early greek mathematics: the heroic age · 2018. 10. 14. · geometric revolution plato and the...

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



Table of contents

1 Geometric Revolution

2 Plato and The Academy

3 Eudoxus and his Students

4 Conclusion




Deductive Reasoning and the Geometric Revolution

Outline




4 Conclusion





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.





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.





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





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.





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.





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




The Academy

Outline




4 Conclusion




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’




The Academy

The Academy

Rafael. The School of Athens. 1509-1511. Fresco. Apostolic Palace, Vatican City.




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




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’




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.




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.




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




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




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’




EudoxusMenaechmusDinostratus

Outline




4 Conclusion





Eudoxus

Eudoxus of Cnidus (c. 355 BCE)Student of Plato’s





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.”





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





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:





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





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





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.”





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:





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





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





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.





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





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




Outline




4 Conclusion




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


early greek mathematics: the heroic age · 2018. 10. 14. · geometric revolution plato and the...

Documents