Mathematics
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
Philosophy
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
Element (mathematics)
In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set. Writing A = means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example, are subsets of A. Sets can themselves be elements. For example, consider the set B =; the elements of B are not 1, 2, 3, 4. Rather, there are only three elements of B, namely the numbers 1 and 2, the set; the elements of a set can be anything. For example, C =, is the set whose elements are the colors red and blue; the relation "is an element of" called set membership, is denoted by the symbol " ∈ ". Writing x ∈ A means that "x is an element of A". Equivalent expressions are "x is a member of A", "x belongs to A", "x is in A" and "x lies in A"; the expressions "A includes x" and "A contains x" are used to mean set membership, however some authors use them to mean instead "x is a subset of A". Logician George Boolos urged that "contains" be used for membership only and "includes" for the subset relation only.
For the relation ∈, the converse relation ∈T may be written A ∋ x, meaning "A contains x". The negation of set membership is denoted by the symbol "∉". Writing x ∉ A means that "x is not an element of A"; the symbol ∈ was first used by Giuseppe Peano 1889 in his work Arithmetices principia, nova methodo exposita. Here he wrote on page X: Signum ∈ significat est. Ita a ∈ b legitur a est quoddam b. So a ∈ b is read; every relation R: U → V is subject to two involutions: complementation R → R ¯ and conversion RT: V → U. The relation ∈ has for its domain a universal set U, has the power set P for its codomain or range; the complementary relation ∈ ¯ = ∉ expresses the opposite of ∈. An element x ∈ U may have x ∉ A, in which case x ∈ U \ A, the complement of A in U; the converse relation ∈ T = ∋ swaps the domain and range with ∈. For any A in P, A ∋ x is true when x ∈ A; the number of elements in a particular set is a property known as cardinality. In the above examples the cardinality of the set A is 4, while the cardinality of either of the sets B and C is 3.
An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets. An example of an infinite set is the set of positive integers. Using the sets defined above, namely A =, B = and C =: 2 ∈ A ∈ B 3,4 ∉ B is a member of B Yellow ∉ C The cardinality of D = is finite and equal to 5; the cardinality of P = is infinite. Halmos, Paul R. Naive Set Theory, Undergraduate Texts in Mathematics, NY: Springer-Verlag, ISBN 0-387-90092-6 - "Naive" means that it is not axiomatized, not that it is silly or easy. Jech, Thomas, "Set Theory", Stanford Encyclopedia of Philosophy Suppes, Axiomatic Set Theory, NY: Dover Publications, Inc. ISBN 0-486-61630-4 - Both the notion of set, membership or element-hood, the axiom of extension, the axiom of separation, the union axiom are needed for a more thorough understanding of "set element". Weisstein, Eric W. "Element". MathWorld
Probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms; these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion. Although it is not possible to predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.
As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics; the mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, by Pierre de Fermat and Blaise Pascal in the seventeenth century. Christiaan Huygens published a book on the subject in 1657 and in the 19th century, Pierre Laplace completed what is today considered the classic interpretation. Probability theory considered discrete events, its methods were combinatorial. Analytical considerations compelled the incorporation of continuous variables into the theory; this culminated on foundations laid by Andrey Nikolaevich Kolmogorov.
Kolmogorov combined the notion of sample space, introduced by Richard von Mises, measure theory and presented his axiom system for probability theory in 1933. This became the undisputed axiomatic basis for modern probability theory. Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately; the measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, more. Consider an experiment that can produce a number of outcomes; the set of all outcomes is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the power set of the sample space of die rolls; these collections are called events. In this case, is the event that the die falls on some odd number.
If the results that occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every "event" a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one. To qualify as a probability distribution, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events, the probability that any of these events occurs is given by the sum of the probabilities of the events; the probability that any one of the events, or will occur is 5/6. This is the same as saying that the probability of event is 5/6; this event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, the event has a probability of 1, that is, absolute certainty; when doing calculations using the outcomes of an experiment, it is necessary that all those elementary events have a number assigned to them. This is done using a random variable.
A random variable is a function that assigns to each elementary event in the sample space a real number. This function is denoted by a capital letter. In the case of a die, the assignment of a number to a certain elementary events can be done using the identity function; this does not always work. For example, when flipping a coin the two possible outcomes are "heads" and "tails". In this example, the random variable X could assign to the outcome "heads" the number "0" and to the outcome "tails" the number "1". Discrete probability theory deals with events. Examples: Throwing dice, experiments with decks of cards, random walk, tossing coins Classical definition: Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see Classical definition of probability. For example, if the event is "occurrence of an number when a die is
Record (computer science)
In computer science, a record is a basic data structure. Records in a database or spreadsheet are called "rows". A record is a collection of fields of different data types in fixed number and sequence; the fields of a record may be called members in object-oriented programming. For example, a date could be stored as a record containing a numeric year field, a month field represented as a string, a numeric day-of-month field. A personnel record might contain a name, a salary, a rank. A Circle record might contain a center and a radius—in this instance, the center itself might be represented as a point record containing x and y coordinates. Records are distinguished from arrays by the fact that their number of fields is fixed, each field has a name, that each field may have a different type. A record type is a data type that describes such variables. Most modern computer languages allow the programmer to define new record types; the definition includes specifying the data type of each field and an identifier by which it can be accessed.
In type theory, product types are preferred due to their simplicity, but proper record types are studied in languages such as System F-sub. Since type-theoretical records may contain first-class function-typed fields in addition to data, they can express many features of object-oriented programming. Records can exist in any storage medium, including main memory and mass storage devices such as magnetic tapes or hard disks. Records are a fundamental component of most data structures linked data structures. Many computer files are organized as arrays of logical records grouped into larger physical records or blocks for efficiency; the parameters of a function or procedure can be viewed as the fields of a record variable. In the call stack, used to implement procedure calls, each entry is an activation record or call frame, containing the procedure parameters and local variables, the return address, other internal fields. An object in object-oriented language is a record that contains procedures specialized to handle that record.
Indeed, in most object-oriented languages, records are just special cases of objects, are known as plain old data structures, to contrast with objects that use OO features. A record can be viewed as the computer analog of a mathematical tuple, although a tuple may or may not be considered a record, vice versa, depending on conventions and the specific programming language. In the same vein, a record type can be viewed as the computer language analog of the Cartesian product of two or more mathematical sets, or the implementation of an abstract product type in a specific language. A record may have zero or more keys. A key is a set of fields in the record that serves as an identifier. A unique key is called the primary key, or the record key. For example an employee file might contain employee number, name and salary; the employee number would be the primary key. Depending on the storage medium and file organization the employee number might be indexed—that is stored in a separate file to make lookup faster.
The department code may not be unique. If it is not indexed the entire employee file would have to be scanned to produce a listing of all employees in a specific department; the salary field would not be considered usable as a key. Indexing is one factor considered; the concept of record can be traced to various types of tables and ledgers used in accounting since remote times. The modern notion of records in computer science, with fields of well-defined type and size, was implicit in 19th century mechanical calculators, such as Babbage's Analytical Engine; the original machine-readable medium used for data was punch card used for records in the 1890 United States Census: each punch card was a single record. Compare the journal entry from 1880 and the punch card from 1895. Records were well established in the first half of the 20th century, when most data processing was done using punched cards; each record of a data file would be recorded in one punched card, with specific columns assigned to specific fields.
A record was the smallest unit that could be read in from external storage. Most machine language implementations and early assembly languages did not have special syntax for records, but the concept was available through the use of index registers, indirect addressing, self-modifying code; some early computers, such as the IBM 1620, had hardware support for delimiting records and fields, special instructions for copying such records. The concept of records and fields was central in some early file sorting and tabulating utilities, such as IBM's Report Program Generator. COBOL was the first widespread programming language to support record types, its record definition facilities were quite sophisticated at the time; the language allows for the definition of nested records with alphanumeric and fractional fields of arbitrary size and precision, as well as fields that automatically format any value assigned to them (e.g. insertion of currency
Medieval Latin
Medieval Latin was the form of Latin used in Roman Catholic Western Europe during the Middle Ages. In this region it served as the primary written language, though local languages were written to varying degrees. Latin functioned as the main medium of scholarly exchange, as the liturgical language of the Church, as the working language of science, literature and administration. Medieval Latin represented, in essence, a continuation of Classical Latin and Late Latin, with enhancements for new concepts as well as for the increasing integration of Christianity. Despite some meaningful differences from Classical Latin, Medieval writers did not regard it as a fundamentally different language. There is no real consensus on the exact boundary where Late Latin Medieval Latin begins; some scholarly surveys begin with the rise of early Ecclesiastical Latin in the middle of the 4th century, others around 500, still others with the replacement of written Late Latin by written Romance languages starting around the year 900.
The terms Medieval Latin and Ecclesiastical Latin are used synonymously, though some scholars draw distinctions. Ecclesiastical Latin refers to the form, used by the Roman Catholic Church, whereas Medieval Latin refers more broadly to all of the forms of Latin used in the Middle Ages; the Romance languages spoken in the Middle Ages were referred to as Latin, since the Romance languages were all descended from Classical, or Roman, Latin itself. Medieval Latin had an enlarged vocabulary, which borrowed from other sources, it was influenced by the language of the Vulgate, which contained many peculiarities alien to Classical Latin that resulted from a more or less direct translation from Greek and Hebrew. Greek provided much of the technical vocabulary of Christianity; the various Germanic languages spoken by the Germanic tribes, who invaded southern Europe, were major sources of new words. Germanic leaders became the rulers of parts of the Roman Empire that they conquered, words from their languages were imported into the vocabulary of law.
Other more ordinary words were replaced by coinages from Vulgar Latin or Germanic sources because the classical words had fallen into disuse. Latin was spread to areas such as Ireland and Germany, where Romance languages were not spoken, which had never known Roman rule. Works written in those lands where Latin was a learned language, having no relation to the local vernacular influenced the vocabulary and syntax of medieval Latin. Since subjects like science and philosophy, including Argumentation theory and Ethics, were communicated in Latin, the Latin vocabulary that developed for them became the source of a great many technical words in modern languages. English words like abstract, communicate, matter and their cognates in other European languages have the meanings given to them in medieval Latin; the influence of Vulgar Latin was apparent in the syntax of some medieval Latin writers, although Classical Latin continued to be held in high esteem and studied as models for literary compositions.
The high point of the development of medieval Latin as a literary language came with the Carolingian renaissance, a rebirth of learning kindled under the patronage of Charlemagne, king of the Franks. Alcuin was an important writer in his own right. Although it was developing into the Romance languages, Latin itself remained conservative, as it was no longer a native language and there were many ancient and medieval grammar books to give one standard form. On the other hand speaking there was no single form of "medieval Latin"; every Latin author in the medieval period spoke Latin as a second language, with varying degrees of fluency and syntax. Grammar and vocabulary, were influenced by an author's native language; this was true beginning around the 12th century, after which the language became adulterated: late medieval Latin documents written by French speakers tend to show similarities to medieval French grammar and vocabulary. For instance, rather than following the classical Latin practice of placing the verb at the end, medieval writers would follow the conventions of their own native language instead.
Whereas Latin had no definite or indefinite articles, medieval writers sometimes used forms of unus as an indefinite article, forms of ille as a definite article or quidam as something like an article. Unlike classical Latin, where esse was the only auxiliary verb, medieval Latin writers might use habere as an auxiliary, similar to constructions in Germanic and Romance languages; the accusative and infinitive construction in classical Latin was replaced by a subordinate clause introduced by quod or quia. This is identical, for example, to the use of que in similar constructions in French. In every age from the late 8th century onwards, there were learned writers who were familiar enough with classical syntax to be aware that these forms and usages were "wrong" and resisted their use, thus the Latin of a theologian like St Thomas Aquinas or of an erudite clerical historian such as William of Tyre tends to avoid most of the characteristics described above, showing its p
Octuple scull
An octuple scull is a racing shell or a rowing boat used in the sport of competitive rowing. The octuple is directed by a coxswain and propelled by eight rowers who move the boat by sculling with two oars, one in each hand. Like a coxed eight, an octuple is 65.2 feet long and weighs 211.2 pounds. Racing boats are long and broadly semi-circular in cross-section in order to reduce drag to a minimum, they have a fin towards the rear, to help prevent roll and yaw. Made from wood, shells are now always made from a composite material for strength and weight advantages; the riggers in sculling apply the forces symmetrically to each side of the boat. When there are eight rowers in a boat, each with only one sweep oar and rowing on opposite sides, the combination is referred to as a "coxed eight." In sweep oared racing the rigging means the forces are staggered alternately along the boat. The symmetrical forces in sculling make the boat more efficient and so the octuple scull is faster than the coxed eight.
"Rowing Evolution in Octuple Sculling". New York Times. 5 November 1905. P. 12. Retrieved 23 June 2010. Thames Ditton Regatta: 11. Thames Ditton, England: www.rowtv.co.uk. 18 May 2008