Story Completeness

Last week I did a little jazz riff on the idea of “story space” — where all the stories live — and how the interesting stories we want to hear are all improbable to the point of having zero chance of actually happening (unless, gasp, statistics can lie).

I thought I’d return to that basic story space idea and, in the process, finally deal with a note that’s been on my idea board for years. My problem has been that, while the idea the note expresses seemed interesting enough, I’ve never quite seen how to turn it into a post. I’m not even sure the idea makes any real sense, let alone is worth trying to write about.

However that’s never stopped me before, and it’s (almost) Chillaxmus, so cue the music, it’s riff time again…

Before I talk about the note, let me back up and explore story space a bit.

It isn’t a new idea and certainly didn’t originate with me. In 1941, Argentine author Jorge Luis Borges wrote about a version in The Library of Babel.

Mathematician Rudy Rucker described the underlying principle in great detail in his 1987 book, Mind Tools. (I wrote about his approach back in 2013. See: L26 and L27 and Beyond.)

The short version is this:

Given some encoding scheme, any book can be seen as a single (huge beyond imagination) unique number. Therefore, every book is a point, a number, on an infinitely long number line.

A consequence of this (introduced last time), is that it works the other way around: every point on that number line is a book! And there are infinitely many those.

Nearly all of which (statistically) are gibberish. The number decoded results in just random letters (not even words). Given our desire to encode all languages, the letters wouldn’t even all be from the Latin alphabet!

Many (an infinite number again) are close to “real” books — varying from a single typo to sweeping editorial changes. Yet close enough to be considered a version rather than a (new!) separate work.

An infinite number of new works exist, too. Lucid, correct stories — some of them really great — that haven’t been written yet. (Or which may never be written. Or never written by humans.)

Very similar is how the transcendental number pi contains every finite number string possible. (See: Here Today; Pi Tomorrow and Happy Pi Day!) That means good ol’ pi also has all the stories, the full Babel Library.

Since I’m telling you a story about story space, this story is there, too. This story (this post) is just a number on that infinite number line. If I go back and fix a typo, its number will change.

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A simple way to think about the story number is to view all the bits of the file containing the story to represent one very long single binary number.

In this view, every image file, every sound file, every video file, is also a huge single binary number.

I said the numbers involved are huge beyond imagination, and that is not an exaggeration.

As an example, I found a PDF of The Library of Babel online, and it is 131,594 bytes in length. That’s a binary number over one-million bits long.

As a decimal number, it’s 316,910 digits long. (It starts with 85112… and ends with …02496. No other file could have that number.)^[1]

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The point of all this is, firstly, to say more about story space, a topic I’ll return to next year, and secondly, to set a mathematical context for stories (and kind of by implication, life).

My note is headed “Godel” (it should be Gödel), for Kurt Gödel, the German mathematician. The idea is in reference to his theories of (mathematical) incompleteness.

Which have been misinterpreted, and misused in arguments, quite a lot.

I’m joining their ranks in almost certainly definitely misusing it, but this is Chillaxmas, so I’m only semi-serious anyway. Take it as metaphorical.

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As briefly as possible, Gödel’s theory applies strictly to mathematical systems (of a certain complexity) and involves two key concepts: consistency and incompleteness.

Consistency means your math system will never betray you by proving 1=2 or 2+2=5 or anything (mathematically) untrue. Consistency means you can trust what your system says is true.

Completeness means your system can tell you every possible truth. This means your system can list all true things, even if that list is infinite and takes forever to enumerate.

Gödel proved two mind-shattering things about mathematical systems:

1. If a system is consistent, it cannot be complete.
2. If you can prove a system complete, it isn’t consistent.

Remember, an inconsistent system can prove 1=0 or 2+2=5, so such a system has no useful value. So the bottom line is that no useful system can tell us all (mathematical) truths (about itself).

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Those parenthetical qualifiers are crucial, and forgetting them is what allows people (heh, like me, now) to misuse Gödel’s theory. You can see the attraction, though: it’s about truth!

The truth here is that Gödel applies to certain mathematical systems only, and not being able to prove all truths about itself doesn’t mean a different mathematical system can’t. In fact, a higher system can prove truths about a lower system just fine.

Still, the elusiveness of truth, its infinity, is an interesting idea to play with metaphorically.

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Here’s the point:

Math (to be interesting and useful) is consistent, but never complete. Truth is particularly elusive mathematically.

Storytelling can be the other way around. It presents truth (through fiction), but it need not be (and often isn’t) consistent. Sometimes, in a story, it’s okay that 2+2=5.

It can be seen as what makes them interesting; math isn’t^[2].

Storytelling is about truth, not consistency.

Some forms take that to extremes; cartoons are a good example. Just consider all the physical inconsistencies in, say, 30 years of The Simpsons.

Compare that to the truths expressed in that time.

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As an aside, while Gödel seems not to apply to real life, it’s interesting to consider it in regard to how our physical theories are all mathematical.

Some theorists take it that reality is just mathematical structures. Max Tegmark is so well-known for this that such theories are called Tegmarkian.

If reality is, at root, purely mathematics, then Gödel may apply!

It does seem certain aspects of our physics may remain forever beyond any grasp, even in principle, no matter how powerful the reach. This is causing some degree of despair in some physics circles.

[We may be stalling on the high-energy theoretical physics front, but fear not, for materials physics is burgeoning! Optical sciences, too.]

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If it is all mathematics, and if there is no metaphysics, then it would seem we’re all part of a vast mathematical structure, and maybe Gödel explains why truth is so damn hard.

Certainly in the casual (not math) sense, life seems rich and complete if not always consistent. Which seems to deny Gödel has any say in our lives.

I’m good with that. Far better a world complete with truths to find than a logically perfect one where some truths can never be known.

The flip side is that an inconsistent world can’t prove its truths. Or, rather, it can prove all truths. Two plus two can equal anything you like!

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I learned a new term the other day: curate’s egg.

It means something is a mix of good and bad with neither dominating.

Ain’t life a curate’s egg?

Whether Gödel has anything at all to do with it or not, whatever we might say about life’s completeness or consistency, it’s all a mix anyway.

Seems like nothing important or interesting is ever just one thing.

A complex multi-variate world.

Interesting Times!

Merry Chillaxmas!

Stay complete, my friends!

^[1] Multiply the number of binary bits by log10(2), which is a hair over 0.3, to get the number of decimal digits. Or by the log of any base to get the number of digits in that base.

For instance, Googol, the famous a 100-digit number, takes log2(10), which is about 3.322, times 100, which equals 333 bits.

You find the conversion factor with logX(Y), where X is the target base and Y is the current base. So binary to decimal is log10(2) and decimal to binary is log2(10).

^[2] Um, er, well, not to most, anyway.