Over at the BBC, Melvyn Bragg presents a weekly radio programme called In Our Time in which are discussed a wide variety of subjects ranging from art and philosophy to politics and drama. Surprisingly, considering the wide range of human culture covered, Bragg devotes a good proportion of episodes to mathematics. He’s just presented a programme on the Negative Numbers and recently presented one on Prime Numbers. If you dig into the science archives you’ll find recordings of programmes on Zero, Infinity, π, Chaos Theory and Renaissance Mathematics. Some of these are conveniently available as podcasts.
(Note that some of the comments about the mathematical topics on the In Our Time website are a little inaccurate, eg. “a team of researchers…calculated the highest prime number”, but the actual guests on these shows are generally mathematicians or historians of mathematics who know what they are talking about.)
The mathematical content may be a bit elementary for some readers here, but this is an entertaining article from American Scientist nonetheless. It’s not every day that there’s an article about group theory in the newsstands, and it’s about a real world problem that I believe many people have pondered over.
There is a great wealth of mathematical fiction. There are obviously works of science fiction with a high mathematical content such as the writings of Greg Egan and Stephen Baxter. But there are popular mainstream works that have a high mathematical content too. For example the excellent The Curious incident of the Dog in the Night-Time has, among other things, some discussion of the Conway’s Soldiers problem and the plays of Tom Stoppard often have non-trivial mathematial content. But if you’d like to have a list of it all, I recently found a fairly comprehensive list of mathematical fiction: MATHFICTION.
Metamath is a project to construct mathematics as proofs in ZFC. This what what we’re all supposed to be doing but in practice proofs tend to be informal arguments that we can convince people could be converted into derivations in ZFC. It looks like a long haul. There are now over 5000 theorems proved but they’ve only just proved a<=b => a<=b+1 in the reals. On the other hand they seem to have several hundred proofs about Hilbert spaces and quantum logic. Meanwhile, Wim Hesselink is hoping to verify a proof of Fermat’s Last Theorem by machine.
There’s a nice little trove of lecture notes on discrete mathematics here. The emphasis is on design theory but other topics are covered too. Design theory isn’t the biggest area in mathematics but the design S(5,8,24) has many interesting properties which have echoes through many branches of mathematics via its connections with certain sporadic simple groups , the Leech lattice and hence subjects like modular forms.
Saul Youssef has a collection of links to papers on exotic variations to probability theory. These are forms of probability theory that share many of the usual axioms of probability theory but in which the probabilities themselves lie in a set other than the non-negative reals eg. the complex numbers, the quaternions, or even the p-adics. The primary motivation is that classical mechanics plus complex probabilities looks a lot like quantum mechanics, and so if you believe in complex probabilities you no longer have to worry about things like wavefunction collapse. Unfortunately it’s all a bit confusing if you’re a frequentist.
Many mathematicians grew up on a diet of puzzles like those set by Raymond Smullyan and Martin Gardner. Unfortunately, ingenious and elegant as these puzzles often are, they frequently have solution methods that don’t give rise to generalisable theory.
So I was recently surprised to find that one of my favourite puzzles of this type, the Muddy Children Problem, is actually an important example that appears in courses on mathematical logic, epistemology, computer science and even quantitative finance. If you haven’t met the problem before then have a go at solving it before looking at the various papers and courses on the subject. A fairly detailed elementary treatment can be found here though there are easier to understand informal arguments in existence.
The main academic approaches to the problem are via modal logic and Kripke models.
In less politically correct days it was known as the unfaithful wives problem and Smullyan’s version of this problem involved logicians with coloured hats.
Did I mention that it’s also a drinking game?