Professor Stout
Office hours: 9-10 MWF, 10-11 TTh
Office
C209C Center for Natural Sciences, 556-3038
Return to L.N. Stout's home page.
Texts:
J.L. Bell A Primer of Infinitesimal Analysis, Cambridge University Press, 1998
A. Robinson Non-standard Analysis , Princeton Landmarks of Mathematics, 1996, original publication 1974
This course invites you to take a fresh look at calculus and the intuitions of its founders through the prism of late 20th century category theory and model theory.
When Newton and Leibniz developed calculus they thought about infinitesimal quantities. When we teach calculus now we use the concept of a limit and the d-e definition of Cauchy-a concept developed well after the main applications of calculus in a successful attempt to found the subject on the properties of numbers.
Two fundamentally different modern approaches to infinitesimals have been developed which recapture the intuition which was so successful for Leibniz and the Bernoulli family: synthetic differential geometry and nonstandard analysis. Both of these approaches use somewhat unconventional reasoning. Changing how you reason about mathematics allows new insights.
The main focus of the course, occupying the first two thirds of the course, will be the development of calculus in a smooth world using infinitesimals which have powers giving 0, the approach of synthetic differential geometry. This approach forces us to use non-classical logic in our thinking; fortunately, there is a good recent book for undergraduates leading the way, J.L. Bell's A Primer of Infinitesimal Analysis. That book also shows how the infinitesimal approach can be used for standard applications in physics: moments, bending of beams, and Kepler's law on equal areas. Showing how higher order infinitesimals relate to theories of multivariate calculus and integration will follow.
In the last third of the course we will explore at infinitesimals with infinite inverses, the approach from non-standard analysis. Our text here is the classic which introduced the subject, Robinson's Non-standard Analysis. In non-standard analysis we are careful about the logical structure of the statements we make and use an extension of the usual reals to include invertible infinitesimals. This extension lives in an ultrapower--a model of the theory of the real numbers somewhat richer than the usual one. Robinson's book takes the subject much further than we will: we will concentrate on the basic construction and its uses in calculus.
Robinson also includes a chapter on the history of calculus and the particular role of thoughts on infinitesimals by the main founders and developers of the subject. We will look at this chapter at the end of the course (after I've gotten all of the original sources translated--Robinson was writing in an era when all mathematicians read French and German, so he gives no translation). We should then be in a position to see how well the modern approaches capture the intuition which was so productive for the founders of calculus.
Prerequisites for the course are Techniques of Proof (Math 200) and either Analysis 2 or Calculus 3. It will count toward the requirement for designated 400 level courses for the major.
Bell's book includes a significant number of appropriate exercises (nearly all are proofs). I will make up similar exercises for the material in Robinson. These will be due on Mondays and will provide one quarter of the points in the course. There will also be three exams in the course, all of equal weight. Tentative times for the exams are on the attached calendar.
I use a straight scale for determining grades. To allow flexibility at boundaries, I reserve the right to change the boundaries, but I will draw them no higher than
A : 90% or over
A- : [87,90)
B+ : [83,87)
B : [78,83)
B- : [75,78)
C+ : [70,75)
C : [65,70)
C- : [60,65)
D : [50,60)
F : below 50%
The line for passing will not move, the others may move downward.
Classes and office hours are what you pay tuition for, so take advantage of them. If you don't come to class you will not learn the material with the same emphasis that I put on it. That will hurt your exam scores and detract from what you learn. I do not deduct points for classes missed. Make-up exams are usually only given when I have accepted a good excuse in advance of the exam.
Work handed in for a grade is expected to be your own work. If you use outside sources, cite them. If you get help from an individual, give credit. Exams will all require all work to be done individually. Homework you may discuss with peers (giving appropriate credit) but the write up should be your own. Any cheating on exams or collaboration on assignments where it has been explicitly prohibited will be treated as a violation of the policy on academic dishonesty in the student handbook and will be reported to the Associate Provost.
Last updated 1/7/2001
lstout@sun.iwu.edu