Lecture 3.1: Euclid and Greek Axiomatics

Homework: rd. Lindberg ch 5: THIS CHAPTER IS FUNDAMENTAL – MAKE SURE YOU KNOW IT WELL
        rd.  Lindberg ch 8 on Arabic Math.: you’ll need it for cocktail parties and you’ll be responsible for it too.
    GS:  rest of the math section (save Ptolemy)

The Rise of Formal Reasoning and Science

The story starts with Heraclitus (floruit i.e. he flourish, was 40 years old in 500 B.C.)
    picked up a distinction between perception and reality
    appearance is in flux: the river always moves
                the flame is always in motion
        their existence depends on their being always in motion
    behind the flux is a logos, a rational order, which is hidden and is more real than the flux

    Parmenides: pursued this rational order
        dismissed the flux as erroneous illusion
        sense perception by which we know the flux is false
    knowledge is got through thought alone
        start from what is known and move by logical necessity from there
    and you will have knowledge all the way

so Parmenides starts from ‘what is, is and must be’ and ‘what is not, is not cannot be’
    since what is is and must be, it cannot not be
        and so cannot cease to exist
        change is impossible

the arguments are highly flawed, but they gave the Greeks the notion of logical necessity

Plato saw a connection bt Parmenidan philosophical argument, i.e. dialectic, and mathematics, which also proceeds by logically necessary argument
    mathematics is a pathway into dialectic
    dialectic is necessary philosophical argument

For Aristotle this kind of necessary deduction formed the demonstrative science
    demonstrative sciences are
        a.  syllogistic: have a “Z” shaped structure
            e.g.  all men are animals
                  all animals are alive
        therefore, all men are alive
        b. they have first principles
            from which conclusions are necessarily deduced
   
Euclid and Axiomatics (see the section in GS)
 Hippocrates of Chios (the doubling the cube guy, 470-400 BCE) (according to Proclus) was the first to compose Elements
Euclid (according to Proclus) writing around 300 B.C.  in the generation before Archimedes
    first Ptolemy asked about easy way to learn geometry, ‘no royal road’

Many Levels of Structure
 
1. Principles
Definitions: some included out of tradition and not used; of some the existence will be proved, e.g. triangle
Postulates : first three are postulates of construction: draw a line; produce a line; draw a circle; equality of right angles; parallels
Common Notions: truth which hold good in geometry, but also hold good outside geometry

2. Proofs and their Parts
1. protasis or enunciation
2. ekthesis or setting out which states the particular data
3. diorismos or definition or specification – the statement of what is  required to do with reference to the particular data
4. kataskeue – construction or machinery used
5.  apodeixis or proof itself
6. sumperasma – conclusion



Elements I. 47: KNOW THIS ONE
    this is the dramatic end of Book I of the Elements

Elements II.11  this is the GOLDEN SECTION (KNOW IT)
    I said that it was the basis of the architecture of the Parthenon
        but Prof. Hurwit of Art History tells me I’m wrong
        and it wouldn’t be the first time.
    however, it is the ‘extraction of the square root”, i.e. how to find a square equal to a given rectilinear area – and that is important, even if Pheidias and Iktinus didn’t care

Elements IX. 20  the infinity of prime numbers:   KNOW THIS ONE
        notice the reliance on line length: geometrical in spite of the numerical nature


Structure of the Books of the Elements

Table of Contents
1. triangles
2. quadrilaterals; parallelograms
3. circles
4. constructions with circles, inscription, subscription
5. theory of proportion
6. application of theory of proportion to geometry
7. earlier theory of proportion
8-9 further arithmetic
    9 ends with how to construct  a  perfect number
10.  commensurables and incommensurables
    introduction of method of exhaustion
11-13 – solids
   
Pappus on Analysis and Synthesis

‘Analysis takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis; for in analysis we admit that which is sought as if it were already done and we inquire what it is from which this results, and again what is the antecedent cause of the latter and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards.
    But in synthesis, reversing the process we take as already done that which was last arrived in the analysis and, by arranging in their natural order as consequences what before were antecedents and successively connecting them one with another, we arrive finally at the construction of what was sought. ‘

Archytas’ doubing of the cube is an example: Hippocrates analyzed the problem of doubling the cube into the finding of the second mean proportional

Archimedes 285-212 (Sicily)
    the ancient Einstein
        the growth of royal patronage
    Eureka, eureka
    mathematical arrogance: give me a place to stand and I shall move the earth
    Sandreckoner – god-like status of the mathematician
    mechanical method in de Methodo

main achievement: quadrature of curved lines and cubature of curved volumes

Diophantus of Alexandria: algebra (c.  A.D. 150)
    we looked at some of his problems