Saturday, February 28, 2015

Self-Referential Paradoxes

Frida: "What Belle is about to say is false." 
Belle: "Frida is telling the truth."

Belle, Frida, Cinder, and Snow got together one Friday night exasperated by the week's events. 

"I'm thoroughly disappointed by what didn't happen," exclaims Cinder, heavy-hearted with the mass of accumulated particles she must now continue to carry until an interaction allows her to release them back out into the universe. 

"You and me, both," Snow said, dropping her head in corresponding behavior to what one might imagine a Princess would do if she were ordinary. 

"I don't believe any of it," thought Frida, as she scrolled through the delights, paradoxes, and possibilities of the week. She also considered Lotería ... the game she loved to play because she appreciated its simple truths. 

"Like I said, Frida is telling the truth...." declared Belle, 
"She has a way of seeing truth I never knew was there." 

The Princess sat with that the nature of truth for awhile, quietly sipping from their bottles of Champagne ... a cloud of rose and lavender hovered above the room. They were in the staff kitchen, hiding far away from the demands of court life. 

"Demands, ... those are too easy," giggled Snow. "More like intrigues ...," added Belle, "the ones where sometimes a Princess just has to sit back and shake her head ... take it all in and experience it." 

"All of it," conceded Cinder. 

"And what comes from this?" asked Frida. 

"Well, Gödel's theorem springs to life," she continued  ... "from a constellation of paradoxes that surround the subject of self-reference. As if giving ourselves a title or a crown has anything to do with sovereignty other than being a reflection of a given trajectory manifest in response to diffused particles migrating through another system."

Take our tangled topic: 

"This statement is a lie."

If the statement is true, then it is false; and if it is false, then it is true. Such self-referential paradoxes are easily constructed and deeply intriguing; they have perplexed Princes and Princesses for centuries. They are the stuff upon which courtly discourse surrounded, but only during late night garden parties ... those absolutely exquisite joies de vivre one gives oneself and one's friends because the universe has allowed for them to be so. In sovereign duty, a Prince or Princess naturally inclines toward welcoming in and allowing for those more universal considerations ... the ones that support the sovereignty of the entire kingdom and all her children. 

The Disneyfied edition of the same conundrum resounds ... 

Frida: "What Belle is about to say is false." 
Belle: "Frida is telling the truth."

Such references leave me wondering if the universe could create itself. The great mathematician and philosopher Bertrand Russell demonstrated that the existence of such paradoxes strikes at the very heart of logic, and undermines any straightforward attempt to construct mathematics rigorously on a logical foundation. 

Gödel went on to adapt these difficulties of self-reference to the subject of mathematics in a brilliant and unusual manner. He considered the relationship between the DESCRIPTION of mathematics and the mathematics itself. 

thinking with my apple pie this morning

I am reminded of Cinder's predicament ... heavy-hearted with the mass of accumulated particles she must now continue to carry until an interaction allows her to release them back out into the universe. 

If there is something to be heavy-hearted, is it independent of the mass it carries? 

These are simple questions to ask, but they actually require a long and very intricate argument. Focusing on the sweets of what is involved, one can imagine listing mathematical propositions by labeling them in dance steps: 1, 2, 3 ... 

Combining a sequence of propositions into a dance video then corresponds to combining the natural numbers that form their label: JUST DANCE. 

In this way, logical operations about mathematics (and pop music) can be made to correspond to mathematical operations themselves. And this is the essence of the self-referential character or Gödel's proof. 

By identifying the subject (JUST DANCE) with the object (WHAT MAKES YOU BEAUTIFUL) - mapping the description of the mathematics onto the mathematics - he uncovered a Russellian paradoxical loop that led directly to the inevitability of undecidable propositions. 

John Barrow wryly remarked that, if a religion is defined to be a system of thought which required belief in unprovable truths, then mathematics is the only religion that can prove it is a religion! 

The key idea at the heart of Gödel's theorem can be explained with the help of a little story. 

In a faraway kingdom (understatement), in a state of dynamic change, in a state of expansion ... a group of Princesses who had never heard of Gödel became convinced that there does indeed exist a systematic procedure to determine the infallibly of a truth or falsity of every meaningful proposition, and they set out to demonstrate it. 

Their system can be operated by a Princess, or a group of Princesses, or anyone from the kingdom, for that matter ... or even a machine, or a combination of any of these. Nobody was quite sure of what the Princesses chose, because it was a secret, locked away in private late night discussions conducted far away from prying eyes and eager ears. 

They called the system  ... EL CORAZON ... 

To test El Corazon's abilities, all sorts of complicated logical and mathematical statements were presented to itself, and, after due time for drinking and other such activities, back came the answers: 

true, true, false, true, false ...

It was not long before El Corazon's wishy washy tendencies spread throughout the kingdom. Many people came to visit the kingdom, and exercised greater and greater ingenuity in formulating ever more difficult problems in an attempt to stump El Corazon. 

But Nobody could. 

So confident grew the Princesses of El Corazon's infallibility that they persuaded their king to offer a prize to anyone who could prove them wrong, who could defeat El Corazon's incredible powers. 

One day a traveler from another country came to the kingdom with an envelope, and asked to challenge El Corazon for the prize. What was the prize? Why, the kingdom, of course! All her mass, all her glories and powers bestowed upon a blossoming kingdom that knows not the origin of its becoming. All recognize the trajectory, but none recognize the impetus. It was kept quiet, shared in whispered hush at the Princesses late night garden party. 

Inside the envelope was a piece of paper with a statement on it, intended for El Corazon. The statement, which we can give the name "S" ("S" for "statement" or "S" for "stupefy") simply read: 

El Corazon cannot prove this statement to be true

S was duly given to El Corazon. Scarcely had a few seconds elapsed before El Corazon began a sort of whimpering. After a half a minute a servant came running from the tent with the news that  El Corazon had been disposed due to serious allegations. 

What had happened? 

Suppose El Corazon were to arrive at the conclusion that S is true. This means that the statement 

El Corazon cannot prove this statement to be true

...will have been falsified, because El Corazon will have just done it. But if S is falsified, S cannot be true. Thus, if El Corazon answers "true" to S, El Corazon will have arrived at a false conclusion, contradicting its much-vaunted infallibility. 

Hence El Corazon cannot answer "true." 

We have therefore arrived at the conclusion of this post ... that S is, in fact, true. But in arriving at this conclusion we have demonstrated that El Corazon cannot arrive at this conclusion. This means we know something to be true that El Corazon can't demonstrate to be true. 

This is the essence of the week the Princesses had and also the essence of Gödel's proof: that there will always exist certain true statements that cannot be proved to be true. 

The traveler, of course, knew this, and had no difficulty in constructing the statement S and claiming the prize. 

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