I’m dragged in the undertow of greater minds, entrained to parrot their thoughts and speak their voice. You’re more than this, you think. You can become more than this.

I feel like I owe this blog something.

Some time ago, I heard Kleene’s Recursion Theorems mentioned in the context of guaranteeing the existence of quine programs in a Turing complete language. That was a while ago, and it took a lot to remember which theorem it was about (“Kleene’s Recursion’s Theorem” is perhaps not the most memorable name). But now I have, and I’m trying to wrap my head around it. To facilitate this, I’m writing these notes, which capture my thoughts as I work through the proof sketches on the Wikipedia page[link to wiki page].

I.

To prove the recursion theorems, we seem to require a little lemma called Roger’s Fixed‍-​Point Theorem (there are so many fixed point theorems!).

Anyway, I don’t like the apparently‍-​canonical statement of the theorem, mostly because I don’t like (meta‍-​)mathematician’s tendency to make everything a natural number. Maybe it’s more elegant, but meaningful types are so much clearer.

According to Wikipedia:

Given a function F(x), a fixed point of F(x) is an index e such that…

Rogers (Rogers 1967: §11.2) describes the following result as “a simpler version” of Kleene’s (second) recursion theorem.

Rogers’s fixed‍-​point theorem. If F is a total computable function, it has a fixed point.

There is a very concrete reason I haven’t been blogging at all.  The immediate reason is that I forgot about this place, but the bigger, over‍-​arching reason is that I don’t have any self‍-​confidence anymore.

I read my old posts, and I cringe.  It’s so much self‍-​appreciative, unsubstantiated hogwash.  It’s so arrogant and useless. But I’m no better than I was then, and so I turned to silence instead of honing my writing skills.

I frequently get into abstract philosophical discussions which a friend of mine.  I have a very contrarian and esoteric philosophical worldview, and in many instances the underlying structure of my thoughts remains inaccessible.  And so she urges me to write an article explaining my thoughts in detail.  And my response is always the same; I don’t have any good ideas to justify a whole article.  Whenever I sit down to write, it suddenly occurs to me how all of my ideas are either good or original, and never both.

This wasn’t supposed to be about me though.  let me just get to talking about epistemology.

i.

I have sometimes wondered, if I slept for a hundred years, what I would dream.

ii.

It’s fair to say I have no ambition.  This is no grand vision of success fixed in my mental sky, no true north.

what do I want

what, if I were given absoute creative power, would I wrought?

…

I could just write

but I do write

the results aren’t fiction, they are schematics, programs.  I write philosophical explorations, not stories

and so it follows that my unfolding is so much technicality

oh

Epistemic Status: mathematical, likely

Tagged in: philosophy

i.

Suppose the universe is both Turing complete and Turing computable.  In this case, by definition, there must exist some (Turing) machine M whose machinations are isomorphic to those of the universe.

The task of science is thus: given an (arbitrarily long) sequence of subsequences of this machine’s tape, produce a description of an equivalent Turing machine.

This is demonstrably uncomputable by a natural diagonalisation argument; suppose such a machine existed, and

$latex \alpha $

$latex \forall $

$latex \implies $

$latex \mathord{\Box} $

$latex \forall \alpha : \necessary (U(\alpha) > 0) \implies U(\alpha) < 0 $

i.

Ockham’s Razor refers to a famous scientific principle, originally rendered in Latin as something like “More things should not be used than are necessary.”, but usually expressed as something like “Do not multiply entities beyond necessity.” or “The simplest explanation is best.”.

The natural formalisation of this concept involves Kolmogorov complexity, the length of the minimal computer program generating a fixed string.  All objects can be expressed as strings of some form (most commonly a plaintext description in a natural language like English.  Mathematical descriptions are preferable, generally).

Computer languages often vary quite a bit in their expressive power.  It seems plausible that a program to calculate some abstrusely‍-​defined mathematical quantity might be shorter in high‍-​level levels (e.g. Haskell) or better yet, languages with natural support for advanced mathematical manipulations (e.g. Mathematica) would wind up several times shorter than a prospective implementation in some assembly dialect.

However, all of these languages are Turing equivalent; there is a program in each one that simulates all the others; thus the shortest program in any one of them is also a valid program in another, plus some constant equal the length of the simulator code.

Thus, because it is invariant up to an additive constant, we can meaningfully speak about the (Kolmogorov) complexity of something somewhat divorced from any specific model of computation.

Kolmogorov complexity canonically measures the complexity of objects, but we can just as well speak of the complexity of functions, so long as they are computable.

Note that complexity w.r.t functions usually means to their computational complexity in time of space.  Whenever we speak of the complexity of functions here, we mean their Kolmogorov complexity.

ii.

One function of particular interest to scientific creatures is the theory of everything, E(n).  For a fixed proper numbering of the states of the world (which will be defined in a moment) E(n) maps the next logical state of the world.

Before we discuss theories of everything, we must define a relation on world‍-​states called succession.  Semantically, succession is exactly the relation of one state leading to another. We can see that succession defines a graph over world‍-​states, with its paths defined by sequences of successive nodes.

We call a state t reachable from s (equivalently, s reaches t) iff there is a path from s to t.  We call the set of nodes reachable from s its descendants.  Symmetrically, we call the set of nodes reaching s its ancestors.

We call two nodes indistinguishable iff their ancestors and descendants are isometric to each other.

Finally, we call a numbering φ of world‍-​states proper if every distinct pair of numbers maps a distinguishable pair of nodes.

iii.

There are a few obvious properties we can assign these universes.  A universe is determinstic if every state has exactly one successor, finite if there are a

I.

I have heard it said that the more deeply engraved an idea or set of ideas is onto your mind, the harder it is to explain them to outsiders.

I won’t say this reflects experience. Firstly because the difficulties I have explaining my ideas could be explained by the massive inferential differences between myself and most people I interact with. We think differently.

The second is bit less facile; teachers, for instance, presumably have ideas engraved upon their minds from years of teaching in way that could only be called ‘deep’.

But, like all superficially insightful ideas, I believe it grasps at a greater pattern.

II.

What do I do with all that rush hour going on inside my head? I write. Some think it impressive that I have written so many books. I appreciate their compliment, but it does not seem so impressive to me. Because I think so much and have so much to say, it all comes very easy to me. It takes no effort, it is simply a natural part of my life, like breathing.

‍—‍ Stan Schmidt, Beyond the Noises: A Poetic Devotional

I first encountered this quote thanks to Scott Alexander. I like it, even if it doesn’t describe me well – in my case the metaphor is less breathing and more a staccato pulse coming unpredictably some days or weeks apart, by turns conveniently coming as I’m sitting in front of a text editor, then as I’m sitting at a desk and undeniably ought to be working, then as I’m falling asleep, tormented by rivers of thought flowing into my mind. I sometimes I feel as though this is what I ancients had in mind when they spoke of muses.

Sometimes, the contents are so compelling I have to eek out the words into my phone’s Spartan word processor because my laptop is so far away this late at night.

III.

In truth, a better metaphor would be nausea. Typically, ideas go through my eyes and are digested by my mind without incident. Only when I am served a particularly disagreeable, perhaps poorly‍-​, or under‍-​cooked, must I vomit forth the contents of my mind to spare myself some form of memetic unpleasantness.

You may find my dysphemistic treatment of creativity vacuous. Be assured, my conceit rests on more than the poetic intuition from the motif of vomit.

IV.

Writing is primarily a work of science, of patter matching and pattern recognition. Great works of blog do not leap forth from the brow of the blogger fully formed.

First you notice something, some amusing coincidence of inscrutable pattern. You file this away as important. You collect a few of these conceptual curiosities. It fascinates, but they are data without pattern, for now. You may not even be conscious of the perceptual stamp collecting, if it ever happens at the conscious level, it is typically under some other pretense. A local dialectical alias for a universal phenomena.

Then you stumbled blindly onto a hook. This comes to you as an urge to blog. Equal parts niggling curiosity and a harder to render emotion metonymous with the call to write, even if you aren’t sure what, entirely.

It starts with a hook, something provocative to catch the interest of both it’s writer and reader. And what follows is a game of what‍-​comes‍-​next as ideas trigger and catalyze and uplift each other in a cacophony of thought.

It’s a feedback loop, but bounded, like a forest‍-​fire. You take your brain off‍-​road and you collect mental underbrush, only that hook, the spark that propels idle musing into a full blown mental paroxysm.

And you step back, and you have worked.

V.

The metaphor with forest fires is satiating, but inadequate. I can feel the outlines of this idea in my mind’s eye and it’s so much more complex and beautiful, like Lovecraft’s amalgam of Cthulhu. This happens a lot.

There’s the forest fire aspect, yeah, but there’s another, where it’s like that miracle move in checkers where you jump over several pieces in a single turn. It’s like a quine, and there’s the alluring prospect that one day I’ll get that snag, the perfect self‍-​aware recursion and write a blog post that never ends.

VI.

But it’s also like underbrush in a difference respect. Blogging works because it’s a single motion that compresses and crystallizes the accumulated mental debris, but produces a product, a consumable as a by‍-​product, a side‍-​effect.

Blogging is flashy, but there are silent equivalents of it. The cowpaths of mind metaphor, for instance, is exactly what I’m reaching for. You walk along those trails in your mind, and it acts to compress and occlude the debris, becoming the inarticulable, inexpressible existential dark matter that attracts you along an accelerating patheway that eventually diverges from everyone you know. You get your identity, at the price of never belonging.

There’s a certain dichotomy‍ ‍‍—‍ really, a hierarchy‍ ‍‍—‍ that we all implicitly accept, or understand: the difference between a man and a rock, between a woman and a computer, between the animate and the inanimate, between the human and the machine.

It’s fiction, of course‍ ‍‍—‍ what is a man but atoms, a woman but squishy computation?  But the words refer to something, and nestled cuckoo‍-​like in that meaning is a value judgment; it is better to be a human than a rock, better to be a human than a machine.

There’s a theme that haunts humanity, a song we can’t get out of our collective heads: death.  Everyone dies, and culture then and now, near and far, is quite preoccupied with it.

Death is something bad; most people, in fact, do not want to die.  It is one of the many hypocrisies of the flesh, but certain transhumanist strain of thought elevates that fault to the level of pathology.

They want to live forever.  They’ve heard the assurances of Culture that death is in fact Good, Natural and Inevitable, and they realize it’s all bunk.  That it’s sour grapes, that it’s delusion.  They adopt that metacontrarian pose, and bravely assert that death is Bad, Actually.

They’ve argued with deathists before.  They’ll argue with deathists again.  They’ll get to the end of this post and decide it’s just more fallicious deathist apologetics.  It is.  You can check out right now, no hard feelings.

One of the common arguments fielded by the naive deathist is the boredom argument.  In the storied tradition of intellectual level scaling, you can grind up your immortalist logic and just get a stat‍-​boosted version of the same argument.  Watch.

The usual immortalist response to the boredom argument (“immortality bad because you’ll get bored”) comes easily to the avid reader, to deep thinker, or anyone whose drunk deeply from the well of existence.  They point to the staggering variety of experiences, the limitless complexity of life, and ask how you could ever grow bored of existence.  It simply abounds.

The argument may impress less those who’ve slaked themselves on a particular art genre. High fantasy books, say, or vanilla pornography.  It grows stale.  Superficial novelty, for sure, abounds, but your tenth or hundredth time around diminishes your returns.  You start to see the skeleton underneath.  See a dozen humans, and you realize we all have essentially the same face, a dozen primates, and you realize we all have essentially the same skull, a dozen mammals, a dozen tetrapods, animals, multicells, life is essentially the same.  Existence is essentially the same.

You’ll recognize this, too.  It’s eternalism, back again. And as good students of meaningness, we realize that both eternalism and its evil twin, nihilism, are essentially mistaken.  The truth is, as always, somewhere in the middle.

Of course, you could pathologize this boredom similar to burnout.  Give it a few years, maybe your memory has rotten enough, your standards backslide enough, that you can pick up another sword and sorcery novel and not be bored to tears.  You’ll get tired of anything eventually, but there’s more genres to read.

But that’s not quite all there is to it, no?  Even as the extraneous memories fade, there’s a harden core of experience, almost crystalline, that sticks with you.  The brain knows how to prioritize.  And as that experiential core build up, as that crystal grows, I don’t think it’s hard to see that you find less and less truly surprising amidst all the superficial novelty.  You start to realize that all surprises are essentially the same.

The immortalist will tell you there’s so many things to do.  But as the centuries wear on, you’ll no doubt begin to notice so many are different in the same essential ways.

Maybe you won’t exhaust the object level, but once you have a meta level theory, it becomes so much less compelling.

And then you get a metameta level theory, and then you realize the meta levels are essentially the same.

It seems a reducio ad absurdum.  It seems the peak of fallacious reasoning.

So you don’t recognize the essential sameness of all experience.  So you run tail between legs at any mean theory that might impinge upon the immortal paradise.

(I find that pathetic.)

So you go on growing and changing (growing and changing in the same essential ways), having new experiences (new in the same essential ways), and living your life forever.

After all, there’s the complexity argument.  You’ll grow bored of things at the same complexity level, sure, but there’s always something more complex (more complex in the same essential ways?)

Still, that utter fear of boredom is what I get hung up on.  Only a machine doesn’t grow bored of performing the same task again and again.  A flag blowing in the wind is animate, yet missing something.

Every Turing machine either halts early or runs long enough to see itself loop forever.

We all die.

One way or another.