What is mathematics?

There is a video doing the rounds that shows that 1 + 2 + 3 + … = -1/12. For those like me who don’t like videos, here is the math. It involves nothing more than simple algebra and it is not a hoax. Apparently, Euler reached the same result.

S1 = 1 – 1 + 1 – 1 + …
1 – S1 = 1 – 1 + 1 – 1 + … = S1
S1 = 1/2
S2 = 1 – 2 + 3 – 4 + 5 – 6 + 7 – 8 + …
S2 =     1 – 2 + 3 – 4 + 5 – 6 + 7 – …
2 S2 = 1 – 1 + 1 – 1 + … = S1
S2 = 1/4
S = 1 + 2 + 3 + 4 + …
S – S2 = 4 + 8 + 12 + 16 + … = 4S
S = -1/12

The reason for this highly non-intuitive result is that all of these sums are divergent series, and the equality used in the second line of the math above doesn’t hold in a strict sense.

Apparently summations like this are useful even if they seem to make no sense. Somewhat like complex numbers.

And that fact raises the question that is the title of the post. More on this later when I get some time to ponder it.

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