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 √ 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 √ 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 Age Geometric Revolution Eudoxus Plato and The Academy Menaechmus Eudoxus and his Students Dinostratus 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 Eudoxus Plato and The Academy Menaechmus Eudoxus and his Students Dinostratus Conclusion Eudoxus
Eudoxus of Cnidus (c. 355 BCE) Student of Plato’s
Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Eudoxus Plato and The Academy Menaechmus Eudoxus and his Students Dinostratus Conclusion Eudoxus
In Plato’s youth, the discovery of the incommensurable was a true issue Theorems involving proportions were now worrisome, for how do you compare ratios of incommensurable magnitudes? Eudoxus would provide an answer. His new definition for proportions, which can be found as Definition 5 of Book V in Elements is: “Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and the third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or are alike less than, the latter equimultiples taken in corresponding order.”
Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Eudoxus Plato and The Academy Menaechmus Eudoxus and his Students Dinostratus Conclusion Proportions
Definition a c Given quantities a, b, c, and d, we declare b = d if and only if, given integers m and n, (i) ma < nb implies mc < nd (ii) ma = nb implies mc = nd (iii) ma > nb implies mc > nd
The true beauty here is that a, b, c, and d don’t have to be whole numbers at all! They can be shapes and objects and the definition still makes sense This definition encompassed incommensurables and hence put ratios back on firm ground
Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Eudoxus Plato and The Academy Menaechmus Eudoxus and his Students Dinostratus Conclusion Menaechmus
Eudoxus had many students, two of which were the brother Menaechmus and Dinostratus (c. 350 BCE) To Menaechmus, we owe the discovery of conic sections and their generated curves:
Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Eudoxus Plato and The Academy Menaechmus Eudoxus and his Students Dinostratus Conclusion Doubling the Cube
Menaechmus used the parabola to solve the ‘Double the Cube’ problem: Consider a 45◦ right circular cone and cut a parabola out of it:
In modern analytic geometry terms, he deduced that such a curve is given by y 2 = `x `, the latus rectum, had an explicit formula derived from classic geometric reasoning
Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Eudoxus Plato and The Academy Menaechmus Eudoxus and his Students Dinostratus Conclusion Doubling the Cube
To double a cube of side-length a: cut two parabolas, each with a latus rectum of a and 2a respectively. Take these parabolas and reorient them at the origin of a 2D plane. Make one in terms of x and the other in terms of y.
Graphs of x2 = ay and y 2 = 2ax. √ The x-coordinate of their intersection is x = a 3 2 and thus one can double the cube
Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Eudoxus Plato and The Academy Menaechmus Eudoxus and his Students Dinostratus Conclusion Menaechmus
Mecaechmus would later go on to mentor Alexander the Great
Legend has it, when Alexander the Great asked for a shortcut to geometry, Menaechmus responded: “O King, for traveling over the country there are royal roads and roads for common citizens; but in geometry there is one road for all.”
Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Eudoxus Plato and The Academy Menaechmus Eudoxus and his Students Dinostratus Conclusion Dinostratus
The other pupil of Eudoxus and brother to Menaechmus was Dinostratus Just as his brother had ‘solved’ the squaring of the circle, he had ‘solved’ the duplication of the cube. Dinostratus had noticed that much more can be deduced from the trisectrix of Hippias:
Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Eudoxus Plato and The Academy Menaechmus Eudoxus and his Students Dinostratus Conclusion Trisectrix Revisted
Recall that the objective to squaring the circle is to construct √ π (equiv. π) Let AB = a. In modern polar notation, the trisectrix of Hippias can be realized by πr sin(θ) = 2aθ
Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Eudoxus Plato and The Academy Menaechmus Eudoxus and his Students Dinostratus Conclusion Squaring the Circle
2a θ 2a Observe that Q = lim r = lim = θ→0 θ→0 π sin(θ) π Thus π can be constructed from Q and therefore the problem is solved For this reason, this curve is sometimes referred to as the quadratrix of Hippias
Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Eudoxus Plato and The Academy Menaechmus Eudoxus and his Students Dinostratus Conclusion Squaring the Circle
θ Of course, Dinostratus did not know the limit lim θ→0 sin(θ) ACı AB Instead, he reasoned that AB = DQ Thus ACˆ could be constructed.
Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Eudoxus Plato and The Academy Menaechmus Eudoxus and his Students Dinostratus Conclusion Squaring the Circle
Dinostratus now argued that, remembering that a = AB, the follows areas are equal:
a
a 2ACÙ In modern notation, we see clearly that 1 π AC˜ = (2πa) = a 4 2 so that indeed: π 2AC˜ · a = 2 · a2 = πa2. 2 Dinostratus’ argument again used Greek geometric properties
Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Eudoxus Plato and The Academy Menaechmus Eudoxus and his Students Dinostratus Conclusion Squaring the Circle
Passing from a rectangle to a square was a matter of applying the geometric mean:
a
q 2ACÙ 2aACÙ
Thus, via the quadratrix of Hippias, Dinostratus was able to square the cube as well Obviously his solution, like others before him, violated the rules of the game, but these mathematicians were enchanted with the puzzle itself
Douglas Pfeffer Early Greek Mathematics: The Heroic Age Geometric Revolution Plato and 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 Eudoxus and his Students Conclusion Conclusion
In 323 BCE, Alexander the Great died, a year later so did Aristotle This fall of an empire also resulted in a great shift in intellectual leadership The city of Alexandria took the place of Athens as the center of the mathematical world The pre-Alexandrian age was an important one for mathematics Overcoming paradoxes and incredible obstacles like incommensurability, mathematicians of this age managed to ground mathematics in the logical world of geometry and deduction It is not a stretch to argue that this age set the foundation for the future of mathematics In particular, we will see how Euclid was influenced and how the subsequent Golden Age of Mathematics handled the scene
Douglas Pfeffer Early Greek Mathematics: The Heroic Age