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Googols and beavers

Lasagna
Posts: 2,440
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10/20/2011 10:29:10 AM
Posted: 5 years ago
http://www.strangehorizons.com...

Have you ever thought of how big a googol (10^100) is? There are less than a googol elementary particles making up everything we can tell is in the entire universe (some estimates at 10^80 total). If there were no voids in space and it was crammed full of matter, that would be in the neighborhood of 10^120. If you increase that number up to 10^googol, it's a googolplex and a googolplexian brings it up again to 10^googolplex.

This is not only beyond our comprehension, it is beyond our imagination. I heard one theory that if one traveled a googolplex yards, you would have traveled through enough possibilities of arrangements of atoms that you would meet another being with your exact atomic makeup.

This article introduces a concept even bigger than googolplexes: busy beavers. Each order of magnitude of a beaver is analogous to counting possibilities of a system - with each increase of one unit to the system, it becomes increasingly complex. I remember in physics being told that you can predict what will happen in systems of one or two particles; once you get to 3 particles, there are so many possibilities of what will happen that it's deemed "chaos." This is similar to how beavers are counted. The article explains exactly how they are counted using a hypothetical machine and some visuals.
Rob
mattrodstrom
Posts: 12,028
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10/20/2011 3:39:31 PM
Posted: 5 years ago
At 10/20/2011 10:29:10 AM, Lasagna wrote:
This article introduces a concept even bigger than googolplexes: busy beavers. Each order of magnitude of a beaver is analogous to counting possibilities of a system - with each increase of one unit to the system, it becomes increasingly complex. The article explains exactly how they are counted using a hypothetical machine and some visuals.
"He who does not know how to put his will into things at least puts a meaning into them: that is, he believes there is a will in them already."

Metaphysics:
"The science.. which deals with the fundamental errors of mankind - but as if they were the fundamental truths."
Ren
Posts: 7,102
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10/24/2011 12:46:44 AM
Posted: 5 years ago
No one has any interest in this?

Dude, this is fascinating. However, I'm not sure what practical use this has. :P

What I mean is, if we've already discovered states that are too large to apply to reality (k = 6 apparently equals 10 ^ 865), then what good does it do us to continue searching for the solution to greater states?

Perhaps, there's a way that we can modify a Turing machine to work on multiple systems simultaneously (say, like a dual-core processor) to partition off computations in order to develop computers with capacities currently beyond conception. In other words, since we know where the 4th state begins, could Turing machine be developed with two boxes, with one that halts at the 4th state and another that begins at the fourth state and continues to effectively reach an 8th state?

Or... I mean, I can't be the only person who's ever thought of that, so perhaps that isn't possible...?

Maybe I'm thinking about it wrong. I mean, you need a function that reaches the 8th state, rather than to that reach the 4th state simultaneously, to truly get to the 8th state, which would be exponentially larger.

Well...

...perhaps, since it requires that you understand the algorithm required to reach a given state, you can determine how to partition them into equal parts and have separate boxes that compute functions that are actually smaller fraction of the overall function to reach a given value K...

...lol, this is fascinating, indeed.
Lasagna
Posts: 2,440
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10/25/2011 12:39:36 PM
Posted: 5 years ago
The largest things we have been studying to date have been the cosmological implications of subatomic particles - which, at its extent, brings us to the scale of about a google.

The ancient Romans considered 10,000 to be about the limit of practical mathematics, and it is logical to conclude that our farthest-reaching ideas are similarly quaint in comparison. But what would we use a number like a super-beaver for?

Computers will continue to get more powerful for at least a few more decades, at which point the graininess of atoms will make it impossible to continue with smaller circuitry. Somehow I doubt that will really stop us (move from binary to trinary or beyond?), and I can indeed imagine practical uses for numbers that are beyond a googol.

You might look back at the Romans and think: "how naive of them, to think of 10,000 as the largest useful numeral, when surely one of them must have looked down and imagined counting the grains of sand right beneath their feet." I'm sure someone thought about it, but probably doubted it's practicality.

We are in a similar situation, even with the super-beavers. Some applications that come to mind:
- weather simulations
- human interaction
- cosmology simulations

How many different things can I do right now? We often say this value is infinite, much like the Romans probably thought that sand was infinite. But there is a limit. With futuristic computers perhaps we can count possibilities of increasingly complex systems.

With that said, I don't personally see how the beavers are going to help us as they seem to be rather unweildy. I would imagine they are only useful as a reference point to avoid cumbersome exponents.
Rob
Ore_Ele
Posts: 25,980
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10/25/2011 1:00:51 PM
Posted: 5 years ago
At 10/24/2011 12:46:44 AM, Ren wrote:
No one has any interest in this?

Dude, this is fascinating. However, I'm not sure what practical use this has. :P

What I mean is, if we've already discovered states that are too large to apply to reality (k = 6 apparently equals 10 ^ 865), then what good does it do us to continue searching for the solution to greater states?

That's exactly why I don't find it interesting.


Perhaps, there's a way that we can modify a Turing machine to work on multiple systems simultaneously (say, like a dual-core processor) to partition off computations in order to develop computers with capacities currently beyond conception. In other words, since we know where the 4th state begins, could Turing machine be developed with two boxes, with one that halts at the 4th state and another that begins at the fourth state and continues to effectively reach an 8th state?

Or... I mean, I can't be the only person who's ever thought of that, so perhaps that isn't possible...?

Maybe I'm thinking about it wrong. I mean, you need a function that reaches the 8th state, rather than to that reach the 4th state simultaneously, to truly get to the 8th state, which would be exponentially larger.

Well...

...perhaps, since it requires that you understand the algorithm required to reach a given state, you can determine how to partition them into equal parts and have separate boxes that compute functions that are actually smaller fraction of the overall function to reach a given value K...

...lol, this is fascinating, indeed.
"Wanting Red Rhino Pill to have gender"
Ore_Ele
Posts: 25,980
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10/25/2011 1:03:57 PM
Posted: 5 years ago
At 10/25/2011 12:39:36 PM, Lasagna wrote:
The largest things we have been studying to date have been the cosmological implications of subatomic particles - which, at its extent, brings us to the scale of about a google.

The ancient Romans considered 10,000 to be about the limit of practical mathematics, and it is logical to conclude that our farthest-reaching ideas are similarly quaint in comparison. But what would we use a number like a super-beaver for?

Well, I suppose you never know. I can imagine if you went back 500 years and said, 1 LY = 9,461,000,000,000 km people might say "what the heck are we gonna need a number that big for." But can you imagin if we measured the distance of stars in km?

However, it is easy enough to create new measuring systems when they are needed, you don't have to make them before hand.


Computers will continue to get more powerful for at least a few more decades, at which point the graininess of atoms will make it impossible to continue with smaller circuitry. Somehow I doubt that will really stop us (move from binary to trinary or beyond?), and I can indeed imagine practical uses for numbers that are beyond a googol.

You might look back at the Romans and think: "how naive of them, to think of 10,000 as the largest useful numeral, when surely one of them must have looked down and imagined counting the grains of sand right beneath their feet." I'm sure someone thought about it, but probably doubted it's practicality.

We are in a similar situation, even with the super-beavers. Some applications that come to mind:
- weather simulations
- human interaction
- cosmology simulations

How many different things can I do right now? We often say this value is infinite, much like the Romans probably thought that sand was infinite. But there is a limit. With futuristic computers perhaps we can count possibilities of increasingly complex systems.

With that said, I don't personally see how the beavers are going to help us as they seem to be rather unweildy. I would imagine they are only useful as a reference point to avoid cumbersome exponents.
"Wanting Red Rhino Pill to have gender"
Ren
Posts: 7,102
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11/2/2011 7:50:28 PM
Posted: 5 years ago
At 10/25/2011 1:03:57 PM, Ore_Ele wrote:
Well, I suppose you never know. I can imagine if you went back 500 years and said, 1 LY = 9,461,000,000,000 km people might say "what the heck are we gonna need a number that big for." But can you imagin if we measured the distance of stars in km?

However, it is easy enough to create new measuring systems when they are needed, you don't have to make them before hand.

Actually, it isn't a measuring system, as far as I can tell. It's a computation system. It basically uses a simple algorithm based on 1's and 0's to automatically run a sequence that has a predetermined ending based on a given function of that sequence.

In other words, the guy invented a machine that essentially runs computations akin to how a computer reads bytes today, but these "programs" actually write their own scripting as they're being executed. It's astoundingly brilliant; moreover, thoroughly understanding would mean that we have the capacity to create any computer program with the same degree of complexity. That would mean that using that machine, we can likely construct a computer program today that would be too complex for any computer in existence to run. Of course, we need computers to construct computer programs (less we did it laboriously by hand, which really isn't possible given the magnitude of scripting in general), and we also need a medium on those computers that works within the capacity of that computer. In other words, that computer is able to lay the blueprints of a computer program that will be impossible to run for decades.

Truly amazing.
Ren
Posts: 7,102
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11/2/2011 7:54:25 PM
Posted: 5 years ago
At 10/25/2011 12:39:36 PM, Lasagna wrote:
The largest things we have been studying to date have been the cosmological implications of subatomic particles - which, at its extent, brings us to the scale of about a google.

The ancient Romans considered 10,000 to be about the limit of practical mathematics, and it is logical to conclude that our farthest-reaching ideas are similarly quaint in comparison. But what would we use a number like a super-beaver for?

Computers will continue to get more powerful for at least a few more decades, at which point the graininess of atoms will make it impossible to continue with smaller circuitry. Somehow I doubt that will really stop us (move from binary to trinary or beyond?), and I can indeed imagine practical uses for numbers that are beyond a googol.

You might look back at the Romans and think: "how naive of them, to think of 10,000 as the largest useful numeral, when surely one of them must have looked down and imagined counting the grains of sand right beneath their feet." I'm sure someone thought about it, but probably doubted it's practicality.

We are in a similar situation, even with the super-beavers. Some applications that come to mind:
- weather simulations
- human interaction
- cosmology simulations

How many different things can I do right now? We often say this value is infinite, much like the Romans probably thought that sand was infinite. But there is a limit. With futuristic computers perhaps we can count possibilities of increasingly complex systems.

With that said, I don't personally see how the beavers are going to help us as they seem to be rather unweildy. I would imagine they are only useful as a reference point to avoid cumbersome exponents.

Something Ore_El said made me come back to this thread and I realize that I completely missed this reply after I replied again.

It seems as though we said essentially the same thing.