This is an informal overview of the Theory of Paradox. I cover what it is, where the logic comes from, and where it's going. This is NOT a scientifically rigorous article; it isn't meant to be. It's meant to explain the concepts, so that when the time comes to start straightening out the details and put it into language that prevents loose logic from causing the 'theory' to be invalid, we will have a reference in layman's terms. 

If this theory (and article, and website) were software, it'd be beta. So test it. Think about it. Write about it. Help it grow. Help the design crew (me) make it something that is comprehensive, accurate, and understandable. Please help me make this something that can change the world for the better. 

Let's begin. 

A while back, I was idly browsing and came across an interesting problem. Science has this term they call, 'emergent behavior.' It's where you take some complicated system, throw a wrench in it, and it throws all predictability out the window. Be it as simple as an atom flying in there, or as complicated as the wingbeats of a butterfly, anything can throw off the predictions; and the more accurate you get in modeling a system, the more these kinds of additions will make you completely wrong.

 Emergent behavior is code for 'we don't know what will happen if you put that there, and we CAN'T know. It's too complicated.' It seems too obvious to invent a phrase for. But... there's a subtle flavor that tastes like: we really SHOULD be able to figure this out... we just haven't yet.

The idea of it got me thinking about systems in general. What IS a system? Any system you look at, if you look really hard at it, starts to get fuzzy around the edges. The limits become softer the more you question them. This got me wondering about the meaning and source of limits. In Psych 101, they teach us that we think by defining boundaries. Any study of words will tell you that we define something by relating it to what it's like, and what it's NOT like. These concepts are created and understood by their limits.

The first eureka that I had was this: What happens when you remove all limits? At first I only meant remove our interpretations and see things as they really are, and without the peyote or LSD. But, that doesn't work. Can't do it, really. Because we need those mental boxes we put things in. If I remove ALL limits, then I have a problem; everything becomes an undefined blob, except...

Sidetrack for a moment. You know how Renee Descartes said, I think, therefore I am? Well, that never really convinced everybody. We say, 'Sure, you're there, but we already knew that... WHY are you there? You give no reasons, only circles.' Well, I had a moment like Des Carte.

That undefined blob I was seeing in my limitless universe was completely blobby and undefined, EXCEPT for the rule there are no limits. That was still there, making everything undefined. But, shouldn't it be blobby too? After all, it's in a universe with no limits, what's holding it together?

The conclusion I reached that blew my mind is: this phrase, 'There Are No Limits'... it cannot, cannot, cannot ever be true. 'There are no limits'.... is a limit.

It goes way beyond my brain or your brain or the words I used to wrap around this concept. It can't be true. Completely independent of anything else, the only thing that makes it fundamentally not true is its own existence and truth. And if I wasn't a human thinking about this... if there were no humans, no planets, no THINGS or limits to be observing this simple statement, it could never, ever be true.

It's the cosmic version of I think, therefore I am. But... opposite. Sort of. At first glance, it defies reason. "I am, therefore I am not."

Drop that for a moment and sidetrack with me again. That place where science and math intersect, especially where science relies on geometry, number theory and Complex Analysis, is based on axioms. 'We hold these truths to be self evident.' However, even though those axioms allow us to create a model of the universe that is highly accurate, they are essentially just semi-safe places where we can say that the buck stops here.

We can imagine a universe where points do not exist, where lines are not straight and where 1=2. We just don't, because they're not as relevant as the system we've got. Mathematicians are fond of saying that math is only a model, it's not really REAL; and axioms inform that opinion. We have no axioms that are truly self-evident by the strictest meaning of the term; they are only evident in context.

This is important because it brings it home the idea that limitlessness is a limit, and in so being, cannot be true IS self-evident. Its evidence is literally itself; and further, I theorize that it cannot be logically disputed. Meditate on that a moment, and ask yourself if that is right.

If so: The Limitless Paradox, which is what I call it-the Limitless Paradox, to my knowledge, is the first truly self-evident axiom ever.

But it moves forward to act as a basis for a whole lot of new ideas, because of the paradoxical nature it has.

This is where things start to get complicated. Bear with me.

Paradoxes are... misunderstood. Most are really just errors in logic. There are a few, however, that defy explanation. Those all have certain traits that never vary:

1. They refer to themselves.
2. By referencing themselves, they theoretically invalidate the entire system they're based on.
3. If the system's already invalid, what do they do next? They VALIDATE it. Or themselves.
4. If they are surrounded by a context of some kind, they can be made to not be paradoxical.

The simplest paradox that does this is called the Liar's paradox. it states,

Z: 'This statement is a lie.'

If you make Z true, then it is false.
If you make Z false, then it is true.

Try it a couple times. Most philosophers eventually give up and say, 'It's the context, stupid.' Because of the words, or the structure, or the person saying it, whatever... the say it's just an error.

They try to make it true. Or false. Hell, some people say it's both at the same time.

This is where I had my second eureka. It's not true. But it's not false, either. Or both true and false. These are all wrong answers; and staying locked within that frame of static truth or falsehood is why there is still a debate about these paradoxes.

A true paradox is an algorithm. It moves... It is true, THEN false, THEN true, in sequence. It's its own cause and effect. Like the limitless paradox, it does not require context; in fact, context, like say, an observer, will screw up the program. (quantum physics anyone?)

Now you may have asked by now, why is There are no limits a paradox? It only goes one way: the wrong way. It's not true, and it's not a true/false/true kind of thing, you say.

Well, I was wrong about that, you see. And that's my third eureka. It CAN be true, under very special circumstances.

Start by assuming it's true. If you look at the rule There are no limits again and look at that universe where only it exists for a second... you'll see that, it limits itself and is false; because it's false, you're now looking at a universe where the only thing that exists is a rule about limits that's not true. Since its truth is its only existence, that means it doesn't exist. And if it doesn't.... then.... well, you now have a universe where there really ARE no limits.

Voila! It's a paradox.

We now have a paradox as the first, fundamental axiom that has ever been discovered. This is the rule that makes all other rules; the non-paradoxical fact that there MUST be rules, MUST be limits, is derived from the paradoxical statement 'There are no limits.' So do all paradoxes blink in and out, true, false, true, like a never changing cosmic stoplight?

No.

How they change will be covered next. Afterwards, I will return to emergent behavior, where I started, and begin to tackle it. In fact, I go on to explain a whole bunch of 'Why's.' Why time, why the 'observer effect' in quantum physics, why a universe and why people. What the self-organizing principle really is, and where it springs from. How to differentiate between systems without those 'fuzzy lines' I talked about in the beginning. Also, for the advanced course--I.E., stuff I'm not 100% on, just maybe 40-60%--things like why relativity and particle physics theory clash, and how to resolve it; a few new tricks for computer programmers to work on, using paradox algorithms, and most importantly right now, how to express paradoxical behavior and its complexities. I.E., a kind of paradox math. Since I'm not a mathematician, that part is really slowing me down, but I'm getting there.

More stuff ('cause I can't make it THAT easy for you)

Sidebar 1: Systems and definitions

Let's use a hard-to-break definition: a door. We know what a door is... so, what is it? What's a door? A door is something that you can open and close to get into a room, right? What's open? What's closed? What's a room? Open is when the door doesn't prevent you from entering... but if you can just unlatch it and walk in, what's to say it's not always open, except for the latch? Well, the latch hooks into the dooframe... so do you have to have a frame to have a door? Not always, it can just be something that's made to fit a doorframe and latch into it. But if that's true, what about open and closed? Well, it could potentially open and close... but then, so could a bottle cap. Why don't we call THOSE doors?

Every answer leads to another question. Sure, it seems like a conversation with a 3 year old, but I'm taking a stand here... it's not a flaw in our thinking. It's not a flaw in the language, it is not because we don't 'know better.' It's not because I'm ignoring the substance of all that is door, and losing my meaning because I've ignored the big picture... and I'm not claiming that the definition of door is meaningless. I only claim that the boundaries of meaning, the limits if you will, are shifty and hazy when you get close to them. And that is a real phenomena, it is NOT a failure to understand.

Limits can be defined two ways: 'hard' and 'soft.' The soft limits can be questioned... and if they are, the hard limits become soft ones that are questionable in turn. Try it. Try thinking like a 3 year old for a moment, without preconceptions handed to you by your parents. Nothing sacred. You'll see that there are no absolute limits, at this level of understanding... just places where we push down a stake and say, THIS IS. Which leads to the ultimate in questions, 'What if there were no words, no names, no boundaries, and no limits...?' Now back to the main dialog.

Sidebar 2: Zen

Zen is the quest to reach 'enlightenment.' Understanding without words, words without exceptions, meaning without boundaries. It uses the tool of the koan to attempt to break the student of his acceptance of normal definitions. The fundamental paradox of Zen is that it must use words and definition to do this.

A zen koan:

The student goes to the master, who is sitting in a temple in meditation. The student says, 'Master, I wish to be enlightened.' The master looks at him and says, 'You cannot.'
The student says, 'Why?'
The master says, 'Because you are not enlightened.'
The student says, 'Why?'
The master says, 'Ah. Good. Perhaps you are enlightened.'
At that moment, the student was enlightened.

(The original version had the master cutting off the student's arms for every 'why'... and when he gets 'enlightened' they're reattached. Meh, it obscures the point in my view, but then, that's probably the reason it's in there. And that's why I'm adding this editorial. Are you enlightened yet? No? Did you really need that left arm of yours, then? hehe....)

Another, my personal favorite; and like all good Zen koans, I don't remember the original, so this is my own version:

A student searches for the master. He's told, 'The master is in the ocean.' He goes to the ocean, and drinks it all; but sees no master, and so he spits out the ocean. He is then told, 'The master is in the rivers.' He goes to the rivers, and drinks it all; then, seeing no master, he spits out the river and it flows again. He is told, 'The master is in the clouds.' So the student breaths deep, and sucks in all the clouds; he sees no master, so he exhales.

Behind him, the Master taps his shoulder. (Insert name of zen master here) says, 'Did you find what you seek?' The student says, 'No, it found me.' The Master becomes enlightened...

But it doesn't end there.

The student says, 'I seek enlightenment.' The master replies, 'Go then, to the shore, and tell me where the ocean is.' The student goes to the shore; sees the ocean, and returns. He says, 'It is where the water stops and the sand begins.' The master then says, 'Go to the river mouth, and tell me where the ocean is.' The student goes to the river mouth; but he cannot find the ocean. He stops, and thinks.... for ten years, he stands contemplating the river and the ocean. Finally, he returns to the master and says, 'It is where the river becomes salty, and does not only flow away, but goes and returns.' The master strikes him, and says, 'You are wrong! Go, then, to the mountains; and tell me where the river begins.' The student goes to the mountains; and watches, as rain becomes trickles, trickles become tributaries, and where they become bigger and bigger. He goes back to the master and says, 'It begins with the rain.' The master strikes him 10 thousand times, and says, 'You are wrong! Look again! Where exactly, does it happen?' The student returns and meditates on the mountain for ten more years, watching the rain, and the snow, and the trickles... After ten years, he stands and returns to the master. He says, 'There is no ocean, and it starts where there is no river; there is no river, and it begins where there is no rain. There is no rain, and it is the will of Buddha that we have nothing, yet everything.' The Buddha says, 'You are enlightened.'

(You're asking, where'd the master go and Buddha come from? Hell if I know. Actually, not knowing where is the correct answer. For the alternative is to be blinded by context... and miss the paradox.) Back to the dialog.

Sidebar 3: Context mucking crap up

Here is a sample context: 'It's not true that...'

It is not true that there are no limits.

That statement right there... is true. It doesn't provoke a paradox, because the statement refers to a situation where 'more than one' is included. This is what messes people up when they think about paradox; because a context can 'resolve' a paradox, they think that any unresolved paradoxes must simply be errors, because we 'ignore' the context that would resolve it.

That is a logically false assumption. Don't take my word for it, study the rules of logic, especially the one called 'petitio principii' or begging the question. Ironically, this is also the logical fallacy that's most effectively used to discredit a paradox, since by nature the paradoxes in question MUST be self-referencing. The difference is, the circles I use are the ones that falsify the original argument; not the ones that reinforce it. Careful analysis of begging the question will see that it does not truly apply to paradoxes such as Russell's paradox or the Nelson-Grelling paradox; or the Goedel Incompleteness Theorem. But those paradoxes DO fall within my informal definition above. 

Back to the dialog. 

Sidebar 4: 'The Barber's Paradox'. The only barber in town has a rule: he won't shave anyone who doesn't shave themselves. So, the paradox is, does he shave himself? A good explanation of the phenomenon can be found here: http://en.wikipedia.org/wiki/Russell's_paradox