of Nature. This book, written in 1686, marked the point where the scene was set for natural philosophy to turn into science. An important distinguishing characteristic of science and natural philosophy is the fact that natural philosophers generally did not feel compelled to test their ideas in a practical way. Instead, they observed phenomena and came up with 'philosophical' conclusions. Proposals for a much more 'inquisitive' and practical approach to the study of nature originated with Francis Bacon. Boyle, while he is the first to fully embrace such an approach in both his experimental endeavours and his writings, shares with Bacon (and Galileo who was the inspiration in these matters for both Bacon and Boyle) a conviction that practical experimental observation was the key to a more satisfactory understanding of nature than would have otherwise been sought through either exclusive reference to received authority or a purely speculative approach. Although Galileo's 'natural philosophy' is hardly distinguishable from science in many ways, the connection between his experiments and his writings about them is characteristically philosophical, rather than being cluttered with the results of meticulously recorded observational detail of practical scientific research, in the way that Boyle (subsequently) advocated. Despite Boyle describing what he was practicing as being Natural Philosophy, the very innovations that Boyle introduced (such as an insistence upon the requirement for publication of detailed experimental results, including the results of unsuccessful experiments and also a requirement for the replication of experiments as a means of validating observational claims) can be seen as a basis for delineating a transition from proto-science to science. However, deciding which side of the proto-science/science divide that you include the term 'natural philosophy'is largely determined by whether you treat Boyle's use of the term as being something he himself invalidated when he applied it to his own work (an anachronistic conflation which is removed by recognising that the distinction between the terms natural philosophy and science only arose (at the earliest) in the post-Newtonian era, after Boyle's passing, and once the term science was in regular use). Thus Boyle could describe his work as Natural Philosophy, whereas we could describe it as science, and yet Boyle's use would be correct (for him) despite the fact that he is in many ways the architect of the modern distinction between the two terms.

Category:Social sciences

]] The social sciences comprise the scientific study of the human aspects of the world. They are also known as social studies. See the list of academic disciplines for a list of social sciences, including some not yet categorized below.
Category:Science Category:Society Category:Society Category:Interdisciplinary fields Category:Academic disciplines Category:Human sciences Category:Environmental science zh-min-nan:Category:Siā-hōe kho-ha̍k ko:분류:사회과학 ja:Category:社会科学

Social science

The social sciences are a group of academic disciplines that study the human aspects of the world. They diverge from the arts and humanities in that the social sciences emphasize the use of the scientific method and rigorous standards of evidence in the study of humanity, including quantitative and qualitative methods. The social sciences are also known pejoratively as the soft sciences in contrast to the hard sciences. Social science theories typically deal with aggregated, not individual, behavior.

Major fields

The main social sciences include:
- Anthropology
- Communication
- Economics
- Education
- History
- Geography
- Linguistics
- Law
- Political Science
- Psychology
- Sociology
- Cultural Studies
- Social Policy Not all institutions recognize these fields as social sciences. For example, communication, cultural studies and history may be classified as humanities depending on how they are taught, and in which country they are taught. Some disciplines have characteristics of both the humanities, social and natural sciences: for example some subfields of anthropology, such as biological anthropology, are closely related to the natural sciences whereas archaeology and linguistics are social sciences. Similarly diverse subjects like geography also traverse the natural and social sciences (e.g., geomorphology and historical geography are often taught in single departments of geography). Some social sciences may converge with certain fields from the natural sciences, and become interdisciplinary. Examples of such fields include sociobiology -- an interdisciplinary field drawing on sociology and biology.

History

In ancient philosophy, there was no difference between the liberal arts of mathematics and the study of history, poetry or politics—only with the development of mathematical proof did there gradually arise a perceived difference between "scientific" disciplines and others, the "humanities" or "liberal arts". Thus, Aristotle studies planetary motion and poetry with the same methods, and Plato mixes geometrical proofs with his demonstration on the state of intrinsic knowledge. This unity of science as descriptive remains, for example, in the time of Thomas Hobbes who argued that deductive reasoning from axioms created a scientific framework, and hence his Leviathan was a scientific description of a political commonwealth. What would happen within decades of his work was a revolution in what constituted "science", particularly the work of Isaac Newton in physics. Newton, by revolutionizing what was then called "natural philosophy", changed the basic framework by which individuals understood what was "scientific". While he was merely the archetype of an accelerating trend, the important distinction is that for Newton, the mathematical flowed from a presumed reality independent of the observer, and working by its own rules. For philosophers of the same period, mathematical expression of philosophical ideals was taken to be symbolic of natural human relationships as well: the same laws moved physical and spiritual reality. For examples see Blaise Pascal, Gottfried Leibniz and Johannes Kepler, each of whom took mathematical examples as models for human behavior directly. In Pascal's case the famous wager, for Leibniz, the invention of binary computation and for Kepler the intervention of angels to guide the planets. In the realm of other disciplines, this created a pressure to express ideas in the form of mathematical relationships. Such relationships, called "Laws" after the usage of the time (see philosophy of science) became the model which other disciplines would emulate. August Comte (1797-1857) argued that ideas pass through three rising stages, Theological, Philosophical and Scientific. He defined the difference as the first being rooted in assumption, the second in critical thinking, and the third in positive observation. This framework, still rejected by many, encapsulates the thinking which was to push economic study from being a descriptive to a mathematically based discipline. Karl Marx was one of the first writers to claim that his methods of research represented a scientific view of history in this model. With the late 19th century, attempts to apply equations to statements about human behavior became increasingly common. Among the first were the "Laws" of philology, which attempted to map the change overtime of sounds in a language. It was with the work of Darwin that the descriptive version of social theory received another shock. Biology had, seemingly, resisted a basis as a mathematical study, and yet the Theory of Natural Selection and the implied idea of Genetic inheritance - later found to have been enunciated by Gregor Mendel, seemed to point in the direction of a scientific biology based, like physics and chemistry, on mathematical relationships. With the early 20th century, a wave of change came to science that saw "statistical" study sufficiently mathematical to be "science". This application of statistics to physics would yield Quantum Dynamics and an increasingly statistical view of biology. The first thinkers to attempt to combine inquiry of the type they saw in Darwin with exploration of human relationships, which, evolutionary theory implied would be based on selective forces, were Freud in Austria and William James in the United States. Freud's theory of the functioning of the mind, and James' work on experimental psychology would have enormous impact on those that followed. Freud, in particular, created a framework which would appeal not only to those studying psychology, but artists and writers as well. One of the most persuasive advocates for the view of scientific treatment of philosophy would be John Dewey (1859-1952). He began, as Marx did, in an attempt to weld Hegelian idealism and logic to experimental science, for example in his "Psychology" of 1887. However, it is when he abandoned Hegelian constructs, and joined the movement in America called Pragmatism, possibly under the influence of William James' "Principles of Psychology" that he began to formulate his basic doctrine, enunciated in essays such as "The Influence of Darwin on Philosophy" (1910). This idea, base on his theory of how organisms respond, states that there are three phases to the process of inquiry: #Problematic Situation, where the typical response is inadequate. #Isolation of Data or subject matter. #Reflective, which is tested empirically. With the rise of the idea of quantitative measurement in the physical sciences, for example Lord Rutherford's famous maxim that any knowledge that one cannot measure numerically "is a poor sort of knowledge", the stage was set for the conception of the humanities as being precursors to "social science" was set. This change was not, and is not, without its detractors, both inside of academia and outside. The range of critiques begin from those who believe that the physical sciences are qualitatively different from social sciences, through those who do not believe in statistical science of any kind, through those who disagree with the methodology and kinds of conclusion of social science, to those who believe the entire framework of scientificizing these disciplines is solely, or mostly, from a desire for prestige and to alienate the public.

Rise

Theodore Porter argued in "The Rise of Statistical Thinking" that the effort to provide a synthetic social science is a matter of both administration and discovery combined, and that the rise of social science was, therefore, marked by both pragmatic needs as much as by theoretical purity. An example of this is the rise of the concept of Intelligence Quotient, or IQ, a test which produces a number which it is not clear what, precisely, is being measured, except that it has pragmatic utility in predicting success in certain tasks. The rise of industrialism had created a series of social, economic, and political problems, particularly in managing supply and demand in their political economy, the management of resources for military and developmental use, the creation of mass education systems to train individuals in symbolic reasoning and problems in managing the effects of industrialization itself. The perceived senselessness of the "Great War" as it was then called, of 1914-1918, now called World War I, based in what were perceived to be "emotional" and "irrational" decisions, provided an immediate impetus for a more "scientific" and easier to manage form of decision making. Simply put, to manage the new multi-national enterprises, private and governmental, required more data. More data required a means of reducing it to information upon which to make decisions. Numbers and charts could be interpreted more quickly and moved more efficiently than long texts. In the 1930s this new model of managing decision making became cemented with the New Deal in the US, and in Europe with the increasing need to manage industrial production and governmental affairs. Institutions such as The New School for Social Research, International Institute of Social History, and departments of "social research" at prestigious universities were meant to fill the growing demand for individuals who could quantify human interactions and produce models for decision making on this basis. Coupled with this pragmatic need was the belief that the clarity and simplicity of mathematical expression avoided systematic errors of holistic thinking and logic rooted in traditional argument. This trend, part of the larger movement known as Modernism provided the rhetorical edge for the expansion of social sciences.

Present state

There continues to be little movement toward consensus on what methodology might have the power and refinement to connect a proposed "grand theory" with the various midrange theories which, with considerable success, continue to provide usable frameworks for massive, growing data banks. See consilience.

Criticism

The social sciences are sometimes criticized as being “less scientific” than the natural sciences, in that they are seen as being less rigorous or empirical in their methods. This claim is most commonly made when comparing social sciences to fields such as physics, chemistry or biology in which direct experimentation and falsification of results is generally carried out in a more direct fashion. Social scientists refute such claims by pointing to the use of a rich variety of scientific processes, mathematical proofs, and other methods in their professional literature. Others, however argue that the social world is much too complex to be studied as one would study static molecules. The actions or reactions of a molecule or chemical substance are always the same when placed in certain situations. Humans, on the other hand, are much too complex for these traditional scientific methodologies. Humans and society do not have certain rules that always have the same outcome and they cannot guarantee to react the same way to certain situations. Another criticism is that social sciences tend to be compromised more frequently by politics, since results from social science may threaten certain centers of power in a society, particularly ones which fund the research institutions. (For example, in the US, corporations and the state are frequently cited as these centers of power.) Further, complexity exacerbates the problems, since observed social data may be the result of factors which are hard to evaluate in isolation.

Reference

The beginnings of the social sciences in the eighteenth century are reflected in the grand encyclopedia of Diderot, with articles from Rousseau and other pioneers. The growth of the social sciences is also reflected in its specialised encyclopedias. The older editions are therefore of strong historical interest while the newest reflects current discussions, methodologies and ideologies.
- 1934, Encyclopedia of the Social Sciences
- 1968, International Encyclopedia of the Social Sciences
- 2001, International encyclopedia of the social & behavioral sciences / ed.-in-chief Neil J. Smelser; Paul B. Baltes, Amsterdam [etc.] : Elsevier, 2001-

External links

- [http://www.dialogical.net/socialsciences/index.html Social Science Virtual Library]
- [http://xlab.berkeley.edu UC Berkeley Experimental Social Science Laboratory]
- [http://www.sosig.ac.uk Social Science Information Gateway] (UK) Category:Humanities occupations ko:사회 과학 ja:社会科学 th:สังคมศาสตร์

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 unwillin