1 + 2 + 3 + 4 + . . . = ?

So my kids have been watching “Numberphile” videos on youtube lately.

Like this one:

https://www.youtube.com/watch?v=w-I6XTVZXww

and this one:

https://www.youtube.com/watch?v=0Oazb7IWzbA

in which we learned that 1 + 2 + 3 + 4 + . . . , that is, the sum of all natural numbers to infinity, equals – 1/12.

Yeah, I know, but watch the video — maybe a couple times.  There’s a step that looks like a “trick” partway along the way, like cheating, but the physicist explaining this says that this result, as “tricky” as it seems (and there are other mathematical proofs that don’t employ this “trick”) is used in such fields as string theory, and, in that sense, has been validated as being “true.”

The second video consists of another mathematician providing more context:  to the objection that there is no way for an infinite, divergent series to have a sum, let alone one as crazy as – 1/12, he says that, basically, the rules are simply different once you add in the element of infinity.  He calls them “regularized sums”, and this page (from a google search) explains this a bit more.

The other interesting thing from the 2nd video is this:  in response to the objection that 1 + 2 + 3 + . . .  = – 1/12 isn’t “real” he provided counter-examples:

i, that is, the imaginary number of √-1, is, by its name, of course, imaginary.  But mathematicians have developed a whole set of useful calculations around it.  And likewise, it was one taken as a given that such numbers as the square root of √2 weren’t “real” because they couldn’t be expressed in a finite or fractional way.

Now, I thought that I would take this idea and use it as a springboard for deep musings on the nature of truth, and the challenges of truths which are counter-intuitive, and the fact that those young people today don’t want to believe in anything that they can’t see right in front of their faces, or at any rate, don’t want to believe in any moral truth that’s not immediately obvious to them.  But that’s not my skill set.  (Though I have been meaning to type something up on how young people are not necessarily godless; it’s more that their god is Science, an all-powerful deity that grants their wishes without caring about their behavior.)  And, at any rate, just because something works well as a metaphor doesn’t mean that it proves anything.

But it’s still interesting, eh?