Two things happened recently, none of them new, but the first time I paid real attention to either of them, to see what they were trying to tell me, instead of just nodding along.
First, we know of some big numbers. Knuth arrows, Ackermann numbers, Graham's number. TREE(3).
These numbers are so big that we have no conception of how big they are. Any adjective, any analogy, falls short.
The Buddha was once giving a loving bashing to a lazy disciple — Imagine a turtle swimming in the ocean, periodically coming out to take a breath. Imagine also, that there is a hollow tube floating around on the same ocean.
Now think of how long it would take before it just so happens that the turtle comes up for air, and finds himself popping out right in the middle of the floating tube.
It takes longer, the Buddha chided, for one to attain a human birth with appropriate conditions for enlightenment. And here you are, writing silly blogs.
As huge as the iterations before turtle or the human soul finds a life buoy are, they are nothing compared to TREE(3).
It is an inexhaustible supply of integers we cannot comprehend, yet I can still make it incomprehensibly bigger — TREE(Graham's number), or TREE(TREE(3)) — just like that, by typing a few letters.
Second, that the sum of all natural numbers, 1 + 2 + 3 + 4 + ..., is
-1/12.
From Euler to Ramanujan, multiple mathematicians have independently arrived at that fact. It is a fact printed in quantum Physics textbooks, and used in routine calculations to get them to agree with “reality”.
I don't know what it means for a divergent sum to circle back on the number line, but what I do know is that it snaps me out of that false sense of understanding I had lulled myself into — that infinity is an unreachably big number.
Infinity is not a big number. We know of numbers that are bigger than infinity.
Infinity is not something I understand. Which is fine; great actually, like re-encountering a sense of wonder at this beautiful mystery. I also don't think it is something un-understandable. Likely not in my lifetime, but if humanity survives, I do think it'll understand what infinity is, or are. And it won't just be a big number.