**What are we going to learn about mathematics from each of these topics?**

Before stating the purpose of this course, I warn you that I am shamelessly promoting a certain viewpoint as to what mathematics is really all about – or *should* be all about. The selection of topics promotes my viewpoint. I invite you the student to listen carefully to my case, learn the topics, but always feel free to disagree, passionately if need be. Maybe you think I should have covered different topics, assigned different readings, or presented different proofs. If you think so — say so. I come from a large loud family and I welcome vigorous debate. You defend your turf, I’ll defend mine, and we’ll learn from each other. We are in for an adventure! So here is what this course is about.

- We start in earnest with Thales who is considered to be the first mathematician in the way we think of a mathematician – somebody who uses deductive reasoning to prove things. He saw the need to not only make mathematical observations but to prove them in a systematic way.
- Pythagoras followed Thales’ desire to prove mathematical observations but also added a passion for mathematics and the need to use mathematics to understand the world. Mathematics for the Pythagoreans was done for its own sake, and as a gateway to the divine, and not for its applications to business, science, economics, and engineering.
- Through Euclid we learn that part of the beauty of mathematics is its structure, its order, and its solid foundation. It all just seems to fit together so nicely. Euclid’s
*Elements*gathered up much of mathematics of that time and put it in good working order – making it perfectly clear as to what you can assume (the axioms) and what you need to prove (the propositions). - The genius and art of mathematics is found in Archimedes. Archimedes not only proved some amazing things but did it in such an elegant way. His works are considered by scholars to be some of the great masterpieces of the written word.
- We learn through the angle trisection problem that, as in science, we need tools and progress can’t be made without them. The tools to finally resolve the angle trisection problem were invented some 2000 years after the initial problem was posed.
- Through Euler’s discussion of polyhedra we see that an effective method of proof is to reduce a complicated problem to an easier one. We also learn that original versions of proofs are not always correct or convincing. Euler’s proof was not as convincing as it should be. In mathematics, as in other subjects, there is a notion of priority as to who really discovered something first. Euler had a slightly flawed proof which was put on firmer ground by Cauchy. However, Descartes had a proof (perhaps flawed) much earlier than both Euler and Cauchy. Who should get the credit and our admiration?
- With the prime number theorem we see that progress in mathematics is often made by a number of people and often at the same time – as there were two who in 1896 independently solved the prime number theorem. We also learn, as we did with the angle trisection problem, that it can take several hundred (even thousands) of years to finally resolve a mathematics problem.
- Contrary to popular belief, not all mathematics is known. The Goldbach conjecture (and its close cousin the weak Goldbach conjecture) remain very much open problems to this day – waiting for somebody to finally solve them and claim the glory. In fact, there is problem closely related to the prime number theorem called The Riemann hypothesis which is one of the most famous unsolved problems in all of mathematics. There is even one million dollar bounty on this problem by the Clay Mathematics Institute.
- Mathematicians are not perfect. We see from the four-color theorem that the first proof was incorrect and nobody noticed until 11 years later! Others had errors in their proofs as well. We also see in the solution of the four-color problem that there is room for controversy in mathematics. The final solution of the four-color problem uses an intricate computer program – which raises some eyebrows. Was is programmed correctly? Is a brute force computation like this really an elegant mathematical solution?
- Fermat’s last theorem is a wonderful surprise. It starts off as a simple number theory problem that should be solved by some clever tricks. Instead, it stumped mathematicians for 300 years and, in the end, wound up inspiring a solution to the profound Taniyama-Shimura conjecture.
- The concept of infinity has been studied for centuries. We learn through Cantor that infinity in mathematics has a precise definition and that there are “levels” of infinity. Indeed there are many surprises when mathematics get their hands on infinity!
- The axiom of choice and the continuum hypothesis, along with the incompleteness theorem shows us that mathematics is always a work in progress and is never complete. There will always be things that we can never prove nor disprove.
- We will read a wonderful essay by the early twentieth century mathematician G. H. Hardy. Hardy beautifully makes the case for why he spent a lifetime doing mathematics. Mathematics, according to Hardy is not supposed to be useful but beautiful and inspiring.