Math 629: History of Mathematics
Frank Sottile

• Opening Remarks: First, this is somewhat late, but the homework is also due in the future, too. This is our last week; my sloth (or over work; I've had a busy week in Texas) makes this necessary. I have decided that we will skip some chapters, which is OK as Stillwell has skipped large swaths of mathematics (where does he talk about electromagnetism and Stokes/Green/Gauss Theorems?). Also, those chapters you may find more technical.
  
      The first for us is on hypercomplex numbers. This shows how even if you wish something to be so, it may not be. Hamilton and others wanted to find a three-dimensional system of numbers that behaved like the real numbers (one-dimensional) and also complex numbers (two-dimensional). Instead, he found a four-dimensional system, called the quaternions, which almost behaves like the others, except that it is noncommutative. It is hard to overstate the importance of this discovery. You all know that many operations, from multuplying matrices to dressing one's self depend upon the order in which they are applied (socks first and shoes second is quite different from shoes first and socks second!). This noncommutativity ended up being crucial for the development of quantum mechanics, which enables, for example, a computer smaller than the size of a room with the capabilities of the laptop I am typing on, not to mention the screen I am looking at, and etc. Also, the unit quaternions form a multiplicative group which is equivalent to the group of rotations in R³. Consequently, every person working in computer animation uses them, and their multiplication is hard-wired into all graphics chips in computers. My knowing this led to my collaborating with two applied mathematicians on a paper inspired by a problem in determining the structure of a certain class of proteins, and then to one of my more influential papers in the past decade.
  
      The development of set theory, especially Cantor's revolutionary ideas, led us to finally develop a theory of infinity as a quantity, a notion that vexed the Greeks. The roots of this were mathematicians trying to understand the very strange sets that came up in understanding aspects of the convergence of Fourier series (another topic mentioned only in passing that led to great mathematical developments in analysis, number theory, and combinatorics–look up the work of Tim Gowers who was a fellow student at Cambridge when I was there). In fact, some have said that Cantor developed the first new branch of mathematics since antiquity; Algebra is as old as counting, and came into its own certainly in China, India, and the Islamic world. Geometry was developed to a high degree by the Greeks, and Analysis, while it was an outgrowth of the Calculus, can be said to have its origins at least in the amazing approximations of Archimedes. Cantor's theory of infinite sets is also remarkably accessible; my high school calculus teacher presented to us the Cantor `middle third' set, as well as his diagonal proof that there are more real numbers than integers–this is a topic I enjoy presenting when I talk to highschool students and in-service teachers. I aso had about 8-10 graduate courses on set theory, logic, and foundations, so I have a soft spot for this.
  In your reading, Stillwell makes an ommission on page 527; the sequence {x_i: i in N} needs to include all real numbers, the construction does not work for every sequence, for example 1/2, 2/3, 3/4, 4/5, .....
  Having re-read this chapter, I now do not like Stillwell's take on these foundational questions.
  Let me add a bit to the ideas presented in Chapter 24. Some of the results, Gödel's incompleteness theorem, the independence of the continuum hypothesis, and the negative solution to the halting problem, are of similar nature in that science has proven that there are limits to our knowledge–some things are not knowable. They have a similar feel to Heisenberg's uncertainty principle.
  
      Lastly, you will read about combinatorics, which is the study of counting or of discrete mathematical structures. While often derided as a mile wide and an inch deep, this mathemartical area has come into its own in the past 100 years, and is increasingly important. Many people argue that it should be taught instead of Calculus (just this week, a friend from graduate school who gave a distingushed lecture at Texas A&M, made this point in a talk she gave–this is Kristin Lauter, the Research Manager for the Cryptography Group at Microsoft Research, and her talk was the reason I was in Texas this week and also why this is coming to you a bit late). I also work in combinatorics, among other mathematical topics, so again I have a soft spot for it. Lest you think that my prejudices are guiding the last few weeks, I am also a huge fan of algebraic number theory, even though I do not work in it.
• Reading:
  Chapters 20, 24, and 25 of Stillwell.
• Assignment: Due Monday, 1 May 2017. (HW 14) To hand in: Email a .pdf to both Bennett Clayton bgclayton@math.tamu.edu and Frank Sottile. fjsteachmath@gmail.com
  1. In Chapter 20, do the exercises 20.5.1, 20.5.2, and 20.5.3
  2. In Chapter 24, do the exercises 24.2.1 and 24.2.2.
  3. In Chapter 25, do the exercises 25.5.2, 25.5.4, and 25.5.5