I think that in the interests of being fair to the creators of the video, you should link to http://www.nottingham.ac.uk/~ppzap4/response.html, the explanation written by (at least one of) the creators of the video, which addresses some of the complaints.
In particular, let me quote the final paragraph:
There is an enduring debate about how far we should deviate from the rigorous academic approach in order to engage the wider public. From what I can tell, our video has engaged huge numbers of people, with and without mathematical backgrounds, and got them debating divergent sums in internet forums and in the office. That cannot be a bad thing and I’m sure the simplicity of the presentation contributed enormously to that. In fact, if I may return to the original question, “what do we get if we sum the natural numbers?”, I think another answer might be the following: we get people talking about Mathematics.
In light of this paragraph, I think a cynical answer to the litmus test is this. Faced with such a ridiculous claim, it’s wrong to engage with it only on the subject level, where your options are “Yes, I will accept this mathematical fact, even though I don’t understand it” or “No, I will not accept this fact, because it flies in the face of everything I know.” Instead, you have to at least consider the goals of the person making the claim. Why are they saying something that seems obviously false? What reaction are they hoping to get?
I think that in the interests of being fair to the creators of the video, you should link to http://www.nottingham.ac.uk/~ppzap4/response.html, the explanation written by (at least one of) the creators of the video, which addresses some of the complaints.
In particular, let me quote the final paragraph:
In light of this paragraph, I think a cynical answer to the litmus test is this. Faced with such a ridiculous claim, it’s wrong to engage with it only on the subject level, where your options are “Yes, I will accept this mathematical fact, even though I don’t understand it” or “No, I will not accept this fact, because it flies in the face of everything I know.” Instead, you have to at least consider the goals of the person making the claim. Why are they saying something that seems obviously false? What reaction are they hoping to get?