The Virtue of an Almost Right Solution

Earlier today, in my final Sondheim post, I was writing in praise of the grotesque.  This post is on the same topic, but I’m code-switching a bit and doing it a more analytic, not aesthetic framework.  Over at LessWrong, there’s a great illustration of how positive bias works:

I am teaching a class, and I write upon the blackboard three numbers: 2-4-6. “I am thinking of a rule,” I say, “which governs sequences of three numbers. The sequence 2-4-6, as it so happens, obeys this rule. Each of you will find, on your desk, a pile of index cards. Write down a sequence of three numbers on a card, and I’ll mark it “Yes” for fits the rule, or “No” for not fitting the rule. Then you can write down another set of three numbers and ask whether it fits again, and so on. When you’re confident that you know the rule, write down the rule on a card. You can test as many triplets as you like.”

Here’s the record of one student’s guesses:

4, 6, 2             No
4, 6, 8             Yes
10, 12, 14      Yes

At this point the student wrote down his guess at the rule. What do you think the rule is? Would you have wanted to test another triplet, and if so, what would it be? Take a moment to think before continuing.

Think of your answer before you pop over and read the whole thing.  I’ll wait.

— — —

 It’s important to practice thinking of negative results that your model predicts (and then coming up with a way to test them).  I think there’s another interesting class of results to try to check: what’s the negative result that comes closest to passing your model’s test?

It’s a little less useful in the example given in Eliezer’s post, but once you’re getting into fuzzier philosophical arguments, I think it could be helpful.  What might pass a good-enough diagnostic, but doesn’t actually belong in the category?  If I’m trying to explain the characteristics of a good relationship, what kind of relationship would take the most work to differentiate from one that met the criteria?

I might not know the answer when I start looking for examples.  This is why I’m grateful for the emphasis my debate group placed on futzing with thought experiments.  It’s easier for me than it used to be to try to isolate and tweak one part of the scenario (if Fosca loved Giorgio in the same way, but didn’t tell him…, if Giorgio dropped the line “love that shuts away the world” from his description of his ideal…, etc).

It can be interesting when a small change pushes a hypothetical clearly into one category or the other, but I’m most energized when I’m pushing a thought-experiment around what I guess is the uncanny valley of concepts.  This is what I’m talking about when I say the love story in Passion is grotesque; it’s very close to the thing we’re looking for, but the divergences are jarring.

Once you’re playing around with tension and paradox, you may realize the fault lay not in your test cases, but in your partitioning of conceptspace.  If the grotesque example really does belong to your rule, maybe you’ve been ignoring some of the dangers of the ideal you were interested in.  If you can figure out what feature bars it from membership in the set, you can look back at some ideas you’ve previously coded as positive results and make sure they didn’t sneak this quality through in some milder form.


And, for an added benefit: this skill also ends up being a drill in modeling other minds.  If you can figure out where an ambiguity or obscurity in your model might lead someone into an honest mistake, maybe that practice will help you defuse your combat reflexes in an argument when you’re about to leap to accusing your interlocutor of willfully misinterpreting or ignoring something they should know.


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  • Ted Seeber

    That one is a little too easy for anybody who took your advice a couple of weeks ago and read HPMOR.

    Especially since it’s a problem that, if taught correctly, can be a bit of a trick, depending on the number of triplets guessed.

    Additional triplets I’d try, given the data so far:

    Which should either be all no, or yes, yes, yes, no in that order.

    • leahlibresco

      Too easy, eh? I notice you only included positive integers in your test cases.

      • Anonymous

        We should probably also make guesses with complex numbers, quaternions, cardinal/ordinal numbers, hyperreals, etc. We wouldn’t want the actual rule to be slightly more general than our approximation. Of course, if the statement of the rule doesn’t apply to these numbers, does it kick back a run-time error? If the statement could be interpreted more broadly but wasn’t intended to be, what happens?

      • Ted Seeber

        Hmmm…That might be due to the fact I’m still trying to teach my special needs 9 year old addition and subtraction. That, and since I write business software, I find very little need for negative numbers or complex numbers (after all, a sum of money can just as easily be represented as cents).

      • Dan F.

        My thought was along the same lines as Ted but added in that I’ve played enough IQ/mind games to start work on a problem like this with the assumption that the answer will be simple, not complex, usually just requiring a look at the data from another direction/angle than the one in which it is presented.

        Now, that assumption could (and does) get me in trouble but because it is successful well more often than not I still use it, particularly if i’m on the clock, knowing that I could be horribly wrong but not particularly caring. Which make me wonder what sort of thought experiment could cause me to be emotionally invested enough in finding the correct answer that I would worry about being wrong.

  • Alexander S. Anderson

    I’m reminded of Thomas Kuhn. Scientific theories, (or, to be more accurate, paradigms) not only determine how scientists think, they also determine where they look for evidence. A paradigm even determines boundary conditions on what can be considered science. Kuhn considered this a good thing, especially considering the underdetermination provided by scientific evidence, it’s good to have a working theory to push science forward. But, it has obvious limitations, as the Less Wrong post shows.

  • Chana Messinger

    This is interesting, but frankly, pretty hard to understand, because it’s written very abstractly, and you don’t give any examples. Is something that comes close to passing a negative test something that happens that your theory doesn’t predict, but barely happens, or whose happening doesn’t matter that much? Or something else entirely?

    • leahlibresco

      I’m talking about when I propose a category and you give an example of something that seems to match my stated criteria but feels really wrong. So then I have to figure out whether I left something out of my specifications or whether I’m not lookin closely enough at the object proposed, or if I can’t define the squick out and have the category still be useful.

      Think about the debate over “human-like intelligence.” As new computer programs pick up new skills we have to keep deciding whether these are necessary and/or sufficient conditions to describe human intelligence. Markov chain tricks make it look like we won’t accept the old Turing Test as a valid test.

      We can get clearer negative results by pointing to a rock or a dog. We get the interesting results when we go in confident of the categorization but unsure about how to defend the intuition. (Though this can also be a situation where you might need to repeat the Litany of Tarksi a few times to yourself before you plunge into the problem.)

  • Jubal DiGriz

    Morality gets interesting when applying a “push until it breaks” approach to generating rules from existing data… which with morality and ethics is actions and their consequences.

    For instance, suppose we have the following “data”:
    Adam is mean, Adam never gets what he wants, Adam is angry (True)
    Bob is angry, Bob is mean, Bob is unhappy (True)
    Cindy is mean, Cindy is calm Cindy never gets what she wants. (True)
    Donna is happy, Donna is nice, Donna sometimes gets what she wants. (True)
    Eric is happy, Eric is calm, Eric is angry (False)

    One moral rule one could construct from this is: Mean people are unhappy and never get what they want, nice people get what they want sometimes. But it could also be: Angry people can never have what they want, while calm people only sometimes get what they want. Or: If a person is nice, they will be happy.

    But then there are people like this:
    Eric is mean, Eric is happy, Eric sometimes gets what he wants. (True)
    Fanny is calm, Fanny is nice, Fanny is unhappy. (True)
    Gerry is unhappy, Gerry is nice, Gerry never gets what he wants. (True)
    Helena is mean, Helena is nice, Helena sometimes gets what she wants (False)

    And we get a strange moral law: Only people who sometimes get want they want are happy, and it doesn’t matter if you are mean or nice.

    And then there are people like this:
    Issac is nice, Issac is unhappy, Issac sometimes gets what he wants (True)

    And we’re left with: People can not be both mean and nice.

    My (perhaps badly illustrated) point is that in the moral realm there will always be exceptions to any useful rule, and any rule that is always true will end up being useless. Perfectly coherent morality and ethics are valueless, and the only useful rules (angry people never get what they want) will always have exceptions. It is better to be useful than consistent.

  • grok87

    Speaking of grotesque, there is not much more grotesque in the gospels than today’s “john the baptist’s head on a plate”
    The family dynamics of the dynasty of the Herods (Hasmoneans) seems pretty grotesque too…

    what strikes me as I read the gospel and the wikipedia is that the Herod’s were caught up in cultural crossfire where the rules were unclear-Jewish rules, Roman rules. Herod (Antipas, the one who killed John the Baptist) doesn’t seem like such a bad guy. He recognizes John the Baptist’s holiness. He likes to listen to him talk. You get the feeling he was roped into arresting him by his wife Herodias in the first place.

    It was out of this cultural crossfire between the Greek/Roman world view and the Jewish worldview that Christianity emerged….

  • The Ubiquitous
    • jenesaispas

      Thankyou great story!

  • Doragoon

    So when a horror is pointed out that is consistent with the rules as they laid them out, and they argue, “That’s not what I was talking about,” they had simply never tried to falsify their idea. Is it okay for people to carve out additional exceptions for their rules by saying, “That’s silly because everyone knows I was only talking about a certain limited set”? In this case, positive real integers.
    If the point of a rule is it’s predictive value, how fare do we allow it to be limited before we decide it’s lost all predictive value. What would we call a rule that only governs an infinity small set of conditions?

    • Doragoon

      If an Examples are needed to clarify my point…
      Nsubx<Nsub(x+1) would govern an infinite set of of real numbers (However I give it no predictive with imaginary numbers or non-numbers, such as the set {0,i,2} or {Blue,4,6}.)
      A rule with infinitely small set of conditions would be something like, "Nsub1=2 & Nsub2=4 & Nsub3=6". In that case, you can't even say that this is ever falsified by the other example sets since those all fall outside the boundary conditions of this rule.