4 ms·

Are there even a Graham's Number of things we could count in the observable universe?

Robotbeat 9y ago
Definitely no. Overwhelmingly and absurdly no. Think about it this way. You can count by just adding. |||| is four. When you're counting small things, that works fine. Then you have multiplication. Instead of writing |||||||||||||||||||| we just write 20. We developed a new abstraction to write large numbers easier. Then you have exponentiation: 10000000000 is just 10^10. This is abstraction is all the further you need to go to talk about the number of atoms in the Universe (which is roughly 10^80), the speed of computers (And exaflop is 10^18 operations per second, there are 10^7 to 10^8 seconds in a year, and the universe is 10^10 years old, so if each atom were an exaflop computer, you could do 10^(80+18+8+10) = 10^116 operations. Easily written in scientific notation. This is really all the level of abstraction needed for roughly writing large numbers in science and technology. But not for math. Certain proofs can require absurdly larger numbers. So what if we abstracted exponentiation? So, instead of writing 10^10^10 (which is 10^10000000000), we write, say, 10^^3. And we can keep doing that. But what if we iterate on the number of abstractions? So the number of "^"s is abstracted, i.e. 10 ^{m} 10. and THAT is abstracted n times (i.e. we replace m with the previous 10 ^{m} 10, n times). Graham's number is n=64. No doubt I probably made a mistake in here... But: tl;dr: no, we need to develop totally new abstractions to represent Graham's number. Numbers which can be merely represented directly in our universe are easily written in exponential format.
godd2 9y ago
No. Even if you wrote a single digit in each planck volume of the observable universe, you'd run out of space. It is safe to say that there are far fewer than a Graham's Number of things in here with us.
amyjess 9y ago
One of the best professors I ever had, when talking about intractability, would always begin his lectures by pointing out that there are only about 10^80 elementary particles in the universe. He has a small essay online about the subject: https://www.utdallas.edu/~tfarage/MyPapers/TheBestComputerEver.htm https://www.utdallas.edu/~tfarage/MyPapers/TheBestComputerEv...

Definitely no. Overwhelmingly and absurdly no. Think about it this way. You can count by just adding. |||| is four. When you're counting small things, that works fine. Then you have multiplication. Instead of writing |||||||||||||||||||| we just write 20. We developed a new abstraction to write large numbers easier. Then you have exponentiation: 10000000000 is just 10^10. This is abstraction is all the further you need to go to talk about the number of atoms in the Universe (which is roughly 10^80), the speed of computers (And exaflop is 10^18 operations per second, there are 10^7 to 10^8 seconds in a year, and the universe is 10^10 years old, so if each atom were an exaflop computer, you could do 10^(80+18+8+10) = 10^116 operations. Easily written in scientific notation. This is really all the level of abstraction needed for roughly writing large numbers in science and technology. But not for math. Certain proofs can require absurdly larger numbers. So what if we abstracted exponentiation? So, instead of writing 10^10^10 (which is 10^10000000000), we write, say, 10^^3. And we can keep doing that. But what if we iterate on the number of abstractions? So the number of "^"s is abstracted, i.e. 10 ^{m} 10. and THAT is abstracted n times (i.e. we replace m with the previous 10 ^{m} 10, n times). Graham's number is n=64. No doubt I probably made a mistake in here... But: tl;dr: no, we need to develop totally new abstractions to represent Graham's number. Numbers which can be merely represented directly in our universe are easily written in exponential format.

godd2 9y ago

No. Even if you wrote a single digit in each planck volume of the observable universe, you'd run out of space. It is safe to say that there are far fewer than a Graham's Number of things in here with us.

amyjess 9y ago

One of the best professors I ever had, when talking about intractability, would always begin his lectures by pointing out that there are only about 10^80 elementary particles in the universe. He has a small essay online about the subject: https://www.utdallas.edu/~tfarage/MyPapers/TheBestComputerEver.htm https://www.utdallas.edu/~tfarage/MyPapers/TheBestComputerEv...