Why large numbers?

Disclaimer: This text is not meant as support of any political institutions or ideas. If you see any, the problem is on your side. This text is meant in complete seriousness. If you disagree, either the author is right (and can laugh at you) or the author is wrong (and you can laugh at him, but can't claim the whole thing is a joke).

Understood? OK. Welcome to the third part of the Large Numbers series.

Just the physical writing of parts one and two took several hours (that doesn't include the several days necessary to make the system). Reading them probably took an hour or so. It might be the time, then, to ask the question "Why?". Remember the point of this site? Things aren't always practical. We can look for a purpose, but the best to be found is that "All the roads lead to Rome". Whatever method we use to create large numbers, the same structure will be present. Take something. Repeat it. Repeat the number of repetitions (remember the up arrows). Repeat all that can be repeated... And so on. Whatever you do, you do essentially the same thing anybody else would do. The only difference is how far you get and the details of the method used. There exists a mathematical structure describing this: ordinal numbers. This is how they work (for our purposes):

Start with 0,1,2,3...

You have just created a sequence. Now, define an ordinal number larger than all of these and call it omega. Rule number two: You can always add one.

ω,ω+1,ω+2...

Look! Another sequence. A number larger than all of these is omega times two. Beware, for reasons outside our level needed it can only be written ω*2, not 2*ω. 2*ω=ω.

In a similar way, we could get to ω*3 etc.

1,ω,ω*2,ω*3,ω*4...

ω*ω=ω² surpasses all of those. Next step is this:

ω,ω²,ω³,ω⁴...

ω^ω

And so on. Next step is this:

ω,ω^ω,ω^ωω,ω^ωωω...

These are surpassed by:

ω^{ωωωωωωωω...}

(An infinity of omegas)

This is a little impractical, which is why it's called epsilon zero (ε₀). We can continue, but the idea is there, hopefully. Whenever we make a sequence, there's an ordinal larger than all of the numbers in the sequence. This is also how arrows work. Take only the part above the arrow now.

1 above the arrow stays a 1. 1V₀0 corresponds to ω. 4V₀7 corresponds to ω*4+7. 1V₀0V₀0 is ω². 7V₀6V₀0V₀8V₀11 is ω⁴*7+ω³*6+ω*8+11. 1V₁0 is ω^ω. A complete translation table would be complicated, but if we get a list under the separator (which is where I left you), we'll get to ω^ωωω. All arrow calculations can be formalised using ordinal numbers. Then, we just put into that some large ordinal like ε₀ or something scary, for example zetta-0 (ε_εεε... ) or complete oddities like Γ₀, the Feferman-Schütte ordinal. It seems like the whole of googology (the study of large numbers) is solved. Mostly, yeah. There are many ideas, but all the good ones go in the same direction.

Where I'm trying to get is that things like playing around with large numbers can have connections to odd parts of mathematics and as such show how all of mathematics is connected. You can find new interpretations of things or interesting ideas anywhere, even in the seemingly useless stuff.

Of course that's not enough. It's not the reason we did this. There is no practical reason. But that isn't as weird as it might seem. Most things we do make no sense. The real question here is related to all those things. The explanation is simple: The base state of every human is boredom. Try it out: stare at the dot behind this sentence for five minutes.

Done? You thought about something. Maybe about anything, just to have something to do while waiting for five minutes. When you can't do anything physical to stop boredom, you start thinking. Apart from stuff like "What's the lunch tomorrow?" (which could be considered practical), you certainly thought about something pointless. That's how it works in life as well. For a while, we stop boredom using practical things (your job). Eventually, we get bored there and move on to something else. Eventually, we find out that we have too much time and too little to do and we do something pointless.

When we do anything we may consider "practical", we're trying to accomplish something. That something has a point as well (otherwise we wouldn't do it, right?), leading to higher level goals. Eventually, we end at basic goals, which include getting food etc. But because it's relatively easy nowadays to fulfill those, we only have the most basic one left. Do anything. Avoid boredom. So the "pointless" things have a more direct point than those we'd call "practical".

Apart from the pretty obvious point (humans do stupid things when bored), there's nothing to see here...

Unless...

The real point of this

G will suffice here. In fact, even g₂ would be enough (whether we take g₁ = 27 or the original definition). The other numbers we created seem unnecessary. We'll return to them later.

We start from an observation first stated as a throwaway remark in part one. G (or any other large number which isn't a power of 10) contains mostly random digits in its decimal representation. Nobody cares enough to give us some conclusion on this matter. While it is possible that over a very long period, the digits have some structure, even if that was the case, there would always be a number very close to G which would have truly random digits. Or we could just write 1234567891011121314151617181920212223... until G and all the number combinations up until G will be there. For sake of simplicity, we'll continue to use G.

Having sorted this out, let's look at what we can find in G. For starters, all the telephone numbers. That's quite lame. It has somewhere around 10 digits depending on where you live. But we can certainly find anything of length up to g₁ (the original, not 27). We could look at G in binary (or look through it until we find a g₂ long section with only ones and zeroes). Any file would be there. For instance, all photos. Any photo you have ever taken and all the photos you haven't. If we looked for long enough, we'd find the photos of Pluto we got from the New Horizons probe. Of course, it wouldn't be very practical, but they were there all the time.

There's also a text file containing the exact procedure necessary to reach immortality. But also all the procedures that lead nowhere. Jorge Luis Borges thought up a massive library containing all of the possible books of 410 pages. By the way, there exists an internet version where you can try out going through random text. You could find the beginning of this text (using the search function, input the beginning of the text, only letters, commas, and spaces). G is such a library. But it goes further. Far, far further. Like a million-year-long movie of 15000x12000 pixel quality about the history of humans. But that's just the beginning.

Putting quantum mechanics aside for now, we can exactly describe a particle (like an electron). Just state a few numbers (position, velocity, mass, charge...) and it's done. We can describe the laws of physics in a similar way. This means we could find descriptions of interacting particles inside G. Quantum mechanics complicates things a bit (but we'll return to that later) and there's insufficient data about some things, but there seems to be no problem with turning the state of some system in physics (a wave function) into a number. Even if it's impossible with absolute precision, we can get extremely close. G, therefore, contains complete simulations of particles interacting.

That doesn't seem very useful. However, we haven't used all of G's capacity. Not at all. We can find large systems. Not just a few particles, but hundreds, thousands, millions, billions of particles interacting. Even a system describing a sphere of perimeter 46 billion light years. The observable universe. We can (again, ignoring quantum mechanics and a little relativity) take the universe apart and describe all the particles. Let's create a filetype .univ (just like .txt or .jpg). It would be a file consisting of many subfiles .particle, each containing the data about a single particle. All in binary. At that point, all we need is a file .lawsofphysics telling us how particles interact (essentially what all the scientists are trying to figure out right now). The file Our_universe.univ will contain a ton of data. It will be an exact copy of our universe, except for two slight issues we have to solve now.

1) Let's assume a universe consisting of just two particles. They are x meters apart. x can have an infinity of decimals with no repeat and no way to be simplified. It wouldn't fit into G. That's easy to solve. Note that all data comes with an error of 1/g₁. When there are g₂ bits, g₁ is just a grain of sand. These tiny imperfections won't have enough time to grow into anything meaningful. We can only measure to twenty significant digits or so. If there's an error at the g₁-th digit, we'll never find it.

2) We've avoided quantum mechanics. If we want to get it back (which is necessary for a functioning universe), we have several options. Either we write the universe from the point of view of humans living in it. This file would be the same (when it comes to human lives), no experiment could differentiate between this approach and the "all particles" approach. While it's possible (and such files exist in G), it smells of avoiding the problem in an indirect way that doesn't really solve it. Or we create another system, which would look like quantum mechanics to the people inside the simulation, but which wouldn't have such trouble. But we have no such system. Neither do we need it. The main problem of quantum mechanics is that it introduces uncertainty. We can still work with the wave function, which is the best description of a quantum system possible. It has oddities like allowing a particle to be at two places at the same time, but that's not a problem. We can't assign a number to every function, but we can keep naming functions in such a way that we get arbitrarily close to any particular one. This allows us to bypass problem number 2 just like we did with number 1. The worst part is that quantum mechanics is complicated and you can never be sure it won't ruin your day. We have to hope it won't ruin ours.

Now look at the situation from the point of view of the humans in this "simulation" in G. The main leap of intuition to make here is realising these people don't know they're simulated. In fact, there's no way to know whether you live in a simulation or not! While this argument has been made for computers (Do you live in Matrix?), nobody has yet made it for G.

Some more questions to deal with:

1) When you see some colours:

You actually see them. It's not just your eyes sending signals to your brain about the colors, then your brain interpreting these signals. There's an actual consciousness involved, which differentiates you from a simulation, right? Right...? Wrong. There's no difference between "actually seeing" and "just eyes and the brain working". There is an old question: Is my red the same as your red? I see something I call "red" (as I'm used to). But what if, if you saw what I'd call "red", you actually saw "green"? Just look at how hard it is to state the question. That points us to its problem. It makes no sense whatsoever. There's no way to state the question correctly and unambiguously. Forget for this moment what you "see" and look at what's actually happening. Photons of visible red light hit your eyes, which react by sending signals through neurons into the brain, which interprets them as "red". If the brain sees the question "Do you see the colour red?", it will think it does based on chemical processes happening in it (which are entirely chemical/physical). Even a brain that doesn't "see" colors will think it does. We have no way of distinguishing between the "fake" and "real" brains/people. Since they can't be distinguished, there's no reason to think about them as different things.

2) We have free will!... No. This is gonna be a bit harder. How does the brain decide? It takes some input and outputs something else. If you can, for example, get from place A to place B either by a road or through thorn bushes, the brain will remember roads tend to be better paths than bushes and will send a signal "take the road". Or it remembers you lost your cell phone in the bushes and sends the signal "go find my cell phone". Or it remembers you think it's a great idea to choose hard ways, so it sends the signal "take the bushes". No matter what you chose, there's no free will involved. Even if we assumed the brain can generate randomness (which it probably can't - or at least not true randomness), it would just be a part of the choice, but it wouldn't "create" free will. When you try to define free will and how it works, you get stuck. It makes no sense whatsoever. Just like point 1.

Now, things point 2 doesn't mean:

It doesn't mean punishment doesn't work. The point of punishing is to fix the criminal (not very reliable, but point 2 doesn't change that), stop them from causing more harm (the same) and send the signal that crime doesn't pay off (point 2 doesn't change anything about that, there's no free will here either).

It doesn't mean you're not in control of your life. It just means your brain has control over you. Your actions have consequences.

It doesn't really mean anything new. It only exposes "free will" as a chimera. Not only it doesn't exist, it can't exist, because nobody knows what it means.

The human brain is like a computer. When it's solving a problem, it always comes up with the same result (unless chance is involved). But that doesn't mean the problem solving is meaningless.

As shown, the humans in the "simulation" in G have no idea they're in a simulation. What does that mean? You have no idea whether you don't happen to be in G! Hard to digest. But even the flow of time is an illusion, it's the result of the sequence of .univ or by the utilisation of the .lawsofphysics file.

What do we do with that?

It's important that any discovery be confirmable and useful. For this purpose, Occam's razor was invented. It says "don't overcomplicate things", or to be more accurate "When two theories can't be distinguished, use the simpler one, that is the one that requires fewer assumptions.". Now, we need to ask if we aren't overcomplicating things. Let's look at the two competing options:

1) Particles (or fields, if we're going quantum), laws of physics, etc. exist. This was all created during the Big Bang (details unclear so far). It requires a description of the universe at some point and of the laws of physics. The data needed is the size of, well, the universe. There are also issues like "Why are constants such as the strength of gravity fine-tuned to allow life?".

2) There exist a few basic axioms of mathematics. They lead to the existence of numbers (and other mathematical objects, but for our purposes we'll disregard those) which contain data, which code the universe containing chemical structures (humans) capable of thinking about the universe. This only requires the existence of axioms, everything else is just their result. In fact, I can put all the data needed to create the universe right here:

You may not understand these, but it's essentially just a bunch of simple assertions (such as "Sets containing the same elements are equal) transformed into symbols (since that's the only way to define them without needing humans to interpret). Some of them may need to be expanded a bit (so that the three dots and the exclamation mark in the sixth row, the "is a subset" mark in the eighth row and most of the ninth row are eliminated), but it's all doable. Anyway, all the data would fit on a sheet of paper. And it creates many universes, which eliminates the fine-tuning issue. The issue of existence is just a matter of definitions. The only difference between 11 and G is that 11 is too simple to contain anything useful.

If the "it's a simulation" idea was the status quo and somebody arrived stating there's no simulation and just particles, not only would it be hard to just state the idea, it would seem nonsensical. The universe popping out of nowhere, or being created by God or otherwise appearing on its own, this all requires too much to begin with. It's not the G hypothesis, but rather the traditional worldview, that requires too many assumptions.

Then, there's an interesting coincidence. All the possible worlds are in G. How come we happened to find the truth when there are many worlds where this never happens? Why did we get to be lucky? And why G? It's quite specific, how did we happen to invent it? It turns out you can create a universe from many different things. In some games (such as Minecraft or infinite 3D chess), you can "program" Turing machines, which can complete any calculation your computer could. When you're inside a Turing machine-simulated universe, it doesn't matter to you where that machine came from. It's not that hard to create a Turing machine. Any civilisation even remotely interested in mathematics will eventually figure it out.

What the world is like

A world described as this is... interesting. For starters: The outer world exists, and you don't. More or less. Existence is complicated. Axioms, sets, numbers, geometrical shapes... all exist. But interpretations like .lawsofphysics get fuzzy. We could either evade the problem by creating a separate mathematical system based around particles, but it wouldn't be very useful. We could just state either everything or nothing exists.

Wait a moment. This is confusing and chaotic. What is existence in the first place? According to Wikipedia existence means being able to interact with the outside world. But does "the outside world" exist? Why would you think that? This requires arbitrarily deciding that some world and things that interact with it exist. Since we see many different worlds as definable but not interacting, either we arbitrarily pick one of them as existing or we get forced to declare each of them separately. At this point, we may be wondering why we're doing this at all. Existence doesn't affect anything and is hard to define satisfyingly. Dividing things into existing and not existing is a physics-based concept. It's not necessary in a mathematical universe. We should just ignore "existence" altogether. The only reason this may be hard to swallow is that we like to think we are real, but numbers and other "lowly" objects aren't. Nonsense. Let's just say for the record that everything's fictional.

Another interesting property of such a world is timelessness. Time, as an illusion, is a result of laws relevant only for our narrow slice. Even it is timeless, time is just an illusion. This means thinking about time outside a very specific context is pointless. It's not timelessness as in no change, it's just that the very concept of time is unapplicable to mathematics. Hard to imagine? Maybe. The mathematical approach requires us not to imagine the unimaginable and just check the consequences. It points to a problem we'll keep facing over and over. The mathematical world is hard to imagine because it's too different from our one. But that's kind of our problem. Humans evolved in an environment where they optimised for wilderness survival, not mathematical-philosophical debates. Not even our vision is optimal, just remember optical illusions. The outside world isn't weird, it's just not what we're trained for. Just like thinking about quantum physics. We have simplified ideas we don't like getting disrupted. The outside world isn't strange, it's us who are.

Where's our place in this? There are two basic ways to create the universe. Either you specify the original state and the laws, or a sequence of states one after another (here we may run into slight causal issues because of relativity, but they may be circumvented by e.g. picking a reference system to which we would relate everything else). The important part is that in the former case, such a world is predictable and it's possible to determine the laws "from the inside". In the latter case, no state is related to the previous one and most such worlds are therefore nonsensical. The problem is that if we select all such worlds that are not different until some point (for example, now that you are reading this sentence), the next state will in most cases be random and unrelated. The Earth will disappear and the universe will be replaced with white noise. But there's a catch. The more a world can be predictable, the less data is needed to describe it. It will require a shorter sequence of digits, which will be more common. As such, most worlds follow the first system with laws and an initial state. The laws of physics as we'll one day (maybe) know them will probably be short enough to be printed on a T-shirt. Definitely no more than a book's worth. The basic ones, at least. Things like boiling points are results of basic laws determining particle interactions. So it's probable you'll stay alive.

There's the question of what exactly in this system you are. If we keep to numbers, "you" are any sequence of numbers (or other information-bearing objects), which, when interpreted in a given way, result in a physics system whose part is an object looking like you. If that object is a part of a greater system, it can be said you're doing something. Your future is not definite, as there are many different future You's. In fact, any possible future of you is somewhere. At every moment, your life forks, regardless of whether you make any decisions. But none of this matters. Existence doesn't matter, the important part is you can't find out what system you're simulated in from within the system (unless something outside the scope of the system happens - if something that clearly isn't a particle appears, you're not in a system simulating through particles only). That's why it doesn't matter.

Conclusion?

Let's return to the question of free will. You can decide that, since there's no free will, you could as well just rest and do nothing, since it's just the result of the past which you can't do anything about. Maybe. But you're a part of the past. If you give a problem to a computer, the solution is predetermined. But that doesn't mean it's logical or inevitable to just do nothing. In a way, you can affect your future. Things you do have consequences. When you're deciding, you're checking the consequences of possible actions. You want to take the best one. All we need to do is redefine choice! Choice doesn't mean taking an arbitrary action, it means being responsible, examining the situation and doing the best you can. Resting and doing nothing is not the best. All your possible futures are happening out there. You can't do anything about that. But (which is probable), if you're living in a universe with some laws, you have some control over what's happening. Out of your possible futures, some are good, some are bad. Pick a good one.