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Italic Type

Italic type

In typography, italic type refers to cursive typefaces based on a stylized form of calligraphic handwriting. The influence from calligraphy can be seen in from their usual slight slanting to the right. Different glyph shapes from roman type are also usually used—another influence from calligraphy. Sometimes the term italic is wrongly applied on oblique fonts (mostly sans-serif), when they are merely distorted into a slanted orientation. Uppercase letters in italic types are usually oblique instead of being true italics. Swash capitals are uppercase letters that have flourishes added to them, originally designed to go with Italic typefaces. Italic type is often used for emphasis to distinguish or otherwise set off certain words within text.

Examples

: An example of normal and true italics text: emphasis : The same example, as oblique text: emphasis

Italics are used for:

- Emphasis.
- The titles of works that stand by themselves, such as books or newspapers. Works that appear within larger works, such as short stories or newspaper articles, are not italicized, but merely set off in quotation marks.
- The names of ships.
- Foreign words.
- Using a word as an example of a word, rather than for its semantic content: see use-mention distinction.
- Introducing terms.
- The latin binary nomenclature (Genus species), in the taxonomy of living organisms.
- Symbols for physical quantities and other mathematical variables. If something within a run of italics needs to be italicized itself (for example, the name of a ship within a sentence already italicized for emphasis), the type is switched back to non-italicized (Roman) type. In media where italicization is not possible, alternatives are used.
- In typewritten or handwritten text, underlining is typically used.
- In plain-text computer files, including e-mail communication, italicised words are often indicated by surrounding them with slashes. For example, "I was /really/ annoyed".

History

The first italic type was employed by Aldus Manutius on his famous edition of Virgil, published at Venice in 1501, and is said to have been cut by Francesco de Bologna to imitate the beautiful hand-writing of Petrarch, famous Italian poet. The form at first consisted only of lower-case, important words being started with small roman capitals and it was independent of any roman font. The italic was used for the complete text, and in the hands of Aldus and the Elzevirs it made an admittedly graceful page medium, but somewhat too informal, in our estimation, and less legible than the more dignified roman style. The use of italic for emphasis in roman text was a later development which some say has marked the loss of its individual character, and not until Garamond made matched romans and italics was associated use considered. At that time italic fonts were considered distinct typefaces in themselves, not the variants they are nowadays. Except for capital letters (which were printed in roman type), entire books were printed in italic type. It was not until later that the modern distinction between italic and roman type was developed. Italic fonts have shown some influences from types and hands common in other regions in the time since they first developed.

Use in web pages

In serifed typefaces like Times New Roman, italic type can be produced in HTML through the use of the <i>...</i> tags (and on sans-serif typefaces they will mostly appear oblique), although this usage is deprecated in favor of Cascading Style Sheets solutions (see below). When an author wants to indicate emphasized text, which is often rendered in italics, they should use the <em>...</em> tags, though this should not be relied on to render in italics. Italic type is achieved in CSS by using the statement font-style: italic;

Use with parentheses

The Chicago Manual of Style suggests that to avoid problems such as overlapping and unequally spaced characters, parentheses and brackets surrounding text that begins and ends in italic or oblique type should also be italicized. An exception to this rule applies when only one end of the parenthetical is italicized (in which case roman type is preferred). Category:Typography

Typography

Typography (from the Greek words typos = form and graphein = to write) is the art and technique of typesetting; that is, of selecting and arranging typefaces, point sizes, line lengths, line leading, character spacing, and word spacing for typeset applications. These applications can be physical or digital. Typography is performed by typographers. It was once exclusively a specialist occupation, but the advent of computers has given many more people the opportunity to experiment with the art. The primary function of typography is the presentation of text in a manner that is both easy to read and visually engaging. Visual interest is achieved through typeface selection, text layout, use of colour, and the interplay of text and graphical elements – all of which combine to give an "atmosphere" or "feel" to the material. Other issues that might interest a typographer involved with physical printed media are paper selection, ink choice, and the printing method. Typographers employ a number of common techniques, or conventions, to achieve eye-pleasing, legible results. Note, however, that these may depend on the culture (language, country). As an example, it is customary in French to insert a non-breaking space before a colon (:) or semicolon (;) in a sentence, while it in English it is not. Contrast typography with orthography (the representation of the sounds of a language by written or printed symbols, and the study of correct spelling according to established usage), and with typeface design. Typography is often an important element of graphic design, and in some applications of typography there is less concern for legibility, and more interest in using type in a purely artistic manner.

References

- Bringhurst, Robert (2002). The Elements of Typographic Style (version 2.5). Vancouver: Hartley & Marks. ISBN 0-88179-133-4. Often referred to simply as Bringhurst, it is widely respected as the modern authority on typographic style for the English language ([http://www.aaronsw.com/2002/typographicStyle excerpts]).
- Lexique des règles typographiques en usage à l'Imprimerie nationale, Imprimerie nationale, 2002, ISBN 2743304820, for French typography

Supporting organizations

- Type Directors Club

External links

- [http://www.faqs.org/faqs/fonts-faq/part4/ Comp.fonts FAQ: General Info] - Section four of six of the newsgroup FAQ
- [http://www.typographi.com Typographica] - a daily journal of typography
- [http://www.typolis.de/version1/indexe.htm Typography, Type and Design]
- [http://www.dmoz.org/Arts/Graphic_Design/Typography/ Typography Directory]
- [http://euro.typo.cz/ Typo.cz] - information on Central European typography and typesetting
- [http://www.flywebmaster.com/webdesign/tips/typography.php Web Typography]
- [http://www.microsoft.com/typography/default.mspx Microsoft Typography page]
- [http://tc.eserver.org/dir/Typography EServer TC Library: Typography]
- [http://www.fontsite.com/ FontSite.com] - Some articles on basic typography for desktop publishers
- [http://diacritics.typo.cz Diacritics Project — All you need to design a font with correct accents]
- [http://www.textism.com/textfaces/ Twenty Faces]
- [http://www.planet-typography.com/ Planet typography] - A magazine on contemporary typography + a directory, a manual and other topics related to typefaces
- [http://www.piggin.net/ Macro-Typography: A Style Guide] Category:Design Category:Typography ja:タイポグラフィ

Oblique type

Oblique type is a form of type that slants slightly to the right, used in the same manner as italic type. Unlike italic type, however, it does not use different glyph shapes; it uses the same glyphs as roman type, except distorted. Oblique and italic type are often confused. The start of this confusion maybe appeared when Adrian Frutiger named the slanted versions of his typefaces Univers and Frutiger as Italic. Following this viewpoint, sans-serif typefaces very rarely have true italic versions. The Gill Sans typeface is one of this rare exceptions.

Sans-serif

In typography, a sans-serif or sans serif typeface is one that does not have the small features called "serifs" at the end of strokes within letters. The term comes from the French word sans (meaning "without"), so the term literally means "without serifs." Sans-serif fonts are typically suited for headlines as opposed to body text. The lack of serifs make sans-serif fonts harder to read in large blocks of text. Before the term “sans serif” became standard in English typography, a number of other terms had been used. One of these outmoded terms for sans serif is gothic, which is still used in Japanese and Korean typography, and sometimes seen in font names like “New Century Gothic”. Sans-serif fonts are sometimes, especially in older books, used as a device for emphasis, due to their typically blacker type color.

Classification

For the purposes of type classification sans-serif designs broadly divide into four major groups:
- Grotesque, early sans-serif designs, such as Grotesque or Royal Gothic. Royal Gothic
- Neo-grotesque or Transitional, modern designs such as Standard, Helvetica, Arial, and Univers. These are the most common sans-serif fonts. They are relatively straight in appearance and have less line width variation than Humanist sans-serif typefaces. Transitional sans-serif is sometimes called "anonymous sans-serif" due to its relatively plain appearance.
Univers
- Humanist (Edward Johnston's Railway type, Gill Sans or Frutiger). These are the most calligraphic of the sans-serif typefaces, with some variation in line width and more readability than other sans-serif fonts.
Frutiger
- Geometric (Futura, Century Gothic, or Spartan). As their name suggests, Geometric sans-serif typefaces are based on geometric shapes. Note the perfectly circular letter "O" and the simple construction of the lowercase letter "a". Geometric sans-serif fonts have a very modern look and feel.
Other commonly-used sans-serif fonts include Optima, Tahoma and Verdana. Note that in some sans-serif fonts I (capital-i) and l (lowercase-L) appear exactly identical. (Arial: Il) Verdana, however, keeps them distinct: Il due to the fact that Verdana's capital-i, as an exception, has serifs.

Emphasis (typography)

In typography, emphasis usually refers to means of stressing parts of a text by using letters in a different style from the rest of the text to make them stand out from the main text body.

Methods and uses of emphasis

typography The human eye is very receptive to differences in brightness within a text body. One can therefore differentiate between types of emphasis according to whether the emphasis changes the "blackness" of text. A means of emphasis that does not have much effect on "blackness" is printing in italics, where the text is written in a script style, or oblique, where the vertical orientation of all letters is slanted to the left or right. With one or other of these techniques (usually only one is available for any typeface), words can be highlighted without making them "stick out" much from the rest of the text (inconspicuous stressing). Traditionally, this is used for marking passages that have a different context, such as words from foreign languages, book titles, etc. By contrast, boldface makes text darker than the surrounding text. With this technique, the emphasized text strongly stands out from the rest; it should therefore be used to highlight certain keywords that are important to the subject of the text, for easy visual scanning of text. For example, printed dictionaries often use boldface for their keywords; Wikipedia follows this convention when the name of each article is marked at the top in bold. If the text body is typeset in a serif typeface, it is also possible to highlight words by setting them in a sans serif face. This is somewhat of an archaic practice.

Emphasis in design

With both italics and boldface, the emphasis is correctly achieved by temporarily replacing the current typeface. Professional typographic systems (which include most modern computers) would therefore not simply tilt letters to the right to achieve italics (that is instead referred to as slanting) or print them darker for boldface, but instead use entirely different typefaces that achieve the effect. As can be seen in Fig. 1, the "w" letter, for example, looks quite different in italics compared to the regular typeface. As a result, typefaces therefore have to be supplied at least fourfold (with computer systems, usually as four font files): as regular, italics, bold, and both bold and italics to provide for all combinations. Professional typefaces sometimes offer even more variations for popular fonts, with varying degrees of blackness. Only if such fonts are not available should the effect of italics or boldface be imitated by tilting or blacking the original font.

Alternative methods for emphasis

Capitalization

The house styles of many U.S. publishers use capitalization or all-uppercase letters, in order to emphasise
- publication titles
- warning messages
- newspaper headlines
- chapter and section headings Capitalization is used much less commonly today by British publishers (usually only for book titles). It is rarely used in other languages. All-uppercase letters are a common form of emphasis where the medium lacks support for boldface, such as old typewriters, plain-text email, SMS and other text-messaging systems.

Letterspacing

SMS In Germany, a different means of emphasis was previously used. To achieve a variance in blackness, instead of making the letters darker, one would increase the spacing between them. This resulted in an effect reverse to boldface: the emphasized text becomes lighter than its environment. This was referred to as sperren in German ("letterspacing" in English), which could here be translated as "spacing out". While sperren normally means "to lock (out)", this particular meaning was figurative: with the older method of typesetting with letters of lead, the spacing would be achieved by inserting additional non-printing slices of metal between the types. The example text reads: "An example of German text in Fraktur in which a portion of the text is spaced out. It is noticed, as with boldface, clearly as opposed to the rest of the text. The reason for this particular German typographic convention can be seen in the traditional use of blackletter typefaces, for which boldface was not feasible, since the letters were very dark in their standard format. The blackletter typefaces were officially abolished in 1942 by Nazi Germany, and after that, its use quickly diminished. As a result, the use of spacing as a means of emphasis in printed materials quickly became obsolete. However, spacing is sometimes still used as a means of emphasis in typographic media where only one typeset is available, e.g. in typewritten communication or on text-only computer terminals.

Special punctuation marks

In Chinese, emphasis in body text is supposed to be indicated by using an "emphasis mark" (着重號), which is a dot placed under each character to be emphasized. This is still taught in schools, but in practice it is not usually done, probably due to the difficulty of doing this in most computer software. Methods used for emphasis in western texts but inappropriate for Chinese, for example underlining and setting text in artificially slanted type (frequently incorrectly called "italics"), are often used instead. Category:typography

Emphasis (typography)

In typography, emphasis usually refers to means of stressing parts of a text by using letters in a different style from the rest of the text to make them stand out from the main text body.

Methods and uses of emphasis

Emphasis in design

Alternative methods for emphasis

Capitalization

Letterspacing

Special punctuation marks

Use-mention distinction

The use-mention distinction is the distinction between using a word (or phrase, etc.) and mentioning it. In written language, mentioned words or phrases often appear between quotation marks or in italics; some authorities insist that mentioned words or phrases always be made visually distinct in this manner. Used words or phrases, being more common than mentioned ones, do not have any typographic distinction. For example, the sentence : Cheese is derived from milk. is a statement about the substance cheese, and involves the use of the word cheese, while the sentence : Cheese is derived from a word in Old English. is a statement about the word cheese, and involves the mention of the word cheese. Putting a statement in quotation marks and attributing it to its originator is a useful way of turning a disputed statement about a subject into an undisputed statement about another statement. Making a statement mention itself is an interesting way of producing logical paradoxes. There are many examples of this phenomenon in the works of Douglas Hofstadter. Violation of the use-mention distinction can produce sentences that sound and appear similar to the original, but have an entirely different meaning. For example, : "The use-mention distinction" is not "strictly enforced here." is literally true because the two phrases in it are not the same.

Use-mention and suppositio

The general property of terms changing their reference depending on the context was called suppositio (substitution) by classical logicians. It describes how one has to substitute a term in a sentence based on its meaning—that is, what referent the term has. In general, a term can be used in several ways. For nouns, they are:
- Properly with a real referent: "That is my cow" (assuming it exists).
- Properly with a generic referent: "Any cow gives milk."
- Properly but with a non-real referent: "Ulysses's cow was big."
- Improperly by way of metaphor: "You are a cow" (assuming the listener is not a cow).
- As a pure term: "Cow has only three letters". The last use is what gives rise to the use-mention distinction.

Use-mention in philosophy

The use-mention distinction is especially important in analytic philosophy. The standard notation for mentioning a term is to put it in single quotes. This form is taken very seriously, as failure to properly distinguish use from mention can produce false or misleading statements. For example: : 'Copper' contains six letters, and is not a metal. : Copper is a metal, and contains no letters.

External links

[http://www.unconventional-wisdom.com/WAW/ROBERT.html More examples] Category:Philosophy

Genus

In biology, a genus (plural genera) is a grouping in the classification of living organisms having one or more related and morphologically similar species. In the common binomial nomenclature, the name of an organism is composed of two parts: its genus (always capitalized) and a species modifier. An example is Homo sapiens, the name for the human species which belongs to the genus Homo. See scientific classification for more details of this system. The type genus of a taxon is usually the first genus to be named and described. Families, and in plants all taxa up to division, are named after the type genus. The genus and these higher taxa are typified by a specimen that shows the characteristics of the genus. The specimen used to describe this species is preserved as the holotype and designated as a generitype in a zoological museum or a herbarium to be available for further study. A generic name in one kingdom is allowed to bear the same name as a genus or other taxon name in another kingdom (though this is discouraged by the International Code of Zoological Nomenclature). For instance, Anura is a genus of plants in the family Asteraceae and the order of frogs; Aotus is the genus of golden peas and night monkeys; Oenanthe is the genus of wheatears and water dropworts, and Prunella is the genus of accentors and self-heal. It is, however, not allowed for two genera within the same kingdom to have the same name. This explains why the platypus genus is Ornithorhynchus — although the name Platypus was chosen by George Shaw in 1799, that name had already been given to the ambrosia beetle by Johann Friedrich Wilhelm Herbst in 1793. Since beetles and platypuses are both member of the kingdom Animalia, the name Platypus could not be used for both. Johann Friedrich Blumenbach published the replacement name Ornithorhynchus in 1800.

Species

In biology, a species is the basic unit of biodiversity. In scientific classification, a species is assigned a two-part name in Latin. The genus is listed first (and capitalized), followed by a specific epithet. For example, humans belong to the genus Homo, and are in the species Homo sapiens. The name of the species is the whole binomial not just the second term (the specific epithet). The binomial, and most other purely formal aspects of the biological codes of nomenclature, were formalized by Carolus Linnaeus in the 1700's and as a result are called the "Linnaean system". At that time, species were thought to represent independent acts of creation by God, and were therefore considered objectively real and immutable. Since the advent of the theory of evolution, the conception of species has undergone vast changes in biology, however no consensus on the definition of the word has yet been reached. The most commonly cited definition of "species" was first coined by Ernst Mayr. By this definition, called the biological species concept or isolation species concept, species are "groups of actually or potentially interbreeding natural populations which are reproductively isolated from other such groups". However, many other species concepts are also used (see other definitions of species below). The scientific name of a species is properly typeset in italics. When an unknown species is being referred to this may be done by using the abbreviation "sp." in the singular or "spp." in the plural in the place of the second part of the scientific name. Note that the word "specie" is not the singular of "species". It refers to coined money.

Definitions of species

The definition of a species given above as taken from Mayr, is somewhat idealistic. Since it assumes sexual reproduction, it leaves the term undefined for a large class of organisms that reproduce asexually. Biologists frequently do not know whether two morphologically similar groups of organisms are "potentially" capable of interbreeding. Further, there is considerable variation in the degree to which hybridization may succeed under natural and experimental conditions, or even in the degree to which some organisms use sexual reproduction between individuals to breed. Consequently, several lines of thought in the definition of species exist: ; Typological species : A group of organisms in which individuals are members of the species if they sufficiently conform to certain fixed properties. The clusters of variations or phenotypes within specimens (ie: longer and shorter tails) would differentiate the species. This method was used as a "classical" method of determining species, such as with Linnaeus early in evolutionary theory. However, we now know that different phenotypes do not always constitute different species (e.g.: a 4-winged Drosophila born to a 2-winged mother is not a different species). Species named in this manner are called morphospecies. ; Morphological species : A population or group of populations that differs morphologically from other populations. For example, we can distinguish between a chicken and a duck because they have different shaped bills and the duck has webbed feet. Species have been defined in this way since well before the beginning of recorded history. This species concept is much criticised because more recent genetic data reveals that genetically distinct populations may look very similar and, contrarily, large morphological differences sometimes exist between very closely-related populations. Nonetheless, most species known have been described solely from morphology. ; Biological / Isolation species : A set of actually or potentially interbreeding populations. This is generally the most useful formulation for scientists working with living examples of the higher taxa like mammals, fish, and birds, but meaningless for organisms that do not reproduce sexually. It does not distinguish between the theoretical possibility of interbreeding and the actual likelihood of gene flow between populations and is thus impractical in instances of allopatric (geographically isolated) populations. The results of breeding experiments done in artificial conditions may or may not reflect what would happen if the same organisms encountered each other in the wild, making it difficult to gauge whether or not the results of such experiments are meaningful in reference to natural populations. ; Mate-recognition species : A group of organisms that are known to recognise one another as potential mates. Like the isolation species concept above, it applies only to organisms that reproduce sexually. Unlike the isolation species concept, it focuses specifically on pre-mating reproductive isolation. ; Phylogenetic / Evolutionary / Darwinian species : A group of organisms that shares an ancestor; a lineage that maintains its integrity with respect to other lineages through both time and space. At some point in the progress of such a group, members may diverge from one another: when such a divergence becomes sufficiently clear, the two populations are regarded as separate species. ; Microspecies : Species that reproduce without meiosis or mitosis so that each generation is genetically identical to the previous generation. See also apomixis. In practice, these definitions often coincide, and the differences between them are more a matter of emphasis than of outright contradiction. Nevertheless, no species concept yet proposed is entirely objective, or can be applied in all cases without resorting to judgement. Given the complexity of life, some have argued that such an objective definition is in all likelihood impossible, and biologists should settle for the most practical definition. For most vertebrates, this is the biological species concept, and to a lesser extent (or for different purposes) the phylogenetic species concept. Many BSC subspecies are considered species under the PSC; the difference between the BSC and the PSC can be summed up insofar as that the BSC defines a species as a consequence of manifest evolutionary history, while the PSC defines a species as a consequence of manifest evolutionary potential. Thus, a PSC species is "made" as soon as an evolutionary lineage has started to separate, while a BSC species starts to exist only when the lineage separation is complete.

Importance in biological classification

The idea of species has a long history. It is one of the most important levels of classification, for several reasons:
- It often corresponds to what lay people treat as the different basic kinds of organism - dogs are one species, cats another.
- It is the standard binomial nomenclature (or trinomial nomenclature) by which scientists typically refer to organisms.
- It is the only taxonomic level which has empirical content, in the sense that asserting that two animals are of different species is saying something more than classificatory about them. After thousands of years of use, the concept remains central to biology and a host of related fields, and yet also remains at times ill-defined and controversial.

Implications of assignment of species status

The naming of a particular species should be regarded as a hypothesis about the evolutionary relationships and distinguishability of that group of organisms. As further information comes to hand, the hypothesis may be confirmed or refuted. Sometimes, especially in the past when communication was more difficult, taxonomists working in isolation have given two distinct names to individual organisms later identified as the same species. When two named species are discovered to be of the same species, the older species name is usually retained, and the newer species name dropped, a process called synonymization, or convivially, as lumping. Dividing a taxon into multiple, often new, taxons is called splitting. Taxonomists are often referred to as "lumpers" or "splitters" by their colleagues, depending on their personal approach to recognizing differences or commonalities between organisms (see lumpers and splitters). Traditionally, researchers relied on observations of anatomical differences, and on observations of whether different populations were able to interbreed successfully, to distinguish species; both anatomy and breeding behavior are still important to assigning species status. As a result of the revolutionary (and still ongoing) advance in microbiological research techniques, including DNA analysis, in the last few decades, a great deal of additional knowledge about the differences and similarities between species has become available. Many populations which were formerly regarded as separate species are now considered to be a single taxon, and many formerly grouped populations have been split. Any taxonomic level (species, genus, family, etc.) can be synonymized or split, and at higher taxonomic levels, these revisions have been still more profound. From a taxonomical point of view, groups within a species can be defined as being of a taxon hierarchically lower than a species. In zoology only the subspecies is used, while in botany the variety, subvariety, and form are used as well.

The isolation species concept in more detail

In general, for large, complex, organisms that reproduce sexually (such as mammals and birds), one of several variations on the isolation or biological species concept is employed. Often, the distinction between different species, even quite closely related ones, is simple. Horses (Equus caballus) and donkeys (Equus asinus) are easily told apart even without study or training, and yet are so closely related that they can interbreed after a fashion. Because the result, a mule or hinny, is not usually fertile, they are clearly separate species. But many cases are more difficult to decide. This is where the isolation species concept diverges from the evolutionary species concept. Both agree that a species is a lineage that maintains its integrity over time, that is diagnosably different to other lineages (else we could not recognise it), is reproductively isolated (else the lineage would merge into others, given the chance to do so), and has a working intra-species recognition system (without which it could not continue). In practice, both also agree that a species must have its own independent evolutionary history—otherwise the characteristics just mentioned would not apply. The species concepts differ in that the evolutionary species concept does not make predictions about the future of the population: it simply records that which is already known. In contrast, the isolation species concept refuses to assign the rank of species to populations that, in the best judgement of the researcher, would recombine with other populations if given the chance to do so.

The isolation question

There are, essentially, two questions to resolve. First, is the proposed species consistently and reliably distinguishable from other species? Secondly, is it likely to remain so in the future? To take the second question first, there are several broad geographic possibilities.
- The proposed species are sympatric—they occupy the same habitat. Observation of many species over the years has failed to establish even a single instance of two diagnostically different populations that exist in sympatry and have then merged to form one united population. Without reproductive isolation, population differences cannot develop, and given reproductive isolation, gene flow between the populations cannot merge the differences. This is not to say that cross breeding does not take place at all, simply that it has become negligible. Generally, the hybrid individuals are less capable of successful breeding than pure-bred individuals of either species.
- The proposed species are allopatric—they occupy different geographical areas. Obviously, it is not possible to observe reproductive isolation in allopatric groups directly. Often it is not possible to achieve certainty by experimental means either: even if the two proposed species interbreed in captivity, this does not demonstrate that they would freely interbreed in the wild, nor does it always provide much information about the evolutionary fitness of hybrid individuals. A certain amount can be inferred from other experimental methods: for example, do the members of population A respond appropriately to playback of the recorded mating calls of population B? Sometimes, experiments can provide firm answers. For example, there are seven pairs of apparently almost identical marine snapping shrimp (Altheus) populations on either side of the Isthmus of Panama, which did not exist until about 3 million years ago. Until then, it is assumed, they were members of the same seven species. But when males and females from opposite sides of the isthmus are placed together, they fight instead of mating. Even if the isthmus were to sink under the waves again, the populations would remain genetically isolated: therefore they are now different species. In many cases, however, neither observation nor experiment can produce certain answers, and the determination of species rank must be made on a 'best guess' basis from a general knowledge of other related organisms.
- The proposed species are parapatric—they have breeding ranges that abut but do not overlap. This is fairly rare, particularly in temperate regions. The dividing line is often a sudden change in habitat (an ecotone) like the edge of a forest or the snow line on a mountain, but can sometimes be remarkably trivial. The parapatry itself indicates that the two populations occupy such similar ecological roles that they cannot coexist in the same area. Because they do not crossbreed, it is safe to assume that there is a mechanism, often behavioral, that is preventing gene flow between the populations, and that therefore they should be classified as separate species.
- There is a hybrid zone where the two populations mix. Typically, the hybrid zone will include representatives of one or both of the 'pure' populations, plus first-generation and back-crossing hybrids. The strength of the barrier to genetic transmission between the two pure groups can be assessed by the width of the hybrid zone relative to the typical dispersal distance of the organisms in question. The dispersal distance of oaks, for example, is the distance that a bird or squirrel can be expected to carry an acorn; the dispersal distance of Numbats is about 15 kilometres, as this is as far as young Numbats will normally travel in search of vacant territory to occupy after leaving the nest. The narrower the hybrid zone relative to the dispersal distance, the less gene flow there is between the population groups, and the more likely it is that they will continue on separate evolutionary paths. Nevertheless, it can be very difficult to predict the future course of a hybrid zone; the decision to define the two hybridizing populations as either the same species or as separate species is difficult and potentially controversial.
- The variation in the population is clinal; at either extreme of the population's geographic distribution, typical individuals are clearly different, but the transition between them is seamless and gradual. For example, the Koalas of northern Australia are clearly smaller and lighter in colour than those of the south, but there is no particular dividing line: the further south an individual Koala is found, the larger and darker it is likely to be; Koalas in intermediate regions are intermediate in weight and colour. In contrast, over the same geographic range, black-backed (northern) and white-backed (southern) Australian Magpies do not blend from one type to another: northern populations have black backs, southern populations white backs, and there is an extensive hybrid zone where both 'pure' types are common, as are crossbreeds. The variation in Koalas is clinal (a smooth transition from north to south, with populations in any given small area having a uniform appearance), but the variation in magpies is not clinal. In both cases, there is some uncertainty regarding correct classification, but the consensus view is that species rank is not justified in either. The gene flow between northern and southern magpie populations is judged to be sufficiently restricted to justify terming them subspecies (not full species); but the seamless way that local Koala populations blend one into another shows that there is substantial gene flow between north and south. As a result, experts tend to reject even subspecies rank in this case.

The difference question

Obviously, when defining a species, the geographic circumstances become meaningful only if the populations groups in question are clearly different: if they are not consistently and reliably distinguishable from one another, then we have no grounds for believing that they might be different species. The key question in this context, is "how different is different?" and the answer is usually "it all depends". In theory, it would be possible to recognise even the tiniest of differences as sufficient to delineate a separate species, provided only that the difference is clear and consistent (and that other criteria are met). There is no universal rule to state the smallest allowable difference between two species, but in general, very trivial differences are ignored on the twin grounds of simple practicality, and genetic similarity: if two population groups are so close that the distinction between them rests on an obscure and microscopic difference in morphology, or a single base substitution in a DNA sequence, then a demonstration of restricted gene flow between the populations will probably be difficult in any case. More typically, one or other of the following requirements must be met:
- It is possible to reliably measure a quantitative difference between the two groups that does not overlap. A population has, for example, thicker fur, rougher bark, longer ears, or larger seeds than another population, and although this characteristic may vary within each population, the two do not grade into one another, and given a reasonably large sample size, there is a definite discontinuity between them. Note that this applies to populations, not individual organisms, and that a small number of exceptional individuals within a population may 'break the rule' without invalidating it. The less a quantitative difference varies within a population and the more it varies between populations, the better the case for making a distinction. Nevertheless, borderline situations can only be resolved by making a 'best-guess' judgement.
- It is possible to distinguish a qualitative difference between the populations; a feature that does not vary continuously but is either entirely present or entirely absent. This might be a distinctively shaped seed pod, an extra primary feather, a particular courting behaviour, or a clearly different DNA sequence. Sometimes it is not possible to isolate a single difference between species, and several factors must be taken in combination. This is often the case with plants in particular. In eucalypts, for example, Corymbia ficifolia cannot be reliably distinguished from its close relative Corymbia calophylla by any single measure (and sometimes individual trees cannot be definitely assigned to either species), but populations of Corymbia can be clearly told apart by comparing the colour of flowers, bark, and buds, number of flowers for a given size of tree, and the shape of the leaves and fruit. When using a combination of characteristics to distinguish between populations, it is necessary to use a reasonably small number of factors (if more than a handful are needed, the genetic difference between the populations is likely to be insignificant and is unlikely to endure into the future), and to choose factors that are functionally independent (height and weight, for example, should usually be considered as one factor, not two).

Historical development of the species concept

In the earliest works of science, a species was simply an individual organism that represented a group of similar or nearly identical organisms. No other relationships beyond that group were implied. Aristotle used the words genus and species to mean generic and specific categories. Aristotle and other pre-Darwinian scientists took the species to be distinct and unchanging, with an "essence", like the chemical elements. When early observers began to develop systems of organization for living things, they began to place formerly isolated species into a context. To the modern mind, many of the schemes delineated are whimsical at best, such as those that determined consanguinity based on color (all plants with yellow flowers) or behavior (snakes, scorpions and certain biting ants). In the 18th century Carolus Linnaeus classified organisms according to differences in the form of reproductive apparatus. Although his system of classification sorts organisms according to degrees of similarity, it made no claims about the relationship between similar species. At the time, it was still widely believed that there is no organic connection between species, no matter how similar they appear; every species was individually created by God, a view today called creationism. This approach also suggested a type of idealism: the notion that each species exists as an "ideal form". Although there are always differences (although sometimes minute) between individual organisms, Linnaeus considered such variation problematic. He strove to identify individual organisms that were exemplary of the species, and considered other non-exemplary organisms to be deviant and imperfect. By the 19th century most naturalists understood that species could change form over time, and that the history of the planet provided enough time for major changes. As such, the new emphasis was on determining how a species could change over time. Jean-Baptiste Lamarck suggested that an organism could pass on an acquired trait to its offspring, i.e., the giraffe's long neck was attributed to generations of giraffes stretching to reach the leaves of higher treetops (this well-known and simplistic example, however, does not do justice to the breadth and subtlety of Lamarck's ideas). Lamarck's most important insight may have been that species can be extraordinarily fluid; his 1809 Zoological Philosophy contained one of the first logical refutations of creationism. With the acceptance of the work of Charles Darwin in the 1860s, Lamarck's view of evolution was quickly eclipsed. It was not until the late 20th century that his work began to be reexamined, and took its place as a fundamental stepping stone to the modern theory of adaptive mutation. Lamarck's long-discarded ideas of the goal-oriented evolution of species, also known the teleological process, have also received renewed attention, particularly by proponents of artificial selection. Charles Darwin and Alfred Wallace provided what scientists now consider the most powerful and compelling theory of evolution. Basically, Darwin argued that it is populations that evolve, not individuals. His argument relies on a radical shift in perspective from Linnaeus: rather than defining species in ideal terms (and searching for an ideal representative and rejecting deviations), Darwin considered variation among individuals to be natural. He further argued that variation, far from being problematic, actually provides the explanation for the existence of distinct species. Darwin's work drew on Thomas Malthus' insight that the rate of growth of a biological population will always outpace the rate of growth of the resources in the environment, such as the food supply. As a result, Darwin argued, not all the members of a population will be able to survive and reproduce. Those that did will, on average, be the ones possessing variations—however slight—that make them slightly better adapted to the environment. If these variable traits are heritable, then the offspring of the survivors will also possess them. Thus, over many generations, adaptive variations will accumulate in the population, while counter-adaptive will be eliminated. It should be emphasized that whether a variation is adaptive or non-adaptive depends on the environment: different environments favor different traits. Since the environment effectively selects which organisms live to reproduce, it is the environment (the "fight for existence") that selects the traits to be passed on. This is the theory of evolution by natural selection. In this model, the length of a giraffe's neck would be explained by positing that proto-giraffes with longer necks would have had a significant reproductive advantage to those with shorter necks. Over many generations, the entire population would be a species of long-necked animals. In 1859, when Darwin published his theory of natural selection, the mechanism behind the inheritance of individual traits was unknown. Although Darwin made some speculations on how traits are inherited (pangenesis), his theory relies only on the fact that inheritable traits exist, and are variable (which makes his accomplishment even more remarkable.) Although Gregor Mendel's paper on genetics was published in 1866, its significance was not recognized. It was not until 1900 that his work was rediscovered by Hugo de Vries, Carl Correns and Erich von Tschermak, who realised that the "inheritable traits" in Darwin's theory are genes. The theory of the evolution of species through natural selection has two important implications for discussions of species -- consequences that fundamentally challenge the assumptions behind Linnaeus' taxonomy. First, it suggests that species are not just similar, they may actually be related. Some students of Darwin argue that all species are descended from a common ancestor. Second, it supposes that "species" are not homogeneous, fixed, permanent things; members of a species are all different, and over time species change. This suggests that species do not have any clear boundaries but are rather momentary statistical effects of constantly changing gene-frequencies. One may still use Linnaeus' taxonomy to identify individual plants and animals, but one can no longer think of species as independent and immutable. The rise of a new species from a parental line is called speciation. There is no clear line demarcating the ancestral species from the descendant species. Although the current scientific understanding of species suggests there is no rigorous and comprehensive way to distinguish between different species in all cases, biologists continue to seek concrete ways to operationalize the idea. One of the most popular biological definitions of species is in terms of reproductive isolation; if two creatures cannot reproduce to produce fertile offspring, then they are in different species. This definition captures a number of intuitive species boundaries, but nonetheless has some problems, however. It has nothing to say about species that reproduce asexually, for example, and it is very difficult to apply to extinct species. Moreover, boundaries between species are often fuzzy: there are examples where members of one population can produce fertile offspring with a second population, and members of the second population can produce fertile offspring with members of a third population, but members of the first and third population cannot produces fertile offspring. Consequently, some people reject this notion of species. In recent years we have witnessed the drastic reduction in the size of breeding populations and the geographical range of many physically large mammals. In earlier times it was assumed that every species existed in at least a few thousand living individuals, except very rare relic, isolated groups. In the present, many well know mammal & bird species are so stressed by habitat loss, and other effects of the modern world, that only a very few breeding males may contribute the genetic material to a small number of breeding females. In these highly stressed conditions, the likelihood of change is very much greater. Mammals may become smaller, have darker fur, more stripes, more cautious behavior, even over time learn to co-exist with the human world. Very likely, evolution is radically accelerated, and we are only beginning to notice it. Species in transition before our eyes. It is possible that this severe stress is essential to the creation of new species, and may have been a prime factor throughout biological history, from other population reducing influences. Richard Dawkins defines two organisms as conspecific if and only if they have the same number of chromosomes and, for each chromosome, both organisms have the same number of nucleotides (The Blind Watchmaker, p. 118). However, most if not all taxonomists would strongly disagree. For example, in many amphibians, most notably in New Zealand's Leiopelma frogs, the genome consists of "core" chromosomes which are mostly invariable and accessory chromosomes, of which exist a number of possible combinations. Even though the chromosome numbers are highly variable between populations, these can interbreed successfully and form a single evolutionary unit. In plants, polyploidy is extremely commonplace with few restrictions on interbreeding; as individuals with an odd number of chromosome sets are usually sterile, depending on the actual number of chromosome sets present, this results in the odd situation where some individuals of the same evolutionary unit can interbreed with certain others and some cannot, with all populations being eventually linked as to form a common gene pool. The classification of species has been profoundly affected by technological advances that have allowed researchers to determine relatedness based on molecular markers, starting with the comparatively crude blood plasma precipitation assays in the mid-20th century and coming into full swing with Charles Sibley's ground-breaking DNA-DNA hybridisation studies in the 1970s. The results of the technique caused revolutionary changes in the higher taxonomic categories (such as phyla and classes), resulting in the reordering of many branches of the phylogenetic tree (see also: molecular phylogeny). For taxonomic categories below genera, the results have been mixed so far; the pace of evolutionary change on the molecular level is rather slow, yielding clear differences only after considerable periods of reproductive separation. Instances of hybridization can result in misleading molecular data, the Pomarine Skua - Great Skua phenomenon being a famous example. Turtles have been determined to evolve with just one-eighth of the speed of other reptiles on the molecular level, and the rate of molecular evolution in albatrosses is half of what is found in the rather closely related storm-petrels. The hybridization technique is however no longer considered a good technique and more reliable computational techniques for sequence comparison are now used for. Molecular taxonomy does not directly measure the evolutionary processes, but rather the overall change brought upon by these processes. The processes that lead to the generation and maintenance of variation such as mutation, crossover and selection are not uniform (see also molecular clock). DNA is only extremely rarely a direct target of natural selection rather than changes in the DNA sequence enduring over generations being a result of the latter; for example, silent transition-transversion combinations would alter the melting point of the DNA sequence, but not the sequence of the encoded proteins and thus are a possible example where, for example in microorganisms, a mutation confers a change in fitness all by itself.

External links

- http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/S/Speciation.html
- [http://www.sciencedaily.com/releases/2003/12/031231082553.htm 2003-12-31, ScienceDaily: Working On The 'Porsche Of Its Time': New Model For Species Determination Offered] Quote: "...two species of dinosaur that are members of the same genera varied from each other by just 2.2 percent. Translation of the percentage into an actual number results in an average of just three skeletal differences out of the total 338 bones in the body. Amazingly, 58 percent of these differences occurred in the skull alone. "This is a lot less variation than I'd expected", said Novak..."
- [http://www.sciencedaily.com/releases/2003/08/030808081854.htm 2003-08-08, ScienceDaily: Cross-species Mating May Be Evolutionarily Important And Lead To Rapid Change, Say Indiana University Researchers] Quote: "...the sudden mixing of closely related species may occasionally provide the energy to impel rapid evolutionary change..."
- [http://www.sciencedaily.com/releases/2004/01/040109064407.htm 2004-01-09 ScienceDaily: Mayo Researchers Observe Genetic Fusion Of Human, Animal Cells; May Help Explain Origin Of AIDS] Quote: "...The researchers have discovered conditions in which pig cells and human cells can fuse together in the body to yield hybrid cells that contain genetic material from both species... "What we found was completely unexpected", says Jeffrey Platt, M.D."
- [http://www.sciencedaily.com/releases/2000/09/000913211733.htm 2000-09-18, ScienceDaily: Scientists Unravel Ancient Evolutionary History Of Photosynthesis] Quote: "...gene-swapping was common among ancient bacteria early in evolution..."
- [http://plato.stanford.edu/entries/species/ Stanford Encyclopedia of Philosophy entry]
- [http://www.barcodinglife.org/ Barcoding of species] rank22 rank22 ms:Spesies ja:種 (生物) th:สปีชีส์

Taxonomy

Taxonomy (from Greek verb tassein = "to classify" and nomos = law, science, cf "economy") may refer to:
- the science of classification (see alpha taxonomy)
- a classification Initially taxonomy was only the science of classifying living organisms, but later the word was applied in a wider sense, and may also refer to either a classification of things, or the principles underlying the classification. Almost anything, animate objects, inanimate objects, places, and events, may be classified according to some taxonomic scheme. Taxonomies are frequently hierarchical in structure. However taxonomy may also refer to relationship schemes other than hierarchies, such as network structures. Other taxonomies may include single children with multi-parents, for example, "Car" might appear with both parents "Vehicle" and "Steel Mechanisms". A taxonomy might also be a simple organization of objects into groups, or even an alphabetical list. In current usage within "Knowledge Management", taxonomies are seen as slightly less broad than ontologies. Mathematically, a hierarchical taxonomy is a tree structure of classifications for a given set of objects. At the top of this structure is a single classification, the root node, that applies to all objects. Nodes below this root are more specific classifications that apply to subsets of the total set of classified objects. So for instance in common schemes of scientific classification of organisms, the root is the Organism (as this applies to all living things, it is implied rather than stated explicitly). Below this are the Domain, Kingdom, Phylum, Class, Order, Family, Genus, and Species, with various other ranks sometimes inserted. Some have argued that the human mind naturally organizes its knowledge of the world into such systems. This view is often based on the epistemology of Immanuel Kant. Anthropologists have observed that taxonomies are generally embedded in local cultural and social systems, and serve various social functions. Perhaps the most well-known and influential study of folk taxonomies is Émile Durkheim's The Elementary Forms of Religious Life. The theories of Kant and Durkheim also influenced Claude Lévi-Strauss, the founder of anthropological structuralism. Lévi-Strauss wrote two important books on taxonomies: Totemism and The Savage Mind. Such taxonomies as those analyzed by Durkheim and Lévi-Strauss are sometimes called folk taxonomies to distinguish them from scientific taxonomies that claim to be disembedded from social relations and thus objective and universal. A recent neologism, folksonomy, should not be confused with Folk Taxonomy (though it is obviously a contraction of the two words). Those who support scientific taxonomies have recently criticized folksonomies by dubbing them fauxonomies. The phrase enterprise taxonomy is used in business to describe a very limited form of taxonomy used only within one organization. The field of solving or best-fitting of numerical equations that characterize all measurable quantities of a set of objects is called cluster analysis; this is a form of taxonomy called numerical taxonomy or taximetrics.

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 col

Italic type

Examples

Italics are used for:

History

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Oblique type

Sans-serif

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Emphasis (typography)

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Emphasis (typography)

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Use-mention and suppositio

Use-mention in philosophy

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Genus

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Species

Definitions of species

Importance in biological classification

Implications of assignment of species status

The isolation species concept in more detail

The isolation question

The difference question

Historical development of the species concept

See also

External links

Taxonomy

See also

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