To P, or to NP? That is the question

I was having a discussion about high school a few days ago, and accidentally stumbled upon half an insight.

I was trying to articulate why I always found English class to be so unenjoyable, and one of the explanations I came up with was that we, as high schoolers, simply weren’t developed enough (as students, or as readers, or as people) to do interesting literary analysis yet. Think of the typical high school essay or book report: “The theme of this book is Death. Because the main character is dying. And even though he was rich, death is something we all must face. By the end of the book he was able to accept death.” One could be forgiven for not being enthralled. At that age our analyses were almost always either superficial, or overly simplistic, or…well, kind of boring. If the theme wasn’t Death (bad, but we should accept it), it was Racism (bad), or Hardship (sort of bad, and apparently endemic to Cape Breton and the Prairies). There was little subtlety and even less nuance. And of course it doesn’t really matter if you’re the one coming up with the boring analysis – boring is still boring. I don’t know if this was the reason I didn’t like English class, but it seems like at least part of the reason (especially since I have a strong feeling that I’d enjoy it much more at my current age).

Anyway, my instinct for self-deprecation served me well here, because I ended up offhandedly summing the whole thing up as, “Basically I wasn’t smart enough to interest myself in high school.”

And that got me thinking. I meant it as a joke, but could something like that actually be true? Could there be times in our lives that we’re more or less interesting to our own selves? Granted, it seems kind of absurd at first glance – you’re the one doing the judging here, after all, and how can you be smart or dumb relative to yourself? But it’s actually not as paradoxical as it sounds.

Consider: much has been made over the years of the analogy between creative problem solving and the class of mathematical problems that complexity theorists call NP. NP problems, recall, are those for which a solution is extremely difficult to find, but very simple to verify. The classic example is factoring large numbers. Given a large number, finding its prime factors is in general a difficult problem; however, if someone gives you a potential solution (i.e. a set of possible factors) you can easily check if the solution is correct – just multiply the factors and see if the right number comes out. Many things people do that we call “creative” (writing a novel, making a work of art, coming up with inventive solutions to problems) have a similar character – it’s much much easier to recognize good work once it has been created than it is to actually create it. Few are the people who can make a show like Breaking Bad, in other words, but many are those who can enjoy it. In that case the “problem” one faces is to create a TV show that people will enjoy, and Breaking Bad represents one of many possible “solutions”. Needless to say, generating solutions to this problem is quite difficult – I’ve never made a TV show, but I’ve heard it’s non-trivial. Verifying solutions, though? Piece of cake! Just sit down and watch the show. If you like it, you’re done. This is why many mathematicians have said that if P (roughly the class of problems that are “easy” to solve) were to turn out to be equal to NP, it would amount to an algorithm for “automating creativity”.

(Of course, you don’t really need P=NP to be able to automate creativity; you can always just do it by creating an AI, assuming you count that as “automating”. P=NP might allow you to dumbly automate creativity in full generality, though)

So let’s go back to my offhand remark. What would it mean to be smart enough (or not smart enough) to interest yourself? Well, in light of the above discussion, I propose that we can think of people as having two different kinds of intelligence (metaphorically speaking, anyway). The first we might call P-intelligence (after the class of “easy” problems, P – you could also call it recognition intelligence, or verification intelligence). It’s the kind of intelligence that allows us to recognize good solutions to problems when we see them, or enjoy good creative works. The second we might call NP-intelligence (or creative intelligence, or generative intelligence). This would be the kind of intelligence that allows us to come up with innovative solutions to problems, or generate great works of art. Of course, the names are merely meant to be suggestive; humans almost certainly can’t actually solve NP problems efficiently. But to whatever extent we can creatively solve difficult problems, I’ll label that ability NP-intelligence.

Viewed from this perspective, it makes perfect sense to talk about people not being smart enough to interest themselves. It simply means that their P-intelligence has developed to a point that they crave novel, interesting stimuli, but their NP-intelligence hasn’t developed enough to provide it. And I think this is what was going on in English class for me – I knew roughly what actually engaging work should look like, and I knew that I wasn’t producing it. Always I wrote at the outermost limits of my ability, but that was no guarantee the results would be deemed compelling by my inner critic, or even acceptable – boredom doesn’t grade on a curve after all, even if the teacher does.

Now, it may seem like overkill to invent a whole new intelligence dichotomy (especially when there’s already like 17 of them) just to explain me not liking one measly class in high school – I mean, maybe I just had bad teachers or something. But the idea rings very true to me. I feel that I really should have liked English class (or could have, anyway), and I feel that I would like it far more today. And even putting English aside, I know that now, at 28, I find my own thoughts interesting in a way that I definitely didn’t when I was younger. I find myself saying “huh” a lot more often when I have some new idea, and then doing that staring-off-into-space thing as I consider the idea in my head, and work out the implications. That didn’t really used to happen, as far as I can remember. And I’m certainly never bored while writing a blog post.

(This all sounds horribly horribly braggy, I know – “I’m so smart I can amaze even myself!” – but I really don’t mean it that way,  I swear. All I’m saying is that there’s been a relative shift between my ability to determine whether a thought is interesting, and my ability to come up with such thoughts. The latter is much closer to the former than it used to be. And that’s interesting, I think – but it’s completely independent of whether or not either ability is actually high on some absolute scale.)

Okay, so let’s grant for the interim that these two types of intelligence make sense as a concept, and actually correspond to something real. What can we say about them? Well, obviously both kinds of intelligence develop gradually over many years as we mature; it’s not simply an on-off switch. As kids we’re entertained by simple stimuli and stumped by straightforward puzzles, but as we get older the degree of complexity we’re comfortable with balloons dramatically, and we start liking novels with intricate plots, and being able to solve more and more difficult problems. Of course, the two types of intelligence need not develop in concert – in fact, it’s pretty much trivially true that your P-intelligence will always be ahead of your NP-intelligence. If you’re capable of generating a solution to a difficult problem, then you’re certainly capable of recognizing it as a solution. But the reverse does not also hold – the ability to verify that a solution works does not imply the ability to come up with the solution in the first place (see the Breaking Bad example above). So regardless of what level of problem you are capable of solving, the level of problem you can recognize a solution to will be higher still. In other words: no matter how smart you are, and no matter how hard you try, your capabilities are always going to fall at least a little short of your standards. This should maybe give you some pause if you tend to be the self-critical type.

So what else can we say about the two kinds of intelligence? Well, here we pretty much reach the limit of how far I’ve taken the idea so far (hey, I said it was half an insight). A number of interesting questions suggest themselves immediately though:

1. How real are the two types of intelligence? Can we come up with a test that can reliably distinguishes between them?

2. Further to (1), how independent are the two types of intelligence? What’s the correlation between them? Could there be individuals who have an unusually large or small gap between the two intelligences? Does the size of the gap correlate with overall intelligence level? More speculatively, could this be related to the Dunning-Kruger effect?

3. Further to (2), if the size of the gap does vary between individuals, are there any interesting correlations between gap size and personality traits? For example, does gap size correlate with extraversion or novelty-seeking behaviour?

4. What are the exact shapes of the development curves for the two types of intelligence? Obviously P-intelligence will always outstrip NP-intelligence as mentioned above, but how does the size of the gap change with age? Are there particular times in our life when the gap is unusually large or small? I alluded to high school as one such possible time – might there be a “plateau” above that age where your P-intelligence stops growing, allowing your NP-intelligence to “catch up”? And perhaps more interestingly, what happens with the development curves at young ages? I suspect that there might be particularly rapid shifts in the intelligence gap during childhood, and that this could potentially explain a number of child behaviours. For instance, it could explain why kids love to play make-believe at young ages but quickly grow out of it (NP-intelligence is initially close enough to P-intelligence to provide captivating scenarios, but eventually falls too far behind as kids get older). Or it could explain why children are so incredibly prone to being bored (the gap is particularly large at that age, so they can’t really interest themselves with their thoughts).

These are all interesting questions, and they’re the kind of questions I think you’d need to answer if you wanted to take the idea further. I have scattered thoughts on some of them, but nothing worth writing down yet.

Anyway, at this point I’m mostly just blogging out loud, so it’s probably time I wrapped this up. I don’t have any grand conclusions or anything – sorry if you were hoping for that. I’d be very interested to hear other people’s thoughts on the whole notion, though – does this dichotomy ring true to you?

If not, hopefully I’ve at least managed to interest you – I know I’ve managed to interest myself.

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5 thoughts on “To P, or to NP? That is the question

  1. I think the main gap is between your ability to recognize something as interesting and your ability to observe your own mind about how exactly it’s being stimulated. I’m not that familiar with literary criticism, but at the level of high school, at least, a lot of it is just unpacking what makes a story hold together and be compelling. If you find the story interesting, then you’re picking up on something, but your self-observation might not be developed enough to figure out what that something is. For me, writing has always been a way to figure out what I actually thought all along more than a way to put down really new ideas. If that counts as creative, then it’s at least a totally different sort than making Breaking Bad.

    So I think you’re onto something, but in many cases the gap has more to do with observation than creativity. Unfortunately, this isn’t that easy to fix, because observation is really hard. There are still a lot of things that go on in my head that I can’t account for.

    There are also plenty of mundane explanations for high school English class being boring: five-paragraph essays are a terrible way to explore and express ideas, the topics you’re allowed to explore are usually limited, you often have to listen to other people say things even more boring than what you’re saying, etc. These simple explanations can probably account for most people not liking it. Maybe you think you’d enjoy it more now, but I wouldn’t be surprised if you’re forgetting how terrible it really was (not that I know much about you or anything about your high school).

    Followed you here from SSC a while back and subscribed because your blog seemed interesting. Can’t say why.

    Like

  2. I like your schema for thinking about creativity.

    How P intelligence (editing mind) sits back anxious to exploit whatever NP
    intelligence can conjure

    Like

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