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Early Greek Mathematics: the Heroic Age

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Early Greek Mathematics: the Heroic Age Geometric Revolution Plato and The Academy Eudoxus and his Students Conclusion Early Greek Mathematics: The Heroic Age Douglas Pfeffer Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy Eudoxus and his Students Conclusion Table of contents 1 Geometric Revolution 2 Plato and The Academy 3 Eudoxus and his Students 4 Conclusion Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy Deductive Reasoning and the Geometric Revolution Eudoxus and his Students Conclusion Outline 1 Geometric Revolution 2 Plato and The Academy 3 Eudoxus and his Students 4 Conclusion Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy Deductive Reasoning and the Geometric Revolution Eudoxus and his Students Conclusion Last Time Last time we saw that the Greek problems of antiquity were attempted by many esteemed mathematicians. Using straight-edge and compass, attempts were made to: Square the Circle Double the Cube Trisect the Angle Additionally, recall that Zeno's paradoxes had highlighted that the Pythagorean ideal of space/time subdivision by rationals was insufficient to explain the real world p Further, the discovery that 2 was incommensurable had rocked the very foundation of mathematics at the time. Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy Deductive Reasoning and the Geometric Revolution Eudoxus and his Students Conclusion Geometry The influence that these paradoxes and incommensurability had on the Greek world was profound Early Greek mathematics saw magnitudes represented by pebbles and other discrete objects By Euclids time, however, magnitudes had become represented by line segments `Number' was still a discrete notion, but the early ideas of continuity was very real and had to be treated separately from `number' The machinery to handle this came through geometry As a result, by Euclids time, geometry ruled the mathematical world and not number. Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy Deductive Reasoning and the Geometric Revolution Eudoxus and his Students Conclusion Deductive Reasoning The origins of deductive reasoning are Greek, but no one is sure who began it Some historians contend that Thales, in his travels to Egypt and Mesopotamia, saw incorrect `theorems' and saw a need for a strict, rational method to mathematics Others claim that its origins date to much later with the discovery of incommensurability Regardless, by Plato's time, mathematics had undergone a radical change Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy Deductive Reasoning and the Geometric Revolution Eudoxus and his Students Conclusion Changes to Mathematics The dichotomy between number and continuous magnitudes meant a new approach to the inherited, Babylonian mathematics was in order No longer could `lines' be added to `areas' With magnitudes mattering, a `geometric algebra' had to supplant `arithmetic algebra' Most arithmetic demonstrations to algebra questions now had to be reestablished in terms of geometry That is, redemonstrated in the true, continuous building blocks of the world The geometric `application of areas' to solve quadratics became fundamental in Euclids Elements. Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy Deductive Reasoning and the Geometric Revolution Eudoxus and his Students Conclusion Geometric Revolution Some examples of this geometric reinterpretation of algebra are the following: a(b + c + d) = ab + ac + ad (a + b)2 = a2 + 2ab + b2 This reinterpretation, despite seeming over complicated, actually simplified a lot of issues p The issues taken with 2 were non existent: If you wanted to find x such that x2 = ab, there was now a geometric way to ‘find’ (read: construct) such a value Incommensurability was not a problem anymore. Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy Deductive Reasoning and the Geometric Revolution Eudoxus and his Students Conclusion Closing the Fifth Century BCE These heroes of mathematics inherited the works of Thales and Pythagoras and did their best to wrestle with fundamental, far-reaching problems The tools they had at their disposal were limited { a testament to their intellectual prowess and tenacity The Greek problems of antiquity, incommensurability, and paradoxes illustrate just how complicated the mathematical scene was in Greece during the fifth century BCE. Moving forward, geometry would form the basis of mathematics and deductive reasoning would flourish as a way toward mathematical accuracy Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy The Academy Eudoxus and his Students Conclusion Outline 1 Geometric Revolution 2 Plato and The Academy 3 Eudoxus and his Students 4 Conclusion Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy The Academy Eudoxus and his Students Conclusion The Academy The fourth century BCE opened with the death of Socrates in 399 BCE Not a mathematician, however his student Plato did care for the subject Plato led the Academy in Athens His appreciation for mathematics is indicated by an inscription placed over the doors of the Academy: \Let no one ignorant of geometry enter here." Plato is not known for being a mathematician, but rather for being a `maker of mathematicians' Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy The Academy Eudoxus and his Students Conclusion The Academy Rafael. The School of Athens. 1509-1511. Fresco. Apostolic Palace, Vatican City. Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy The Academy Eudoxus and his Students Conclusion The Academy Our present investigation will take us through some of the work of: Eudoxus of Cnidus (c. 355 BCE) Menaechmus (c. 350 BCE) Dinostratus (c. 350 BCE) Each of the above were mathematicians in attendence at the Academy Their relationships are: Eudoxus was a student of Plato's Menaechmus and Dinostratus were brothers that were students of Eudoxus Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy The Academy Eudoxus and his Students Conclusion Plato Reportedly, it was Archytas that converted Plato to the appreciation of mathematics As with the Pythagoreans, Plato drew a sharp distinction between arithmetic and logistic Logistic: The technique of computation Deemed appropriate for the businessman or for the man of war who `must learn the art of numbers or he will not know how to array his troops' Arithmetic: The theory of numbers Appropriate for the philosopher `because he has to arise out of the sea of change and lay hold of true being' Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy The Academy Eudoxus and his Students Conclusion Plato Plato seems to have adopted the Pythagorean number mysticism Support for this claim come in part from his writings in two dialoges { Republic and Laws. Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy The Academy Eudoxus and his Students Conclusion Plato Plato seems to have adopted the Pythagorean number mysticism Support for this claim come in part from his writings in two dialoges { Republic and Laws. Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy The Academy Eudoxus and his Students Conclusion Plato In Republic, he refers to a number that he calls `the lord of better and worse births' No one knows for sure what number he is referring to { it has been dubbed the `Platonic Number' One theory is that the number is 604, an old Babylonian number important to numerology Another theory is the number 5040, since in Laws, he notes that the ideal number of citizens in the ideal state is 7 · 6 · 5 · 4 · 3 · 2 · 1 Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy The Academy Eudoxus and his Students Conclusion Plato The separation of arithmetic into number theory and logistic (i.e., pure and applied) extended to geometry as well Plato seemingly revered `pure' geometry In Plutarch's Life of Marcellus (75 CE), he references Plato's regard to mechanical intrusions into geometry as: \the mere corruption and annihilation of the one good of geometry, which was thus shamefully turning its back upon the unembodied objects of pure intelligence." As a result, it may very well have Plato that perpetuated the straight-edge and compass restrictions to the Greek problems of antiquity Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and The Academy The Academy Eudoxus and his Students Conclusion Foundations Influenced by Archytas, Plato would eventually add stereometry (the study of solid geometry) to the quadrivium Additionally, Plato would revisit the foundations of mathematics as well He would emphasize that geometric reasoning does not refer to the visible figured that are drawn in the argument, but to the absolute ideas they represent It is due to him that the following interpretations exist: A point is the beginning of a line A line has `breadthless length' A line `lies evenly with the points on it' Douglas Pfeffer Early Greek Mathematics: The Heroic
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