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What, When, Where, How, Who?


Introduction, Important Definitions and Related Concepts:

Definition: A set S1 is a superset of another set S2 if every element in S2 is in S1. S1 may have elements which are not in S2. A set is a collection of distinct objects considered as a whole. Sets are one of the most fundamental concepts in mathematics. The study of the structure of sets, set theory, is rich and ongoing. Having only been invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics education, being introduced from primary school in many countries.[citation needed] Set theory can be viewed as a foundation from which nearly all of mathematics can be derived. In philosophy, sets are ordinarily considered to be abstract objects [1][2] [3] [4] the physical tokens of which are, for instance; three cups on a table when spoken of together as "the cups", or the chalk lines on a board in the form of the opening and closing curly bracket symbols along with any other symbols in between the two bracket symbols. However, proponents of mathematical realism including Penelope Maddy have argued that sets are concrete objects. The term "concept" is traced back to 1550–60 (conceptum something conceived), but what is today termed "the classical theory of concepts" is the theory of Aristotle on the definition of terms. As the term is used in mainstream cognitive science and philosophy of mind, a concept or conception is an abstract idea or a mental symbol, typically associated with a corresponding representation in a language or symbology. Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".[2] Other practitioners of mathematics maintain that mathematics is the science of pattern, and that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere.[3][4] Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.[5] 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. Knowledge and use of basic mathematics have always been an inherent and integral part of individual and group life. Refinements of the basic ideas are visible in mathematical texts originating in the ancient Egyptian, Mesopotamian, Indian, Chinese, Greek and Islamic worlds. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. The development continued in fitful bursts until the Renaissance period of the 16th century, when mathematical innovations interacted with new scientific discoveries, leading to an acceleration in research that continues to the present day.[6] Today, mathematics is used throughout the world in many fields, including natural science, engineering, medicine, and the social sciences such as economics. Applied mathematics, the application of mathematics to such fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although applications for what began as pure mathematics are often discovered later.[7] The word theory has a lot of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. In science a theory is a testable model of the manner of interaction of a set of natural phenomena, capable of predicting future occurrences or observations of the same kind, and capable of being tested through experiment or otherwise verified through empirical observation. It follows from this that for scientists "theory" and "fact" do not necessarily stand in opposition. For example, it is a fact that an apple dropped on earth has been observed to fall towards the center of the planet, and the theories commonly used to describe and explain this behavior are Newton's theory of universal gravitation (see also gravitation), and the theory of general relativity. In common usage, the word theory is often used to signify a conjecture, an opinion, or a speculation. In this usage, a theory is not necessarily based on facts; in other words, it is not required to be consistent with true descriptions of reality. This usage of theory leads to the common incorrect statements. True descriptions of reality are more reflectively understood as statements which would be true independently of what people think about them. According to the National Academy of Sciences,

Some scientific explanations are so well established that no new evidence is likely to alter them. The explanation becomes a scientific theory. In everyday language a theory means a hunch or speculation. Not so in science. In science, the word theory refers to a comprehensive explanation of an important feature of nature that is supported by many facts gathered over time. Theories also allow scientists to make predictions about as yet unobserved phenomena.[1] The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar. During the 19th century, the Spanish, Portuguese, Chinese, and Ottoman empires began to crumble and the Holy Roman and Mughal empires ceased. Following the Napoleonic Wars, the British Empire became the world's leading power, controlling one quarter of the world's population and one third of the land area. It enforced a Pax Britannica, encouraged trade, and battled rampant piracy. During this time the 19th century was an era of widespread invention and discovery, with significant developments in the understanding or manipulation of mathematics, physics, chemistry, biology, electricity, and metallurgy largely setting the groundworks for the comparably overwhelming and very rapid technological innovations which would take place the following century. Modest advances in medicine and the understanding of human anatomy and disease prevention were also applicable to the 1800s, and were partly responsible for rapidly accelerating population growth in the western world. The introduction of Railroads provided the first major advancement in land transportation for centuries, and their placement and application radically altered the ways people could live and rapidly and reliably obtain necessary commodities, fueling major urbanization movements in countries across the globe. Numerous cities worldwide surpassed populations of 1,000,000 or more during this century, the first time which cities surpassed the peak population of ancient Rome. The last remaining undiscovered landmasses of Earth, largely pacific island chains and atolls, were discovered during this century, and with the exception of the extreme zones of the Arctic and Antarctic, accurate and detailed maps of the globe were available by the 1890s. Slavery was greatly reduced around the world. Following a successful slave revolt in Haiti, Britain forced the Barbary pirates to halt their practice of kidnapping and enslaving Europeans, banned slavery throughout its domain, and charged its navy with ending the global slave trade. Britain abolished slavery in 1834, America's 13th Amendment following their Civil War abolished slavery there in 1865, and in Brazil slavery was abolished in 1888 (see Abolitionism). Similarly, serfdom was abolished in Russia. The 19th century was remarkable in the widespread formation of new settlement foundations which were particularly prevalent across North America and Australasia, with a significant proportion of the two continents' largest cities being founded at some point in the century. Education encompasses teaching and learning specific skills, and also something less tangible but more profound: the imparting of knowledge, positive judgment and well-developed wisdom. Education has as one of its fundamental aspects the imparting of culture from generation to generation (see socialization). Education means 'to draw out', facilitating realization of self-potential and latent talents of an individual. It is an application of pedagogy, a body of theoretical and applied research relating to teaching and learning and draws on many disciplines such as psychology, philosophy, computer science, linguistics, neuroscience, sociology —often more profound than they realize—though family teaching may function very informally. A primary school (from French école primaire[1]) is an institution where children receive the first stage of compulsory education known as primary or elementary education. Primary school is the preferred term in the United Kingdom and many Commonwealth Nations, and in most publications of the United Nations Educational, Scientific, and Cultural Organization (UNESCO).[2] In some countries, and especially in North America, the term elementary school is preferred. Children generally attend primary school from around the age of four or five until the age of eleven or twelve. Philosophy is the discipline concerned with questions of how one should live (ethics); what sorts of things exist (metaphysics); the nature of knowledge (epistemology); and the principles of reasoning (logic).[1][2] The word is of Ancient Greek origin: φιλοσοφία (philosophía), meaning "love of knowledge", "love of wisdom".[3][4][5] In philosophy it is commonly considered that every object is either abstract or concrete. Abstract objects are sometimes called abstracta (sing. abstractum) and concrete objects are sometimes called concreta (sing. concretum). The abstract-concrete distinction is often introduced and initially understood in terms of paradigmatic examples of objects of each kind:

Examples of Abstract and Concrete Objects
Abstracta Concreta
Tennis Tennis player
Redness A particular inscription of the word "red"
5 (number) Five cats
Justice Court
humanism human
The type versus token distinction is a distinction that separates an abstract concept from the objects which are particular instances of the concept. For example, the particular apple in your pocket is a token of the type of thing known as "an apple." In logic, the distinction is used to clarify the meaning of symbols of formal languages. Types are abstract objects. They do not exist anywhere in particular because they are not physical objects. Types may have many tokens. However, types are not directly producible as tokens are. You may, for instance, show someone the apple in your pocket, but you cannot show someone "the apple." Tokens always exist at a particular place and time and may be shown to exist as a concrete physical object.

Symbols are objects, pictures, or other concrete representations of ideas, concepts, or other abstractions. For example, in the United States, Canada, Australia and Great Britain, a red octagon is a symbol for "STOP". Common examples of symbols are the symbols used on maps to denote places of interest, such as crossed sabres to indicate a battlefield, and the numerals used to represent numbers. Common psychological symbols are the use of a gun to represent a penis or a tunnel to represent a vagina. [1] See: phallic symbol and yonic symbol. All languages are made up of symbols. The word "cat", whether spoken or written, is not a cat but a sequence of symbols that represent a cat. Mathematical realism, like realism in general, holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. In this point of view, there is really one sort of mathematics that can be discovered: Triangles, for example, are real entities, not the creations of the human mind. Many working mathematicians have been mathematical realists; they see themselves as discoverers of naturally occurring objects. Examples include Paul Erdős and Kurt Gödel. Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (e.g., for any two objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true, but some conjectures, like the continuum hypothesis, might prove undecidable just on the basis of such principles. Gödel suggested that quasi-empirical methodology could be used to provide sufficient evidence to be able to reasonably assume such a conjecture. Within realism, there are distinctions depending on what sort of existence one takes mathematical entities to have, and how we know about them.

Penelope Maddy UCI Distinguished Professor of Logic & Philosophy of Science and of Mathematics

Research Interests:

My work begins from methodological questions in the foundations of set theory, especially how new axioms can be properly criticized or defended.  This focused inquiry leads to more general questions in the metaphysics and epistemology of mathematics and logic, and the relations of these subjects to natural science.  Progress on all this seems to me to require some attention to the methodology of philosophy itself; I’ve been especially interested in naturalism, but also logical positivism, ordinary language philosophy and its descendents, various ‘therapeutic’ approaches, etc. -- and in radical skepticism as a diagnostic tool for comparing and contrasting these schools of meta-philosophical thought.  This is coupled with an amateur’s historical interest in such figures as Kant, Frege, Wittgenstein, Moore, Austin, Carnap and Quine.

Concrete is a construction material composed of cement (commonly Portland cement) as well as other cementitious materials such as fly ash and slag cement, aggregate (generally a coarse aggregate such as gravel limestone or granite, plus a fine aggregate such as sand), water, and chemical admixtures. The word concrete comes from the Latin word "concretus", which means "hardened" or "hard". Concrete solidifies and hardens after mixing with water and placement due to a chemical process known as hydration. The water reacts with the cement, which bonds the other components together, eventually creating a stone-like material. The reactions are highly exothermic and care must be taken that the build-up in heat does not affect the integrity of the structure. Concrete is used to make pavements, architectural structures, foundations, motorways/roads, bridges/overpasses, parking structures, brick/block walls and footings for gates, fences and poles. More concrete is used than any other man-made material in the world.[1] As of 2006, about seven billion cubic meters of concrete are made each year—more than one cubic meter for every person on Earth.[2] Concrete powers a US$35-billion industry which employs more than two million workers in the United States alone. More than 55,000 miles of highways in America are paved with this material. The People's Republic of China currently consumes 40% of the world's cement [concrete] production. The term cognition is used in different ways by different disciplines. In psychology, it refers to an information processing view of an individual's psychological functions. Other interpretations of the meaning of cognition link it to the development of concepts; individual minds, groups, organizations, and even larger coalitions of entities, can be modelled as societies which cooperate to form concepts. The autonomous elements of each 'society' would have the opportunity to demonstrate emergent behavior in the face of some crisis or opportunity. Cognition can also be interpreted as "understanding and trying to make sense of the world".[citation needed]. In its broadest sense, science (from the Latin scientia, meaning "knowledge") refers to any systematic knowledge or practice. In its more usual restricted sense, science refers to a system of acquiring knowledge based on scientific method, as well as to the organized body of knowledge gained through such research.[1][2] Experimental science to differentiate it from applied science, which is the application of scientific research to specific human needs, though the two are often interconnected. Dualism and monism are the two major schools of thought that attempt to resolve the mind-body problem. It can be traced back to Plato,[2] Aristotle[3][4][5] and the Sankhya and Yoga schools of Hindu philosophy,[6] but it was most precisely formulated by René Descartes in the 17th century.[7] Substance dualists argue that the mind is an independently existing substance, whereas Property dualists maintain that the mind is a group of independent properties that emerge from and cannot be reduced to the brain, but that it is not a distinct substance.[8] Monism is the position that mind and body are not ontologically distinct kinds of entities. This view was first advocated in Western Philosophy by Parmenides in the 5th century BC and was later espoused by the 17th century rationalist Baruch Spinoza.[9] Physicalists argue that only the entities postulated by physical theory exist, and that the mind will eventually be explained in terms of these entities as physical theory continues to evolve. Idealists maintain that the mind is all that exists and that the external world is either mental itself, or an illusion created by the mind. Neutral monists adhere to the position that there is some other, neutral substance, and that both matter and mind are properties of this unknown substance. The most common monisms in the 20th and 21st centuries have all been variations of physicalism; these positions include behaviorism, the type identity theory, anomalous monism and functionalism.[10] Many modern philosophers of mind adopt either a reductive or non-reductive physicalist position, maintaining in their different ways that the mind is not something separate from the body.[10]



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