
Griechische Mathematik, Zeitlinie, Ereignisse Around 600 BC the Cretan poet Epimenides (Επιμενίδης o Κρης) is attributed to have invented the linguistic paradox with his phrase "Cretans are ever liars"  the Liar's Paradox. 2500 years later, the mathematician Kurt Gödel invents an adaptation of the Liar's Paradox that reveals serious axiomatic problems at the heart of modern mathematics. Thales of Miletus (Θαλής ο Μιλήσιος) He brings Babylonian mathematical knowledge to Greece and uses geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. About 530 BC Pythagoras (Πυθαγόρας ο Σάμιος) leaved Samos and went to Italy. He discovered the dependence of musical intervals on the arithmetical ratios of the lengths of string at the same tension, 2:1 giving an octave, 3:2 the fifth, and 4:3 the fourth. He is also credited with a general formula for finding two square numbers the sum of which is also a square, namely (if m is any odd number), m2+{1/2(m21)}2={1/2(m2+1)}2. "The Pythagoreans and Plato [as well as the Renaissance NeoPlatonists] noted that the conclusions they reached deductively agreed to a remarkable extent with the results of observation and inductive inference. Unable to account otherwise for this agreement, they were led to regard mathematics as the study of ultimate, eternal reality, immanent in nature and the universe, rather than as a branch of logic or a tool of science and technology" (Boyer 1949:1). Consequently, when the Pythagoreans developed the theory of geometric magnitudes, by which they were able to compare two surfaces' ratio, they were led, for lack of a system which could handle irrational numbers, to the 'incommensurability problem': Applying the side of a square to the diagonal, no common rational measure is discoverable. About 480 BC Corollary to Parmenides' rejection of the existence of 'nothing' is the Greek number system which, like the later Roman system, refused to use the Babylonian positional number system with its marker for 'nothing.' Making no clear distinction between nature and geometry, "mathematics, instead of being a science of possible relations, was to [the Greeks] the study of situations thought to subsist in nature" (Boyer 1949:25). Moreover, "almost everything in [Greek] philosophy became subordinated to the problem of change.... All temporal changes observed by the senses were mere permutations and combinations of 'eternal principles,' [and] the historical sequence of events (which formed part of the 'flux') lost all fundamental significance" (Toulmin and Goodfield 1965:40). Death of Pythagoras 480411 BC According to Heath Antiphon (Αντιφών ο Αθηναίος) (a sophist) .. deserves an honourable place in the history of geometry as having originated the idea of exhausting an area by means of inscribed regular polygons with an ever increasing number of sides, an idea upon which ... Eudoxus founded his epochmaking method of exhaustion. About 465 BC About 450 BC Zeno of Elea (Ζήνων ο Ελεάτης) propounded forty paradoxes probably to point out inconsistencies in Pythagorean positions. One of the most famous is this: The fleeing and slower runner can never be overtaken by the faster, pursuer because the faster must first reach the point where the slower is at a that time, but by then the slower will be some distance ahead. Other paradoxes made the same or apposite points, but, in fact, mathematical analysis shows that infinite aggregates and the nature of the continuum are not selfcontradictory but only counter to intuition. Oenopides of Chios (Οινοπίδης ο Χίος) probably created the first three of what became Euclid's 'postulates' or assumptions. What is postulated guarantees the existence of straight lines, circles, and points of intersection. That they needed to be postulated is because they require 'movement,' the possibility of which was challenged by the Eleatics (Szabó 1978:276279). About 440430 BC Hippias of Elis (Ιππίας ο Ηλείος) invents the quadratrix which may have been used by him for trisecting an angle and squaring the circle. About 425 BC About 387 BC After about 380 BC Plato said, in the Timaeus, that "as being is to becoming, so is truth to belief" (Plato 1929:29c). In other words, we can only believe, not know, on the basis of experience. Like, Parmenides, he held being and truth, indeed the world, to be timeless and unchanging, an ideal of which man can only hold the idea. This permitted him a certain amount of flexibility: He was willing to accept objections to his view of the universe, for example, if the new hypothesis would provide a rational explanation or 'save the appearance' presented by the planets. In the Timaeus, he also held that the 'world soul' was constructed according to mathematical principles, and, therefore, these principles are already fixed in the individual. (Forms or ideas that have existence independent of any particular mind came to be called archetypes.) He scattered reflections on mathematical issues throughout his dialogues; e.g., in the Meno, he illustrates the difference between a class and its members by reference to the difference between defining 'figure' and enumerating specific figures. References to ratios and proportions are everywhere. The five regular polygons he ascribed to the four elements plus the "decoration" of the universe (Plato 1929:55c), probably the animals of the zodiac. About 375 BC About 370360 BC About 340 BC Aristotle (Αριστοτέλης ο Σταγειρίτης) settles in Athens, founds Lyceum. He said that universals are abstractions from particulars and that we "have knowledge of a scientific fact when we can prove that it could not be otherwise." But "since observation never shows whether this is the case," he established "reason rather observation at the center of scientific effort" (Park 1990:32). A deductive argument is "a 'demonstration' when the premises from which the reasoning starts are true and primary.... Things are 'true' and 'primary' which are believed on the strength not of anything else but themselves" (Aristotle 1928:100a100b). Aristotle defined the syllogism as a formal argument in which the conclusion necessarily follows from the premises, and said that the four most common statements of this sort are 'all Subject is Predicate,' 'no S is P,' 'some S is P,' and 'some S is not P.' He also discerned four sorts of 'cause.' The 'formal cause' is the design of a thing. The 'material cause' is that of which it is made. The 'efficient cause' is the maker. And the 'final cause' is the purpose of the thing. Aristotle also insisted on the operational character of mathematics and rejected any metaphysical character of number. About 330310 BC 322 BC About 320 BC About 300 BC Eukleides, better known as Euclid (Ευκλείδης ο Αλεξανδρεύς), published his Elements, a reorganized compilation of geometrical proofs including new proofs and a much earlier essay on the foundations of arithmetic. Elements conclude with the construction of Plato's five regular solids. Euclidean space has no natural edge, and is thus infinite. In his Optica, he noted that light travels in straight lines and described the law of reflection. About 290260 BC Aristarchus of Samos (Αρίσταρχος ο Σάμιος), in On the Sizes and Distances of the Sun and Moon, used trigonometry to estimate the size of the Moon and its distance by the Earth's shadow during a lunar eclipse. Archimedes and others said that he maintained that the Moon revolved around the Earth and the Earth around the Sun which remained stationary like the stars. 287 BC 285 BC Philetas of Cos  died from considering the Liar Paradox. About 280 BC 280206 BC 276 BC About 270 BC About 260250 BC Archimedes of Syracuse contributed numerous advances to science including the principle that a body immersed in fluid is buoyed up by a force equal to the weight of the displaced fluid and the calculation of the value of pi. "His method was to select definite and limited problems. He then formulated hypotheses which he either regarded, in the Euclidean manner, as selfevident axioms or could verify by simple experiments. The consequences of these he then deduced and experimentally verified" (Crombie 1952:278). Description of the Loculus of Archimedes; Archimedean Polyhedra; Volume of Intersection of Two Cylinders; Archimedes' Cattle Problem. About 230 BC About 230 BC In the early second century BC About 225210 BC? About 225 BC Around 212 BC Around 200 BC About 60 About 90 About 110 About 250 About 300 About 325 About 385 Aurelius Augustinus, later known as Augustine, a Christian saint, writing in Latin, found the Platonist notion of eternal ideas a certain basis for knowledge which he promulgated in his books Confessiones and Civitas Dei. [["The fourth and fifth centuries saw the intellectual triumph of [Roman] Christianity in Europe.... In 389 Christian monks sacked the great Greek library in Alexandria.... Since Greek was the language of a literature whose most famous works expressed a pagan culture [and] by 425 Saint Jerome's [official Latin or] Vulgate Bible was being copied and distributed..., Western scholars no longer needed Hebrew or Greek" (Park 1990:7879).]] About 390 About 400 About 450 or later About 500 See also: T. L. Heath. Astronomy and Mathematics from the Legacy of Greece Timeline: Mathematicians from http://wwwgroups.dcs.stand.ac.uk/~history/Timelines/index.html http://wwwhistory.mcs.standrews.ac.uk/history/index.html Ancient Greek mathematical texts Preliminary version A History of Mathematics, 2nd Edition Carl B. Boyer, Uta Merzbach, John Wiley & Sons, Inc. 1989 
