early greek mathematics: the heroic age · 2018. 10. 14. · geometric revolution plato and the...
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Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
Early Greek Mathematics: The Heroic Age
Douglas Pfeffer
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
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 RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
Deductive Reasoning and the Geometric Revolution
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 RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
Deductive Reasoning and the Geometric Revolution
Last Time
Last time we saw that the Greek problems of antiquity wereattempted by many esteemed mathematicians. Usingstraight-edge and compass, attempts were made to:
Square the CircleDouble the CubeTrisect the Angle
Additionally, recall that Zeno’s paradoxes had highlighted thatthe Pythagorean ideal of space/time subdivision by rationalswas insufficient to explain the real world
Further, the discovery that√
2 was incommensurable hadrocked the very foundation of mathematics at the time.
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
Deductive Reasoning and the Geometric Revolution
Geometry
The influence that these paradoxes and incommensurabilityhad on the Greek world was profound
Early Greek mathematics saw magnitudes represented bypebbles and other discrete objects
By Euclids time, however, magnitudes had become representedby line segments
‘Number’ was still a discrete notion, but the early ideas ofcontinuity 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 mathematicalworld and not number.
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
Deductive Reasoning and the Geometric Revolution
Deductive Reasoning
The origins of deductive reasoning are Greek, but no one issure who began it
Some historians contend that Thales, in his travels to Egyptand Mesopotamia, saw incorrect ‘theorems’ and saw a need fora strict, rational method to mathematicsOthers claim that its origins date to much later with thediscovery of incommensurability
Regardless, by Plato’s time, mathematics had undergone aradical change
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
Deductive Reasoning and the Geometric Revolution
Changes to Mathematics
The dichotomy between number and continuous magnitudesmeant a new approach to the inherited, Babylonianmathematics was in order
No longer could ‘lines’ be added to ‘areas’With magnitudes mattering, a ‘geometric algebra’ had tosupplant ‘arithmetic algebra’
Most arithmetic demonstrations to algebra questions now hadto be reestablished in terms of geometry
That is, redemonstrated in the true, continuous building blocksof the world
The geometric ‘application of areas’ to solve quadraticsbecame fundamental in Euclids Elements.
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
Deductive Reasoning and the Geometric Revolution
Geometric Revolution
Some examples of this geometric reinterpretation of algebraare 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 tofind 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 RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
Deductive Reasoning and the Geometric Revolution
Closing the Fifth Century BCE
These heroes of mathematics inherited the works of Thalesand Pythagoras and did their best to wrestle withfundamental, far-reaching problems
The tools they had at their disposal were limited – a testamentto their intellectual prowess and tenacity
The Greek problems of antiquity, incommensurability, andparadoxes illustrate just how complicated the mathematicalscene was in Greece during the fifth century BCE.
Moving forward, geometry would form the basis ofmathematics and deductive reasoning would flourish as a waytoward mathematical accuracy
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
The Academy
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 RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
The Academy
The Academy
The fourth century BCE opened with the death of Socrates in399 BCE
Not a mathematician, however his student Plato did care forthe subjectPlato led the Academy in Athens
His appreciation for mathematics is indicated by an inscriptionplaced over the doors of the Academy: “Let no one ignorantof geometry enter here.”Plato is not known for being a mathematician, but rather forbeing a ‘maker of mathematicians’
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
The Academy
The Academy
Rafael. The School of Athens. 1509-1511. Fresco. Apostolic Palace, Vatican City.
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
The Academy
The Academy
Our present investigation will take us through some of thework of:
Eudoxus of Cnidus (c. 355 BCE)Menaechmus (c. 350 BCE)Dinostratus (c. 350 BCE)
Each of the above were mathematicians in attendence at theAcademy
Their relationships are:
Eudoxus was a student of Plato’sMenaechmus and Dinostratus were brothers that were studentsof Eudoxus
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
The Academy
Plato
Reportedly, it was Archytas that converted Plato to theappreciation of mathematics
As with the Pythagoreans, Plato drew a sharp distinctionbetween arithmetic and logistic
Logistic: The technique of computation
Deemed appropriate for the businessman or for the man of warwho ‘must learn the art of numbers or he will not know howto array his troops’
Arithmetic: The theory of numbers
Appropriate for the philosopher ‘because he has to arise out ofthe sea of change and lay hold of true being’
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
The Academy
Plato
Plato seems to have adopted the Pythagorean numbermysticism
Support for this claim come in part from his writings in twodialoges – Republic and Laws.
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
The Academy
Plato
Plato seems to have adopted the Pythagorean numbermysticismSupport for this claim come in part from his writings in twodialoges – Republic and Laws.
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
The Academy
Plato
In Republic, he refers to a number that he calls ‘the lord ofbetter and worse births’
No one knows for sure what number he is referring to – it hasbeen dubbed the ‘Platonic Number’
One theory is that the number is 604, an old Babyloniannumber important to numerologyAnother theory is the number 5040, since in Laws, he notesthat the ideal number of citizens in the ideal state is7 · 6 · 5 · 4 · 3 · 2 · 1
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
The Academy
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’sregard to mechanical intrusions into geometry as:
“the mere corruption and annihilation of the one good ofgeometry, 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 thestraight-edge and compass restrictions to the Greek problemsof antiquity
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
The Academy
Foundations
Influenced by Archytas, Plato would eventually addstereometry (the study of solid geometry) to the quadrivium
Additionally, Plato would revisit the foundations ofmathematics as well
He would emphasize that geometric reasoning does not referto the visible figured that are drawn in the argument, but tothe absolute ideas they representIt is due to him that the following interpretations exist:
A point is the beginning of a lineA line has ‘breadthless length’A line ‘lies evenly with the points on it’
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
EudoxusMenaechmusDinostratus
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 RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
EudoxusMenaechmusDinostratus
Eudoxus
Eudoxus of Cnidus (c. 355 BCE)Student of Plato’s
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
EudoxusMenaechmusDinostratus
Eudoxus
In Plato’s youth, the discovery of the incommensurable was atrue issue
Theorems involving proportions were now worrisome, for howdo you compare ratios of incommensurable magnitudes?
Eudoxus would provide an answer. His new definition forproportions, which can be found as Definition 5 of Book V inElements is:
“Magnitudes are said to be in the same ratio, the first to thesecond and the third to the fourth, when, if any equimultiples
whatever be taken of the first and the third, and anyequimultiples whatever of the second and fourth, the formerequimultiples alike exceed, are alike equal to, or are alike lessthan, the latter equimultiples taken in corresponding order.”
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
EudoxusMenaechmusDinostratus
Proportions
Definition
Given quantities a, b, c , and d , we declare ab = c
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 bewhole numbers at all! They can be shapes and objects andthe definition still makes sense
This definition encompassed incommensurables and hence putratios back on firm ground
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
EudoxusMenaechmusDinostratus
Menaechmus
Eudoxus had many students, two of which were the brotherMenaechmus and Dinostratus (c. 350 BCE)
To Menaechmus, we owe the discovery of conic sections andtheir generated curves:
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
EudoxusMenaechmusDinostratus
Doubling the Cube
Menaechmus used the parabola to solve the ‘Double theCube’ problem:
Consider a 45◦ right circular cone and cut a parabola out of it:
In modern analytic geometry terms, he deduced that such acurve is given by y2 = `x
`, the latus rectum, had an explicit formula derived fromclassic geometric reasoning
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
EudoxusMenaechmusDinostratus
Doubling the Cube
To double a cube of side-length a: cut two parabolas, eachwith a latus rectum of a and 2a respectively.
Take these parabolas and reorient them at the origin of a 2Dplane.
Make one in terms of x and the other in terms of y .
Graphs of x2 = ay and y2 = 2ax .
The x-coordinate of their intersection is x = a 3√
2 and thusone can double the cube
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
EudoxusMenaechmusDinostratus
Menaechmus
Mecaechmus would later go on to mentor Alexander the Great
Legend has it, when Alexander the Great asked for a shortcutto geometry, Menaechmus responded:
“O King, for traveling over the country there are royal roadsand roads for common citizens; but in geometry there is oneroad for all.”
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
EudoxusMenaechmusDinostratus
Dinostratus
The other pupil of Eudoxus and brother to Menaechmus wasDinostratus
Just as his brother had ‘solved’ the squaring of the circle, hehad ‘solved’ the duplication of the cube.
Dinostratus had noticed that much more can be deduced fromthe trisectrix of Hippias:
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
EudoxusMenaechmusDinostratus
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 ofHippias can be realized by πr sin(θ) = 2aθ
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
EudoxusMenaechmusDinostratus
Squaring the Circle
Observe that Q = limθ→0
r = limθ→0
2a
π
θ
sin(θ)=
2a
πThus π can be constructed from Q and therefore the problemis solved
For this reason, this curve is sometimes referred to as thequadratrix of Hippias
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
EudoxusMenaechmusDinostratus
Squaring the Circle
Of course, Dinostratus did not know the limit limθ→0
θ
sin(θ)
Instead, he reasoned thatıACAB = AB
DQ
Thus AC could be constructed.
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
EudoxusMenaechmusDinostratus
Squaring the Circle
Dinostratus now argued that, remembering that a = AB, thefollows areas are equal:
2ÙAC a
a
In modern notation, we see clearly that
AC =1
4(2πa) =
π
2a
so that indeed:
2AC · a = 2 · π2a2 = πa2.
Dinostratus’ argument again used Greek geometric properties
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
EudoxusMenaechmusDinostratus
Squaring the Circle
Passing from a rectangle to a square was a matter of applyingthe geometric mean:
2ÙAC a
√2aÙAC
Thus, via the quadratrix of Hippias, Dinostratus was able tosquare the cube as well
Obviously his solution, like others before him, violated therules of the game, but these mathematicians were enchantedwith the puzzle itself
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
Geometric RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
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 RevolutionPlato and The Academy
Eudoxus and his StudentsConclusion
Conclusion
In 323 BCE, Alexander the Great died, a year later so didAristotle
This fall of an empire also resulted in a great shift inintellectual leadershipThe city of Alexandria took the place of Athens as the centerof the mathematical world
The pre-Alexandrian age was an important one formathematics
Overcoming paradoxes and incredible obstacles likeincommensurability, mathematicians of this age managed toground mathematics in the logical world of geometry anddeductionIt is not a stretch to argue that this age set the foundation forthe future of mathematics
In particular, we will see how Euclid was influenced and howthe subsequent Golden Age of Mathematics handled the scene
Douglas Pfeffer Early Greek Mathematics: The Heroic Age
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