Saturday, August 21, 2010


Love is the emotion of strong affection and personal attachment.[1] In philosophical context, love is a virtue representing all of human kindness, compassion, and affection. In religious context, love is not just a virtue, but the basis for all being ("God is love"[2]), and the foundation for all divine law (Golden Rule).

The word love can refer to a variety of different feelings, states, and attitudes, ranging from generic pleasure ("I loved that meal") to intense interpersonal attraction ("I love my wife"). "Love" can also refer specifically to the passionate desire and intimacy of romantic love, to the sexual love of eros (cf. Greek words for love), to the emotional closeness of familial love, or to the platonic love that defines friendship,[3] to the profound oneness or devotion of religious love. [4] This diversity of uses and meanings, combined with the complexity of the feelings involved, makes love unusually difficult to consistently define, even compared to other emotional states.

Love in its various forms acts as a major facilitator of interpersonal relationships and, owing to its central psychological importance, is one of the most common themes in the creative arts.

Definitions

Part of a series on love

Basic aspects
Charity
Human bonding
Chemical basis
Religious views
Philosophy of love
Historically
Courtly love
Types of emotion
Eroticism
Platonic love
Familial love
Romance
See also
Puppy love
Limerence
Love sickness
Love addiction
Human sexuality
Unrequited love
Valentine's Day
Sexual intercourse
Interpersonal relationship
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The word "love" can have a variety of related but distinct meanings in different contexts. Often, other languages use multiple words to express some of the different concepts that English relies mainly on "love" to encapsulate; one example is the plurality of Greek words for "love." Cultural differences in conceptualizing love thus make it doubly difficult to establish any universal definition.[5]

Although the nature or essence of love is a subject of frequent debate, different aspects of the word can be clarified by determining what isn't love. As a general expression of positive sentiment (a stronger form of like), love is commonly contrasted with hate (or neutral apathy); as a less sexual and more emotionally intimate form of romantic attachment, love is commonly contrasted with lust; and as an interpersonal relationship with romantic overtones, love is commonly contrasted with friendship, although other definitions of the word love may be applied to close friendships in certain contexts.



Fraternal love (Prehispanic sculpture from 250–900 A.D., of Huastec origin). Museum of Anthropology in Xalapa, Veracruz, Mexico.
When discussed in the abstract, love usually refers to interpersonal love, an experience felt by a person for another person. Love often involves caring for or identifying with a person or thing, including oneself (cf. narcissism). In addition to cross-cultural differences in understanding love, ideas about love have also changed greatly over time. Some historians date modern conceptions of romantic love to courtly Europe during or after the Middle Ages, although the prior existence of romantic attachments is attested by ancient love poetry.[6]



Two hands forming the outline of a heart shape.
Because of the complex and abstract nature of love, discourse on love is commonly reduced to a thought-terminating cliché, and there are a number of common proverbs regarding love, from Virgil's "Love conquers all" to the Beatles' "All you need is love". St. Thomas Aquinas, following Aristotle, defines love as "to will the good of another."[7] Bertrand Russell describes love as a condition of "absolute value," as opposed to relative value. Philosopher Gottfried Leibniz said that love is "to be delighted by the happiness of another."[8]

Love is sometimes referred to as being the "international language", overriding cultural and linguistic divisions.

Impersonal love

A person can be said to love an object, principle, or goal if they value it greatly and are deeply committed to it. Similarly, compassionate outreach and volunteer workers' "love" of their cause may sometimes be borne not of interpersonal love, but impersonal love coupled with altruism and strong political convictions. People can also "love" material objects, animals, or activities if they invest themselves in bonding or otherwise identifying with those things. If sexual passion is also involved, this condition is called paraphilia.[9]

Interpersonal love

Interpersonal love refers to love between human beings. It is a more potent sentiment than a simple liking for another. Unrequited love refers to those feelings of love that are not reciprocated. Interpersonal love is most closely associated with interpersonal relationships. Such love might exist between family members, friends, and couples. There are also a number of psychological disorders related to love, such as erotomania.

Throughout history, philosophy and religion have done the most speculation on the phenomenon of love. In the last century, the science of psychology has written a great deal on the subject. In recent years, the sciences of evolutionary psychology, evolutionary biology, anthropology, neuroscience, and biology have added to the understanding of the nature and function of love.

Chemical basis
Main article: Love (scientific views)
Biological models of sex tend to view love as a mammalian drive, much like hunger or thirst.[10] Helen Fisher, a leading expert in the topic of love, divides the experience of love into three partly overlapping stages: lust, attraction, and attachment. Lust exposes people to others; romantic attraction encourages people to focus their energy on mating; and attachment involves tolerating the spouse (or indeed the child) long enough to rear a child into infancy.



Simplified overview of the chemical basis of love.
Lust is the initial passionate sexual desire that promotes mating, and involves the increased release of chemicals such as testosterone and estrogen. These effects rarely last more than a few weeks or months. Attraction is the more individualized and romantic desire for a specific candidate for mating, which develops out of lust as commitment to an individual mate forms. Recent studies in neuroscience have indicated that as people fall in love, the brain consistently releases a certain set of chemicals, including pheromones, dopamine, norepinephrine, and serotonin, which act in a manner similar to amphetamines, stimulating the brain's pleasure center and leading to side effects such as increased heart rate, loss of appetite and sleep, and an intense feeling of excitement. Research has indicated that this stage generally lasts from one and a half to three years.[11]

Since the lust and attraction stages are both considered temporary, a third stage is needed to account for long-term relationships. Attachment is the bonding that promotes relationships lasting for many years and even decades. Attachment is generally based on commitments such as marriage and children, or on mutual friendship based on things like shared interests. It has been linked to higher levels of the chemicals oxytocin and vasopressin to a greater degree than short-term relationships have.[11] Enzo Emanuele and coworkers reported the protein molecule known as the nerve growth factor (NGF) has high levels when people first fall in love, but these return to previous levels after one year.[12]

Psychological basis
Further information: Human bonding
Psychology depicts love as a cognitive and social phenomenon. Psychologist Robert Sternberg formulated a triangular theory of love and argued that love has three different components: intimacy, commitment, and passion. Intimacy is a form in which two people share confidences and various details of their personal lives, and is usually shown in friendships and romantic love affairs. Commitment, on the other hand, is the expectation that the relationship is permanent. The last and most common form of love is sexual attraction and passion. Passionate love is shown in infatuation as well as romantic love. All forms of love are viewed as varying combinations of these three components. American psychologist Zick Rubin seeks to define love by psychometrics. His work states that three factors constitute love: attachment, caring, and intimacy.[13] [14]



Grandmother and grandchild, Sri Lanka
Following developments in electrical theories such as Coulomb's law, which showed that positive and negative charges attract, analogs in human life were developed, such as "opposites attract." Over the last century, research on the nature of human mating has generally found this not to be true when it comes to character and personality—people tend to like people similar to themselves. However, in a few unusual and specific domains, such as immune systems, it seems that humans prefer others who are unlike themselves (e.g., with an orthogonal immune system), since this will lead to a baby that has the best of both worlds.[15] In recent years, various human bonding theories have been developed, described in terms of attachments, ties, bonds, and affinities.

Some Western authorities disaggregate into two main components, the altruistic and the narcissistic. This view is represented in the works of Scott Peck, whose work in the field of applied psychology explored the definitions of love and evil. Peck maintains that love is a combination of the "concern for the spiritual growth of another," and simple narcissism.[16] In combination, love is an activity, not simply a feeling.

Comparison of scientific models
Biological models of love tend to see it as a mammalian drive, similar to hunger or thirst.[10] Psychology sees love as more of a social and cultural phenomenon. There are probably elements of truth in both views. Certainly love is influenced by hormones (such as oxytocin), neurotrophins (such as NGF), and pheromones, and how people think and behave in love is influenced by their conceptions of love. The conventional view in biology is that there are two major drives in love: sexual attraction and attachment. Attachment between adults is presumed to work on the same principles that lead an infant to become attached to its mother. The traditional psychological view sees love as being a combination of companionate love and passionate love. Passionate love is intense longing, and is often accompanied by physiological arousal (shortness of breath, rapid heart rate); companionate love is affection and a feeling of intimacy not accompanied by physiological arousal.

Studies have shown that brain scans of those infatuated by love display a resemblance to those with a mental illness. Love creates activity in the same area of the brain where hunger, thirst, and drug cravings create activity. New love, therefore, could possibly be more physical than emotional. Over time, this reaction to love mellows, and different areas of the brain are activated, primarily ones involving long-term commitments.

Cultural views

Persian
Rumi, Hafez and Sa'di are icons of the passion and love that the Persian culture and language present. The Persian word for love is eshgh, derived from the Arabic ishq, however is considered by most to be too stalwart a term for interpersonal love and is more commonly substituted for 'doost dashtan' ('liking'). In the Persian culture, everything is encompassed by love and all is for love, starting from loving friends and family, husbands and wives, and eventually reaching the divine love that is the ultimate goal in life. Over seven centuries ago, Sa'di wrote:

The children of Adam are limbs of one body
Having been created of one essence.
When the calamity of time afflicts one limb
The other limbs cannot remain at rest.
If you have no sympathy for the troubles of others
You are not worthy to be called by the name of "man."
Chinese and other Sinic cultures

This article contains Chinese text. Without proper rendering support, you may see question marks, boxes, or other symbols instead of Chinese characters.


"Ai," the traditional Chinese character for love (愛) consists of a heart (middle) inside of "accept," "feel," or "perceive," which shows a graceful emotion. It can also be interpreted as a hand offering ones heart to another hand.
Two philosophical underpinnings of love exist in the Chinese tradition, one from Confucianism which emphasized actions and duty while the other came from Mohism which championed a universal love. A core concept to Confucianism is Ren ("benevolent love", 仁), which focuses on duty, action and attitude in a relationship rather than love itself. In Confucianism, one displays benevolent love by performing actions such as filial piety from children, kindness from parent, loyalty to the king and so forth.

The concept of Ai (愛) was developed by the Chinese philosopher Mozi in the 4th century BC in reaction to Confucianism's benevolent love. Mozi tried to replace what he considered to be the long-entrenched Chinese over-attachment to family and clan structures with the concept of "universal love" (jiān'ài, 兼愛). In this, he argued directly against Confucians who believed that it was natural and correct for people to care about different people in different degrees. Mozi, by contrast, believed people in principle should care for all people equally. Mohism stressed that rather than adopting different attitudes towards different people, love should be unconditional and offered to everyone without regard to reciprocation, not just to friends, family and other Confucian relations. Later in Chinese Buddhism, the term Ai (愛) was adopted to refer to a passionate caring love and was considered a fundamental desire. In Buddhism, Ai was seen as capable or being either selfish or selfless, the latter being a key element towards enlightenment.

In contemporary Chinese, Ai (愛) is often used as the equivalent of the Western concept of love. Ai is used as both a verb (e.g. wo ai ni 我愛你, or "I love you") and a noun (such as aiqing 愛情, or "romantic love"). However, due to the influence of Confucian Ren, the phrase ‘Wo ai ni’ (I love you) carries with it a very specific sense of responsibility, commitment and loyalty. Instead of frequently saying "I love you" as in some Western societies, the Chinese are more likely to express feelings of affection in a more casual way. Consequently, "I like you" (Wo xihuan ni, 我喜欢你) is a more common way of expressing affection in Chinese; it is more playful and less serious.[17] This is also true in Japanese (suki da, 好きだ). The Chinese are also more likely to say "I love you" in English or other foreign languages than they would in their mother tongue.

Japanese
In Japanese Buddhism, ai (愛) is passionate caring love, and a fundamental desire. It can develop towards either selfishness or selflessness and enlightenment. Amae (甘え), a Japanese word meaning "indulgent dependence," is part of the child-rearing culture of Japan. Japanese mothers are expected to hug and indulge their children, and children are expected to reward their mothers by clinging and serving. Some sociologists have suggested that Japanese social interactions in later life are modeled on the mother-child amae.

Ancient Greek
Greek distinguishes several different senses in which the word "love" is used. For example, Ancient Greek has the words philia, eros, agape, storge, and xenia. However, with Greek (as with many other languages), it has been historically difficult to separate the meanings of these words totally. At the same time, the Ancient Greek text of the Bible has examples of the verb agapo having the same meaning as phileo.

Agape (ἀγάπη agápē) means love in modern-day Greek. The term s'agapo means I love you in Greek. The word agapo is the verb I love. It generally refers to a "pure," ideal type of love, rather than the physical attraction suggested by eros. However, there are some examples of agape used to mean the same as eros. It has also been translated as "love of the soul."

Eros (ἔρως érōs) (from the Greek deity Eros) is passionate love, with sensual desire and longing. The Greek word erota means in love. Plato refined his own definition. Although eros is initially felt for a person, with contemplation it becomes an appreciation of the beauty within that person, or even becomes appreciation of beauty itself. Eros helps the soul recall knowledge of beauty and contributes to an understanding of spiritual truth. Lovers and philosophers are all inspired to seek truth by eros. Some translations list it as "love of the body."

Philia (φιλία philía), a dispassionate virtuous love, was a concept developed by Aristotle. It includes loyalty to friends, family, and community, and requires virtue, equality, and familiarity. Philia is motivated by practical reasons; one or both of the parties benefit from the relationship. It can also mean "love of the mind."

Storge (στοργή storgē) is natural affection, like that felt by parents for offspring.

Xenia (ξενία xenía), hospitality, was an extremely important practice in Ancient Greece. It was an almost ritualized friendship formed between a host and his guest, who could previously have been strangers. The host fed and provided quarters for the guest, who was expected to repay only with gratitude. The importance of this can be seen throughout Greek mythology—in particular, Homer's Iliad and Odyssey.

Turkish (Shaman & Islamic)
In Turkish, the word "love" comes up with several meanings. A person can love a god, a person, parents, or family. But that person can "love" just one person from the opposite sex, which they call the word "aşk." Aşk is a feeling for to love, or being "in love" (Aşık), as it still is in Turkish today. The Turks used this word just for their loves in a romantic or sexual sense. If a Turk says that he is in love (Aşık) with somebody, it is not a love that a person can feel for his or her parents; it is just for one person, and it indicates a huge infatuation. The word is also common for Turkic languages, such as Azerbaijani (eşq) and Kazakh (ғашық).

Ancient Roman (Latin)
The Latin language has several different verbs corresponding to the English word "love." Amāre is the basic word for to love, as it still is in Italian today. The Romans used it both in an affectionate sense as well as in a romantic or sexual sense. From this verb come amans—a lover, amator, "professional lover," often with the accessory notion of lechery—and amica, "girlfriend" in the English sense, often as well being applied euphemistically to a prostitute. The corresponding noun is amor (the significance of this term for the Romans is well illustrated in the fact, that the name of the City, Rome—in Latin: Roma—can be viewed as an anagram for amor, which was used as the secret name of the City in wide circles in ancient times),[18] which is also used in the plural form to indicate love affairs or sexual adventures. This same root also produces amicus—"friend"—and amicitia, "friendship" (often based to mutual advantage, and corresponding sometimes more closely to "indebtedness" or "influence"). Cicero wrote a treatise called On Friendship (de Amicitia), which discusses the notion at some length. Ovid wrote a guide to dating called Ars Amatoria (The Art of Love), which addresses, in depth, everything from extramarital affairs to overprotective parents.

Complicating the picture somewhat, Latin sometimes uses amāre where English would simply say to like. This notion, however, is much more generally expressed in Latin by placere or delectāre, which are used more colloquially, the latter used frequently in the love poetry of Catullus. Diligere often has the notion "to be affectionate for," "to esteem," and rarely if ever is used for romantic love. This word would be appropriate to describe the friendship of two men. The corresponding noun diligentia, however, has the meaning of "diligence" or "carefulness," and has little semantic overlap with the verb. Observare is a synonym for diligere; despite the cognate with English, this verb and its corresponding noun, observantia, often denote "esteem" or "affection." Caritas is used in Latin translations of the Christian Bible to mean "charitable love"; this meaning, however, is not found in Classical pagan Roman literature. As it arises from a conflation with a Greek word, there is no corresponding verb.

Religious views

Abrahamic religions


Robert Indiana's 1977 "LOVE sculpture" spelling ahava in Israel
Judaism
In Hebrew, Ahava is the most commonly used term for both interpersonal love and love of God. Judaism employs a wide definition of love, both among people and between man and the Deity. Regarding the former, the Torah states, "Love your neighbor like yourself" (Leviticus 19:18). As for the latter, one is commanded to love God "with all your heart, with all your soul and with all your might" (Deuteronomy 6:5), taken by the Mishnah (a central text of the Jewish oral law) to refer to good deeds, willingness to sacrifice one's life rather than commit certain serious transgressions, willingness to sacrifice all of one's possessions, and being grateful to the Lord despite adversity (tractate Berachoth 9:5). Rabbinic literature differs as to how this love can be developed, e.g., by contemplating divine deeds or witnessing the marvels of nature. As for love between marital partners, this is deemed an essential ingredient to life: "See life with the wife you love" (Ecclesiastes 9:9). The biblical book Song of Solomon is considered a romantically phrased metaphor of love between God and his people, but in its plain reading, reads like a love song. The 20th-century Rabbi Eliyahu Eliezer Dessler is frequently quoted as defining love from the Jewish point of view as "giving without expecting to take" (from his Michtav me-Eliyahu, Vol. 1).

Christianity


Sacred Love Versus Profane Love (1602–03) by Giovanni Baglione
The Christian understanding is that love comes from God. The love of man and woman—eros in Greek—and the unselfish love of others (agape), are often contrasted as "ascending" and "descending" love, respectively, but are ultimately the same thing.[19]

There are several Greek words for "love" that are regularly referred to in Christian circles.

Agape: In the New Testament, agapē is charitable, selfless, altruistic, and unconditional. It is parental love, seen as creating goodness in the world; it is the way God is seen to love humanity, and it is seen as the kind of love that Christians aspire to have for one another.
Phileo: Also used in the New Testament, phileo is a human response to something that is found to be delightful. Also known as "brotherly love."
Two other words for love in the Greek language, eros (sexual love) and storge (child-to-parent love), were never used in the New Testament.
Christians believe that to Love God with all your heart, mind, and strength and Love your neighbor as yourself are the two most important things in life (the greatest commandment of the Jewish Torah, according to Jesus; cf. Gospel of Mark chapter 12, verses 28–34). Saint Augustine summarized this when he wrote "Love God, and do as thou wilt."

The Apostle Paul glorified love as the most important virtue of all. Describing love in the famous poem in 1 Corinthians, he wrote, "Love is patient, love is kind. It does not envy, it does not boast, it is not proud. It is not rude, it is not self-seeking, it is not easily angered, it keeps no record of wrongs. Love does not delight in evil but rejoices with the truth. It always protects, always trusts, always hopes, and always perseveres." (1 Cor. 13:4–7, NIV)

The Apostle John wrote, "For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life. For God did not send his Son into the world to condemn the world, but to save the world through him." (John 3:16–17, NIV) John also wrote, "Dear friends, let us love one another for love comes from God. Everyone who loves has been born of God and knows God. Whoever does not love does not know God, because God is love." (1 John 4:7–8, NIV)

Saint Augustine says that one must be able to decipher the difference between love and lust. Lust, according to Saint Augustine, is an overindulgence, but to love and be loved is what he has sought for his entire life. He even says, “I was in love with love.” Finally, he does fall in love and is loved back, by God. Saint Augustine says the only one who can love you truly and fully is God, because love with a human only allows for flaws such as “jealousy, suspicion, fear, anger, and contention.” According to Saint Augustine, to love God is “to attain the peace which is yours.” (Saint Augustine's Confessions)

Christian theologians see God as the source of love, which is mirrored in humans and their own loving relationships. Influential Christian theologian C.S. Lewis wrote a book called The Four Loves. Benedict XVI wrote his first encyclical on "God is love". He said that a human being, created in the image of God, who is love, is able to practice love; to give himself to God and others (agape) and by receiving and experiencing God's love in contemplation (eros). This life of love, according to him, is the life of the saints such as Teresa of Calcutta and the Blessed Virgin Mary and is the direction Christians take when they believe that God loves them.[19]

In Christianity the practical definition of love is best summarised by St. Thomas Aquinas, who defined love as "to will the good of another," or to desire for another to succeed.[7] This is the explanation of the Christian need to love others, including their enemies. As Thomas Aquinas explains, Christian love is motivated by the need to see others succeed in life, to be good people.

Islam and Arab
In a sense, love does encompass the Islamic view of life as universal brotherhood that applies to all who hold the faith. There are no direct references stating that God is love, but amongst the 99 names of God (Allah), there is the name Al-Wadud, or "the Loving One," which is found in Surah 11:90 as well as Surah 85:14. It refers to God as being "full of loving kindness." All who hold the faith have God's love, but to what degree or effort he has pleased God depends on the individual itself.

Ishq, or divine love, is the emphasis of Sufism. Sufis believe that love is a projection of the essence of God to the universe. God desires to recognize beauty, and as if one looks at a mirror to see oneself, God "looks" at itself within the dynamics of nature. Since everything is a reflection of God, the school of Sufism practices to see the beauty inside the apparently ugly. Sufism is often referred to as the religion of love. God in Sufism is referred to in three main terms, which are the Lover, Loved, and Beloved, with the last of these terms being often seen in Sufi poetry. A common viewpoint of Sufism is that through love, humankind can get back to its inherent purity and grace. The saints of Sufism are infamous for being "drunk" due to their love of God; hence, the constant reference to wine in Sufi poetry and music.

Eastern religions
Buddhism
In Buddhism, Kāma is sensuous, sexual love. It is an obstacle on the path to enlightenment, since it is selfish. Karuṇā is compassion and mercy, which reduces the suffering of others. It is complementary to wisdom and is necessary for enlightenment. Adveṣa and mettā are benevolent love. This love is unconditional and requires considerable self-acceptance. This is quite different from ordinary love, which is usually about attachment and sex and which rarely occurs without self-interest. Instead, in Buddhism it refers to detachment and unselfish interest in others' welfare.

The Bodhisattva ideal in Mahayana Buddhism involves the complete renunciation of oneself in order to take on the burden of a suffering world. The strongest motivation one has in order to take the path of the Bodhisattva is the idea of salvation within unselfish, altruistic love for all sentient beings.

Hinduism


Kama (left) with Rati on a temple wall of Chennakesava Temple, Belur.
In Hinduism, kāma is pleasurable, sexual love, personified by the god Kamadeva. For many Hindu schools, it is the third end (artha) in life. Kamadeva is often pictured holding a bow of sugar cane and an arrow of flowers; he may ride upon a great parrot. He is usually accompanied by his consort Rati and his companion Vasanta, lord of the spring season. Stone images of Kamadeva and Rati can be seen on the door of the Chennakeshava temple at Belur, in Karnataka, India. Maara is another name for kāma.

In contrast to kāma, prema – or prem – refers to elevated love. Karuna is compassion and mercy, which impels one to help reduce the suffering of others. Bhakti is a Sanskrit term, meaning "loving devotion to the supreme God." A person who practices bhakti is called a bhakta. Hindu writers, theologians, and philosophers have distinguished nine forms of bhakti, which can be found in the Bhagavata Purana and works by Tulsidas. The philosophical work Narada Bhakti Sutras, written by an unknown author (presumed to be Narada), distinguishes eleven forms of love.

Friday, August 20, 2010

mathematics


"Maths" and "Math" redirect here. For other uses of "Mathematics" or "Math", see Mathematics (disambiguation) and Math (disambiguation).



Euclid, Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.[1]
Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns,[2][3] formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.[4]

There is debate over whether mathematical objects such as numbers and points exist naturally or are human creations. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions".[5] Albert Einstein, on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[6]

Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics continued to develop, for example in China in 300 BC, in India in AD 100, and in the Muslim world in AD 800, until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that continues to the present day.[7]

Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered.[8]


Etymology

The word "mathematics" comes from the Greek μάθημα (máthēma), which means learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times.[9] Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant the mathematical art.

The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle, and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from the Greek.[10] In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.

History

Main article: History of mathematics


Pythagoras (c.570-c.495 BC) has commonly been given credit for discovering the Pythagorean theorem. Well-known figures in Greek mathematics also include Euclid, Archimedes, and Thales.
The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals,[11] was probably that of numbers: the realization that a collection of two apples and a collection two oranges (for example) have something in common, namely quantity of their members.

In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time – days, seasons, years.[12] Elementary arithmetic (addition, subtraction, multiplication and division) naturally followed.

Since numeracy pre-dated writing, further steps were needed for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data.[citation needed] Numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.[citation needed]



Mayan numerals
The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy.[13] The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.[14]

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."[15]

Inspiration, pure and applied mathematics, and aesthetics

Main article: Mathematical beauty


Sir Isaac Newton (1643-1727), an inventor of infinitesimal calculus.
Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.[16] 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. A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications, e.g. number theory in cryptography. This remarkable fact that even the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics".[17] As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages.[18] Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.

For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.[19] Mathematicians often strive to find proofs of theorems that are particularly elegant, a quest Paul Erdős often referred to as finding proofs from "The Book" in which God had written down his favorite proofs.[20][21] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

Notation, language, and rigor



Leonhard Euler, who created and popularized much of the mathematical notation used today
Main article: Mathematical notation
Most of the mathematical notation in use today was not invented until the 16th century.[22] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.[23] Euler (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax (which to a limited extent varies from author to author and from discipline to discipline) and encodes information that would be difficult to write in any other way.

Mathematical language can also be hard for beginners. Words such as or and only have more precise meanings than in everyday speech. Moreover, words such as open and field have been given specialized mathematical meanings. Mathematical jargon includes technical terms such as homeomorphism and integrable. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".



The infinity symbol ∞ in several typefaces.
Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject.[24] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.[25]

Axioms in traditional thought were "self-evident truths", but that conception is problematic. 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 (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible. 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.[26]

Mathematics as science



Carl Friedrich Gauss, himself known as the "prince of mathematicians",[27] referred to mathematics as "the Queen of the Sciences".
Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".[28] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means (field of) knowledge. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to natural science is of later date. If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. Albert Einstein stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[6]

Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper.[29] However, in the 1930s important work in mathematical logic convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."[30] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself.

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.[31] 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. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method.[citation needed] In his 2002 book A New Kind of Science, Stephen Wolfram argues that computational mathematics deserves to be explored empirically as a scientific field in its own right.

The opinions of mathematicians on this matter are varied. Many mathematicians[who?] feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others[who?] feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics.[citation needed]

Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the Fields Medal,[32][33] established in 1936 and now awarded every 4 years. It is often considered the equivalent of science's Nobel Prizes. The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize, was introduced in 2003. These are awarded for a particular body of work, which may be innovation, or resolution of an outstanding problem in an established field. A famous list of 23 such open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Solution of each of these problems carries a $1 million reward, and only one (the Riemann hypothesis) is duplicated in Hilbert's problems.

Fields of mathematics



An abacus, a simple calculating tool used since ancient times.
Mathematics can, broadly speaking, be subdivided 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), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.

Quantity
The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem. Number theory also holds two problems widely considered to be unsolved: the twin prime conjecture and Goldbach's conjecture.

As the number system is further developed, the integers are recognized as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchy of numbers that goes on to include quarternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of "infinity". Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.


Natural numbers Integers Rational numbers Real numbers Complex numbers
Structure
Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Moreover, it frequently happens that different such structured sets (or structures) exhibit similar properties, which makes it possible, by a further step of abstraction, to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups, rings, fields and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of abstract algebra. By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory, which involves field theory and group theory. Another example of an algebraic theory is linear algebra, which is the general study of vector spaces, whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of geometry and algebra have very strong interactions in modern mathematics. Combinatorics studies ways of enumerating the number of objects that fit a given structure.


Combinatorics Number theory Group theory Graph theory Order theory
Space
The study of space originates with geometry – in particular, Euclidean geometry. Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known Pythagorean theorem. 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 and calculus on manifolds, in particular, vector and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of polynomial 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 have been the greatest growth area in 20th century mathematics; it includes point-set topology, set-theoretic topology, algebraic topology and differential topology. In particular, instances of modern day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory. Topology also includes the now solved Poincaré conjecture and the controversial four color theorem, whose only proof, by computer, has never been verified by a human.


Geometry Trigonometry Differential geometry Topology Fractal geometry Measure Theory
Change
Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. 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.


Calculus Vector calculus Differential equations Dynamical systems Chaos theory Complex analysis
Foundations and philosophy
In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. Category theory, which deals in an abstract way with mathematical structures and relationships between them, is still in development. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930.[34] Some disagreement about the foundations of mathematics continues to present day. The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory and the Brouwer-Hilbert controversy.

Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. As such, it is home to Gödel's incompleteness theorems which (informally) imply that any formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proven are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). Whatever finite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a complete axiomatization of full number theory.[citation needed] Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science.


Mathematical logic Set theory Category theory
Theoretical computer science
Theoretical computer science includes computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most powerful known model – the Turing machine. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with rapid advance of computer hardware. A famous problem is the "P=NP?" problem, one of the Millennium Prize Problems.[35] Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as compression and entropy.


Theory of computation Cryptography
Applied mathematics
Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the sciences, business, and other areas.

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments; the design of a statistical sample or experiment specifies the analysis of the data (before the data be available). When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians "make sense of the data" using the art of modelling and the theory of inference – with model selection and estimation; the estimated models and consequential predictions should be tested on new data.[36]

Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using ideas of functional analysis and techniques of approximation theory; numerical analysis includes the study of approximation and discretization broadly with special concern for rounding errors. Other areas of computational mathematics include computer algebra and symbolic computation.


Mathematical physics

Fluid dynamics

Numerical analysis

Optimization

Probability theory

Statistics

Financial mathematics

Game theory

Mathematical biology

Mathematical chemistry

Mathematical economics

Control theory