A Midsummer Night's Dream
A Midsummer Night's Dream is a comedy written by William Shakespeare in 1595/96. It portrays the events surrounding the marriage of the Duke of Athens, to Hippolyta; these include the adventures of four young Athenian lovers and a group of six amateur actors who are controlled and manipulated by the fairies who inhabit the forest in which most of the play is set. The play is one of Shakespeare's most popular works for the stage and is performed across the world; the play consists of four interconnecting plots, connected by a celebration of the wedding of Duke Theseus of Athens and the Amazon queen, which are set in the woodland and in the realm of Fairyland, under the light of the moon. The play opens with Hermia, in love with Lysander, resistant to her father Egeus's demand that she wed Demetrius, whom he has arranged for her to marry. Helena, Hermia's best friend, pines unrequitedly for Demetrius, who broke up with her to be with Hermia. Enraged, Egeus invokes an ancient Athenian law before Duke Theseus, whereby a daughter needs to marry a suitor chosen by her father, or else face death.
Theseus offers her another choice: lifelong chastity as a nun worshipping the goddess Artemis. Peter Quince and his fellow players Nick Bottom, Francis Flute, Robin Starveling, Tom Snout and Snug plan to put on a play for the wedding of the Duke and the Queen, "the most lamentable comedy and most cruel death of Pyramus and Thisbe". Quince bestows them on the players. Nick Bottom, playing the main role of Pyramus, is over-enthusiastic and wants to dominate others by suggesting himself for the characters of Thisbe, the Lion, Pyramus at the same time, he would rather be a tyrant and recites some lines of Ercles. Bottom is told by Quince that he would do the Lion so as to frighten the duchess and ladies enough for the Duke and Lords to have the players hanged. Snug remarks that he needs the Lion's part because he is "slow of study". Quince ends the meeting with "at the Duke's oak we meet". In a parallel plot line, king of the fairies, Titania, his queen, have come to the forest outside Athens. Titania tells Oberon that she plans to stay there until she has attended Theseus and Hippolyta's wedding.
Oberon and Titania are estranged because Titania refuses to give her Indian changeling to Oberon for use as his "knight" or "henchman", since the child's mother was one of Titania's worshippers. Oberon seeks to punish Titania's disobedience, he calls upon Robin "Puck" Goodfellow, his "shrewd and knavish sprite", to help him concoct a magical juice derived from a flower called "love-in-idleness", which turns from white to purple when struck by Cupid's arrow. When the concoction is applied to the eyelids of a sleeping person, that person, upon waking, falls in love with the first living thing they perceive, he instructs Puck to retrieve the flower with the hope that he might make Titania fall in love with an animal of the forest and thereby shame her into giving up the little Indian boy. He says, "And ere I take this charm from off her sight,/As I can take it with another herb,/I'll make her render up her page to me." Hermia and Lysander have escaped to a forest in hopes of running away from Theseus.
Helena, desperate to reclaim Demetrius's love, tells Demetrius about the plan and he follows them in hopes of finding Hermia. Helena continually makes advances towards Demetrius, promising to love him more than Hermia. However, he rebuffs her with cruel insults against her. Observing this, Oberon orders Puck to spread some of the magical juice from the flower on the eyelids of the young Athenian man. Instead, Puck mistakes Lysander for Demetrius, not having seen either before, administers the juice to the sleeping Lysander. Helena, coming across him, wakes him while attempting to determine whether he is asleep. Upon this happening, Lysander falls in love with Helena. Helena, runs away with Lysander following her; when Hermia wakes up, she sees that Lysander goes out in the woods to find him. Oberon sees Demetrius still following Hermia, who thinks Demetrius killed Lysander, is enraged; when Demetrius goes to sleep, Oberon sends Puck to get Helena. Upon waking up, he sees Helena. Now, both men are in love with Helena.
However, she is convinced that her two suitors are mocking her. Hermia finds Lysander and asks why he left her, but Lysander claims and denies he never loved Hermia, but Helena. Hermia accuses Helena of stealing Lysander away from her while Helena believes Hermia joined the two men in mocking her. Hermia tries to attack Helena. Lysander, tired of Hermia's presence, tells her to leave. Lysander and Demetrius decide to seek a place to duel to prove; the two girls go their own separate ways, Helena hoping to reach Athens and Hermia chasing after the men to make sure Lysander doesn't get hurt or killed. Oberon orders Puck to keep Lysander and Demetrius from catching up with one another and to remove the charm from Lysander so Lysander can return to love Hermia, while Demetrius continues to love Helena. Meanwhile and his band of six labourers have arranged to perform their play about Pyramus and Thisbe for Theseus' wedding and venture into the forest, near Titania's bower, for their rehearsal. Bottom is spotted by Puck.
When Bottom returns for his next lines, the other workmen run screaming in terror: They claim that they are haunted, much to Bot
Philosophy is the study of general and fundamental questions about existence, values, reason and language. Such questions are posed as problems to be studied or resolved; the term was coined by Pythagoras. Philosophical methods include questioning, critical discussion, rational argument, systematic presentation. Classic philosophical questions include: Is it possible to know anything and to prove it? What is most real? Philosophers pose more practical and concrete questions such as: Is there a best way to live? Is it better to be just or unjust? Do humans have free will? "philosophy" encompassed any body of knowledge. From the time of Ancient Greek philosopher Aristotle to the 19th century, "natural philosophy" encompassed astronomy and physics. For example, Newton's 1687 Mathematical Principles of Natural Philosophy became classified as a book of physics. In the 19th century, the growth of modern research universities led academic philosophy and other disciplines to professionalize and specialize.
In the modern era, some investigations that were traditionally part of philosophy became separate academic disciplines, including psychology, sociology and economics. Other investigations related to art, politics, or other pursuits remained part of philosophy. For example, is beauty objective or subjective? Are there many scientific methods or just one? Is political utopia a hopeful dream or hopeless fantasy? Major sub-fields of academic philosophy include metaphysics, ethics, political philosophy and philosophy of science. Traditionally, the term "philosophy" referred to any body of knowledge. In this sense, philosophy is related to religion, natural science and politics. Newton's 1687 Mathematical Principles of Natural Philosophy is classified in the 2000s as a book of physics. In the first part of the first book of his Academics, Cicero introduced the division of philosophy into logic and ethics. Metaphysical philosophy was the study of existence, God, logic and other abstract objects; this division has changed.
Natural philosophy has split into the various natural sciences astronomy, chemistry and cosmology. Moral philosophy still includes value theory. Metaphysical philosophy has birthed formal sciences such as logic and philosophy of science, but still includes epistemology and others. Many philosophical debates that began in ancient times are still debated today. Colin McGinn and others claim. Chalmers and others, by contrast, see progress in philosophy similar to that in science, while Talbot Brewer argued that "progress" is the wrong standard by which to judge philosophical activity. In one general sense, philosophy is associated with wisdom, intellectual culture and a search for knowledge. In that sense, all cultures and literate societies ask philosophical questions such as "how are we to live" and "what is the nature of reality". A broad and impartial conception of philosophy finds a reasoned inquiry into such matters as reality and life in all world civilizations. Western philosophy is the philosophical tradition of the Western world and dates to Pre-Socratic thinkers who were active in Ancient Greece in the 6th century BCE such as Thales and Pythagoras who practiced a "love of wisdom" and were termed physiologoi.
Socrates was a influential philosopher, who insisted that he possessed no wisdom but was a pursuer of wisdom. Western philosophy can be divided into three eras: Ancient, Medieval philosophy, Modern philosophy; the Ancient era was dominated by Greek philosophical schools which arose out of the various pupils of Socrates, such as Plato, who founded the Platonic Academy and his student Aristotle, founding the Peripatetic school, who were both influential in Western tradition. Other traditions include Cynicism, Greek Skepticism and Epicureanism. Important topics covered by the Greeks included metaphysics, the nature of the well-lived life, the possibility of knowledge and the nature of reason. With the rise of the Roman empire, Greek philosophy was increasingly discussed in Latin by Romans such as Cicero and Seneca. Medieval philosophy is the period following the fall of the Western Roman Empire and was dominated by the ris
The Epimenides paradox reveals a problem with self-reference in logic. It is named after the Cretan philosopher Epimenides of Knossos, credited with the original statement. A typical description of the problem is given in the book Gödel, Bach, by Douglas Hofstadter: Epimenides was a Cretan who made one immortal statement: "All Cretans are liars."A paradox of self-reference arises when one considers whether it is possible for Epimenides to have spoken the truth. Thomas Fowler states the paradox as follows: "Epimenides the Cretan says,'that all the Cretans are liars,' but Epimenides is himself a Cretan, but if he is a liar, what he says is untrue, the Cretans are veracious. Thus we may go on alternately proving that Epimenides and the Cretans are truthful and untruthful."The Epimenides paradox in this form, can be solved. There are two options: it is either true or false. First, assume that it is true, but Epimenides, being a Cretan, would be a liar, making the assumption that liars only make false statements, the statement is false.
So, assuming the statement is true leads us to conclude that the statement is false. This is a contradiction, so the option of the statement being true is not possible; this leaves the second option:. If we assume the statement is false and that Epimenides is lying about all Cretans being liars there must exist at least one Cretan, honest; this does not lead to a contradiction. This means that Epimenides can say the false statement that all Cretans are liars while knowing at least one honest Cretan and lying about this particular Cretan. Hence, from the assumption that the statement is false, it does not follow that the statement is true. So we can avoid a paradox as seeing the statement "all Cretans are liars" as a false statement, made by a lying Cretan, Epimenides; the mistake made by Thomas Fowler above is to think that the negation of "all Cretans are liars" is "all Cretans are honest" when in fact the negation is "there exists a Cretan, honest", or "not all Cretans are liars". The Epimenides paradox can be modified as to not allow the kind of solution described above, as it was in the first paradox of Eubulides but instead leading to a non-avoidable self-contradiction.
Paradoxical versions of the Epimenides problem are related to a class of more difficult logical problems, including the liar paradox, Socratic paradox, the Burali-Forti paradox, all of which have self-reference in common with Epimenides. Indeed, the Epimenides paradox is classified as a variation on the liar paradox, sometimes the two are not distinguished; the study of self-reference led to important developments in logic and mathematics in the twentieth century. In other words, it is not a paradox once one realizes "All Cretans are liars" being untrue only means "Not all Cretans are liars" instead of the assumption that "All Cretans are honest". Better put, for "All Cretans are liars" to be a true statement, it does not mean that all Cretans must lie all the time. In fact, Cretans could tell the truth quite but still, all be liars in the sense that liars are people prone to deception for dishonest gain. Considering that “All Cretans are liars” has been seen as a paradox only since the 19th century, this seems to resolve the alleged paradox.
Of course, if ‘all Cretans are continuous liars’ is true asking a Cretan if they are honest would always elicit the dishonest answer ‘yes’. So arguably the original proposition is not so much paradoxical as invalid. A contextual reading of the contradiction may provide an answer to the paradox; the original phrase, "The Cretans, always liars, evil beasts, idle bellies!" Asserts not an intrinsic paradox, but rather an opinion of the Cretans from Epimenides. A stereotyping of his people not intended to be an absolute statement about the people as a whole. Rather it is a claim made about their position regarding their religious beliefs and socio-cultural attitudes. Within the context of his poem the phrase is specific to a certain belief, a context that Callimachus repeats in his poem regarding Zeus. Further, a more poignant answer to the paradox is that to be a liar is to state falsehoods, nothing in the statement asserts everything said is false, but rather they're "always" lying; this is not an absolute statement of fact and thus we cannot conclude there's a true contradiction made by Epimenides with this statement.
Epimenides was a 6th-century BC philosopher and religious prophet who, against the general sentiment of Crete, proposed that Zeus was immortal, as in the following poem: They fashioned a tomb for thee, O holy and high oneThe Cretans, always liars, evil beasts, idle bellies! But thou art not dead: thou livest and abidest forever,For in thee we live and move and have our being. Denying the immortality of Zeus was the lie of the Cretans; the phrase "Cretans, always liars" was quoted by the poet Callimachus in his Hymn to Zeus, with the same theological intent as Epimenides: O Zeus, some say that thou wert born on the hills of Ida. -- “Cretans are liars.” Yea, a tomb, O Lord, for thee the Cretans builded. The logical inconsistency of a Cretan asserting all Cretans are always liars may not have occurred to Epimenides, nor to Callimachus, who both used the phrase to emphasize their point, without irony meaning that all Cretans lie but not
Italo Calvino was an Italian journalist and writer of short stories and novels. His best known works include the Our Ancestors trilogy, the Cosmicomics collection of short stories, the novels Invisible Cities and If on a winter's night a traveler. Admired in Britain and the United States, he was the most-translated contemporary Italian writer at the time of his death. Italo Calvino was born in Santiago de las Vegas, a suburb of Havana, Cuba, in 1923, his father, was a tropical agronomist and botanist who taught agriculture and floriculture. Born 47 years earlier in Sanremo, Mario Calvino had emigrated to Mexico in 1909 where he took up an important position with the Ministry of Agriculture. In an autobiographical essay, Italo Calvino explained that his father "had been in his youth an anarchist, a follower of Kropotkin and a Socialist Reformist". In 1917, Mario left for Cuba to conduct scientific experiments, after living through the Mexican Revolution. Calvino's mother, Giuliana Luigia Evelina "Eva" Mameli, was a university professor.
A native of Sassari in Sardinia and 11 years younger than her husband, she married while still a junior lecturer at Pavia University. Born into a secular family, Eva was a pacifist educated in the "religion of civic duty and science". Eva gave Calvino his unusual first name to remind him of his Italian heritage, although since he wound up growing up in Italy after all, Calvino thought his name sounded "belligerently nationalist". Calvino described his parents as being "very different in personality from one another", suggesting deeper tensions behind a comfortable, albeit strict, middle-class upbringing devoid of conflict; as an adolescent, he found it hard relating to poverty and the working-class, was "ill at ease" with his parents' openness to the laborers who filed into his father's study on Saturdays to receive their weekly paycheck. In 1925, less than two years after Calvino's birth, the family returned to Italy and settled permanently in Sanremo on the Ligurian coast. Calvino's brother Floriano, who became a distinguished geologist, was born in 1927.
The family divided their time between the Villa Meridiana, an experimental floriculture station which served as their home, Mario's ancestral land at San Giovanni Battista. On this small working farm set in the hills behind Sanremo, Mario pioneered in the cultivation of exotic fruits such as avocado and grapefruit obtaining an entry in the Dizionario biografico degli italiani for his achievements; the vast forests and luxuriant fauna omnipresent in Calvino's early fiction such as The Baron in the Trees derives from this "legacy". In an interview, Calvino stated that "San Remo continues to pop out in my books, in the most diverse pieces of writing." He and Floriano would climb the tree-rich estate and perch for hours on the branches reading their favorite adventure stories. Less salubrious aspects of this "paternal legacy" are described in The Road to San Giovanni, Calvino's memoir of his father in which he exposes their inability to communicate: "Talking to each other was difficult. Both verbose by nature, possessed of an ocean of words, in each other's presence we became mute, would walk in silence side by side along the road to San Giovanni."
A fan of Rudyard Kipling's The Jungle Book as a child, Calvino felt that his early interest in stories made him the "black sheep" of a family that held literature in less esteem than the sciences. Fascinated by American movies and cartoons, he was attracted to drawing and theatre. On a darker note, Calvino recalled that his earliest memory was of a Marxist professor, brutally assaulted by Benito Mussolini's Blackshirts: "I remember that we were at dinner when the old professor came in with his face beaten up and bleeding, his bowtie all torn, asking for help."Other legacies include the parents' beliefs in Freemasonry, Republicanism with elements of Anarchism and Marxism. Austere freethinkers with an intense hatred of the ruling National Fascist Party and Mario refused to give their sons any education in the Catholic Faith or any other religion. Italo attended the English nursery school St George's College, followed by a Protestant elementary private school run by Waldensians, his secondary schooling, with a classical lyceum curriculum, was completed at the state-run Liceo Gian Domenico Cassini where, at his parents' request, he was exempted from religion classes but asked to justify his anti-conformism to teachers and fellow pupils.
In his mature years, Calvino described the experience as having made him "tolerant of others' opinions in the field of religion, remembering how irksome it was to hear myself mocked because I did not follow the majority's beliefs". In 1938, Eugenio Scalfari, who went on to found the weekly magazine L'Espresso and La Repubblica, a major Italian newspaper, came from Civitavecchia to join the same class though a year younger, they shared the same desk; the two teenagers formed a lasting friendship, Calvino attributing his political awakening to their university discussions. Seated together "on a huge flat stone in the middle of a stream near our land", he and Scalfari founded the MUL. Eva managed to delay her son's enrolment in the Party's armed scouts, the Balilla Moschettieri, arranged that he be excused, as a non-Catholic, from performing devotional acts in Church, but on, as a compulsory member, he could not avoid the assemblies and parades of the Avanguardisti, was forced to participate in the Italian invasion of the French Riviera in June 1940.
In 1941, Calvino enrolled at the University of Turin, choosing th
Deoxyribonucleic acid is a molecule composed of two chains that coil around each other to form a double helix carrying the genetic instructions used in the growth, development and reproduction of all known organisms and many viruses. DNA and ribonucleic acid are nucleic acids; the two DNA strands are known as polynucleotides as they are composed of simpler monomeric units called nucleotides. Each nucleotide is composed of one of four nitrogen-containing nucleobases, a sugar called deoxyribose, a phosphate group; the nucleotides are joined to one another in a chain by covalent bonds between the sugar of one nucleotide and the phosphate of the next, resulting in an alternating sugar-phosphate backbone. The nitrogenous bases of the two separate polynucleotide strands are bound together, according to base pairing rules, with hydrogen bonds to make double-stranded DNA; the complementary nitrogenous bases are divided into two groups and purines. In DNA, the pyrimidines are cytosine. Both strands of double-stranded DNA store the same biological information.
This information is replicated as and when the two strands separate. A large part of DNA is non-coding, meaning that these sections do not serve as patterns for protein sequences; the two strands of DNA are thus antiparallel. Attached to each sugar is one of four types of nucleobases, it is the sequence of these four nucleobases along the backbone. RNA strands are created using DNA strands as a template in a process called transcription. Under the genetic code, these RNA strands specify the sequence of amino acids within proteins in a process called translation. Within eukaryotic cells, DNA is organized into long structures called chromosomes. Before typical cell division, these chromosomes are duplicated in the process of DNA replication, providing a complete set of chromosomes for each daughter cell. Eukaryotic organisms store most of their DNA inside the cell nucleus as nuclear DNA, some in the mitochondria as mitochondrial DNA, or in chloroplasts as chloroplast DNA. In contrast, prokaryotes store their DNA only in circular chromosomes.
Within eukaryotic chromosomes, chromatin proteins, such as histones and organize DNA. These compacting structures guide the interactions between DNA and other proteins, helping control which parts of the DNA are transcribed. DNA was first isolated by Friedrich Miescher in 1869, its molecular structure was first identified by Francis Crick and James Watson at the Cavendish Laboratory within the University of Cambridge in 1953, whose model-building efforts were guided by X-ray diffraction data acquired by Raymond Gosling, a post-graduate student of Rosalind Franklin. DNA is used by researchers as a molecular tool to explore physical laws and theories, such as the ergodic theorem and the theory of elasticity; the unique material properties of DNA have made it an attractive molecule for material scientists and engineers interested in micro- and nano-fabrication. Among notable advances in this field are DNA origami and DNA-based hybrid materials. DNA is a long polymer made from repeating units called nucleotides.
The structure of DNA is dynamic along its length, being capable of coiling into tight loops and other shapes. In all species it is composed of two helical chains, bound to each other by hydrogen bonds. Both chains are coiled around the same axis, have the same pitch of 34 angstroms; the pair of chains has a radius of 10 angstroms. According to another study, when measured in a different solution, the DNA chain measured 22 to 26 angstroms wide, one nucleotide unit measured 3.3 Å long. Although each individual nucleotide is small, a DNA polymer can be large and contain hundreds of millions, such as in chromosome 1. Chromosome 1 is the largest human chromosome with 220 million base pairs, would be 85 mm long if straightened. DNA does not exist as a single strand, but instead as a pair of strands that are held together; these two long strands coil in the shape of a double helix. The nucleotide contains both a segment of the backbone of a nucleobase. A nucleobase linked to a sugar is called a nucleoside, a base linked to a sugar and to one or more phosphate groups is called a nucleotide.
A biopolymer comprising multiple linked nucleotides is called a polynucleotide. The backbone of the DNA strand is made from alternating sugar residues; the sugar in DNA is 2-deoxyribose, a pentose sugar. The sugars are joined together by phosphate groups that form phosphodiester bonds between the third and fifth carbon atoms of adjacent sugar rings; these are known as the 3′-end, 5′-end carbons, the prime symbol being used to distinguish these carbon atoms from those of the base to which the deoxyribose forms a glycosidic bond. When imagining DNA, each phosphoryl is considered to "belong" to the nucleotide whose 5′ carbon forms a bond therewith. Any DNA strand therefore has one end at which there is a phosphoryl attached to the 5′ carbon of a ribose and another end a
Recursion occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic; the most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this defines an infinite number of instances, it is done in such a way that no loop or infinite chain of references can occur. In mathematics and computer science, a class of objects or methods exhibit recursive behavior when they can be defined by two properties: A simple base case —a terminating scenario that does not use recursion to produce an answer A set of rules that reduce all other cases toward the base caseFor example, the following is a recursive definition of a person's ancestors: One's parents are one's ancestors; the ancestors of one's ancestors are one's ancestors. The Fibonacci sequence is a classic example of recursion: Fib = 0 as base case 1, Fib = 1 as base case 2, For all integers n > 1, Fib:= Fib + Fib.
Many mathematical axioms are based upon recursive rules. For example, the formal definition of the natural numbers by the Peano axioms can be described as: 0 is a natural number, each natural number has a successor, a natural number. By this base case and recursive rule, one can generate the set of all natural numbers. Recursively defined mathematical objects include functions and fractals. There are various more tongue-in-cheek "definitions" of recursion. Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be'recursive'. To understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps based on a set of rules; the running of a procedure involves following the rules and performing the steps. An analogy: a procedure is like a written recipe. Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure.
For instance, a recipe might refer to cooking vegetables, another procedure that in turn requires heating water, so forth. However, a recursive procedure is where one of its steps calls for a new instance of the same procedure, like a sourdough recipe calling for some dough left over from the last time the same recipe was made; this creates the possibility of an endless loop. If properly defined, a recursive procedure is not easy for humans to perform, as it requires distinguishing the new from the old invocation of the procedure. For this reason recursive definitions are rare in everyday situations. An example could be the following procedure to find a way through a maze. Proceed forward until reaching either an exit or a branching point. If the point reached is an exit, terminate. Otherwise try each branch in turn, using the procedure recursively. Whether this defines a terminating procedure depends on the nature of the maze: it must not allow loops. In any case, executing the procedure requires recording all explored branching points, which of their branches have been exhaustively tried.
Linguist Noam Chomsky among many others has argued that the lack of an upper bound on the number of grammatical sentences in a language, the lack of an upper bound on grammatical sentence length, can be explained as the consequence of recursion in natural language. This can be understood in terms of a recursive definition of a syntactic category, such as a sentence. A sentence can have a structure in which what follows the verb is another sentence: Dorothy thinks witches are dangerous, in which the sentence witches are dangerous occurs in the larger one. So a sentence can be defined recursively as something with a structure that includes a noun phrase, a verb, optionally another sentence; this is just a special case of the mathematical definition of recursion. This provides a way of understanding the creativity of language—the unbounded number of grammatical sentences—because it predicts that sentences can be of arbitrary length: Dorothy thinks that Toto suspects that Tin Man said that.... There are many structures apart from sentences that can be defined recursively, therefore many ways in which a sentence can embed instances of one
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to