Does AI Need Conceptual Understanding to Utilize Predictive Power?



The Knowledge Problem in Soteriology: Risk-Reward Paradigms and the Montefeltro Metric

“A man…called his servants and entrusted his wealth to them. To one he gave five talents of gold, to another two talents, and to another one talent, each according to his ability. Then he went on his journey. The man who had received five talents went at once and put his money to work and gained five more. So also, the one with two talents gained two more. But the man who had received one talent went off, dug a hole in the ground and hid his master’s money. After a long time the master of those servants returned and settled accounts with them. The man who had received five talents brought the other five. “Master,” he said, “you entrusted me with five talents. See, I have gained five more.”

His master replied, “Well done, good and faithful servant! You have been faithful with a few things; I will put you in charge of many things. Come and share your master’s happiness!”

The man with two talents also came. “Master,” he said, “you entrusted me with two talents; see, I have gained two more.”

His master replied, “Well done, good and faithful servant! You have been faithful with a few things; I will put you in charge of many things. Come and share your master’s happiness!”

Then the man who had received one talent came. “Master,” he said, “I knew that you are a hard man, harvesting where you have not sown and gathering where you have not scattered seed. So I was afraid and went out and hid your gold in the ground. See, here is what belongs to you.”

His master replied, “You wicked, lazy servant! So you knew that I harvest where I have not sown and gather where I have not scattered seed? Well then, you should have put my money on deposit with the bankers, so that when I returned I would have received it back with interest. So take the talent from him and give it to the one who has ten. For whoever has will be given more, and they will have an abundance. Whoever does not have, even what they have will be taken from them. And throw that worthless servant outside, into the darkness, where there will be weeping and gnashing of teeth.”

-Matthew 25:14-30

The parable of the talents is universally recognized as one of the most famous of Jesus’ stories, and has generated commentary so exhaustive and profound that I can’t offer anything new on the topic. Echoed in every commentary I’ve read, though, is the condemnation of the final servant. The general consensus is that his attribution of selfishness and property seizure to his master was a last-minute excuse to obfuscate the true reason for his failure: a lack of motivation to serve the master.

Yet I urge my readers to consider, for the time being, compassion for the unfortunate servant. I see one argument according to which the unfortunate, tooth-gnashing man should be spared.

The Argument

There’s something weird about the above parable, which is that the Rule of Three is invoked and then sort of not exploited. That is to say: there are three servants, but only two meaningfully different outcomes. There is an important distinction between Servant 1 and Servant 2 in that 2 has less to invest, but behaviorally, Servants 1 and 2 are isomorphic, and teleologically, the master treats them the same.

So, what if Servant 2 had lost the money? This is seemingly a crucial counterfactual that the parable ignores. Given that the two scrupulous servants invested the talents in some economic pursuits that could have gone awry, they might have lost the talents in the process, a situation allegorical, in all ways except its conclusion, for living the pious life. I see three possible explanations for why Jesus didn’t reveal the counterfactual:

  1. The counterfactual would’ve changed nothing (the master still would’ve received the servants with rejoicing if they’d lost his money) and thus isn’t important.
  2. The counterfactual would’ve changed everything (the master would’ve rebuked the servants if they’d lost his money), which would’ve made a less compelling story.
  3. The counterfactual is an unimportant situation for the allegorical meaning of the story (it’s impossible to “lose” by investing spiritual capital).

I think Option 3 is most likely, but Options 1 and 2 still bear considering. So set aside for a moment the spiritual implications of the allegory. In 30 A.D., a talent of gold was worth about two years’ salary for a skilled laborer. Investing even that much money–let alone five times that value!–was clearly risky, especially since the money did not belong to the servants themselves. In the absence of a clear heuristic or algorithm for taking risks with items that do not belong to us, burying the talents is the only “safe” choice. To work up the chutzpah to invest the money–even in a bank–we need some sort of utility theory, and some notion of safeness, and the two need to be connected.

But EUT doesn’t take us very far. We can calculate an expected payoff, but expected utility is risk-neutral; a high-risk algorithm and a low-risk algorithm with the same average payoffs will integrate to the same expected result. We can establish an Ellsberg-type theory around salience to tell us how marginal a difference in risk must be to justify preferencing a “risky” option over a less risky one.

In general, though, there are attributes of probability and risk that we have good reason to believe are conceptually orthogonal. That probably sounds crazy–you might argue that a machine that has a 50% chance of killing you and a 50% chance of giving you $1m is riskier than a machine that effects those outcomes 1% of the time each, and does nothing the other 98%. But that’s begging the question, by already taking the span of outcomes as given. Before we can decide how “risky” an outcome is based on its probability, we need to understand what the symmetric span of outcomes looks like. First, how terrible a world can obtain? Only second can we bring the likelihood of such occurrences into the calculus. So I claim there are two parts of risk: one dealing with likelihood of bad outcomes, and one, “symmetric span,” dealing with the possibility that those bad outcomes might obtain. The latter part looks unrelated to probability.

And it is–until we account for statistical entropy. Both are obviously related to it: the lower the average probability of an outcome, the higher the entropy; as entropy increases across a symmetric distribution, risk lowers (in that new, less extreme outcomes pop up on both the good and bad sides). Much like von Neumann eigenfunctions, probability and risk can be treated as independent, but they certainly aren’t unrelated.

In behavioral economics, we tend to see EUT-type preferencing as largely monotone and risk-preferencing as indicating attributes of the consumer, much like preference for one-shot versus gradual resolution of uncertainty. We have to account for these factors in our utility functions themselves. And even a bounded rationality, Kahneman/Tversky-type model does not differentiate between risks taken with things belonging to us and things of which we are merely stewards. Thus I conclude that even we, out of the Bronze Age thousands of years hence, have little machinery with which to advise the wayward servant.

Salience and Infinity: the Montefeltro Metric

Okay, so maybe the original “wayward servant problem” as I posed it isn’t as hard as I’ve made it out to be. The servant doesn’t need all that much counsel in expected utility theory. The master pretty much gets it right: “You knew that I harvest where I have not sown and gather where I have not scattered seed? Well then, you should have put my money on deposit with the bankers, so that when I returned I would have received it back with interest.” In other words, the servant’s prior that the master will eviscerate him if he doesn’t grow the investment is probably high enough to achieve the requisite activation energy for him to invest the money. But there’s a generalization of the wayward servant problem in which the answer really isn’t so simple. The talents in the parable allegorically represent an infinite investment, so let’s drop the allegory.

As my readers surely know, I’m a Catholic, combinatorialist, effective altruist Dantista with a Borgean bent. Naturally, my utility functions look bizarre. In my paradigm, infinite risks and rewards aren’t a Dutch-books trick–they’re a constant reality. As such, I lean on salience a lot in decision-making. With all else finite, infinite things are salient. With multiple infinite things on the table, ordinal numbers and well-orderings (e.g. infinite money < infinite DALYs; quantifiable infinities < unquantifiable infinities) help somewhat, but not tremendously.

I think a lot about the Large Hadron Collider. It seems ridiculously obvious to me that it shouldn’t exist. And before you start accusing me of being a Luddite (that’s fine, I’ve been called worse), really? How are our priors so vastly disparate that you think any chance of destroying the known universe is worth taking, for the sake of knowledge alone–knowledge that is inapplicable, and will likely remain so for a long time? Why does it matter how small the odds are, when an uncountable penalty is at play? Bear with me; I’m not conservative with risk-taking in general. I think the high-risk, often life-saving surgeries my father performs every day are incredibly worthy. I think AI researchers should proceed with caution, and that the world’s generally been better since the invention of nukes. But note that in all these situations, there is a positive to balance out the danger. Successful surgery? Many DALYs. Aligned superintelligent AI? Potentially infinite DALYs. No more war with another nuclear power? Potentially infinite DALYs. But is the good that emerges from LHC research seriously unquantifiable?

It’s not that I’m a consequentialist, and it’s obviously not that I hate knowledge. There’s a kind of meta-intentionalism at work here, which sounds complicated but is actually probably the least confusing, most Kantian consistency metric I’ve ever presented on this blog. I’m going to call it the Montefeltro metric, after the unfortunate counselor of fraud: You evaluate what your intent isIf the system you’re using aligns with your intent, proceed. If it doesn’t, don’t. In this way, you can see whether the object of your will is in contradiction with the method of your will. For instance, if the world is sucked into a microscopic black hole, there will be no world to study. The ascertainment of knowledge is empirical; without data, there is no studying. This meta-operator doesn’t shut down the counterexamples I posed, because the nonexistence of the world in particular disallows the discovery of knowledge about it. It doesn’t disallow making it more peaceful, or glorifying it, because existence isn’t a predicate. Maybe this is a sketchy argument. My point is, at least the contradiction of will and outcome isn’t as obvious in the latter cases.

The Montefeltro metric has its flaws. It’s weak in the sense that it can only tell you what risks not to take. Also, ideally I’d like an operator that reflects my sensibilities more in its evaluations (e.g. would reject the LHC because it’s unreasonable, not because it’s contradictory). But the nice thing about Montefeltro is that it can tell us not to bet on a symmetrically infinite distribution: even if infinite knowledge can be gleaned, the LHC fails the Montefeltro metric.

Knowledge and Soteriology

I mentioned Dante, and EUT + infinity $\implies$ Pascal’s Wager, so you’re probably waiting for the part where I talk about Heaven and Hell in the value calculus. I posted earlier about certainty and the human condition, but again, infinity throws EUT out of whack. I’ll start with the following question, which I get asked a lot: if Heaven is an unquantifiable good, and Hell is an unquantifiable bad, how on earth do I and other Christians not spend our entire lives single-mindedly devoted to getting as many people into Heaven and out of threat of Hell as possible?

Think about it this way. For any event you have prior probability P(A obtains). Now consider the prior for your prior–P(my assessment of P(A obtains) is accurate). For instance, my prior that I will eat ramen tonight is .9, but my prior for that prior is only .1, because I literally made up the number .9 while typing, so now I’ll stick with it, but I could have picked any other single-digit number.

(Now I have of course thought of a question that no one has asked, and that no one will, but that I’ll answer anyway: “But Tessa, doesn’t the existence of “priors of priors” imply that probability is real, which is false?” No, it doesn’t, as long as you sum over classes of events. Just like, in theory, you could repeat the conditions for event A a bunch of times and come up with a good approximation of P(A), you could introspect a bunch of times about your priors for various similar events, run trials, and plug into Bayes to get posteriors. Oh, and also, the existence of free will means that counterfactuals can obtain in the realm of human mental state.)

As I get up into higher and higher levels of meta-priors, the one for “Heaven and Hell work the way I expect them to” shrinks faster than my priors for any other events–much faster than my priors for non-soteriological elements of Christian doctrine. I think Hell is empty. What’s my sureness that I’m right? Almost zero. (Of course, it can’t get to zero, because then all my subsequent priors would have to be 1.) Soteriology has the optimal mix of divine unknowability, human unpredictability, and a generous sprinkling of Will that makes it utterly impossible to consider in a Bayesian framework. Knowledge doesn’t make any sense when applied to soteriology–which isn’t true for other theological disciplines. 

There’s a common misconception, especially in the Age of Reason, that knowledge spells doom for religion. Detractors point to the oft-repeated aphorism that our faith should be like that of the children–wrongly, because the reason we seek to espouse children’s faith is not because of its blindness but because of its sincerity. We’re told to “know thyself,” so obviously, unless introspection is an exercise in futility, knowledge can permit us to grow in faith. Knowledge, when applied correctly, generally helps theology, just like it helps all other fields. Why, then, am I claiming this isn’t true for the theory of salvation?

Consider Guido da Montefeltro. “No power can the impenitent absolve,/Nor to repent, and will, at once consist,/By contradiction absolute forbid.” One can’t will and repent simultaneously, of course–doing so precludes one from repenting at all. But then what’s different about doing something you know you can repent for later, and counting on that contingency? You’re simultaneously intending to sin and intending to repent. You intend to sin only because you know it is feasible to repent later. But then you can’t truly repent for the sin–you can’t be sorry you did it. One must rue all the consequences of sin in order to repent. But it’s impossible to rue debauchery if you enjoyed it, when you know that it didn’t cause you any harm because you could later repent! We then have the following system:

If you do something for which you know you can later repent, and for which you intend to count on later repenting, you cannot sincerely regret it. Therefore you cannot repent.

But then you can repent if and only if you can’t repent. Is this a theological Russell’s paradox? Do we need a new ZF(+/- C to taste) for Catholic doctrine? Let’s see where we went wrong. Decision theory got us into this mess, so it has to get us out of it. Recall:

You intend to sin only because you know it is feasible to repent later.

I emphasized the wrong words in that sentence. The word that ought to have been emphasized was “know.” Clearly, though, you don’t know you can repent later, because based on the paradox, it turns out you can’t. But the only reason you intend to sin in the first place is because you can later repent. If you know–or at least very much suspect–that you can’t repent later, then you can repent later, because you can absolutely rue the consequences you wrought upon yourself as architects of your inherent damnation! But this seems to have only worsened the paradox, because now neither “sins that are redeemable” nor “sins that are irredeemable” is a viable category. Which means sin doesn’t exist. I’m digging myself into a hole to China here.

Consider what the black cherub tells Francis and Guido:

  1. Absolution requires repentance.
  2. One cannot repent and will simultaneously.
  3. Therefore, Boniface’s absolution of Guido was illegitimate.

The fact that one can’t repent and will at once does not preclude further repentance after the fact. Thus this only works because Guido did not further repent for the fraudulent counsel (he thought he had already been absolved). This is an important distinction. Guido didn’t particularly enjoy counseling Boniface to trick the Colonnas. Being a Franciscan, he got pretty much nothing out of it, so he experienced no later gratification that he couldn’t rue because of its consequences for him. If, then, his sin was not “redeemable,” that occurred inasmuch as his reasoning (about whether he had repented) was mistaken. He erroneously assumed he’d repented.

Redeemable isn’t a qualifier that can be applied to sin at all. It’s only one that can be applied to people, post hoc, based on whether they repent. The fact that absolution requires repentance implies that redemption is posterior to repentance, which means that considering sins repentable or not begs the question. The problem, then, isn’t that “sins for which you can repent” and “sins for which you can’t repent” cause contradictions as categories; that was a red herring all along. Of course, there’s no such thing as a sin for which one can’t repent, but not because of this “contradiction.” All sins belong to the category “sins for which one can repent,” but the tricky word “know” is a game-changer.

How can this be true? If all sins are “sins for which you can repent,” and you know that, then aren’t all sins “sins for which you know you can repent”? How can changing the location of the word “know” change a set from enormous to empty?

When will or intent is involved, knowing you can do something changes whether related things are doable (cf. Toxin Puzzle). This isn’t just about uncertainty–even thinking you can do something can have that effect. The Will plus uncertainty plus infinite risk and reward plus a notion of technical predestination that, when taken too far, spells despair, make repentance, absolution, and soteriology the kind of thing I cannot profess to know anything about. My priors that whatever methodologies I’m using to help people attain salvation are the correct ones are constantly changing. Ergo, I devote my time not to yelling at strangers on the street that they can be saved, but to yelling at strangers on the Internet that they can be saved but I don’t know if they’re saved and if I knew that they were saved, that might somehow mean they aren’t saved.

Q: “But shouldn’t you tell people to repent? It can’t possibly hurt!”

I do tell people to repent. What I can’t do is tell people, “If you repent, then you’ll be saved.” Because while from a divine perspective that must make sense, from a human perspective it’s true if and only if it’s false.

Q: “Why don’t you devote every possible free second to trying to get more people to repent, then?”

Obviously, it’s not very effective. I make a good-faith effort to convert people, but I also think it’s dangerous for me to spend too much time trying to convert atheists, because they might convert me.

Q: “Wait, what? Don’t you want to believe what’s true? If an atheist convinced you out of faith, wouldn’t that mean you came around to believe that atheism is true? Why do you want to avoid truth? Are you a Trump voter? You’re a coward! You’ll never understand the enveloping curves of sequences of ratios of 1-periodic functions, you ignorant Catholic! RELIGIOUS PEOPLE LIVE UNDER ROCKS AND AVOID INTELLIGENT DISCUSSION!” *smashes table*

So, some Freudian slips in there, but glad you asked. Not sufficiently explaining the answer I gave above provokes cries precisely to that effect. First off, I converted from atheism, and I still have some atheist sensibilities, so there are atheist arguments that strike emotional sympathy in me even though I think they’re dumb (e.g. “How can you believe in virgin birth?”), and when I feel that way, it causes me to worry that I’m not being sufficiently rational in my evaluation of arguments, and ergo that arguing with atheists brings out my knee-jerk tendencies that prove to have negative consequences for my analytic thinking. I’m not living under rocks in the meantime; I’m working to improve on that intellectual flaw that I have.

Secondly, there is a majorimportant, and tremendously culturally ignored difference between “something that someone is saying sounds believable” and “something that someone is saying is true.” When I was a freshman entering a debate society, I went back and forth between two upperclassmen for a week arguing about populism. I’d talk to the first one, who’d convince me populism is a good ideology, and then the second one would promptly talk me out of it. “While i < k, i = k + 1. While i > k, i = k – 1. End. Print i.”

I don’t want to get too deep into algorithms here (although I do feel like I’m supposed to, because my imaginary opponent questioned my mathematical abilities), but here’s an analogy: For any input I’ve tried, the Collatz process terminates. This boosts my priors that the Collatz conjecture is true, but the returns diminish as I keep trying inputs, because I’m either exhausting small inputs (which doesn’t tell me anything about the conjecture in general) or I’m trying random big inputs (which doesn’t tell me anything about the conjecture in general). It seems obvious that if I keep multiplying odd numbers by 3 and adding 1, I should eventually hit a power of 2, but that’s intuition, not truth, and truth is not inductive. This isn’t to bash inductive reasoning. Empirical evidence is helpful, but not that helpful, and the slope of the graph of prior vs. dataset size decreases as x increases. Abstruse reasoning works much the same way, replacing any pretensions to universal empiricism with the anecdotal, and often justifying premises ex post facto.

This is a big problem, given that:

  1. We live in societies that attempt and have always attempted to systematically apprehend truth by arguing;
  2. Everyone’s a missionary, and fringe theorists abound; and
  3. The more you argue with a fringe theorist, the better he will get at winning arguments.

The fact that large swaths of people believe something is not always a good reason to believe it. Or here’s a better distinction: the fact that large swaths of people believe something that does not have a perceptible effect on their daily lives is not a good reason to believe it. (If an overwhelming majority of employees really likes its boss, that’s evidence that the boss is a good boss. If an overwhelming majority of employees believe dinosaurs are our immediate ancestors, that is not good evidence for anything except maybe that that particular company has been indoctrinating its workers with wrong ideas about evolutionary biology.) Even the fact that large swaths of smart people believe something is not a good reason to believe it. Very intelligent people are prone to different, but no less insidious, fallacies than people in general, myopia being the one that comes to mind first, and intelligence signaling second. Many apocryphal texts are very convincing, which illustrates that there’s a crucial disparity between “compelling” and “correct.”

I can’t offer much by way of solution. The same notion–that charismatically arguing tends to convince people–is the notion I’m using to try to convince you right now. But I do think that separation from face-time helps somewhat. While I don’t argue with atheists all the time, I read a lot of Dawkins, listen to Harris, etc., and doing so rather than firsthand discussion, I think, permits me to focus more exclusively on the logical.

The Upshot

Q: “Okay, Tessa, so you went off on a ridiculously long tangent about algorithms, Bayesian priors, and rational argumentation theory. Are you ready to do that thing you do that you think is so clever where it turns out that your weird tangent is somehow related to the problem at hand?”

You got me.

As I expressed above, every empirical experience we have changes our priors ever-so-slightly toward 1 or 0 or, perchance, some limiting value bounded away from 1 and 0. Perhaps human uncertainty can actually help us here. It’s impossible to repent for something we know we can repent for. But repenting for something we’re only .9999 sure we can repent for neatly avoids the paradox because not repenting never wins the day under any expected utility model. Finite positive times infinite is infinite. Our repentance is sincere.

I hope this is an unexpected upshot, because it was unexpected to me: under the lens of the knowledge-and-soteriology paradox, we can justify two things that seem unsavory in theology.

First, the fundamental uncertainty of God’s existence makes sense in light of the paradox I described. I think this examination actually presents a very elegant notion of why the unknowability of God is a good thing. In the human mind, probability 0 times infinite risk is indeterminate. But any small epsilon times infinite risk is infinite. Uncertainty is good insofar as it avoids the logical paradox of intent and repentance. Unknowability is crucial to Pascal’s wager, which is what makes it so compelling in the first place. With total faith in God, there’s no need for any analysis whatsoever. But the lack of perfect knowledge permits both the presentation of elegant mathematical ideas in the realm of theology and the avoidance of complete sureness of the doctrine that repentance is in any way related to redemption.

Second, the model of “I’m only .999 sure that I’ve repented adequately/that my repentance was sincere/that I can repent at all” provides some understanding of why human self-centeredness isn’t altogether terrible. The fear of eternal damnation isn’t a good reason to repent (having strayed from the Good is a much better reason). But the latter isn’t as easily quantified–or rather, as easily categorizable as unquantifiable–as the former. I’ve spoken before of the binary notion of virtue and sin; there isn’t a readily emergent way to order how much we’ve strayed from the Good. Say what you want about Hell, but you can’t say it’s not the salient risk in pretty much any outcome spread it finds itself a part of. There will be weeping and gnashing of teeth…


Religion and Humility: Rationality, Diagonalization, and the Hardness Criterion

This summer I had a good old-fashioned Crisis of Faith.

It became apparent that I’ve let myself go a little in terms of having a ready retort on hand for spontaneous atheist arguments. I spent some time this summer at a conservative think tank, full of minds like mine (if significantly more libertarian) and blest with a high degree of Catholic literacy. Although I was regaled there daily with requests to mathematically prove God’s existence, thankfully the majority of the religious arguments my classmates took up with me ran along the lines of “Is the seat of Peter empty?” rather than “How can you believe in miracles?”

My return to New Haven engulfed me in the world of mathematics, leaving little time for theological debate. Mathematics departments nationwide run the gamut from very religious to Dawkinsian (it’s hard to be a Humean mathematician, and impossible to be a Humean statistician). Ours enjoys a variety of religious viewpoints, with the majority falling secular agnostic. Thus, when a mentor posed a new and unusual atheist argument to me, I was caught unprepared.

The Problem

I’ve seen all the inane, readily neutralized atheist claims–“Do you really believe in virgin birth?,” “Haven’t terrible things been done in the name of Christ?,” or “Religion was established to keep citizens complicit.” SSC raises a hilarious one about whales not being fish.

Nonetheless, arguments from epistemology are more compelling. Not the “argument from unknowability,” per se. I’ve long considered the existence-of-God problem undecidable. This doesn’t bother me, because I’m not a logical positivist; physical facts are not the only important components of a system. I don’t care that atheists and I don’t disagree on any physical matters that can be finite-time decided, and I don’t think the criterion of falsifiability is useful.

The counterargument with which I was presented was much slicker, and imbued with all that meta-level, logically contradictory, late-Inferno-style contrapasso of which I am so fond: “Throughout history, people have realized how much they don’t know. The more we learn, the more there is to learn. Religion, in presupposing the ultimate answers, is the Platonic form of hubris.” Steelmanned: “Religion is prideful, but prides itself in being humble.”

That got to me. My discipline of choice is a field in which we constantly know less than we did before, in a certain sense, because every answer prompts questions that didn’t previously occur to us. We learn “calculus” in high school and think we know what integration is, then learn vector analysis in college and think this time we really know what integration is, then learn Lebesgue theory and realize we’ll never know what integration is. Humility is both necessary and proper to the discipline of mathematics, as it is to the discipline of theology. But mathematicians don’t claim to have solved the (perhaps undecidable) Collatz conjecture, whereas theologians do claim to have solved the (probably undecidable) God problem.

Religious sensibilities are more insidious than religious confession. My mother, an evolutionary biologist and enthusiastic Dawkinsian atheist, is terrified by The Exorcist and has admitted to me that she’d never attend a LaVeyan meetup because she could not sign her soul over to Satan even though she believes him nonexistent. She’s one of many nonreligious I know with religious sensibilities ranging from the theological to the social to the moral, yet I know no believers who have the faith but lack the sensibilities. I believe these inclinations precede confession; they are a necessary but not sufficient prerequisite to genuine faith. So what happens when religious sensibilities undermine religious conviction? What happens when the truth claim of religion is at least in some sense hubristic, but the sensibilities of religion are humble?

I go to a highly-regarded research university and thus constantly make use of the immediately available option to knock on the office door of one of the smartest people in the world and demand answers. My theology thesis advisor wasn’t in, so I stopped in at the first office I could find: that of a professor specializing in the intersection of religion and political theory. Perfect–exactly the kind of person who’d know all about the theory of the “opiate of the masses.” I walked in, introduced myself, and explained my problem: the priors for religion are heavily dependent on humility, but the truth claims of religion are hubristic. How can I be both Bayesian and Catholic? Help!

A Helpful Digression

The paradox with which I confronted the professor is related to signaling theory and what I’ll describe as the “hardness criterion.”

Definition 1. Hardness Criterion. A map F defined from the set of tuples on the space of choices to the space itself, where F(a,b,…) = argmax{difficulty(a), difficulty(b),…}.

In other words, the Hardness Criterion is the belief: “When presented with multiple options of action, I should do the one that is most difficult.” Naturally, “difficult” can mean a bunch of different things, some of which may be contradictory. For example, being a doctor is more technically difficult than being a garbage disposal worker, but the latter is more psychologically difficult for an Alpha on the alpha island in Brave New World.

The Hardness Criterion seems obviously wrong at a first glance, but I urge my readers to consider it more carefully. Steelmanned, it tells us that Man has a duty to pursue the highest spheres of work, self-analysis, and the search for truth, and to reject hedonism, which seems observably true. It doesn’t beget any of the silly fallacies detractors would like–“But everyone can’t do the universal hardest thing; some people have to do something else, or else we have a society of doctors” and “If everyone does the hardest thing, no one will be good at his job”–because what’s difficult differs by person, and how hard something is, in my experience, is orthogonal to how good I am at doing it. I’ve never been able to gauge how good I am at mathematics because it seems roughly equally difficult no matter how good you are at it, like cross-country running but unlike music or politics.

Those who deride religion for providing cushiness and a “Heavenly Daddy” figure are unknowingly, implicitly employing the Hardness Criterion in a way similar to Occam’s Razor. The argument goes like this: Religion permits an emotional solace in the form of the promise of eternal life, whereas atheism does not permit such solace. Therefore atheism is more difficult and I should do it.

Of course this requires the Hardness Criterion, because there is no other grounds for rejecting religion on the basis of its provision of emotional solace. One can only reject this solace if they believe the solace to be bad, which requires the Hardness Criterion, because in theory, whether a belief provides emotional solace is orthogonal to whether it is true. Sure, emotional solace might discredit the epistemic honesty of one’s acceptance of the framework, but it bears no consequences for the truthfulness of the framework itself–unless you’re willing to categorize “things that provide emotional solace” as “things I should not believe,” which utilizes the Hardness Criterion.

To reject the Hardness Criterion properly requires diagonalization. It’s noticeable that “hardness” generalizes to the meta-level, which prompts the question, “Is the algorithm ‘do the action that is hardest’ the hardest algorithm? Doesn’t doing the easiest thing all the time place me in opposition to the Hardness Criterion, which is, if I believe in the Hardness Criterion, an intellectually difficult space in which to operate?” This counterargument works beautifully, because at the meta-level, “choose the most difficult thing all the time” is a very easy algorithm, in that there aren’t any hard choices, given that your options are well-ordered. It seems to me that one could prove the Hardness Criterion is not well-defined in much the same way one can prove the halting problem is undecidable.

This is the reason the Hardness Criterion argument against religion is easily deflated. On the meta-level, “believing the thing that is harder” provides a degree of emotional solace that stems from finding one’s beliefs to be in accordance with the Hardness Criterion, while being religious is “harder” in that sense. Similarly, while atheism is “harder” than religion in terms of lacking the component of emotional solace, religion is “harder” than atheism in terms of a meta-level hardness factor: the difficulty the religious face in rationally justifying their beliefs given their first-order apparent rejection of the criterion. This ultimate point–that under the Hardness Criterion, the most contrarian seems always to win–deals a death blow to its acceptance as a useful algorithm.

The Solution

I ended up talking to the professor for about thirty minutes, and she did not disappoint (how I love this school!). We had a fruitful discussion about the fallacy described in the digression section, and she forwarded me an article she’d written arguing that support of Islamic political parties in Muslim-majority countries is rational insofar as the emotional support provided by religion eases stress. Naturally, I and my Dante-meets-Borges-meets-Bostrom mindset loved this because of its seeming counterintuitiveness: as strange as it is to accept, the emotionally easier option is of course the more rational one, in the sense of utility maximization.

I went home and thought about this for hours. Hours turned into weeks, which turned into months. And finally I figured it out. Would it be possible to create an inverted Hardness Criterion labeled the Ease Criterion, affiliated with a straightforward Kahneman/Tversky-type utility function, yielding a bijective relationship between Ease Criterion rankings and outputs of some rational choice function? Definitely. Pick the option with minimal difficulty.

But does this Ease Criterion collapse as obviously as its negation does? In one sense, the Ease Criterion is easy on the meta-level because the choices it provides are well-defined. There’s no simple, Berry-paradox-type situation in which the Ease Criterion falls apart. For all intents and purposes, the Ease Criterion is at least as good as Occam’s Razor, because I can imagine some situation exists in which the algorithm that uses simplicity to pick a course of action is not the simplest algorithm. Does there exist one in which an algorithm that uses ease isn’t the easiest (if we admit “emotional solace” as a stand-in for “ease”)?

Indeed there does. The Ease Criterion on two variables always picks what the Hardness Criterion doesn’t pick, so the inverse diagonalization produces a contradiction. I can readily imagine somebody emotionally tortured by the notion that he’s always choosing the easiest option! A theorem of this flavor feels like it ought to follow:

Theorem 2. No operator that definitionally outputs a single choice from a choice set by a metric of difficulty or complexity is consistently defined.

I don’t think the generalization to multidimensional operators works, but that isn’t really relevant here, as no one claims two religions. The conclusion: if we allow difficulty, ease, simplicity, or complexity to serve as a stand-in for “rationality,” then we cannot consistently behave rationally. (Aside: I know rationality isn’t everything, but it still benefits us to create a more nuanced notion of what rationality is.)

The contradiction my mentor voiced was, as you may have by now realized, isomorphic to the problem with the univariate criteria described above. I could now see that the problem he had presented was that the Humility Criterion is inconsistent, and his claim was definitely legitimate. The Humility Criterion makes truth claims! Of course, on the meta-level, it isn’t humble.

Central Question: Does Christianity actually make use of the Humility Criterion?

Naturally, the only way to disarm the paradox of the humility of faith versus the pride of faith is to reject the notion that Christianity uses a so-called “Humility Criterion”–e.g. while humility is a virtue, it is not the methodology one uses to arrive at Christian conclusions.

Virtues are not algorithms. Consider the algorithm “Do the thing that is virtuous, or if multiple virtuous options exist, the one that is most good” (so phrased because I don’t like the notion of “most virtuous”). If you’re an effective altruist, it’s clear this algorithm is virtuous, which is not self-contradictory. But performing this algorithm is not a virtue any more than entering a convent is a virtue. They’re both methodologies used to pursue virtue. (This is why I love that Christianity enumerates so specifically what the virtues actually are.)

Not convinced? Consider the following argument why virtue is not meta-level. Take the action of “cultivating an environment in which I can better pursue the virtue of almsgiving.” It’s clear that an almsgiving person who cultivated such an environment and an almsgiving person who didn’t are both almsgiving, and thus are both manifesting the virtue of charity. The person who didn’t cultivate such an environment might even be a better person, by dint of emerging triumphant against more temptation.

Similarly, I recently posed the following thought experiment to a Catholic close friend: Mr. Brown doesn’t want to give alms. Which is worse: for Mr. Brown to falsely tell mendicants he doesn’t carry his wallet, or for Mr. Brown to deliberately leave his wallet at home so he doesn’t have to lie when he tells that to mendicants? We agreed that it was the latter, because it eliminates the possibility of repentance (cf. Guido da Montefeltro in Inferno XXVII).

Humility is not an algorithm; it is consistent for Christians to use algorithms that are not themselves humble in order to maximize their humility. And because humility is not an algorithm, it is not used to discern truth, and thus it cannot be a contradiction that the “lux” part of faith is so glorious. The centrality of the nonexistence of a Humility Criterion is paramount! Without it, “do the things that are humble” does not imply “believe the things that are humble.”

“Humble yourselves before the Lord, and He will lift you up.” -James 4:10


The Libertarian Theodicy

There is an eminent tautology that emerges when we consider where federal power ends. All powers that are not claimed by the national government are left to and reserved for the subsidiary bodies of government, especially the states. The question of where, exactly, federal power ends has been one of much controversiality in American history, from the Virginia and Kentucky Resolutions to the Civil War to the Seventeenth Amendment to the civil rights movement.

Tremendous swaths of people have fought bitterly for legal decisions to be made closer to home, whether home was Texas or Massachusetts. Indeed, it’s a common misconception that states’ rights were the Southern battle cry during the Civil War. The Northerners used those arguments too–especially in their rally against the Dred Scott decision and Fugitive Slave Act, which they saw as federal infringements upon their states’ illegalization of slavery. The claim “on our land, you follow our rules” has further manifested in recent times, surrounding the Obergefell decision. Even today, the legality of state nullification and secession is an open question.

States’ rights’ advocates will tell you that the argument against these extensions of federal power is orthogonal to any metric of virtue. Small-government enthusiasts claim not to oppose the federal enfranchisement of minorities or gay people out of racism or fear or even the belief that these laws are wrong. They won’t defend their desires to have the matters relegated to state courts for any ideologically-based reason. Rather, they’ll tell you they fight such measures because of their nefarious sweeping scope. Libertarians detest the notion that the national government is settling what they see as intra-state affairs on states’ behalf.

1865 libertarians wanted the issue of slavery left to the states. 1965 libertarians wanted the same for civil rights. Abortion, capital punishment, gay marriage, marijuana? Advocates and detractors alike of the individual issues have argued for the legality of these actions and products to be determined on a smaller scale.

In a world where righteous indignation has always been the motivator of activism, it seems remarkable that so many have crusaded for the right to be wrong–not, in fact, believing themselves to be correct, but rather seeking the opportunity to determine their own stance; to have moral choices not prescribed unto them.

Why fight for states’ right to decide, even if that means some states won’t legislate the way I want them to? At first blush, the problem looks isomorphic to that of non-natural theodicy, the consideration of why men commit acts of evil in the world. The canonical answer is that free will and perfect goodness are mutually exclusive. God had to pick one.

While the natural world is an object-level entity, the realm of the divine ought be considered as the meta-level whose principles manifest in the world that we see. In the eyes of the divine, when is free will better than goodness? Clearly, if there is either something good in free will exercised in and of itself, or something bad in permanent and immutable goodness. Object-level goodness is not the same as meta-level goodness; choosing the option that maximizes the amount of good is the correct solution, and free will both allows individual choice (meta-level good) and results in some people who perform acts of virtue (further instantiation of object-level good). Instilling perfect goodness allows only the latter. It is better for Man and for God’s glory that humans have ownership and possession of the acts they perform.

Now is the part where I show my cards and say that there’s an enormous logical fallacy taking place in the application of human theodicy to government. The two are not isomorphic, because states’ rights constitute a case in which ownership can differ greatly from possession, involvement, and agency, all of which need to be present to justify free will.

Can a Texan claim that he has more ownership of local politics than of federal–that a court decision made locally is more his than one made by SCOTUS? Probably. He has a small chance of sitting on a Texas jury, and none of sitting on a federal one. He has some tiny margin of influence on ballot propositions in his home state, again narrowly beating out his influence vector for federal propositions. Any positive epsilon is greater than zero; given infinite time, our Texan will, sooner or later, find himself the determining vote in an issue important to him. So his state’s decisions belong to him more viscerally. So far, so good.

Is his home state more likely to side with him in terms of deciding legal matters? Definitely. Statistically, the population of Texas is likely to look more like him than the voter base of the nation at large, as there is some self-segregation due to shared state values. His ten neighbors are more likely to agree with him politically than ten people selected at random from the United States voter base, by sheer dint of the fact that they chose to live near him, which they probably wouldn’t have done if his politics were anathema to them. This also makes his state’s decisions more his, because they are more likely to resemble his own decisions.

But does the Texan have more involvement in his state’s decisions than in national ones–that is, more of a say? Not really. It is more likely that the outcome will turn out in a way he’d find agreeable, but his participation had near nothing to do with it. In terms of issues, he has narrowly more of a say in state matters, but not so in elections. In Boolean terms, the influence of individual voters within a state (especially a small one) is just about as negligible as that of voters in a federal election.

Does the Texan have more agency in his state than in the nation? No–that one’s just ridiculous, because agency is concerned more with actions than results. Not that the results of his political activities are really that different in the discrete spheres. He can write an angry letter to his Congressman, to his governor, or to the President. None of those people is significantly more likely to read the letter than the others. (One of his congressman’s staffers will probably read it, but won’t be able to do anything about it. So in each case, he took the same actions and got the same null results.)

So the Texan has more ownership of his state’s politics, but no more involvement or agency in their determination. The free will argument doesn’t hold up, because even if God predetermines our actions, they still belong to us, and the culpability is ours. Calvin, the father of determinist theology, argued in Institutes of the Christian Religion that in the predeterminist worldview, Man retains full ownership of his actions. For Man to have free will, he needs something greater than ownership–involvement and agency, which the Texan gets no more out of Texas than out of the U.S.A.

Thus I conclude that he wants the issue at hand to be left to the states not out of a want of self-sovereignty but because he concludes, not incorrectly, that the population of his state is more likely to vote his way. That seems pretty likely when you realize that none of the approximately three Green Party members in Texas wants the issue of abortion determined closer to home.

Is there a saving grace for the states’ rights argument? Yes! The empirically validated belief that communities themselves make the rules that work best for them. But this is clearly an argument from epistemic modesty, rather than from the lofty rhetoric of freedom as its own good. Groups of people should be self-governing because they are more likely to do a good job than their detached overseers, not because their freedom is viscerally important in the abstract.

This understanding, radically different in its first principles, doesn’t change political results all that much. It does, however, alter the approach with which we view the negative externalities I mentioned above: that some states won’t legislate the way want them to. For issues that are one-size-fits-all, we no longer need gaze upon our fellowmen in different territories with sympathy at the unfortunate political results their freedom has prompted. We might instead assume their model works better for them, and wonder whether it would for us, too.

Error, Noise, and Data: The Allegory of the Lightning Strike

The Allegory

Adam and Eve lived with their children Cain and Abel in an otherwise empty world. Cain and Abel did not go hungry, for their parents farmed the land; they were not bored, for they had each other for amusement. But they were uncertain, for, having been expelled from God’s presence, they had no heavenly voice to tell them what things they did were good or bad.

“Hey, Cain!” Abel cried. “I want to roll this marble, but I fear that if I do so, it will roll under the bed, and Mom will be annoyed.”

Cain responded, “I wonder if God could tell us whether you should roll the marble or not. Maybe he’d send a sign–a lightning strike, or a sudden gust of wind.”

“Good point.” Abel took a deep breath. Then he bellowed, “Heavenly Father! If the marble will roll under the bed, then send a lightning strike!”

No lightning strike emerged from the cerulean blue sky. It was a pleasant spring midday, after all. So Abel went ahead and rolled the marble, and it rolled under the bed.

“Huh,” said Cain. “We know God exists, is benevolent, and is listening. But he did not warn us that the marble would roll under the bed. Therefore, I conclude that the consequences of the marble’s rolling weren’t important enough to get God’s attention.”

It became something of a game between the brothers: whenever a small decision weighed on their minds, they’d ask God to send a lightning strike to elucidate their thinking. No such strikes occurred, and so they went ahead with their actions, and got the expected outcomes most of the time.

As they grew older, Cain and Abel found themselves facing more challenging and important decisions. What should they get Mother and Father for their joint birthday celebration? How much corn should they plant this year? Which of the suddenly emergent womenfolk should they marry? And it was with regards to this last question that a strange event proceeded to occur.

“Heavenly Father, send a lightning strike if you wish me to marry Wilhemina!” cried Cain, and to his immense surprise, lightning came screaming from the blue skies.

“Wow!” Abel yelled. “Heavenly Father, send me a lightning strike if you wish me to marry Philomena!” No lightning. Cain married, and Abel remained a bachelor.

Later that year, they were playing with marbles when Abel wondered if he might roll a marble under the bed. “Heavenly Father,” he cried, “send down lightning if this marble will roll under the bed!”


“Wait, Abel,” cautioned Cain as Abel prepared to put his marble away. “God only responds to important questions, remember? This lightning was just a fluke.”

The Interpretation

And so Abel rolled the marble. The story ends there, and it doesn’t matter whether the marble rolled under the bed. Cain and Abel grew so used to the absence of lightning caterwauling down in their (comparably many) small decisions that they felt that it should only be used as a value metric in their (comparably few) large decisions. Rendering this in probabilistic terms, we can establish the following. First, lightning is very unlikely to occur from clear skies; secondly, if it does occur, it’s more likely that God sent it than that it was random.

  1. P(no lightning strike) >> P(lightning strike)
  2. P(God sent it|lightning strike) >> P(It was random|lightning strike)
  3. P(God sends lightning strike|important question) > P(God sends lightning strike|unimportant question)

Cain and Abel assumed this third, non-verifiable relationship at the get-go. Ultimately, out of the myriad “unimportant” questions they asked God, only one was ever answered by a lightning strike (Abel’s last marble roll). Let’s say Cain and Abel asked 1,000 unimportant questions. Out of the maybe ten “important” questions they asked God, one was also answered by a lightning strike. It’s obvious that 10% > 0.1%, so the brothers’ initial assumption that God was considerably more likely to send a lightning strike in response to an important question seems empirically verified. Nonetheless, in concluding that the final lightning strike was accidental, they make a logical error.

Cain and Abel’s assumption that the lack of lightning strikes about unimportant issues means that the ultimate lightning strike was a fluke is unfounded. It’s a reductio argument: if God didn’t discriminate between important and unimportant questions, and if one lightning strike occurred in each, then the respective presence or absence of lightning strikes in each category up to that point is incredibly unlikely.

For instance, assuming that God does not discriminate in response between important and unimportant queries, then if we use the unimportant question set to gain an approximation of the likelihood of question response, the odds that one of the first ten important questions would be answered by a lightning strike (given that the approximate odds of response are roughly one in a thousand) are approximately 1%. This does not, however, mean that the lightning strike was a fluke; it merely means that we are faced with two unlikely-looking explanations, one of which has to be true:

  1. God responded to a question within the first ten, which is very unusual.
  2. The lightning strike was random.

How likely is a random lightning strike? If P(It was random|lightning strike) < 1%, then Option 1 is favorable.

The margin at which we decide it is no longer more likely that God engineered the lightning strike varies based on which set we use as the probability determiner. If we instead use the important-question set (which makes sense, because a lightning strike answered a question out of that set first), then we can assume that the odds are about 1/10, so assuming God is equally willing to answer unimportant questions, the probability of his not doing so for 1000 questions is a whopping 2E-44%. But the operative question, still, is not How unlikely is that? but Is that less likely than the alternative (a random lightning strike occurrence)?

Importantly, since both of the possible implications of a lightning strike (randomness or divine intervention’s occurrence seeming very non-Bayesian) are fairly unlikely in this framework, the situation itself–lightning strike occurring once in the large data set and once in the small–is highly improbable. Given that it has obtained, though, the more likely explanation is probably the correct one.


Something about these calculations might feel off to you, and if you think that, you’re right. The problem with the above reasoning is that when considering the overall probabilities of God-sent lightning strikes via the totality of data, I didn’t weigh in the possibility that one of the two lightning strikes Cain and Abel experienced might have been a random fluke.

That probability is small, but given that we have no way of knowing after the fact whether any individual lightning strike was God-sent or random (only that both have a greater-than-zero probability of occurring), it must be considered.

Random lightning strikes can here be considered a stand-in for noise in data sets in cases where that noise could significantly confound our prior probabilities. We could liken this to an arbitrarily long Boolean string in which a value of 0 in the kth place in the original string is highly unlikely, and the string then undergoes some fuzziness after which the k-value is, with some degree of likelihood, altered. The question that this poses to statisticians, which is more intuitively apparent in the lightning strike example, is the following:

If we attain our prior probabilities through an empirical analysis of the data that we have, knowing that noise can be a factor, and then find that our distributions are odd, in what cases does this make it likely that random noise altered some of the strings in the original set?

If noise is known to be less or more probable than the result we are looking for, we would conclude accordingly; the problem emerges in cases where we cannot see the original data set, and thus have no way of knowing what our priors should be that noise occurred. In the lightning strike allegory, there is a solution to this problem, that perhaps could be generalized.

The Solution

Cain and Abel should repeat their experiment arbitrarily many times–let’s say, until they have achieved sufficiently many lightning strikes that the number of strikes is suitable for data analysis. Perhaps after one hundred years of asking questions, Cain and Abel receive one hundred lightning strikes. In this case, there is an obvious mechanism through which the brothers can discern whether lightning strikes are God-sent: whether the lightning strikes make correct truth claims. (E.g. if Abel asked for a lightning strike if the marble would roll under the bed, a lightning strike occurred, and the marble did not roll under the bed, then that lightning strike was not divine in origin.) The following claims seem to be reasonable:

  1. P(lightning strike is accurate|lightning strike was God-sent) = 1
  2. P(lightning strike is accurate|lightning strike was random) = appx. .5, barring cases in which one of the outcomes was very unlikely, in which case the brothers wouldn’t have needed divine advice anyway).

Now we have a set of probabilities through which we can come to ascertain the approximate proportion of random lightning strikes to divine lightning strikes.

The same can be said of noise in data sets. For cryptography, there is an obvious, immediate parallel to the case I describe: does the resultant string make sense? If not, then it is much more likely to have been noise-altered. By ascribing sufficiently accurate prior probabilities to these cases, we can discern the relative likelihood of noise to that of the genuine obtainment of unlikely outcomes, and then can proceed with the analysis described in “Interpretation.”