1. Spirals: From Theodorus To Chaos In his introduction to Spirals From Theodorus to Chaos, Phil Davis writes, "To me around the study of a difference equation that Davis dubs theodorus of cyrene, the book takes http://www.ibuki-trading-post.com/dir_akp/akp_spifro.html

Spirals: From Theodorus to Chaos Philip J. Davis ISBN 1-56881-010-5 AK Peters - 1993 Hardcover 248pp In his introduction to Spirals: From Theodorus to Chaos, Phil Davis writes, "To me, mathematics has always been more than its form, or its content, its logic, its strategies, or its applications. Mathematics is one of the greatest of human intellectual experiences, and as such merits and requires a rather liberal approach." He takes just such an approach in this book inspired by the Hedrick Lectures of the Seventy-Fifth Anniversary of the Mathematical Association of America. Although loosely organized around the study of a difference equation that Davis dubs Theodorus of Cyrene, the book takes us on an eclectic whirlwind tour of history, philosophy, anecdote and, of course, mathematics. Incorporating the old and the new, the proved and the conjectural, Davis examines Theodorus in light of mathematical concerns that have grown and changed over the past twenty-five hundred years. info@ibuki.com

2. A Lesson On Spirals exercise below has been attributed to theodorus of cyrene (~465399 BC). Theodorus was Platos tutor Socrates makes reference to Theodorus proving that the square roots of 3, 5 http://courses.wcupa.edu/jkerriga/Lessons/A Lesson on Spirals.html

A Lesson on The Root Spiral Kate Long The Shipley School Klong@shipleyschool.org Objectives Practice with compass and straight edge Explore a geometric representation of square roots, deepening understanding Introduce students to spirals, curves that are seldom studied in traditional textbooks Develop an awareness of the historical context for the study of irrational numbers and spirals Recognize spirals in nature and appreciate the mathematics inherent in them Historical Perspectives Theaetus , Socrates makes reference to Theodorus proving that the square roots of 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, and 17 were irrational. Plato raises the question "Why did Theodorus stop at 17?". One possible answer is that the 17 hypotenuse belongs to the last triangle that does not overlap the figure. In 1958 E. Teuffel proved several more interesting facts about the root spiral. If the procedure for generating the spiral is continued indefinitely so that the figure overlaps, no two hypotenuses will coincide. In other words, they will never lie directly on top of each other. Also, if the sides of unit "one" length are extended forever, they will not pass through any of the other vertices of the total figure. The Spiral of Theodorus approximates the Logarithmic Spiral. By the early 1600’s the logarithmic spiral was being studied in depth. See Historical Perspectives II in Appendix A for additional information.

3. History Of Mathematics: Greece 430) *SB. Hippias of Elis (c. 425) theodorus of cyrene (c. 425 c. 250) Nicoteles of Cyrene (c. 250) Strato (c http://aleph0.clarku.edu/~djoyce/mathhist/greece.html

Greece Cities Abdera: Democritus Alexandria : Apollonius, Aristarchus, Diophantus, Eratosthenes, Euclid , Hypatia, Hypsicles, Heron, Menelaus, Pappus, Ptolemy, Theon Amisus: Dionysodorus Antinopolis: Serenus Apameia: Posidonius Athens: Aristotle, Plato, Ptolemy, Socrates, Theaetetus Byzantium (Constantinople): Philon, Proclus Chalcedon: Proclus, Xenocrates Chalcis: Iamblichus Chios: Hippocrates, Oenopides Clazomenae: Anaxagoras Cnidus: Eudoxus Croton: Philolaus, Pythagoras Cyrene: Eratosthenes, Nicoteles, Synesius, Theodorus Cyzicus: Callippus Elea: Parmenides, Zeno Elis: Hippias Gerasa: Nichmachus Larissa: Dominus Miletus: Anaximander, Anaximenes, Isidorus, Thales Nicaea: Hipparchus, Sporus, Theodosius Paros: Thymaridas Perga: Apollonius Pergamum: Apollonius Rhodes: Eudemus, Geminus, Posidonius Rome: Boethius Samos: Aristarchus, Conon, Pythagoras Smyrna: Theon Stagira: Aristotle Syene: Eratosthenes Syracuse: Archimedes Tarentum: Archytas, Pythagoras Thasos: Leodamas Tyre: Marinus, Porphyrius Mathematicians Thales of Miletus (c. 630-c 550)

4. Theodorus theodorus of cyrene. theodorus of cyrene was a pupil of Protagoras and himself thetutor of Plato, teaching him mathematics, and also the tutor of Theaetetus. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Theodorus.html

Theodorus of Cyrene Born: 465 BC in Cyrene (now Shahhat, Libya) Died: 398 BC in Cyrene (now Shahhat, Libya) Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Theodorus of Cyrene was a pupil of Protagoras and himself the tutor of Plato , teaching him mathematics, and also the tutor of Theaetetus Plato travelled to and from Egypt and on such occasions he spent time with Theodorus in Cyrene. Theodorus, however, did not spend his whole life in Cyrene for he was certainly in Athens at a time when Socrates was alive. Theodorus, in addition to his work in mathematics, was [5]:- ... distinguished ... in astronomy, arithmetic, music and all educational subjects. A member of the society of Pythagoras , Theodorus was one of the main philosophers in the Cyrenaic school of moral philosophy. He believed that pleasures and pains are neither good nor bad. Cheerfulness and wisdom, he believed, were sufficient for happiness. Our knowledge of Theodorus comes through Plato who wrote about him in his work Theaetetus.

5. References For Theodorus 7 (1969), 359379. L Giacardi, On theodorus of cyrene s problem, Arch.Internat. Hist. Sci. 27 (101) (1977), 231-236. TL Heath, A http://www-gap.dcs.st-and.ac.uk/~history/References/Theodorus.html

References for Theodorus Biography in Dictionary of Scientific Biography (New York 1970-1990). Articles: B Artmann, A proof for Theodorus' theorem by drawing diagrams, J. Geom. M S Brown, Theaetetus : Knowledge as Continued Learning, Journal of the History of Philosophy L Giacardi, On Theodorus of Cyrene's problem, Arch. Internat. Hist. Sci. T L Heath, A History of Greek Mathematics I (Oxford, 1921), 203-204, 209-212. R L McCabe, Theodorus' irrationality proofs, Math. Mag. A Wasserstein, Theaetetus and the History of the Theory of Numbers, Classical Quarterly Main index Birthplace Maps Biographies Index History Topics ... Anniversaries for the year JOC/EFR January 1999 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/References/Theodorus.html

6. Spirals: From Theodorus To Chaos, By P. Davis around the study of a difference equation that Davis dubs theodorus of cyrene, the book takes the conjectural, Davis examines Theodorus in light of mathematical concerns that have http://www.webbooks.net/books/_peters/Davis.html

Spirals: From Theodorus to Chaos Philip Davis In his introduction, the author writes, "To me, mathematics has always been more than its form, or its content, its logic, its strategies, or its applications. Mathematics is one of the greatest of human intellectual experiences, and as such merits and requires a rather liberal approach." He takes just such an approach in this book inspired by the Hedrick Lectures of the Seventy-Fifth Anniversary of the Mathematical Association of America. Although loosely organized around the study of a difference equation that Davis dubs Theodorus of Cyrene, the book takes us on an eclectic whirlwind tour of history, philosophy, anecdote and, of course, mathematics. Incorporating the old and the new, the proved and the conjectural, Davis examines Theodorus in light of mathematical concerns that have grown and changed over the past 2,500 years. A. K. Peters, Ltd., 1993 ISBN 1-56881-010-5 248 pages List price: $34.00 WebBooks Price: $30.50 Handling/shipping: Surface shipping included; express/overnight at cost. Jump to WebBooks Home Page

7. GO.HRW.COM Lesson 12.4 The Distance Formula. theodorus of cyrene, theodorus of cyrene wasa mathematician, astronomer, musician, philosopher, and the tutor of Plato. http://go.hrw.com/hrw.nd/gohrw_rls1/pKeywordResults?MA1 Theodorus

8. Theodorus Biography of Theodorus (465BC398BC) theodorus of cyrene. Born 465 BC in Cyrene (now Shahhat, Libya) theodorus of cyrene was a pupil of Protagoras and himself the tutor of Plato, teaching him mathematics http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Theodorus.html

Theodorus of Cyrene Born: 465 BC in Cyrene (now Shahhat, Libya) Died: 398 BC in Cyrene (now Shahhat, Libya) Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Theodorus of Cyrene was a pupil of Protagoras and himself the tutor of Plato , teaching him mathematics, and also the tutor of Theaetetus Plato travelled to and from Egypt and on such occasions he spent time with Theodorus in Cyrene. Theodorus, however, did not spend his whole life in Cyrene for he was certainly in Athens at a time when Socrates was alive. Theodorus, in addition to his work in mathematics, was [5]:- ... distinguished ... in astronomy, arithmetic, music and all educational subjects. A member of the society of Pythagoras , Theodorus was one of the main philosophers in the Cyrenaic school of moral philosophy. He believed that pleasures and pains are neither good nor bad. Cheerfulness and wisdom, he believed, were sufficient for happiness. Our knowledge of Theodorus comes through Plato who wrote about him in his work Theaetetus.

9. GO.HRW.COM The activity also provides links to interesting Internet sites related to physics.MA1 Theodorus Learn about theodorus of cyrene and the golden spiral. http://go.hrw.com/ndNSAPI.nd/gohrw_rls1/pKeywordResults?MA1 CH12

10. Theodorus theodorus of cyrene. Born 465 BC in Cyrene (now Shahhat,Libya) Died 398 BC in Cyrene (now Shahhat, Libya). http://intranet.woodvillehs.sa.edu.au/pages/resources/maths/History/Thdrs.htm

Theodorus of Cyrene Born: 465 BC in Cyrene (now Shahhat, Libya) Died: 398 BC in Cyrene (now Shahhat, Libya) Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Welcome page Theodorus was a tutor of Plato and Theaetetus and is best remembered by mathematicians for his contribution to the development of irrational numbers. Theodorus was also one of the main philosophers in the Cyrenaic school of moral philosophy. He believed that pleasures and pains are neither good nor bad. Cheerfulness and wisdom, he believed, were sufficient for happiness. References (7 books/articles) Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Welcome page History Topics Index Famous curves index ... Search Suggestions JOC/EFR December 1996 The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Theodorus.html

11. Table Of Contents For Gazalé, M.: Number: From Ahmes To Cantor. The Ladder of theodorus of cyrene and Diophantine Equations 169. A Variation on the Ladder of Theodorus 171. Fermat's Last Theorem 172 http://www.pupress.princeton.edu/TOCs/c6794.html

PRINCETON University Press SEARCH: Keywords Author Title More Options Power Search Search Hints E-MAIL NOTICES NEW IN PRINT E-BOOKS ... HOME PAGE Number: From Ahmes to Cantor Book Description TABLE OF CONTENTS: Preface xi INTRODUCTION 3 CHAPTER 1 The Genesis of Number Systems 9 Foundations 9 Matching 10 Naming 11 Counting 12 Grouping 15 Archaic Number Systems 16 The Egyptians 17 The Mesopotamians 28 The Greeks 34 The Mayas 38 Two Current Number Systems 41 The Hindus 42 The Arabs 43 The Decimal Number System 45 Fractional Numbers 50 Uttering versus writing 52 Units 54 The Binary Number System 55 CHAPTER 2 Positional Number Systems 59 The Division Algorithm 59 Codes 61 Mixed-Base Positional Systems 64 Finding the Digits of an Integer 69 Addition 72 Uniform-Base Multiplication 75 Mixed-Base Multiplication 77 Construction 1: A Parallel Adder 78 Construction 2: A Digital-to-Analog Converter 80 Construction 3: A Reversible Binary-to-Analog Converter 80 Positional Representation of Fractional Numbers 82 Going to Infinity 89 How Precise Is a Mantissa? 93

12. Full Alphabetical Index Miletus (404*) Theaetetus of Athens (82) theodorus of cyrene (58) Theodosius http://intranet.woodvillehs.sa.edu.au/pages/resources/maths/History/Flllph.htm

Full Alphabetical Index Click on a letter below to go to that part of this file. A B C D ... XYZ Click below to go to the separate alphabetical indexes A B C D ... XYZ The number of words in the biography is given in brackets. A * indicates that there is a portrait. A Abbe , Ernst (602*) Abel , Niels Henrik (2899*) Abraham bar Hiyya (240) Abraham, Max Abu Kamil Shuja (59) Abu'l-Wafa al'Buzjani (243) Ackermann , Wilhelm (196) Adams, John Couch Adams, J Frank Adelard of Bath (89) Adler , August (114) Adrain , Robert (79) Aepinus , Franz (124) Agnesi , Maria (196*) Ahlfors , Lars (725*) Ahmed ibn Yusuf (60) Ahmes Aida Yasuaki (114) Aiken , Howard (94) Airy , George (313*) Aitken , Alexander (825*) Ajima , Chokuyen (144) Akhiezer , Naum Il'ich (248*) al'Battani , Abu Allah (194) al'Biruni , Abu Arrayhan (306*) al'Haitam , Abu Ali (269*) al'Kashi , Ghiyath (73) al'Khwarizmi , Abu (123*) Albanese , Giacomo (282) Albert, Abraham Adrian (158*) Albert of Saxony Alberti , Leone (181*) Albertus Magnus, Saint (109*) Alcuin of York (237*) Aleksandrov , Pave (160*) Alembert , Jean d' (291*) Alexander , James (130*) Amringe , Howard van (354*) Amsler , Jacob (82) Anaxagoras of Clazomenae (169) Anderson , Oskar (67) Andreev , Konstantin (117) Angeli , Stefano degli (234) Anstice , Robert (209) Anthemius of Tralles (55) Antiphon the Sophist (125) Apollonius of Perga (276) Appell , Paul (1377) Arago , Dominique (345*) Arbogast , Louis (87) Arbuthnot , John (251*) Archimedes of Syracuse (467*) Archytas of Tarentum (103) Arf , Cahit (1452*) Argand , Jean (81) Aristaeus the Elder (44) Aristarchus of Samos (183)

13. Square Roots Chronology About 425BC. theodorus of cyrene shows that the square roots of 3, 5, 6, 7, 8, 10, 11 and 17 are irrational. Theodorus approximates R(3) as 7/4 since http://rpimath.topcities.com/irrationals/squareroots.html

A Chronology of Square Roots [ Back to Danielle's Math Links Page ] Note: This page uses R(n) to indicate the square root of n (this notation was commonly used before the modern day notation was adopted). This page also uses pi for the irrational number 3.14159..., phi for the golden mean (which is the irrational number 1.618...), and ^ to indicate exponents. About 1750BC The Babylonians compile tables of square and cube roots. They have a very accurate value for R(2). About 425BC Theodorus of Cyrene shows that the square roots of 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15 and 17 are irrational. Theodorus approximates R(3) as 7/4 since 49/16 is almost equal to 3. About 375BC Archytas gives a method for approximating R(n). This method was known to the Babylonians. About 350BC Plato approximates R(2) as 7/5 since 49/25 is almost 2. About 325BC Aristotle gives a proof that R(2) is irrational. This proof was known long before him. About 300BC Euclid, in The Elements , says that the line AB is divided in "extreme and mean ratio" (phi) by C if AB:AC = AC:CB. The Elements also deals with the idea of incommensurability, although the proof that R(2) is irrational was probably added in later editions of the work.

14. Index Of Persons And Locations © 1998 Bernard SUZANNE. Last updated December 24, 2001. Plato and his dialogues Home Biography - Works and links to them - History of interpretation - New hypotheses - Map of dialogues table version or non tabular version. Aristides * Aristippus of Cyrene * Aristophanes Themistocles * theodorus of cyrene. Thera. Thermopylæ http://www.plato-dialogues.org/tools

Bernard SUZANNE Last updated December 24, 2001 Plato and his dialogues : Home Biography Works and links to them History of interpretation New hypotheses - Map of dialogues : table version or non tabular version . Tools : Index of persons and locations - Detailed and synoptic chronologies - Maps of Ancient Greek World . Site information : About the author This page is part of the "tools" section of a site, Plato and his dialogues , dedicated to developing a new interpretation of Plato's dialogues. The "tools" section provides historical and geographical context (chronology, maps, entries on characters and locations) for Socrates, Plato and their time. This page provides an index to the entries on persons (*) and locations of interest in the study of the historical context of Socrates and Plato that are available on other pages of this site ( names in italic are names for wich there is no specific entry, but which are delt with through another entry By clicking on a name in the index, you can go to individual entries on famous Greek leaders, writers, thinkers of the Vth and IVth centuries B. C., and also on characters staged in Plato's dialogues, or on the main cities and locations of Ancient Greece that are of interest in the study of Plato's dialogues, either as the location of noteworthy historical events of that time, or as the birthplace of famous writers or philosopher, or as locations mentioned in one or another dialogue. By clicking on the minimap below a city's name, you can go to a full size map for a better viewing of the city's location. You may also click on the area number at the beginning of the text to go to a director map that will show you where the specific portion of the full size map shown in the minimap is located in the larger map (this option is not available for locations in Attica, the relationship between the minimap and the full size map being obvious in that case).

15. Plato FAQ: The Allegory Of The Cave in a ditch while looking at the stars in the caricature of philosopher he putsin the mouth of Socrates for his friend theodorus of cyrene, the scientist, in http://plato-dialogues.org/faq/faq002.htm

Bernard SUZANNE Last updated May 1st, 1999 Plato and his dialogues : Home Biography - Works and links to them History of interpretation New hypotheses - Map of dialogues : table version or non tabular version . Tools : Index of persons and locations Detailed and synoptic chronologies - Maps of Ancient Greek World . Site information : About the author F requently A sked Q uestions about Plato The allegory of the cave "Could you tell me in what work of Plato I might find his "cave analogy". It is a story of men chained in a cave only able to see their own shadows and delude into thinking that this was all there was to reality." The story this letter refers to, usually called "the allegory of the cave", is found at the beginning of book VII of Plato's dialogue called The Republic . The Stephanus references (the universal way of quoting Plato, available in all editions of his works) for the section telling the allegory are Republic , VII, 514a-517a. It is followed by an interpretation of the allegory put by Plato in the mouth of Socrates, as is the allegory itself. The text of this section of the Republic is available in various English translations on the web, including :

16. Plato FAQ: The Allegory Of The Cave he puts in the mouth of Socrates for his friend theodorus of cyrene, the scientist, in the Theætetus caricature along the lines of what Theodorus sees as the proper behavior http://www.plato-dialogues.org/faq/faq002.htm

Bernard SUZANNE Last updated May 1st, 1999 Plato and his dialogues : Home Biography - Works and links to them History of interpretation New hypotheses - Map of dialogues : table version or non tabular version . Tools : Index of persons and locations Detailed and synoptic chronologies - Maps of Ancient Greek World . Site information : About the author F requently A sked Q uestions about Plato The allegory of the cave "Could you tell me in what work of Plato I might find his "cave analogy". It is a story of men chained in a cave only able to see their own shadows and delude into thinking that this was all there was to reality." The story this letter refers to, usually called "the allegory of the cave", is found at the beginning of book VII of Plato's dialogue called The Republic . The Stephanus references (the universal way of quoting Plato, available in all editions of his works) for the section telling the allegory are Republic , VII, 514a-517a. It is followed by an interpretation of the allegory put by Plato in the mouth of Socrates, as is the allegory itself. The text of this section of the Republic is available in various English translations on the web, including :

17. Biography-center - Letter T Theodore II, www.knight.org/advent/cathen/14570b.htm; theodorus of cyrene,wwwhistory.mcs.st-and.ac.uk/~history/Mathematicians/Theodorus.html; http://www.biography-center.com/t.html

Visit a random biography ! Any language Arabic Bulgarian Catalan Chinese (Simplified) Chinese (Traditional) Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Norwegian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Turkish T 361 biographies Tabern, Donalee L. www.invent.org/hall_of_fame/142.html Tacca, Ferdinand www.getty.edu/art/collections/bio/a451-1.html Tacquet, Andrea www-history.mcs.st-and.ac.uk/~history/Mathematicians/Tacquet.html Taft, Helen Herron www.whitehouse.gov/WH/glimpse/firstladies/html/ht27.html Taft, William Howard odur.let.rug.nl/~usa/P/wt27/about/taftbio.htm Tagore, Rabindranath www.nobel.se/literature/laureates/1913/tagore-bio.html Tahir, ibn www-history.mcs.st-and.ac.uk/~history/Mathematicians/Al-Baghdadi.html Taiga, Marton www.kirjasto.sci.fi/taiga.htm Taillandier, Vincent www.getty.edu/art/collections/bio/a1075-1.html Tait, Peter Guthrie www-history.mcs.st-and.ac.uk/~history/Mathematicians/Tait.html Takacs, Karoly

18. Math History - Pre-historic And Ancient Times About 425BC, theodorus of cyrene shows that certain square roots are irrational.This had been shown earlier but it is not known by whom. http://lahabra.seniorhigh.net/pages/teachers/pages/math/timeline/MpreAndAncient.

Math History Timeline Pre-historic and Ancient Times 1,000,000 B.C. - 500 A.D. Math History Prehistory and Ancient Times Middle Ages Renaissance Reformation ... 20th Century ... non-Math History Prehistory and Ancient Times Middle Ages Renaissance Reformation ... External Resources About 30000BC Palaeolithic peoples in central Europe and France record numbers on bones. About 25000BC Early geometric designs used. About 4000BC Babylonian and Egyptian calendars in use. About 3400BC The first symbols for numbers, simple straight lines, are used in Egypt. About 3000BC Babylonians begin to use a sexagesimal number system for recording financial transactions. It is a place-value system without a zero place value. About 3000BC Hieroglyphic numerals in use in Egypt. About 3000BC The abacus is developed in the Middle East and in areas around the Mediterranean. A somewhat different type of abacus is used in China. About 1950BC Babylonians solve quadratic equations.

19. A History Of Irrational Numbers Square roots. Pythagoras of Samos. Born about 569 BC in Samos, Ionia Someclaim he had proofed that the sqrt(2) is irrational. theodorus of cyrene. http://home.zonnet.nl/mathematics/Geschiedenis/Getallen/sub4.htm

A History of irrational numbers The history of irrational numbers is almost as complex as the history of rational numbers. I have diffided it in three sectens. 1. Square roots 2. Pi 3. e Square roots Pythagoras of Samos Born: about 569 BC in Samos, Ionia Some claim he had proofed that the sqrt(2) is irrational. Theodorus of Cyrene Born: 465 BC in Cyrene (now Shahhat, Libya) Theodorus of Cyrene was a pupil of Protagoras and himself the tutor of Plato, Our whole knowledge of Theodorus's mathematical achievements are given by this passage from Plato. Yet there are points of interest which immediately arise. The first point is that Plato does not credit Theodorus with a proof that the square root of two was irrational. This must be because 2 was proved irrational before Theodorus worked on the problem, as stated before, some claim this was proved by Pythagoras himself. Theaetetus of Athens Born: about 417 BC in Athens, Greece Euclid of Alexandria Born: about 325 BC Book ten deals with the theory of irrational numbers and is mainly the work of Theaetetus. Euclid changed the proofs of several theorems in this book so that they fitted the new definition of proportion given by Eudoxus. Pi e The number e was first studied by the Swiss mathematician Leonhard Euler in the 1720s, although its existence was more or less implied in the work of John Napier, the inventor of logarithms, in 1614. Euler was also the first to use the letter e for it in 1727 (the fact that it is the first letter of his surname is coincidental). As a result, sometimes e is called the Euler Number, the Eulerian Number, or Napier's Constant (but not Euler's Constant).

20. Ancient Greece Mathematics Timeline About 425 BC theodorus of cyrene shows that certain square roots are irrational.This had been shown earlier but it is not known by whom. http://www.mlahanas.de/Greeks/TLMathematics.htm

Timeline Ancient Greece Mathematics Around 600 BC the Cretan poet Epimenides 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. About 600 BC Thales of Miletus , 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 no common rational measure is discoverable About 480 BC Parmenides of Elea founded the Eleatic School where he taught that 'all is one,' not an aggregation of units as Pythagoras had said, and that to arrive at a true statement, logical argument is necessary. Truth "is identical with the thought that recognizes it" (Lloyd 1963:327). Change or movement and non-being, he held, are impossibilities since everything is 'full' and 'nothing' is a contradiction which, as such, cannot exist. "Parmenides is said to have been the first to assert that the Earth is spherical in shape...; there was, however, an alternative tradition stating that it was Pythagoras" (Heath 1913:64). 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).

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