Required Books:

1. Proclus, A Commentary on Euclid's Elements I, trans. by G. Morrow, with intro. by Ian Mueller (Princeton:  Princeton University Press, 1992)
2. Gottlob Frege, Foundations of Arithmetic, 2nd ed., trans. by J.L. Austin (Evanston:  Northwestern University Press, 1990)
3. Stewart Shapiro, Thinking about Mathematics (Oxford:  Oxford University Press, 2000)
4. Mark Balaguer, Xerox Packet
5. Henry Mendell, Xerox Packet

Class Notes

Introduction to the third of the course

Reading for last 3 sessions of Mendell's part.

• Times of Arrival

Note that readings may be added from time to time; these additions will be posted below.

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• Outline of the First Four Weeks of the Course

1. Babylonian/Egyptian style:  mathematical problems with actual numbers and a veneer of being practical (in the Babylonian case, the veneer is often very thin).  Some Babylonian problems have the form of a geometrically based algebra.

I assume that this conception of mathematics poses nothing philosophically more puzzling than any other ordinary part of human discourse.  In any case, it does not seem to have given rise to any.  Characteristic of Babylonian mathematics is a sexagesimal (base-60) number system, which (especially with the incorporation of a ciper or 0 around the 3rd cent. B.C.E.) allows for the easy computation  of rational numbers.  When the ideas behind this are incorporated into a decimal (base 10), ca. 500 C.E. in India, we get the foundation of the modern decimal system.

2. Greek style:  mathematical propositions expressed in abstract terms (without actual numbers) are demonstrated by deductive methods.  In geometry, actual numbers are replaced by identifying letters.  Problems, still centrally important, are solved with procedures of construction whose resulting properties still need to be proved deductively.  Note that Greek mathematicians, in this higher mathematics, treat numbers either as whole numbers, but that proofs about whole numbers use sample numbers (and so until the time of Descartes).  In his algebra, however, Diophantus (ca. 3-4 cent. C.E.) treats of positive rational numbers.  Note also that Greek mathematicians used Egyptian parts or unit fractions (2/5 might be expressed as a third, fifteenth).

3. Algebra and the development of analysis.  This tradition emerges out of the Babylonian tradition, mediated by Indian mathematics, and tempered by te Greek.  It is generally associated with medieval Arabic mathematics (Al-Khwarismi and Omar al-Khayyam), and later with Cardano, Vieta, and Descartes.  An important characteristic is the development of the concept of positive real number and solutions to problems which are neutral between arithmetic (so expanded) and geometry.  Important philosophically is al-Khayam's treatment of ratios as numerical quantities, hence the germ of the notion of positive real numbers.

4. Analysis and the development of calculus, 1600-1800.  The period is characterized by the use of infinitary techniques (both large and small) and the gradual reduction of geometrical problems to algebraic problems.  Characteristic is a free wheeling attitude towards the infinitely small and occasional attempts to justify it and not occasional attempts to retrench in a 'geometrical' style.  Two general approaches to the small are notions of infinitesimal but unequal quantities and bringing in time and velocity at an instant as an essential parts of geometry (time is already crucial in some constructions of Archimedes).

5. Foundationalism (1800-).  Characteristic is the establishment of fundamental theorems of algebra and number theory and the assumptions underlying them.  This leads to the exploration of axiomatic theories.
Some Philosophical Issues
 

Some Philosophical Issues in Ancient Greek Mathematics

2. Epistemological Issues:  these are different depending on the going epistemological questions.  For example, in the time of Plato and Aristotle, a central issue is what conception of deduction allows us to say that mathematics is a form of knowledge in what sense.  This then determines certain conceptions of proof.  We find similar concerns in almost every subsequent period in the history of mathematics.

3. The infinite:  how, if at all, can the infinite be allowed into mathematics?  There are five basic issues.

i) Can proofs be infinite in length?  Aristotle attempts to produce an argument against this thesis.
ii) Suppose that we have a infinite procedure which approximates a figure.  Are we then allowed to say that the approximated figure has the properties of the approximating figures?
iii) In what sense can we, if at all, speak of an infinite division of a figure?
iv) Is there a sense in which there is a figure which is infinitely large?
v) Does it make sense to think of a figure of n-dimensions as composed of an infinite number of figures of n-1 dimensions, e.g. of a solid as composed of an infinity of planes.