Math 629: History of Mathematics
Frank Sottile

• Opening Remarks: A prominent 20th century British mathematician described the Greeks as `fellows of another college'. Their accomplishments and methods remain valid and inspiring to many, even after over 2 millennia. This illustrates a major feature of mathematics, which sets it apart from other human endeavors: Mathematics is additive. While topics may go in or out of style and what is important may change over the generations, the discoveries of the past remain valid and they form part of the foundation of current work. It is common to cite work published in the 19th century in current research.
      I believe that Stillwell starts with the Greeks because of their modern outlook (hence the above quote), they were the first to use the deductive method and require that mathematical truths be proven. They also used abstraction (think Platonic ideals, or potentially infinite Euclidean lines, or...) There is an enormous amount written about Greek accomplishments in mathematics and culture. Stillwell presents a selection of their more elementary accomplishments.
  Chapter 1 introduces us to two of the great theorems in mathematicians: First, the Pythagorean Theorem. Stillwell does this justice, as it brings together geometry, algebra, and numbers. Famously, there are many distinct proofs of it, and several of the more elementary are given (note that the proofs in Figures 1.7 and 1.8 require the unproven fact that area is additive: The area of a disjoint union is the sum of the areas. This innocent statement hides remarkable subtleties, check out the Banach–Tarski paradox and Hilbert's third problem. The second is of course, the irrationality of the square root of 2, which revealed that completely new class of numbers must necessarily be, and upended the (then) current notions of number. Chapter 2 is a sketch of Greek geometry, and the start of the deductive method. It is worthwhile to reflect on the profound influence of Euclid on Human intellectual culture.     The discussion of curves in Chapter 2 of Stillwell is informed by advances made nearly two millennia after the Greeks; namely René Déscartes' cartesian coordinates in the first half of the 17th century.
• Reading:
  • Please watch The History of Mathematics, from minute 35 to minute 53, on Greek mathematics.
  • Chapters 1 and 2 of Stillwell.
    In your readings of Stillwell, I'd like it if you could include reading the exercises, as well as the text. In this text, like many, some of the material is developed in the exercises.
        The historical notes at the end of the chapter provide some interesting color about the lives of the protagonists; they are likewise worth reading.
  • While we appear to have a great understanding of Greek mathematicians and their mathematics, this knowledge has not come to us as easily as, say our knowledge of 19th century physicists. To appreciate this, please read the St. Andrews page How do we know about Greek mathematics?.
  • In Plato's dialog Meno type "inquisitive" into your web browser's text search window to get quickly to the scene with the slave boy. Read that part of the dialog.
  • Try your hand at Euclidea, an on-line straightedge and compass constructor.
• Assignment: Due Monday, January 29, 2024. (HW 2 and Concept Quiz 2)  
  The quiz (on Gradescope) is four exceptionally short questions with even shorter answers that you need to take after you have completed the reading.
  It is OK to discuss the homework among yourselves.
        Some of these problems will require you to find other material than just what is in the readings.
  Here is a .pdf and a LaTeX source of the assignment.
  To hand in: We are using Gradescope for homework submission. While the point totals on this homework are different than those on last homework (this is to make it easier on the grader), each homework assignment will have the same weight.
  1. Who was the British mathematician who made the remark at the top of the page? What do you think he meant by this?
  2. Greek mathematics deeply affected at least two US presidents. Research and write about the influence of Euclid and Pythagoras on US Presidents (There is one US president for each of these Greek mathematicians). For each, there is at least one very interesting detail/story. Find it, and describe it.
  3. What were the three geometric problems of antiquity? (Sometimes called the Three Classical Problems). For one, describe it origins (at least the best that you can find out). Were any solved by the Greeks (in any fashion)?
  4. Do Exercise 1.3.4 in Stillwell. Note that t is the vertical coordinate of the point where the longish secant meets the vertical axis. How is this related to the `world's sneakiest substitution' from Calculus?
  5. Do Exercise 1.4.2 of Stillwell. This is one of the easiest proofs of the Pythagorean Theorem, and is often presented in elementary geometry classes.
  6. Do Exercises 2.4.1 and 2.4.2 from Stillwell.

Last modified: Tue Feb 06 20:10:39 GMT 2024 by sottile