Metamagical Themas: Snags & Snarls
Exploring self-referencing
This is a brain dump about Section One of Hofstadter’s collection of essays - Metamagical Themas. For an introduction to the book and hyperlinks to all the sections, see HERE.
Let’s look at the concept of reflexivity: i.e., the way some systems (language, biology, government, mathematical structures, etc.) problematically twist back onto themselves to form a conceptual loop. Reflexivity can create interesting phenomena, and Hofstadter goes through upwards of a hundred ways in which language can be self-referential and create paradoxes that are difficult to untangle. A classic example is the following: “This sentence is false”. This yields two options:
1) If the sentence is false, then the sentence true.
2) If the sentence is true, then the sentence false.
By referring to itself, it makes an unsolvable paradox.
There are many variations of the snags and snarls that arise with self-referencing in language. Not all are necessarily paradoxical, but they are interesting and clever at the least.
Let us make a new convention: that anything enclosed in triple quotes - for example, ‘"No, I have decided to change my mind; when the triple quotes close, just skip directly to the period and ignore everything up to it"’ - is not even to be read (much less paid attention to or obeyed).
Only the fool would take trouble to verify that this sentence was composed of ten a's, three b's, four c's, four d's, forty-six e's, sixteen f's, four g's, thirteen h's, fifteen i's, two k's, nine l's, four m's, twenty-five n's, twenty-four o's, five p's, sixteen r's, forty-one s's, thirty-seven t's, ten u's, eight v's, eight w's, four x's, eleven y's, twenty-seven commas, twenty-three apostrophes, seven hyphens, and, last but not least, a single !
In this sentence, the number of occurrences of 0 is ___, of 1 is __ , of 2 is ___, of 3 is ___, of 4 is ___, of 5 is ___, of 6 is ___, of 7 is ___, of 8 is ___, and of 9 is ___. (there are two solutions. My attempts keep looping and I give up)
One day during a walk, a son tells a big lie. His father direly warns him about the "Liar’s Bridge", which they are approaching. This bridge always collapses when a liar walks across it. After hearing this, the boy admits his lie and confesses the truth. When I told a ten-year-old boy this story, he asked me what happened when they eventually came to the bridge. I replied, "It collapsed of course. After all, the father lied when he said there is a Liar’s Bridge."
The ‘theme’ here is that self-referential sentences can cause a ruckus for those who study logic. Hofstadter claims these examples (specifically the “this sentence is false” example) to be language analogs to the mathematical proof of Gödel's Incompleteness Theorem (GIT). GIT is a series of logical steps that concludes the following: any mathematically powerful and consistent axiomatic system contains an endless series of true but unprovable formulas that can be constructed by the technique of self-reference. GIT is really interesting, but as a non-mathematician or philosopher, it’s kinda hard to internalize what it means. Fortunately, we have a 5 minute TED Talk to help us.
Beyond language paradoxes, reflexivity (i.e. self-referencing and its subsequent snarls) can be found in (marginally) less abstract things. He makes some suggestions that the DNA replication process from DNA to RNA to protein back to DNA to RNA to protein contains a self-encapsulated loop. He also has this funky paragraph that is clever.1 And this sentence.2 But these all seem like bathwater. What’s the baby?
Reflexive, viral, self-replicating ideas.
Viruses are inert pieces of information that replicate by commandeering the more advanced complex facilities of their hosts. A meme is an idea or cluster of ideas. Thus, viral memes are ideas that inertly leap from brain to brain being copied and regurgitated.3 Just like in the biosphere, in the ideaosphere ideas must compete with each other for resources (both space and time devoted to the meme by minds), find a fertile home to parasitize (naïve versus already parasitized minds), and jump from host to host (via language, art, etc.) in order to stay afloat. Naturally, memes develop ways to resist being destroyed. They evolve slowly as they jump from host to host as to prevent outright rejection. They develop hooks that pierce deeply into people's psychology. As in biology, some memes will co-exist well together and may depend on each other. Some will attempt to latch themselves onto preexisting memes as a way to help themselves survive in the meme pool. It is a meme-eat-meme world out there, and consequently, some memes have traits that make them more likely to be passed on.
Some examples:
Blind faith: ‘Trust Meme Z in the absence of evidence, even in the face of evidence. Trusting in Meme Z in this way is a sign of your strength. It is a weakness to not have blindfaith.’4
The Game: The point of the game is to not think of the game. The one rule of The Game is that if you think of it, you must declare it publically. (Sadly, I have been deeply infected by this meme. I apologize for now infecting you)
Remember those emails that used to go around? ‘DON’T BREAK THE CHAIN. SALLY DIDN’T SEND THIS MESSAGE TO 3 OTHER PEOPLE AND HER DOG DIED THE NEXT DAY’. YOU HAVE 72 HOURS TO FWD THIS. JUST HIT FWD NOW!5
What do all of these examples have in common - self-referencing
Let’s have a more formal approach:
Idea X: It is your duty to convince others that this sentence is true.
If this idea were to establish roots in your mind, you would attempt to convince your friends that X is true, after all it is your duty to do so! If they were foolish, they would convince their friends, and so on until every human mind contained a copy of X. Thus, X is a self-replicating idea. It takes control of the mind's intellectual machinery and use it to produce hundreds of copies of itself in other minds. An infected mind will plan, scheme, and deliberate on how to most effectively spread the news of X. The kicker is, it may even enjoy doing so! What started as an abstract intangible idea manifests in the physical world as billboards, Facebook posts, divisions of social groups, and more. This is the danger and power of self-referencing!
If Idea M is ‘Pizza tastes good’, there is no desire to propagate it. Why? It’s not self-referencing!
But surely no one would actually believe it is their duty to convince the world of the literal version of Idea X. In life, it’s not that simple. Idea X will be attached to a host of others in a large symbiotic cluster in the ideaosphere. Straight from Hofstadter,
Consider the following:
System S:
Begin:
S₁: Blah.
S₂: Blah blah.
S₃: Blah blah blah.
S₉₉: Blah blah blah blah blah blah...
S₁₀₀: It is your duty to convince others that System S is true.
End.
Here, S₁ through S₉₉ are meant to be statements that constitute a belief system having some degree of coherency. If System S taken as a whole were convincing, then the entire system would be self-replicating. System S would be especially convincing if S₁₀₀ were not stated explicitly but held as a logical consequence of the other ideas in the system.
Statements S₁-S₉₉ are the bait which attracts the fish and conceals the hook.
No bait-no bite. If the fish is fool enough to swallow the baited hook, it will have little enough time to enjoy the bait. Once the hook takes hold, the fish will lose all its fishiness and become instead a busy factory for the manufacture of baited hooks.
Are there any real idea systems that behave like System S? I know of at least three. Consider the following:
Begin:
X₁: Anyone who does not believe System X will suffer in hell.
X₂: It is your duty to save others from suffering.
End.
If you believed in System X, you would attempt to save others from hell by convincing them that System X is true. Thus System X has an implicit 'hook'
Clearly, it doesn’t take a lot for ideas to become ‘viral’. It took only 2 sentences and System X became a self-replicating system!
Thinking of ideas as memes and beliefs as viruses with varying degrees of infectivity has a dystopian flavor. Do I like Taco Bell, or am I just a vessel who has been infected by the idea of Taco Bell? But perhaps that is because your worldview contains a System in which S₄₂ says something like ‘Your meaning in life depends on having agency and independence from the world and its conditions’. If we can remove that deeply nestled ‘hook’ out, maybe swallowing pill labeled ‘I am simply a conglomeration of parasitic ideas that are using my mind and its marvelous mechanics to propagate themselves in an eternal battle of survival of the fittest in the ideaosphere’ would be easier.
Hoofstafder explores one last way in which reflexivity shows spreads its wings in this section on Snags and Snarls - government. Governments both make and follow the law - a tad recursive if you think about it. Hofstadter’s example is a local government that establishes an Internal Affairs Bureau to investigate police brutality (this was written in 1980). The issue is, this group is still within the governmental system. Who checks on the work of the Internal Affairs Bureau? A secondary department? Who checks on them? And can’t they just change the rules? No legal system seems to have any rules that are absolutely immune to legal change (and if there were, what would be the rules on changing those rules). It’s self-referencing all the way down; turtles as far as one can see. Even revolutions are still working within this paradigm. Revolutionary change is just more of the same: In a revolution, rules that have been assumed to be totally immutable simply are rendered mutable by other rules that are more deeply immutable. Law and government are a self-created, self-referencing, self-propagating loop. This cynicism leads some to quip that "Government is just a game."
A guy named Peter Suber took this seriously and invented a tangled law-making quasi-government game called Nomic.
The essential activity of the game, like an actual government, is law-making. The game exists in a specific state until a move (i.e. changing a law/rule) is made. The game starts with a set of conditions, but after that evolves as players do whatever they want. Standard games (like Monopoly, basketball, and chess) possess continuity by having a set of unchanging rules. When you play, you know the boundaries of the game from the start. Nomic, however, is a self-referential system: it is a rule-governed set of systems, directives, and processes undergoing constant rule-governed change. Sound reflexive? It is! The whole point of the game is to propose, debate, and vote on rule changes in a self-amending loop.
The victory condition in Suber's initial ruleset is the accumulation of 100 points, but "this rule is deliberately boring [and vague] so that players will quickly amend it to please themselves". It’s chaos, but a fun example of the strange loops of self-referential activity. 6
This Is the Title of This Story, Which Is Also Found Several Times in the Story Itself
This is the first sentence of this story. This is the second sentence. This is the title of this story, which is also found several times in the story itself. This sentence is questioning the intrinsic value of the first two sentences. This sentence is to inform you, in case you haven't already realized it, that this is a self-referential story, that is, a story containing sentences that refer to their own structure and function. This is a sentence that provides an ending to the first paragraph.
‘The whole point of this sentence is to make clear what the point of this sentence is’
I’m not making any normative claim about whether this ‘good’ or ‘bad’; it just seems to be what actually happens. Ethics be damned
Nothing is more lethal for a meme than a tendency to look for evidence against it. The meme for blind faith secures its own perpetuation by encouraging not looking for evidence against it. A clever technique!
Easy propagation attached to a perceived high risk of negative consequences for not acting is another powerful evolved-meme-trait
Nomic was invented in the 1980’s, before computers were ubiquitous. If someone wanted to today, I can imagine there’s a way to incorporate blockchain technology here.