Gödel, Arrow, Preference, Spirit, and Dictatorship


2022-03-17 journal entry #2

3:29pm, Waltham Panera Bread

Letter to send to Jim [my friend and employer] and Daxe [my brother David].

Hi Jim and Daxe,

I’m following up on our thread based on Jim’s question, which I paraphrased as: What does Arrow’s Impossibility Theorem (AIT) imply about the tendency for societies to evolve to dictatorship? I’ll also touch upon the issue of “digital democracy” which Jim emailed about yesterday.


  • AIT says: If an algorithm to aggregate individual rankings over a set of (at least three) choices into an aggregate societal ranking satisfies some specific, simply stated axioms (that intuitively should be satisfied by a fair voting system), then that algorithm must always return the ranking of some specific individual, the “dictator”.
  • There is a huge literature on the success and failure of applying and extending AIT.
  • We’re using AIT here to motivate creative riffing that could, hopefully, lead to disciplined results in the future.
  • I pointed out that AIT only gives an abstract constraint on possible ways decision making can be structured. It says nothing about human behavior. For that we need some additional source of thinking, like behavioral economics.
  • Reacting to Jim’s and my input, Daxe riffed, incorporating ideas he has had over the course of many years.

Daxe Prompts

  • I will phrase a direction following up on my last email as a reaction to two of Daxe’s points:
    • “[AIT] pairs well with Gödel’s impossibility theorem, in understanding the theoretical limits of what can be said logically/politically;” and
    • “[There is] another layer to AIT, the spiritual/religious.”
      “There are preferences over what we do, and how we do them; there are preferences over why we do them.”

Incompleteness and The Underground Man

  • Dostoevsky’s “underground man” essentially says that any attempt to describe or constrain man to be a purely rational creature is doomed to failure: “man everywhere and at all times, whoever he may be, has preferred to act as he chose and not in the least as his reason and advantage dictated.”
    • Gödel-like version: Any rational explanation of Man must be incomplete.
    • Arrow-like version: A complete rational explanation of Man is impossible.
  • I interpret Daxe as implying that: A “psychological+rational” explanation of Man must be incomplete; it is impossible to find a complete such explanation.
  • Perhaps there is a loophole, as there is with Gödel’s incompleteness theorem: We could possibly add an infinite collection of Ptolemaic circles (i.e., narrow and specific axioms) to make the rational man hypothesis correct.
  • I believe Daxe suggests that rather than adding increasingly arcane circular reasoning, we need a new kind of celestial reasoning.
    • Just as Newton put the sun at the center to explain the orbiting of the planets, Daxe would put spirit at the center to explain the actions of people.
  • Obviously, no sequence of words or mathematical symbols can capture the human soul in full.
    • Like Gödel’s unprovable statements, unconquerable spirits exist.
    • Gödel proved unprovability by finding the words to give meaning to the sentence “I am unprovable”.
    • Dostoevsky struggled with unconquerability by finding words to give meaning to the sentence “I am unconquerable.”

The Unconquerable Spirit

  • Jim and I have discussed Thomas Paine’s paragraph starting “These are the times that try men’s souls”, which ends “Tyranny, like hell, is not easily conquered; yet we have this consolation with us, that the harder the conflict, the more glorious the triumph. What we obtain too cheap, we esteem too lightly: ’tis dearness only that gives every thing its value. Heaven knows how to put a proper price upon its goods; and it would be strange indeed if so celestial an article as FREEDOM should not be highly rated.”
  • My self-referential metaphorical instincts are not up to the task, but somehow I think I am about to launch a line of reasoning about unreasonable humans to try to conquer the tyranny of the unconquerable spirit.
    • I pre-declare that I will give more glory (meaning) to unconquerability, by trying and failing to conquer it.
    • I also acknowledge that this line of reasoning can be thought of as a version of Daxe’s talk about self-reference and preference.
    • It is here I will also implicitly pull in a version of digital democracy.

Imagined Reasoning

  • Imagine a society where everyone has preferences over everything, including the nature of the algorithm for converting individual preference to society wide preferences. We assume that each individual wants their preferences to seem fair, but has a bias toward advantaging themselves.
  • Let us also assume a mechanism that determines society wide actions based on societal preferences.
  • Imaginings I1 and I2 determine a system of co-evolving preferences and actions.
  • In particular, it determines the evolution of the algorithm for converting individual preferences to societal wide preferences.
  • By (some generalized) AIT, there can be no stable “fair” solution for aggregation of preferences. The small biases toward self-advantage will increasingly self-reinforce and battle each other until a single dictator evolves.

The Power of Cheating

  • Cheat 1. I point out here, that my “imaginings” I1 and I2 are the crucial place where I sweep chaos under the rug, and merely imagine controlling the uncontrollable by critical thought.
  • Cheat 2. I make vague statements that I take to have whatever meaning I need when I use them.
  • Cheat 3. I use magical reasoning; I wave my wand and … hypnotize you into believing … .
  • Cheat 4. I’m kind of BS’ing, hoping to lure you in. Though you may be bored or angry this time, I can iterate and, with practice, am bound to eventually come up with arguments that overcome your worn down resistance.
  • Cheat 5. Whether we (not-quite-Karamazov) brothers fight or cooperate for credit on this approach, if we provide enough intellectual force and bluster, we will gain a following and conquer all. If not, someone else will conquer us and ascend further up the hierarchy.

The real power behind authoritarian pushes is not the imagined reasoning but the power of cheating.

Take care,

stopped at 6:30 pm

~ Scott Axelrod ~