Math 252 -- History of Mathematics
(Old course number: 260)

Other relevant web pages:

• My web page of rules about homework, absences, etc.
• list of homework assignments
• an explanation of my grade printouts
• Common errors in undergraduate mathematics
• (under construction) a web tour related to my lectures
• a collection of links to other people's web sites about math history.
• a brief essay by me, about calculus and the Age of Enlightenment.

Course description and syllabus (version of August 25, 1999). This course is offered by the Mathematics Department. It is a mathematics course, with some historical commentary, devoted to mathematical topics chosen for their historical interest. It is not a history course; such courses can be found only in the History Department. Prerequisites for Math 252 are completion of 170b or 175 or their equivalent and some algebra -- preferably both linear algebra and abstract algebra -- or consent of instructor.

Grades will be based on many homework exercises and a few quizzes and tests. I do not yet have a schedule of quizzes and tests or an accounting of what percentage of your grade will be taken up by each of these items. This course is somewhat experimental, so I'm making it up as I go along. However, I can tell you at least this much now:

There will be no surprise quizzes and tests; for all quizzes and tests I will give you advance warning of both the date and the subject matter. The quizzes, tests, and homework will not contain any subjective material (such as essays or posters); it will all be objective material (such as solving math problems of types that we will study in class). I have not yet decided whether there will be some quizzes on names and dates in mathematical history; if there are any such quizzes, I'll warn you of them in advance. The homework will count for a large percentage of the grade, so take your homework seriously.

Makeups for missed quizzes or tests or late homework will not be accepted without a good excuse, and (as teacher) I am the sole judge of what is a good excuse. All excuses must be submitted in writing. I will automatically accept two absences for illness; those two notes needn't be signed by anyone but yourself.

Textbook. The book that is required for the course is

David M. Burton, The History of Mathematics: An Introduction. 4th edition. WCB/McGraw-Hill. Approx. $81.

Topics. Burton's book is organized chronologically -- i.e., century by century. However, we will not go through the book in that order, and we will not attempt to cover all of mathematics. Instead, we will go through a few topics, and for each topic we will cover a somewhat chronological approach. This approach will entail skipping around in Burton's book and in the supplements that I will supply in class; I'll try to make it clear at all times just which pages you should be reading.

My criteria for choosing topics are similar to the criteria given in the book by Laubenbacher and Pengelley: "the importance of the [concept] as a milestone of progress and its accessibility without extensive prior preparation." Accessibility includes the requirement that the teacher should be able to find enough examples, exercises, and theorems that are nontrivial but are elementary enough to be suitable for Math 252. "Nontrivial" means that we'll probably skip things like Babylonian arithmetic.

The following list of topics is still somewhat tentative, and probably will change a bit as the semester progresses.

What is a Number?

• A modern answer is that a "number" is a member of some field. Many of the historical results mentioned below will be easier to understand if we have some understanding of what a field is, so early in the semester I'll distribute a brief introduction to fields, including definitions, examples, and exercises.
• (6th century BC) The Pythagoreans, an early Greek religious sect, believed that everything in the universe could be explained in terms of the ratios of integers. Their confidence was buoyed up by their discoveries of some principles of musical scales. Their beliefs are still reflected in our language today, which uses the same word for "quotient of integers" and "sane." They were dismayed when they discovered that the diagonal of a unit square is not rational. (Selections from Burton's chapter 3, especially pages 86-89 and 104-105.)
• Later Greek geometers tried unsucessfully to square the circle, trisect the angle, and double the cube, using just straightedge and compass (Burton, Section 3.4.). The problem of those constructions went unsolved until the 19th century AD, when algebraists proved all three constructions to be impossible. The numbers attainable by classical geometric construction lie in a subfield of the reals that does not contain p, sin(p/9), or the cube root of 2. We may prove one or more of these results, following a handout distributed in class, based largely on the exposition in chapter 1 of Hadlock's book.
• Ancient Egyptian geometers knew how to solve quadratic equations, and 16th century algebraists also figured out how to answer cubic and quartic equations -- but they didn't entirely understand the answers they were getting, which involved negative numbers and also "imaginary" numbers. (Some or all of Burton's Chapter 7.) The acceptance of negative numbers was just beginning around that time; imaginary numbers were not fully understood until the early 19th century (Burton's section 11.4; also a handout distributed in class). You'll learn how to solve 3rd and 4th degree polynomial equations -- it's like the quadratic formula, but more complicated.
• Abel and Galois proved that there is no analogous formula for solving polynomial equations of degree greater than 4 by extraction of roots. But even if there is no analogous formula for the roots, the roots are still there -- the Fundamental Theorem of Algebra says every nth degree polynomial has n roots. A proof was given in 1746 by d'Alembert; several other proofs were later given by Gauss. Until recently, most history books attributed the theorem to Gauss, but some recent books give d'Alembert the credit. We'll go through d'Alembert's proof, which (from our modern viewpoint) is probably the easiest proof. Then we'll use modern computers to find approximate solutions to some polynomial equations.
• Around 1700, Newton and Leibniz and other mathematicians figured out how to answer many kinds of calculus problems -- but they didn't entirely understand the methods they were using, which seemed to involve "infinitesimal" numbers. Indeed, George Berkeley is most remembered for his 1734 criticism of infinitesimals -- they lacked any rigorous foundation and seemed like mere superstition. (This difficulty actually can be traced back to the ancient Greeks, who went to great lengths to avoid using limit arguments that they did not know how to justify.) It was a century later that Cauchy started using the epsilon-delta proofs we find in calculus books today, thus making the infinitesimals unnnecessary.
          Even Cauchy's arguments were not complete. It was in 1872 that Dedekind published the first clear and rigorous explanation of what the real numbers really are, thus making it possible to finally fill in the holes in the reasoning of d'Alembert, Gauss, Cauchy, et al. Ironically, in 1960 Robinson finally found a rigorous foundation for infinitesimals, justifying the original intuition of Newton and Leibniz. (However, the epsilon-delta proofs still seem simpler to most mathemticians.) Although we won't go through Robinson's complicated arguments, we can easily show that there do exist fields with infinitesimals -- i.e., non-Archimedean fields. One such field is given by the collection R(t) of all rational functions in one variable t, with real coefficients. In retrospect, there is not really a great deal of difference between
  • the infinitesimals of Newton, Leibniz, and Robinson, which were positive constants smaller than any ordinary positive number
  • the delta of Cauchy, which is a positive variable that is shrinking and eventually becomes smaller than any fixed positive number
  • the infinitesimals in R(t), which are just functions g(t) that become smaller than any fixed positive number when t becomes sufficiently large.
  All three small quantities are used in nearly the same fashion, for nearly the same purpose. -- Some of this material will be covered in notes distributed in class; a little of it can also be found in parts of Burton's section 11.3.

Historical topics related to calculus

That discovery had tremendous cultural significance (though I don't know if we'll spend any time on it). For millenia, humans had been at the mercy of forces of nature, particularly from the heavens: too much or too little rain would destroy the crops, and the times of planting and harvest could be read from the calendar that could be read in the motions of the stars. Nature sometimes seemed cruel and arbitrary, but Kepler's Laws showed God to be a geometer, and Newton's Laws showed that even the complicated laws of the universe follow from simple ones. This discovery was a major contributing factor in the start of the so-called Age of Enlightenment, an age in which philosophers (Voltaire, Rousseau, Jefferson, etc.) began to believe that the universe is not chaotic and arbitrary, and that humans can come to understand it and -- to an increasing degree -- control it. Thus, Newton's calculus may have contributed to a world-view that made possible democracy, the steam engine, and the industrial revolution.

Perhaps we'll also spend a few days looking at the Henstock integral. It's only an accident of history that college students nowadays learn the Riemann integral (1880's) rather than the Henstock integral (1950's). The Henstock integral is much more powerful, and its definition is only slightly more complicated.

Geometry

Non-Euclidean geometry is not just a dead and unchanging part of mathematical history. New discoveries are still being made in this branch of mathematics. Perhaps even more exciting, new discoveries in astronomy give new significance to non-Euclidean geometry. Astronomers soon hope to have observations and measurements accurate enough to decide what kind of curvature our physical universe has -- a question raised centuries ago, by Gauss. Also, is the universe finite or infinite? Also, physicists soon hope to decide how many dimensions our universe has. (It's a lot more than 3.) An article about the shape of the universe appeared in the December 1998 issue of the Notices of the AMS. And here are a few more links about the shape of the universe: a b c d e f g h i j k l

In 1868 Beltrami showed that hyperbolic geometry can be modeled within Euclidean geometry, thus proving that if the axioms of Euclidean geometry are consistent, then the axioms of hyperbolic geometry are consistent. Thus, geometry was the first branch of mathematics in which the modern role of axioms became clear. It served as an example a few decades later, when mathematicians began to axiomatize set theory.

Infinity, Set Theory, Logic, Constructivism

In mathematics, the Age of Enlightenment may have reached its highest point in Hilbert's plan (stated most clearly in the 1920's) to unify all of mathematics on a firm foundation of logic and set theory. But Hilbert's program turned out to be unfeasible --- around 1930 mathematician Gödel and physicist Heisenberg demonstrated that some uncertainty is unavoidable. (The rise and fall of certainty is traced particularly well in a history book by Kline.) It is ironic that Gödel, himself a Platonist, may have done more than anyone else to weaken the Platonist view of mathematics. Gödel strengthened the Formalist view -- i.e., that what really exists is not the collection of mathematical objects, but rather the language that mathematicians build and use to describe those objects. The Formalist view is not entirely satisfactory -- e.g., it suggests that mathematical truth is somehow subjective; it does not explain the certainty that mathematicians still usually feel about their results. At present, there does not seem to be a satisfactory philosophy of mathematics, though some philosophers and mathematicians are still searching for one.

Mathematics has gradually become less controversial, over the centuries, as mathematicians have gained a clearer understanding of what constitutes a proof. Today there is very little room for opinion about the truthfulness or "legitimacy" of anything in mathematics; mathematicians are in full agreement about it. Probably the last great controversy or disagreement in mathematics was that over constructive versus nonconstructive mathematics. I think the controversy ended when the language of mathematics evolved enough to permit the two viewpoints to coexist; but I think that was around 1930, when Gödel proved that the Axiom of Choice was consistent with the rest of set theory. If time permits, we may also look at earlier instances of the controversy over constructivism and similar issues, and the interesting biographies of Cantor and Brouwer.

Other topics

• Slide rules, nomographics, and other pre-electronic calculating devices.
• Number theory, with particular attention to Fermat's Last Theorem.
• Projective geometry.