the rise of deductive mathematics: early greek mathematics · pdf file ·...

The rise of deductive mathematics: Early Greek mathematics

The rise of deductive mathematics:Early Greek mathematics

Waseda University, SILS,History of Mathematics


Outline

Introduction

The myths of the foundersSources of the mythsA modern reconstructionAn ancient reconstruction

The first mathematical argumentsHippocrates Lunes (late 5th century)

Conclusion


Introduction

History and myth

Both early Greek history and history of mathematics are full ofmany legendary stories that are told many times over the years,but which may have no basis in what actually happened.(Examples: The murder of Hippasus at sea, Euclid and KingPtolemy discussing the absence of a royal road in geometry,1

Archimedes burning the Roman ships in the harbor ofSyracuse.)

These stories are part of the lore of the discipline. They play animportant role in helping form the identity of the field byproducing a notion of heritage, but they are not history.

1mŸ eÚnai basilikŸn Ćsrapän âpi gewmetrÐan.


Introduction

History and heritage, 1

I We can take as an examplethe so-called Pythagoreantheorem (N):

I From the perspective ofheritage we can say this isa2 + b2 = c2.

I Euclid himself talks aboutthe square on AB, thesquare on AC and thesquare on BC.

I How are these different?


Introduction


Mathematicians have a vested interest in telling the story ofpast mathematics in a particular way – one that sheds direct lighton their own activity. In order to do this, they transform a pastresult, N, into their own conception.

Feature History Heritagemotivation for N important issue unimportant

types of influence can be negativeor positive; bothare noted

clean them up(only direct posi-tive influences)


Introduction


Feature History Heritagesuccessful devel-opments

important, butalso studyfailures, delays,etc.

the main concern

determined? generally notclaimed

may be determin-ist, “we had to gethere”

foundations of atheory

dig down tothem, build onwhat you find

lay them downand build fromthis solid ground

level of impor-tance of N

will vary overtime

not considered


Introduction

The evidence for Greek mathematics

I We have some direct material evidence for arithmetic,surveying, etc., but almost none for theoretical mathematics.

I Almost all of our evidence for Greek mathematics comesfrom the medieval manuscript tradition.

I The theoretical texts are written mostly in Ancient Greek incopies of copies of copies many times removed. Some arein Arabic, Latin, Persian, Hebrew, etc.

I The full manuscripts that we use to make a printed editionare from as early as the 9th century CE to as late as the 16thcentury. (Around 1,000 to 1,500 years later than the textswere written.)

I We have almost no direct sources. We have no ancientautographs. (We have a number of medieval ones.)


Introduction

Papyrus fragment of Euclid’s Elements II 52

2From the Egyptian city of Oxyrhynchus, dated around 1st or 2nd century CE. Thepassage is from Elements Book II proposition 5.


Introduction

The oldest manuscript of Euclid’s Elements II 53

3Written in 888 CE. From the Bodleian Library, D’Orvill 301.


Introduction

Al-Harawı̄’s version of Menelaus’ Spherics4

4Ahmet III, 3464 (1227 CE). Propositions I 57-61.


Introduction

Ancient cultures around the Mediterranean


Introduction

The Hellenic city states, around 550 BCE


Introduction

Ancient Greek political and intellectual culture

I The political organization was generally around small,independent city-states.

I The form of government differed from state to state, butmany of them were democratic,5 or participatory.

I As a result of this, Greek intellectual culture centeredaround open, public debates.

I They developed techniques of rhetoric and dialectic – thepractice of using emotional or logical verbal constructionsto convince others of one’s position.

5In Greece, “democracy” had a limited definition since only malespossessing a certain amount of wealth, or land, were defined as “citizens”and could participate.


Introduction

The public forum

The theater


Introduction

The sophists6

The sophists were a group of traveling intellectuals whoclaimed that they could give the answer to any question.(Protagoras, Gorgias, Antiphon, etc.)

I They collected money for giving classes, in which theyshowed how to use rhetoric and dialectic to convinceothers that the speaker held the correct view on any giventopic.

I They were especially common in Athens, which was ahighly litigious city-state.

Although they were criticized by Socrates, Plato, Aristotle, etc.,the philosophers adopted many of their methods.

Deductive mathematics was born in the context of theseargumentative, public spaces.

6From ”sophia” (sofÐa), skill, intellegence.


Introduction

The place of mathematicians in Ancient Greek culture

I Greco-Roman culture was not a scribal culture.I Most citizens (and some slaves and women) could read

and write. (Nevertheless, this was a small percentage of thetotal population.)

I Literacy was a form of participation in higher political andintellectual culture.

I Although there was a class of people carrying out theaccounting-style mathematics we saw in the Egyptian andMesopotamian cultures, they were socially and culturallydistinct from the theoretical mathematicians.

I By and large, theoretical mathematics was carried out as aform of high culture – it was not meant to be useful, orimmediately beneficial. It was something like music,theater or poetry.


Introduction

An example of mathematics in public: Plato’s MenoI Plato’s Meno is a dialog in which Socrates tries to answer

the question, “What is (human) excellence (Ćretă)?”I The main characters are Socrates and Meno. (Meno is

young, good-looking and well-born, he may be a sophist.)I Along the way, they are trying to figure out whether or not

excellence can be taught, and, further, whether anything canbe taught.

I Socrates believes that all true knowledge is innate – that it isalready there inside us, we just need to properly rememberwhat it is.

I In order to make his point, he asks Meno to lend him oneof Meno’s slaves. He then attempts to show that the slaveis capable of “remembering” mathematics, without beingtold any answers.


Introduction

Reading the Meno

I Three students will read out the part of Socrates, the SlaveBoy and Meno.

I I will draw figures on the board. Note that the figurereferred to in the text has no letter-names.

I The presence of letter-names, in later texts, is an indicationof the transformation from an oral to a textual tradition.


The myths of the founders

Sources of the myths

Early history of science

I It was characteristic of Greek historians to attributeeverything to the work of some individual inventor ororiginator.

I In fact, many things whose origins were unknown wereattributed to famous people of the past, merely becausethey were famous.

I Early Greek histories are a sort of blend of facts and fiction,woven together in what the author considers will make agood story.

I In the case of mathematics, we have reports of, andquotations from, books called the History of Geometry andthe History of Arithmetic, by Eudemus of Rhodes (late 4thcentury BCE).



Sources of the myths

The doxographical tradition

I Later, during the Hellenistic period, the history ofintellectual traditions were collected in a type of work thatwe call doxography – reporting and writing about thethoughts of others.

I Based on the work of Eudemus, and other sources, in laterperiods there developed a body of work expounding thehistory of early Greek mathematics.

I Our actual sources for this material often come fromauthors as late as Simplicius and Eutocius (6th century CE).

I Hence, our idea of the history of early Greek mathematicsis based on a great deal of (re)construction – both ancientand modern.



A modern reconstruction

Thales: the angle in a semicircle is a right angle

Here we look at a reconstruction by Dunham. Note that Thaleshimself left no writings. What might his proof have been like?(Euclid proves the same thing in Elements III 31.)

I A direct proof by (simple) construction.I Preliminaries: (1) Radii of a circle are equal (definition). (2)

The bases of an isosceles triangle are equal (Elem. I 5). (3)The sum of the angles of a triangle are two right angles(180◦).

I We start with a semicircle on diameter BC. We take centerO, and a random point, A, on the arc. We join line AO. . .



An ancient reconstruction

Pythagoreans: Incommensurability in the square7

We want to show that there is no line, no matter how small,that will measure both the side and the diameter of the square.This is directly related to the our notion of irrational.

I An indirect proof by construction.I Preliminaries: (1) Any ratio can be expressed in least

terms, (2) if n2 is even, then n is even.I We start with square ABCD and assume that side AB and

diameter AC can be expressed by some least ratio in twointegers, ef and g. We will then argue that, if this is the casethen g is both even and odd – which is absurd. . .

7This proof comes from Euclid’s Elements X 117. Even in antiquity, it was attributed tothe Pythagoreans.


The first mathematical arguments

Hippocrates Lunes (late 5th century)

Quadrature of the lunes

I Our first evidence for logical arguments come from areport written in the 6th c. CE by Simplicius about workdone by Hippocrates, some eight centuries earlier.

I Much of what we can say about this must be reconstructedfrom the text.

I Hipparchus’s argument relies on a number ofpreliminaries: (1) The so-called Phythagorean theorem, (2)angles in a semicircle are right (attributed to Thales), (3)the possibility of squaring any rectilinear figure, (4) theareas of similar segments of circles are as the squares ontheir chords (A1 : A2 = c2

1 : c22).




Quadrature of a simple lune (Alexander)

I A direct proof by construction.I We begin with a lune, AECF, formed on the side of a

square drawn in circle ABC.I We draw in the internal triangles, 4AOC and 4COB, and

argue that the areas can be related based on the geometryof the figure. . .




Interpreting an ancient text

I We have seen Dunham’s reading of the semicircular lune.He presents a similar argument for the hexagonal lune.Both of these reconstructions are drawn from Alexander ofAphrodisias (end of 2nd centure CE).

I Simplicius argues that Hippocrates thought he couldsquare every lune, because he squares one on thesemicircle, one on a segment greater than a semicircle andone on a segment lesser than a semicircle.

I But in fact, the second two lunes that Simplicius discussesare “special cases,” and Hippocrates must have knownthis.




Two quadratures attributed to Hippocrates bySimplicius

I In fact, Hippocrates arguments are simple and elegant –based directly on the claims that circular segments are asthe squares of their bases.

I He begins by considering the lune with an outercircumference that is divided into two equal parts andstands on a base whose square is equal to the sum of thesquares on the sides.

I Then he examines a lune with an outer circumference thatis divided into three equal parts.

I (The final lune in the reading is based on the same generalprinciple but is more complicated.)


Conclusion

The logical structure of early proofs

I What was the idea of the completeness of the logicalargument?

I What kinds of things could serve as starting points(assumptions)?

I What was the overall logical organization of early proofs?

I Were there theories, or just loose collections of theorems?

the rise of deductive mathematics: early greek mathematics · pdf file ·...

Documents