Lecture 5.2  Theory of Mathematics and Mechanics

Read for Mechanics: 6.5; 6.6; 6.7; 6.11
Read for Optics: 7.1; 7.4
Read for Astrology: 4.4; 4.6; 4.7

I.  Theory of Mathematics

a)  Platonists
    theory of Forms
    Forms are unique, unchanging, immaterial, objects of knowledge
    they are opposed to multiple objects of sense perception, changing and material
    mathematical entities are in between
        they are unchanging, immaterial objects of knowledge
        but they are multiple and manipulable

b) Aristotelians
    substance is material
    mathematics is the study of the abstracted quantity of material substance
    Aristotle’s favorite example:
        snub nose is curvature in nose flesh
        curvative is shape abstracted from matter


II. Mixed Sciences and Axiomatics

Aristotle recognized various levels of abstraction in “pure and applied mathematics”
    e.g. general mathematics, geometry, optics, iridology, material physics
    Almagest, Euclid’s Optics, etc. are geometrical studies of nature

Dispute between Aristotle and Plato important for Galileo and Descartes
    G and D. side with Plato: the physics is mathematical
    and against Aristotle: physical things have physical, not mathematical properties
    -still it is a somewhat bogus question

III.  Mechanics

technology, science and economic development
    slavery
    aristocratic cooption for intellectual elites
    kingly patronage for military purposes

[we didn't cover this material in class; it won't be on the exam BUT IT IS SO COOL
Mechanical Problems cited by Aristotle and Pappus (Mathematical Collection, VIII. 1-5)
    -to find the force necessary to draw a weight up an inclined plane at a given angle
    Heron discusses a study of friction and inclined planes (GS 6.11 pp. 168-9)
    -given a wheel with x cogs, find the diameter of a second wheel with y cogs which will engage with it
Archimedes’ Equilibrium of Planes
    method of determine the inverse proportional law for balance
    difference between dynamic and static methods (compare the mechanical method Archimedes used for determining the relative size of a triangle inscribed in a parabola (GS pp. 29-30)
    see Plane Equilibrium (GS 6.5 p.156-7)
    On Floating Bodies treated as application of equilibrium]

Heron (writing AD 55-68)
    from preface on Pneumatics: “We are further led to write this work from the consideration that it is fitting that the treatment of this subject should correspond with the method given by us in our treatise, in four books, on water-clocks. For, by the union of air, earth, fire and water, and the concurrence of three, or four, elementary principles, various combinations are effected, some of which supply the most pressing wants of human life, while others produce amazement and alarm.”
    on the ataraxia brought on by possessing war engines.