Mathematical thought from Euclid to Cantor
In the fall of 2011, I taught a seminar on the history and philosophy
of mathematics to sixteen freshmen in the Dietrich College of Humanities
and Social Sciences at Carnegie Mellon University. We relied almost
exclusively on primary sources, focusing on important mathematical
developments and on philosophical reflection on those developments. Along
the way, we read Plato, Aristotle, Euclid, Cardano, Descartes, Galileo,
Newton, Leibniz, Dedekind, Cantor, and Hilbert, among others.
The experience was extremely rewarding, and I highly recommend it to
anyone who has the opportunity. Feel free to contact me for more
information or any of the materials I used. The following list of
readings conveys a sense of what we discussed.
Euclid, The Elements, Book I, through Proposition 3
Euclid, The Elements, Book I, through Proposition 10
Excerpts from Plato, The
Excerpts from the Republic
Continue reading Book I of the Elements
Finish reading Book I of the Elements.
Lear, Jonathan, "Aristotle's Philosophy of Mathematics"
Euclid's "geometric algebra" in Book II.
Eudoxus' theory of Propositions in Book V.
(Various secondary sources as well.)
Read the definitions in Book VII of Euclid, and excerpts from VII and
IX (number theory).
Read the section on Al-Khwarizmi in Struik's sourcebook.
Read the section on Cardano from the Laubenbacher and Pengelley book.
Read the introduction to the section on Viète in the Struik
Start reading Descartes' "Rules for the direction of the mind."
Henk Bos on Descartes' rules
Descartes' Rules 13-21.
Excerpts from Descartes' Geometry.
Excerpt from Galileo's Two New Sciences, third day
Excerpts from Descartes' Geometry (from Struik)
Bos, on Descartes (excerpt from Redefining Geometrical Exactness)
Continue reading Descartes.
Continue reading Descartes.
Start reading Newton's Principia: the Central Argument:
Read the Forward and Preliminaries.
Read Newton's Preface to the Reader, on page 3.
Skim the definitions from pages 5-27.
Continue reading the guide to the Principia:
Read the laws of motion on pages 29 to 30, and the first two
corollaries on 31 and 32.
Start reading Book I, from pages 47 to 71.
Continue reading through the Principia. In particular:
Read Lemma 8 on page 91.
Read Lemma 9 on page 94.
Read Lemma 10 on page 99.
Read Lemma 11 on page 108.
Read the Scholium on pages 119 to 121, and notice that Spencer was
right and I was wrong in the last class.
Start reading Section 2 on page 123, and see if you can understand
Read Proposition 6 on page 178 of Densmore, and Corollary 1 on page 182.
(See also the excerpt from the book by Guicciardini.)
This is the big day! We will work our way through Proposition 11 on
page 227 of Densmore.
Read the Leibniz excerpt from Struik.
Dedekind, "Continuity and irrational numbers."
Read Dedekind's "Letter to Keferstein"
Following the sketch there, read the following excerpts from "The
Nature and Meaning of Numbers" (also in the Essays on Numbers):
Sections 1-3, on "systems," or sets of objects
The definition of a "chain" in Section 37
The "proof" that there exists an infinite system, in Section 66
The notion of a "simply infinite system" in Sections 71 and 73
Read the first few sections of Cantor's "Theory of transfinite numbers."
Read as much of Chapter I of Hilbert's Foundations of Geometry as you
can. Focus on:
The introduction, and the very first definition.
Theorems 3 and 5.
Axiom III, 1.