Plato was an Athenian philosopher during the Classical period in Ancient Greece, founder of the Platonist school of thought, the Academy, the first institution of higher learning in the Western world. He is considered the pivotal figure in the history of Ancient Greek and Western philosophy, along with his teacher and his most famous student, Aristotle. Plato has often been cited as one of the founders of Western religion and spirituality; the so-called Neoplatonism of philosophers like Plotinus and Porphyry influenced Saint Augustine and thus Christianity. Alfred North Whitehead once noted: "the safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato."Plato was the innovator of the written dialogue and dialectic forms in philosophy. Plato appears to have been the founder of Western political philosophy, his most famous contribution bears his name, the doctrine of the Forms known by pure reason to provide a realist solution to the problem of universals.
He is the namesake of Platonic love and the Platonic solids. His own most decisive philosophical influences are thought to have been along with Socrates, the pre-Socratics Pythagoras and Parmenides, although few of his predecessors' works remain extant and much of what we know about these figures today derives from Plato himself. Unlike the work of nearly all of his contemporaries, Plato's entire oeuvre is believed to have survived intact for over 2,400 years. Although their popularity has fluctuated over the years, the works of Plato have never been without readers since the time they were written. Due to a lack of surviving accounts, little is known about education. Plato belonged to an influential family. According to a disputed tradition, reported by doxographer Diogenes Laërtius, Plato's father Ariston traced his descent from the king of Athens and the king of Messenia, Melanthus. Plato's mother was Perictione, whose family boasted of a relationship with the famous Athenian lawmaker and lyric poet Solon, one of the seven sages, who repealed the laws of Draco.
Perictione was sister of Charmides and niece of Critias, both prominent figures of the Thirty Tyrants, known as the Thirty, the brief oligarchic regime, which followed on the collapse of Athens at the end of the Peloponnesian War. According to some accounts, Ariston tried to force his attentions on Perictione, but failed in his purpose; the exact time and place of Plato's birth are unknown. Based on ancient sources, most modern scholars believe that he was born in Athens or Aegina between 429 and 423 BC, not long after the start of the Peloponnesian War; the traditional date of Plato's birth during the 87th or 88th Olympiad, 428 or 427 BC, is based on a dubious interpretation of Diogenes Laërtius, who says, "When was gone, joined Cratylus the Heracleitean and Hermogenes, who philosophized in the manner of Parmenides. At twenty-eight, Hermodorus says, went to Euclides in Megara." However, as Debra Nails argues, the text does not state that Plato left for Megara after joining Cratylus and Hermogenes.
In his Seventh Letter, Plato notes that his coming of age coincided with the taking of power by the Thirty, remarking, "But a youth under the age of twenty made himself a laughingstock if he attempted to enter the political arena." Thus, Nails dates Plato's birth to 424/423. According to Neanthes, Plato was six years younger than Isocrates, therefore was born the same year the prominent Athenian statesman Pericles died. Jonathan Barnes regards 428 BC as the year of Plato's birth; the grammarian Apollodorus of Athens in his Chronicles argues that Plato was born in the 88th Olympiad. Both the Suda and Sir Thomas Browne claimed he was born during the 88th Olympiad. Another legend related that, when Plato was an infant, bees settled on his lips while he was sleeping: an augury of the sweetness of style in which he would discourse about philosophy. Besides Plato himself and Perictione had three other children; the brothers Adeimantus and Glaucon are mentioned in the Republic as sons of Ariston, brothers of Plato, though some have argued they were uncles.
In a scenario in the Memorabilia, Xenophon confused the issue by presenting a Glaucon much younger than Plato. Ariston appears to have died in Plato's childhood, although the precise dating of his death is difficult. Perictione married Pyrilampes, her mother's brother, who had served many times as an ambassador to the Persian court and was a friend of Pericles, the leader of the democratic faction in Athens. Pyrilampes had a son from a previous marriage, famous for his beauty. Perictione gave birth to Pyrilampes' second son, the half-brother of Plato, who appears in Parmenides. In contrast to his reticence about himself, Plato introduced his distinguished relatives into his dialogues, or referred to them with some precision. In addition to Adeimantus and Glaucon in the Republic, Charmides has a dialogue named after him; these and other references suggest a considerable amount of family pride and enable us to reconstruct Plato's family tree. According to Burnet, "the opening scene of the Ch
In probability theory and related fields, a stochastic or random process is a mathematical object defined as a collection of random variables. The random variables were associated with or indexed by a set of numbers viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes are used as mathematical models of systems and phenomena that appear to vary in a random manner, they have applications in many disciplines including sciences such as biology, ecology and physics as well as technology and engineering fields such as image processing, signal processing, information theory, computer science and telecommunications. Furthermore random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes.
Examples of such stochastic processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse, the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time; these two stochastic processes are considered the most important and central in the theory of stochastic processes, were discovered and independently, both before and after Bachelier and Erlang, in different settings and countries. The term random function is used to refer to a stochastic or random process, because a stochastic process can be interpreted as a random element in a function space; the terms stochastic process and random process are used interchangeably with no specific mathematical space for the set that indexes the random variables. But these two terms are used when the random variables are indexed by the integers or an interval of the real line. If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space the collection of random variables is called a random field instead.
The values of a stochastic process are not always numbers and can be vectors or other mathematical objects. Based on their mathematical properties, stochastic processes can be divided into various categories, which include random walks, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, branching processes; the study of stochastic processes uses mathematical knowledge and techniques from probability, linear algebra, set theory, topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, functional analysis. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications. A stochastic or random process can be defined as a collection of random variables, indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set.
The set used to index. The index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time; each random variable in the collection takes values from the same mathematical space known as the state space. This state space can be, for example, the integers, the real n - dimensional Euclidean space. An increment is the amount that a stochastic process changes between two index values interpreted as two points in time. A stochastic process can have many outcomes, due to its randomness, a single outcome of a stochastic process is called, among other names, a sample function or realization. A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the cardinality of the state space; when interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers the stochastic process is said to be in discrete time.
If the index set is some interval of the real line time is said to be continuous. The two types of stochastic processes are referred to as discrete-time and continuous-time stochastic processes. Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge due to the index set being uncountable. If the index set is the integers, or some subset of them the stochastic process can be called a random sequence. If the state space is the integers or natural numbers the stochastic process is called a discrete or integer-valued stochastic process. If the state space is the real line the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space. If the state space is n -dimensional Euclidean space the stochastic process is called a n -dimensional vector process or n -vector process; the word stochastic in English was used as an adjective with the definition "pertaining to conjecturing", stemming from a Greek word meaning "to aim at a mark, guess", the Oxford English Dictionary gives the year 16
A phenomenon is any thing which manifests itself. Phenomena are but not always, understood as "things that appear" or "experiences" for a sentient being, or in principle may be so; the term came into its modern philosophical usage through Immanuel Kant, who contrasted it with the noumenon. In contrast to a phenomenon, a noumenon cannot be directly observed. Kant was influenced by Gottfried Wilhelm Leibniz in this part of his philosophy, in which phenomenon and noumenon serve as interrelated technical terms. Far predating this, the ancient Greek Pyrrhonist philosopher Sextus Empiricus used phenomenon and noumenon as interrelated technical terms. In modern philosophical use, the term phenomena has come to mean'what is experienced is the basis of reality'. In Immanuel Kant's inaugural dissertation, On the Form and Principles of the Sensible and Intelligible World, Kant theorizes that the human mind is restricted to the logical world and thus can only interpret and understand occurrences according to their physical appearances.
He wrote that humans could infer only as much as their senses allowed, but not experience the actual object itself. Thus, the term phenomenon refers to any incident deserving of inquiry and investigation events that are unusual or of distinctive importance. According to the Columbia Encyclopedia, "Modern philosophers have used'phenomenon' to designate what is apprehended before judgment is applied." This may not be possible. In scientific usage, a phenomenon is any event, observable, however common it might be if it requires the use of instrumentation to observe, record, or compile data concerning it. For example, in physics, a phenomenon may be described by a system of information related to matter, energy, or spacetime, such as Isaac Newton's observations of the moon's orbit and of gravity, or Galileo Galilei's observations of the motion of a pendulum. In natural sciences, a phenomenon is event; this term is used without considering the causes of a particular event. Example of a physical phenomena is an observable phenomenon of the lunar orbit or the phenomenon of oscillations of a pendulum.
A mechanical phenomenon is a physical phenomenon associated with the equilibrium or motion of objects. Some examples are Newton's cradle and double pendulums. Group phenomena concern the behavior of a particular group of individual entities organisms and most people; the behavior of individuals changes in a group setting in various ways, a group may have its own behaviors not possible for an individual because of the herd mentality. Social phenomena apply to organisms and people in that subjective states are implicit in the term. Attitudes and events particular to a group may have effects beyond the group, either be adapted by the larger society, or seen as aberrant, being punished or shunned. In popular usage, a phenomenon refers to an extraordinary event; the term is most used to refer to occurrences that at first defy explanation or baffle the observer. According to the Dictionary of Visual Discourse, "In ordinary language'phenomenon/phenomena' refer to any occurrence worthy of note and investigation an untoward or unusual event, person or fact, of special significance or otherwise notable."
Condition of possibility Essence Electrical phenomena List of geological phenomena List of Internet phenomena List of natural phenomena Observation Optical phenomena
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
Pragmatism is a philosophical tradition that began in the United States around 1870. Its origins are attributed to the philosophers William James, John Dewey, Charles Sanders Peirce. Peirce described it in his pragmatic maxim: "Consider the practical effects of the objects of your conception. Your conception of those effects is the whole of your conception of the object."Pragmatism considers words and thought as tools and instruments for prediction, problem solving and action, rejects the idea that the function of thought is to describe, represent, or mirror reality. Pragmatists contend that most philosophical topics—such as the nature of knowledge, concepts, meaning and science—are all best viewed in terms of their practical uses and successes; the philosophy of pragmatism "emphasizes the practical application of ideas by acting on them to test them in human experiences". Pragmatism focuses on a "changing universe rather than an unchanging one as the Idealists and Thomists had claimed". Pragmatism as a philosophical movement began in the United States in the 1870s.
Charles Sanders Peirce is given credit for its development, along with twentieth century contributors, William James and John Dewey. Its direction was determined by The Metaphysical Club members Charles Sanders Peirce, William James, Chauncey Wright, as well as John Dewey and George Herbert Mead; the first use in print of the name pragmatism was in 1898 by James, who credited Peirce with coining the term during the early 1870s. James regarded Peirce's "Illustrations of the Logic of Science" series as the foundation of pragmatism. Peirce in turn wrote in 1906 that Nicholas St. John Green had been instrumental by emphasizing the importance of applying Alexander Bain's definition of belief, "that upon which a man is prepared to act". Peirce wrote. John Shook has said, "Chauncey Wright deserves considerable credit, for as both Peirce and James recall, it was Wright who demanded a phenomenalist and fallibilist empiricism as an alternative to rationalistic speculation."Peirce developed the idea that inquiry depends on real doubt, not mere verbal or hyperbolic doubt, said, in order to understand a conception in a fruitful way, "Consider the practical effects of the objects of your conception.
Your conception of those effects is the whole of your conception of the object", which he called the pragmatic maxim. It equates any conception of an object to the general extent of the conceivable implications for informed practice of that object's effects; this is the heart of his pragmatism as a method of experimentational mental reflection arriving at conceptions in terms of conceivable confirmatory and disconfirmatory circumstances—a method hospitable to the generation of explanatory hypotheses, conducive to the employment and improvement of verification. Typical of Peirce is his concern with inference to explanatory hypotheses as outside the usual foundational alternative between deductivist rationalism and inductivist empiricism, although he was a mathematical logician and a founder of statistics. Peirce further wrote on pragmatism to make clear his own interpretation. While framing a conception's meaning in terms of conceivable tests, Peirce emphasized that, since a conception is general, its meaning, its intellectual purport, equates to its acceptance's implications for general practice, rather than to any definite set of real effects.
Peirce in 1905 coined the new name pragmaticism "for the precise purpose of expressing the original definition", saying that "all went happily" with James's and Schiller's variant uses of the old name "pragmatism" and that he nonetheless coined the new name because of the old name's growing use in "literary journals, where it gets abused". Yet in a 1906 manuscript he cited as causes his differences with Schiller. And, in a 1908 publication, his differences with James as well as literary author Giovanni Papini. Peirce in any case regarded his views that truth is immutable and infinity is real, as being opposed by the other pragmatists, but he remained allied with them on other issues. Pragmatism enjoyed renewed attention after Willard Van Orman Quine and Wilfrid Sellars used a revised pragmatism to criticize logical positivism in the 1960s. Inspired by the work of Quine and Sellars, a brand of pragmatism known sometimes as neopragmatism gained influence through Richard Rorty, the most influential of the late twentieth century pragmatists along with Hilary Putnam and Robert Brandom.
Contemporary pragmatism may be broadly divided into a strict analytic tradition and a "neo-classical" pragmatism that adheres to the work of Peirce and Dewey. Inspiration for various pragmatists included: Francis Bacon who coined the saying ipsa scientia potestas est David Hume for his naturalistic account of knowledge and action Thomas Reid, for his direct realism Immanuel Kant, for his idealism and from whom Peirce derives the name "pragmatism" G. W. F. Hegel who introduced temporality into philosophy J. S. Mill for his nominalism and empiricism George Berkeley for his project to eliminate all unclear concepts from philosophy Henri Bergson who influenced William James to renounce intellectualism and logical methods A few of the various but interrelated positions characteristic
Evidence, broadly construed, is anything presented in support of an assertion. This support may be weak; the strongest type of evidence is that. At the other extreme is evidence, consistent with an assertion but does not rule out other, contradictory assertions, as in circumstantial evidence. In law, rules of evidence govern the types of evidence. Types of legal evidence include testimony, documentary evidence, physical evidence; the parts of a legal case which are not in controversy are known, in general, as the "facts of the case." Beyond any facts that are undisputed, a judge or jury is tasked with being a trier of fact for the other issues of a case. Evidence and rules are used to decide questions of fact that are disputed, some of which may be determined by the legal burden of proof relevant to the case. Evidence in certain cases must be more compelling than in other situations, which drastically affects the quality and quantity of evidence necessary to decide a case. Scientific evidence consists of observations and experimental results that serve to support, refute, or modify a scientific hypothesis or theory, when collected and interpreted in accordance with the scientific method.
In philosophy, the study of evidence is tied to epistemology, which considers the nature of knowledge and how it can be acquired. The burden of proof is the obligation of a party in an argument or dispute to provide sufficient evidence to shift the other party's or a third party's belief from their initial position; the burden of proof must be fulfilled by both establishing confirming evidence and negating oppositional evidence. Conclusions drawn from evidence may be subject to criticism based on a perceived failure to fulfill the burden of proof. Two principal considerations are: On whom does the burden of proof rest? To what degree of certitude must the assertion be supported? The latter question depends on the nature of the point under contention and determines the quantity and quality of evidence required to meet the burden of proof. In a criminal trial in the United States, for example, the prosecution carries the burden of proof since the defendant is presumed innocent until proven guilty beyond a reasonable doubt.
In most civil procedures, the plaintiff carries the burden of proof and must convince a judge or jury that the preponderance of the evidence is on their side. Other legal standards of proof include "reasonable suspicion", "probable cause", "prima facie evidence", "credible evidence", "substantial evidence", "clear and convincing evidence". In a philosophical debate, there is an implicit burden of proof on the party asserting a claim, since the default position is one of neutrality or unbelief; each party in a debate will therefore carry the burden of proof for any assertion they make in the argument, although some assertions may be granted by the other party without further evidence. If the debate is set up as a resolution to be supported by one side and refuted by another, the overall burden of proof is on the side supporting the resolution. In scientific research evidence is accumulated through observations of phenomena that occur in the natural world, or which are created as experiments in a laboratory or other controlled conditions.
Scientific evidence goes towards supporting or rejecting a hypothesis. The burden of proof is on the person making a contentious claim. Within science, this translates to the burden resting on presenters of a paper, in which the presenters argue for their specific findings; this paper is placed before a panel of judges where the presenter must defend the thesis against all challenges. When evidence is contradictory to predicted expectations, the evidence and the ways of making it are closely scrutinized and only at the end of this process is the hypothesis rejected: this can be referred to as'refutation of the hypothesis'; the rules for evidence used by science are collected systematically in an attempt to avoid the bias inherent to anecdotal evidence. Evidence forms the foundation of a legal system, without which law would be subject to the whims of those with power. In law, the production and presentation of evidence depends first on establishing on whom the burden of proof lies. Admissible evidence is that which a court receives and considers for the purposes of deciding a particular case.
Two primary burden-of-proof considerations exist in law. The first is on. In many Western, the burden of proof is placed on the prosecution in criminal cases and the plaintiff in civil cases; the second consideration is the degree of certitude proof must reach, depending on both the quantity and quality of evidence. These degrees are different for criminal and civil cases, the former requiring evidence beyond a reasonable doubt, the latter considering only which side has the preponderance of evidence, or whether the proposition is more true or false; the decision maker a jury, but sometimes a judge, decides whether the burden of proof has been fulfilled. After deciding who will carry the burden of proof, evidence is first gathered and presented before the court: In criminal investigation, rather than attempting to prove an abstract or hypothetical point, the evidence gatherers attempt to determine, responsible for a criminal act; the focus of criminal evidence is to connect physical evidence and reports of witnesses to a specific person.
The path that physical evidence takes from the scene of a crime or the arrest of a suspect to the courtroom is called the chain of custody. In a criminal case, this path must be cle