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Non-logical Axioms
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. As used in modern logic, an axiom is a premise or starting point for reasoning. As used in mathematics, the term ''axiom'' is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms (e.g., ) are actuall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statement (logic)
In logic, the term statement is variously understood to mean either: #a meaningful declarative sentence that is true or false, or #a proposition. Which is the ''assertion'' that is made by (i.e., the meaning of) a true or false declarative sentence. In the latter case, a statement is distinct from a sentence in that a sentence is only one formulation of a statement, whereas there may be many other formulations expressing the same statement. Overview Philosopher of language, Peter Strawson advocated the use of the term "statement" in sense (b) in preference to proposition. Strawson used the term "Statement" to make the point that two declarative sentences can make the same statement if they say the same thing in different ways. Thus in the usage advocated by Strawson, "All men are mortal." and "Every man is mortal." are two different sentences that make the same statement. In either case a statement is viewed as a truth bearer. Examples of sentences that are (or make ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Verbal Noun
A verbal noun or gerundial noun is a verb form that functions as a noun. An example of a verbal noun in English is 'sacking' as in the sentence "The sacking of the city was an epochal event" (''sacking'' is a noun formed from the verb ''sack''). Verbal nouns are morphologically related to non-finite verb forms, but they are not themselves non-finite verbs. Non-finite verb forms are forms such as gerunds, infinitives and participles in English. Some grammarians use the term "verbal noun" to cover verbal noun, gerund, and nominal infinitive. Some may use the term "gerund" to cover both verbal noun and gerund. "Verbal noun" has often been treated as a synonym for "gerund". This article includes only gerundial nouns within the scope of "verbal nouns", excluding gerunds, nominal infinitives, and nouns formed from verbs through derivational processes. Outside of English, the term "verbal noun" may be used for 1) the citation form of verbs such as the masdar in Arabic and the verbal no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Science
Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for scientific reasoning is tens of thousands of years old. The earliest written records in the history of science come from Ancient Egypt and Mesopotamia in around 3000 to 1200 Common Era, BCE. Their contributions to mathematics, astronomy, and medicine entered and shaped Greek natural philosophy of classical antiquity, whereby formal attempts were made to provide explanations of events in the Universe, physical world based on natural causes. After the fall of the Western Roman Empire, knowledge of History of science in classical antiquity, Greek conceptions of the world deteriorated in Western Europe during the early centuries (400 to 1000 CE) of the Middle Ages, but was preserved in the Muslim world during the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a '' geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical Greece, Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of philosophy within the Lyceum (classical), Lyceum and the wider Aristotelianism, Aristotelian tradition. His writings cover many subjects including Physics (Aristotle), physics, biology, zoology, metaphysics, logic, ethics, aesthetics, Poetics (Aristotle), poetry, theatre, music, rhetoric, psychology, linguistics, economics, politics, meteorology, History of geology, geology, and government. Aristotle provided a complex synthesis of the various philosophies existing prior to him. It was above all from his teachings that Western culture, the West inherited its intellectual lexicon, as well as problems and methods of inquiry. As a result, his philosophy has exerted a unique influence on almost every form of knowledge in the West a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the |
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