On Numbers And Mysteriousness

I saw a cool proof that the sum of all the positive integers, 1+2+3+4+5+..., is equal to -1/12. Here it is.

This math proof is weird, and I'm not sure I buy it. (Wikipedia seems to agree with it.)
However, I seem more willing to accept it than other people. Why?

We typically conceive of numbers in terms of discrete objects. When I think of "1", I think of a single item. This helps me wrap my head around the concept, and generally it works very well. However, numbers are not discrete objects. They are conventions, constructions, things that do not exist anywhere but our minds and math publications. You're not going to go out and find 2 somewhere out in the universe, floating or otherwise. This can be more clearly seen when we look at complex numbers. The square root of negative 1, i, is unimaginable. i objects won't ever exist. Numbers are ideas.

Yet numbers' lack of objective, solid-state existence does not preclude them from forming the foundation of any technology we have. Numbers give us everything from the convenience of microwaves, to the exchange of goods for currency, to space travel. My point is simply this: Ideas are powerful, especially when you can transcribe them into solid-state existence. The most complicated, intricate achievements and innovations we have as a species come from the most seemingly basic idea of numbers. This basic idea, too, has grown into bizarre esotericisms, seemingly divorced from the objective realities we experience through our sensory perceptions.

This brings me back to the idea that the sum of all positive integers is equal to -1/12. Apparently, through mathematics I'm nowhere near smart enough to comprehend, this sum has actually proven useful in fields such as complex analysis and string theory. I'm not sure what "proven useful" even means; I'm just going with Wikipedia on this. Perhaps more astoundingly, the proof does not require anything other than the most rudimentary understanding of arithmetic and algebra. It's remarkably elegant and concise. It's also a little -- no, a lot -- mysterious, and I'm not sure if it ever won't be a lot mysterious -- to me, or to the most brilliant mathematician in existence.

I think there is some element of this world that will always remain a lot mysterious to us. I'm fine with that. In fact, I think I kind of like it.

-Me