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Classical forms

2007-06-09 16:40:23

Biography
, Ancient Greeks developed the biographical tradition which we have inherited, although until the 5th century AD, when the word 'biographia' first appears, in Damascius' Life of Isodorus, biographical pieces were called simply "lives" (????: "bioi"). It is quite likely that the Greeks were drawing on a pre-existing eastern tradition; certainly Herodotus' Histories contains more detailed biographical information on Persian kings and subjects than on anyone else, implying he had a Persian source for it. The earliest surviving pieces which we would identify as biographical are Isocrates' Life of Evagoras and Xenophon's Life of Agesilaos, both from the fifth century BC. Both identified themselves as encomia, or works of praise, and that biography was regarded as a discrete entity from historiography is evidenced by the fact that Xenophon treated King Agesilaos of Sparta twice in his works, once in the above-mentioned encomium and once in his Greek History; evidently the two genres were conceived as making different demands of authors who enrolled in them. Xenophon could present his Cyropaedia, an account of the childhood of the Persian King Cyrus the Great now regarded as so fabulous that it falls rather into a novelistic tradition than a biographical one, as a serious work, without any disclaimers or caveats. Whereas Thucydides set the benchmark for a historiographical tradition comprising 'conclusions ... drawn from proofs quoted ... [which] may safely be relied upon' (Thuc. 1.21), and offering little explicit judgement on the men with whom he dealt, biographers were quite often more concerned with drawing a moral point from their investigations of their subjects. Parallel Lives by Plutarch, a Greek writing under the Roman empire, is a series of short biographies of eminent men, ancient and contemporary, arranged in pairs comprising one Greek, one Roman, in order that a broad educative point might be extraced from the comparison (for example Mark Anthony and Demetrius were paradigms of tyranny, Lysander and Sulla examples of great men degenerating into blood-thirsty corruption). However, although their moralising approach is not in fashion in the current intellectual climate, Greek biographies still have much to offer the modern reader, and for the most part it is reasonable to assume that while authors may have suppressed details which did not fall in with the general theme which they wished to convey, they are unlikely to have fabricated much. Not least, they were instrumental in developing the modern idea of the person. The traditional Greek attitude to individuals was to 'reduce them to types'; the Peripatetic tradition records various categories into which men might fall: the flatterer, the superstitious man and so on. Greek rhetorical handbooks give advice on 'ethopoia', that is creating a character, one of a recognised type, to win favour in the law courts. The biographical tradition does draw on these types, but it also gives explicit recognition to the importance of individual ideosyncrasies in defining a man, and places the emphasis firmly on a man's personality rather than merely listing his accomplishments. As Plutarch says in the introduction to his Life of Alexander the Great, 'in the most illustrious deeds there is not always a manifestation of virtue and vice, but a slight thing like a phrase or a jest often makes a greater revelation than battles where thousands fall, or the greatest armaments, or sieges of cities'. Thus the individual is recognised as having some value and interest irrespective of the impact of his actions on the broader sweep of history. Under the Roman Empire, the biographical and historiographical traditions converged somewhat, likely due to the nature of government, whereby the state was dominated by a single emperor with totalitarian power and whose character and actions set the tone for the period; Tacitus's History and his Annals, as well as Dio's History contain much of the same material as the biographer Suetonius's Lives of the Twelve Caesars. However, although Tacitus in particular was extremely critical of the regime, his disapproval emerges in subtle characterisation and arrangement of his material, in contrast with Suetonius' vicious authorial comment.
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The various arts

2007-06-09 15:09:03

A precise definition of the arts can be contentious, but the following areas of activity usually are included:

Historically, the arts included the Artes Liberales (liberal arts) taught in medieval universities as part of the Trivium (grammar, rhetoric, and logic) and the Quadrivium (arithmetic, geometry, music, and astronomy.) In modern academia, the arts are usually grouped with or a subset of the Humanities. Some subjects in the Humanities are history, linguistics, literature, philosophy, women's studies. Newspapers such as the New York Times and The Times of London typically include a section on the arts.
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Mathematics during the 1800s up to the mid-1900s

2007-06-09 14:09:58

Symbols and rules: In rapid succession the mathematics of George Boole (1847, 1854), Gottlob Frege (1879), and Giuseppe Peano (1888–1889) reduced arithmetic to a sequence of symbols manipulated by rules. Peano's The principles of arithmetic, presented by a new method (1888) was "the first attempt at an axiomatization of mathematics in a symbolic language" (Heijenoort, p. 81ff). But Heijenoort gives Frege (1879) this kudos: Frege’s is "perhaps the most important single work ever written in logic. ... in which we see a " 'formula language', that is a lingua characterica, a language written with special symbols, "for pure thought", that is, free from rhetorical embellishments ... constructed from specific symbols that are manipulated according to definite rules"( p. 1). The work of Frege was further simplified and amplified by Alfred North Whitehead and Bertrand Russell in their Principia Mathematica (1910–1913). The paradoxes: At the same time a number of disturbing paradoxes appeared in the literature, in particular the Burali-Forti paradox (1897), the Russell paradox (1902–03), and the Richard Paradox (1905, Dixon 1906), (cf Kleene (1952) p. 36–40). The resultant considerations led to Kurt Gödel’s paper (1931) — he specifically cites the paradox of the liar — that completely reduces rules of recursion to numbers. In rapid succession the following appeared: Church-Kleene's ?-calculus (cf footnote in Alonzo Church's paper, Undecidable p. 90), Church's (1936) theorem (Undecidable, p. 88ff), Emil Post's (1936) "process" (Undecidable, p. 289–290), Alan Turing's (1936–1937) "a- [automatic-] machine" (Undecidable, p. 116ff), J. Barkley Rosser's (1939) definition of "effective method" in terms of "a machine" (Undecidable, p. 226), and S. C. Kleene's (1943) proposal of the "Church-Turing thesis" (Undecidable, p. 273–274) Emil Post (1936) and Alan Turing (1936, 1937) Here is a remarkable coincidence of two men not knowing each other but describing a process of men-as-computers working on computations — and they yield virtually identical definitions. Emil Post (1936) described the actions of a "computer" (human being) as follows:

"...two concepts are involved: that of a symbol space in which the work leading from problem to answer is to be carried out, and a fixed unalterable set of directions.
His symbol space would be
"a two way infinite sequence of spaces or boxes... The problem solver or worker is to move and work in this symbol space, being capable of being in, and operating in but one box at a time.... a box is to admit of but two possible conditions, i.e. being empty or unmarked, and having a single mark in it, say a vertical stroke.
"One box is to be singled out and called the starting point. ...a specific problem is to be given in symbolic form by a finite number of boxes [i.e. INPUT] being marked with a stroke. Likewise the answer [i.e. OUTPUT] is to be given in symbolic form by such a configuration of marked boxes....
"A set of directions applicable to a general problem sets up a deterministic process when applied to each specific problem. This process will terminate only when it comes to the direction of type (C ) [i.e. STOP]." (U p. 289–290) See more at Post-Turing machine
Alan Turing’s work (1936–1937) preceded that of Stibitz (1937); it is unknown if Stibitz knew of the work of Turing. Turing’s biographer believed that Turing’s use of a typewriter-like model derived from a youthful interest: “Alan had dreamt of inventing typewriters as a boy; Mrs. Turing had a typewriter; and he could well have begun by asking himself what was meant by calling a typewriter 'mechanical'" (Hodges, p. 96) Given the prevalence of Morse code and telegraphy, ticker tape machines, and Teletypes we might conjecture that all were influences. Turing — his model of computation is now called a Turing machine — begins, as did Post, with an analysis of a human computer that he whittles down to a simple set of basic motions and "states of mind". But he continues a step further and creates his machine as a model of computation of numbers (Undecidable p. 116):
"Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book....I assume then that the computation is carried out on one-dimensional paper, i.e. on a tape divided into squares. I shall also suppose that the number of symbols which may be printed is finite....
"The behavior of the computer at any moment is determined by the symbols which he is observing, and his "state of mind" at that moment. We may suppose that there is a bound B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations. We will also suppose that the number of states of mind which need be taken into account is finite...
"Let us imagine that the operations performed by the computer to be split up into 'simple operations' which are so elementary that it is not easy to imagine them further divided" (Undecidable p. 136).
Turing's reduction yields the following:
"The simple operations must therefore include:
"(a) Changes of the symbol on one of the observed squares
"(b) Changes of one of the squares observed to another square within L squares of one of the previously observed squares.
"It may be that some of these change necessarily invoke a change of state of mind. The most general single operation must therefore be taken to be one of the following:
"(A) A possible change (a) of symbol together with a possible change of state of mind.
"(B) A possible change (b) of observed squares, together with a possible change of state of mind"
"We may now construct a machine to do the work of this computer."(Undecidable p. 137)

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