Wowing without warping

I'm usually a big fan of Numberphile but their most recent video seems to me to, at best, miss the point, and at worst, trade on dishonesty. If you haven't seen it, it's titled "ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12". You should definitely watch it, because it's really interesting. What's the problem?

The thing is, there are lots of counterintuitive results in maths and physics. There are lots of moments of wonder that our students should experience and enjoy and understand. This is potentially one of them. But the wow in this video pivots on a deceipt that is never addressed: that there is a consistent meaning of sum for a divergent sequence which corresponds either to our intuition or to our regular definition of adding.

It took a fair amount of work in mathematical analysis to put the sums of convergent series on a rigorous footing, but we do have a definition (the limit of the partial sums) that matches up quite nicely with our intuition about what should happen if we add, for example, $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}...$

By contrast, there is an extensive menu of summation methods for divergent series and which one you choose depends on what properties you want it to have. The various flavours do have features in common. They give the expected answers to finite or convergent sums, for example, and they satisfy linearity, in the sense that we want the sum of any two series to be the sum of the single series generated by adding individual terms together pairwise, which is the property that the calculation in the video hinges on.

Some definitions don't give an answer to the series in question. Some, like the Ramanujan sum, assign it a value of -1/12. But neither Ramanujan, nor any other mathematician or physicist, would interpret this as meaning 'if you keep adding the natural numbers for ever you get -1/12'.

One fairly significant property of finite or convergent summation, that a sum of positive terms is bigger than any one of them, hasn't survived this definition, but rather than encouraging students to think about this as a compromise solution and explore its implications or consider alternative approaches, they're supposed to just accept that it's a bit weird and move on.

The second deceit is an unnecessary and lamentable appeal to authority: "physicists have proved this using string theory!" No. There is a result in physics that uses a certain summation method to rule out some possible canditates for the structure of the universe. Which is awesome and interesting in its own right, but not the same thing at all.

The idea that we can make up rules for doing impossible sums and it sort of works in some senses but not in others, and that the technique has an application to theoretical physics that might constrain the number of dimensions out of which our universe is built is... brilliant. And it's brilliant without recourse to a tabloid headline like "Physics proves maths is cray-cray".

There is an opportunity here to think about how mathematics works and how it gets its legitimacy; about how mathematical structures and operations are built to satisfy some desirable properties often at the expense of others; and how miraculous it is that these choices later find application in the physical world. It's sad if these opportunities aren't taken.

There's a tendency in education toward the well-intentioned lie. It's understandable - deploying the odd Amazing Factoid generates interest and motivation. But the truth, though sometimes more complicated, is usually more rewarding. In any case, if we're in the business of developing critical thinking in our students, we're doing them a disservice if we treat wow-moments as an end in themselves rather than as a starting point for a deeper discussion.