"Lecture" for Week 10

I have been striving mightily for the past few days to find some "value added" that I can provide you this week, because recent lectures have been rather skimpy and Dr. Allen's on-line material is very weak on the 18th century. Unfortunately, I have been unable to find references that provide as much information as I want on the two topics I picked out, so what I say will be slightly vague. The best resource has been a book by J. Stillwell, "Mathematics and Its History" (2nd edition, Springer, 2002). (It might make a better text for this course, but let's not go there today.) Also useful were C. B. Boyer's "A History of Mathematics" and the MacTutor (St. Andrews) and Wikipedia pages about Bézout.

Non-Euclidean geometry: The case of the obtuse angle

In my opinion Katz missed the boat in his discussions of the parallel postulate, Secs. 15.1 and 19.1. (THIS CRITICISM IS WRONG; SEE LECTURE 10 OF SPRING 2015.) He gives the impression that Saccheri and Lambert showed that "the hypothesis of the obtuse angle" can be disproved from Euclid's first four postulates. But it is notorious that the geometry of a sphere is a consistent model of that alternative (no parallel lines exist; the angles of a triangle add to more than 180 degrees; etc.). Indeed, it seems from p. 393 that Lambert was in some sense aware of that fact (usually attributed to Gauss in the next century).

Apparently this seeming contradiction hinges on an ambiguity in one of Euclid's postulates, the assumption that a straight line can be indefinitely extended. This is true of a great circle, in the sense that it just keeps winding around the sphere as far as you want to go. But it is false if interpreted as saying that you keep encountering new points, or that the total length of the line is infinite. That latter interpretation was used, at least tacitly, in reaching a reductio ad absurdum from the obtuse-angle hypothesis.

Bézout

Our text tells us in Secs. 14.1 and 14.2 about linear equations in n unknowns and polynomial equations in one unknown. At around the same time there was a landmark in the history of polynomial equations in more than one unknown, a subject that is receiving great attention in mathematics departments today, partly because it still contains important unsolved problems and partly because the development of computer technology has put the development of algorithms for studying such equations back near the top of the agenda (both for practical purposes and because numerical examples can now cast a whole new light on pure-mathematical problems).

Étienne Bézout (1730-1783) was a professor of mathematics in naval and artillery academies. He took his teaching and research responsibilities there quite seriously, so his writings have a more elementary and practical flavor than those of, say, Fermat (a lawyer for whom mathematics was always a hobby) or Euler (a full-time pure mathematician at the elite academic level). His name is forever attached to the fundamental theorem about the solutions of simultaneous polynomial equations -- although the main idea goes back at least to Newton and a completely rigorous formulation and proof did not appear until decades later (19th century). Restricted to two unknowns, and expressed in the geometrical language of the time, the simplest form of Bézout's theorem is:

A curve of degree m intersects a curve of degree n in at most mn points. Here it is assumed that the curves do not have components that coincide (so that their number of points in common is infinite).

For example, a conic section is a curve of degree 2, represented by an equation of second order in two variables. It is clear that a noncircular ellipse intersects 4 times with the 90 degree rotation of itself; this is the expected generic behavior. A circle rotated coincides with itself; this case is regarded as "cheating" because the two equations are not independent. An ellipse and a displaced rotated ellipse may intersect in only two points (or 3, or 1, or not at all).

Just as for polynomial equations in one variable, it is possible to improve the theorem to:

A curve of degree m intersects a curve of degree n in EXACTLY mn points, if one knows how to count points correctly.

To understand what this means, consider the cases we already know:

  1. The polynomial x2 + 1 has 2 roots, including complex roots.
  2. The polynomial (x-1)2 has 2 roots, which coincide.
  3. The linear system x+y=2, x+y=5 can be said to describe two (parallel) lines that intersect at infinity.
The last of these may seem to be a shameless example of proof by definition, but there is a subject called "projective geometry" that makes the notion of intersections at infinity both rigorous and nontautological.

Euler

You have more than enough to read, so I will not list this even as "optional" reading, but I have to say it: The special issue of Mathematics Magazine (vol. 56, no. 5, 1983) cited by Katz on p. 362 is full of interesting articles about Leonhard Euler.

Euler worked at the very beginning of modern ideas of mathematical rigor. He helped to create those ideas, but at the same time his own tastes and talents seem to have been as close to those of a theoretical physicist like Paul Dirac or Richard Feynman as to those of a typical modern pure mathematician. By this I mean that he trusted symbols and their formal manipulation far beyond the realm in which they could be logically justified (at the time). If an infinite series diverged, his reaction was not to discard it as undefined, but to assume that it did have a meaning and to attack the mystery of what its meaning was. This attitude often yielded true and useful results that were not rigorously justified until later centuries.