Our second day of class went fairly well, I thought. Students appeared to be working well together, building consensus (or at least near-consensus) on the truth or falsity of four statements. As a refresher, here they are:
- Given any triangle, there is always a circle that can be inscribed in it.
- For ever integer it is possible to choose points on a circle in such a way that if you connect every pair of points with straight line segments, the circle will be divided into regions.
- Let be the product of the first primes. For every positive integer , is prime.
- Every even number greater than 2 can be written as , where and are both prime numbers.
Since this class is all about proving things, I offer the following image as proof that, indeed, students were working on these true or false statements:
As we discussed in class, no one yet has an answer to the fourth statement. I don’t mean no one in our class, I mean no one, anywhere. (I suppose its possible that someone out there has an argument that demonstrates whether this is true or false and has chosen not to share it… let’s ignore that unlikely possibility.) Indeed, this is a famous conjecture, one called Goldbach’s Conjecture.
I think there is a good lesson here. Perhaps even multiple lessons. For instance, I think its important for FOM students to keep in mind that I might be trolling you. Yes, I had heard of this conjecture before compiling this assignment. Yes, I intentionally included it amongst other statements that can, actually, be argued true or false. Another lesson (hopefully?) learned here: mathematics is incomplete. There are lots and lots of mathematical statements that we can’t yet decide are true or false. In fact, this is what mathematical research is all about, finding new statements and proving whether or not they are true or false. What’s even better — and we’ll talk about this later in the semester — there are even mathematical statements that can never be proven true or false! Its crazy, I know, but there it is: sometimes a statement can never be proven true nor can it ever be proven false.
I will probably post another entry concerning my Blue Eyed Faculty story, but for now I raise my glass to this year’s FOM class(es). Cheers to an intoxicating semester!