What do mathematicians do? Most people don’t have a good or fair idea of what it is we do. The mathematics learnt at school fails to capture what the subject is actually about. No we don’t just add large numbers together all day, or recite to hundreds of decimal places. On the other hand, a subject like physics has much better public perception. Plently of non-physists have thought about black holes or the Higgs boson, or at least heard about them. I believe that mathematicians are partly to blame for this disparity – failing to communicate what is an inherently beautiful subject. We have our own versions of the Higgs boson or string theory that we could be sharing with the public. Here I attempt to put my money where my mouth is.
Cardinality: Understanding Infinity
The idea of infinity is an old one, stretching back at least as far as the ancient Greeks. However, it took mathematicians a long time to get a handle on these big ideas. For example, it wasn’t until 1874 that Greorg Cantor showed that there were different sizes of infinity – i.e., that some infinities were bigger than others! I discuss what that means and show how he proved it:
We all know what a cube is. If we saw one we would all recognize a tetrahedron too. Some of us might be able to pick out an octahedron, or even a dodecahedron and an icosahedron. It is a remarkable fact that these are the only five, so called, Platonic solids! How would one go about proving that there were no other such solids? I explain in this article on Euler’s formula and the five Platonic solids
Math Explorer’s Club
Cornell runs an outreach program called the Math Explorer’s Club aimed at introducing new mathematical concepts to high school students grades 8 – 12. In Fall 2015 I ran two workshops with the theme ‘What is proof?’. Here’s what we covered.
- What is mathematical proof?
- Proof by contradiction
- Graph theory and Euler Characteristic
- Proof by induction
- The square root of 2 and a brief history of numbers
Alternatively, all the worksheets have been bundled together for convenience: Worksheets on Proof.