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The social world is (in many ways) continuous but people’s mental models of the world are Boolean


Raghu Parthasarathy points me to this post and writes:

I wrote after seeing one too many talks in which someone bases boolean statements about effects “existing” or “not existing” (infuriating in itself) based on “p < 0.05” or “p > 0.5”. Of course, you’ve written tons of great things on the pitfalls, errors, and general absurdity of NHST [null hypothesis significance testing], but I’m not sure if you’ve ever called out the general error of “binary” thinking, and how NHST enables this.

In reply, I pointed him to these old posts:

Thinking like a statistician (continuously) rather than like a civilian (discretely)

Message to Booleans: It’s an additive world, we just live in it

Confirmationist and falsificationist paradigms of science

Whither the “bet on sparsity principle” in a nonsparse world?

Raghu responded:

Your second link contains a very interesting sentence, that “We live in an additive world that our minds try to model Booleanly.” Often, when people criticize science, a common complaint is that science and scientists want to see complex issues as “black and white.” However, science that’s done well doesn’t do this, as you’ve written many times—it recognizes and quantifies uncertainty, complexity, and all the rest. One could argue that the “our” in “that our minds try to model Booleanly” is the view not of the non-scientist lay-person, nor of a “good” scientist, but rather that of a naive scientist who hasn’t moved beyond the simple textbook picture of science that we teach people at young ages.

I replied that I do think it’s a human tendency to understand things as Boolean, maybe because such rules are simpler to remember and compute.

To which Raghu responded:

You’re probably right. There must be some interesting psychological / anthropological / historical work out there on when people (either as individuals or culturally) start to, at least sometimes, adopt continuous rather than binary measures of causes & effects.


  1. Eric Loken says:

    The binary thinking about which variables “work” and “don’t work”, about what’s there and is rampant, in my opinion, among social scientists. I’ve worked with lots of smart, energetic, thoughtful scientists who nevertheless are unaware that their reasoning can be stuck in these binary assessments. As we’ve been discussing recently, in cases of clear data and strong effects this isn’t necessarily an issue, and many smart scientists get themselves to these areas. But I’ve listened to some painful conversations about what did and didn’t work in noisier data sets. And then those assessments become the basis of future analyses (no need to model X because it doesn’t do anything, but do include Z because it’s important), and future designs.

  2. anon says:

    Actually, isn’t the world discrete and continuity is just a convenient approximation?

    • Andrew says:


      Sure, there are discrete aspects of the social world—we are individual people, transactions have specific times and prices, etc.—but in the above post I was thinking of models such as X causes Y or X predicts Y. Here, anything worth thinking about will have an effect that varies, so I don’t think it’s so useful to frame research around yes/no questions, I don’t like talking about false positives and type 1 and type 2 errors, and so forth.

    • Keith O'Rourke says:

      Depends what you mean by _world_.

      Our experiences of the world are discrete but the reality that presumably caused those experiences may well be continuous.

      The question perhaps becomes is it more profitable to think about that reality being continuous. More profitable that is in leading to more actions that are less frustrated by reality.

    • Rahul says:

      >>>the general error of “binary” thinking, and how NHST enables this.<<<

      Isn't this discrete nature of decisions just an pervasive feature of the world we live in?

      e.g. Income tax rates come in brackets. Cellphone plans have discrete usage steps. Bankers refer to interest tables than use a function. Games have winners & losers. We give students discrete letter grades instead of just summing up all their test scores. I need to select either vendor A or vendor B. We either select a new catalyst or stay with the existing one.

      Somehow people prefer the discrete to the continuous?

      • When we act, sometimes we act continuously (decide how much to invest, how much to eat, how much to exercise), and sometimes we act discretely (decide which grad school to go to). Sometimes our data is discrete (we get dealt a hand of cards or we decide how many beers to drink at a bar), and sometimes it’s continuous (I draw a straw of a continuous length or decide how much wine to pour into my glass at a party).

        But I think Andrew’s argument is that the components that influence how we act are usually continuous. In social science, it’s not usually a question of whether something has an effect (does where I grow up influence where I go to college? does income affect my voting preference?) so much as how big is the effect and in which direction and how does it interact with other predictors and effects.

        For instance, I live or die at any given point in time (discrete), but my diet and the amount I exercise is likely to have a continuous effect on when I die. My measurement of blood pressure may look discrete (a pair of integers) but is actually continuous. Income tax is incredibly complicated, but let’s just look at the tax bracket cutoff as implemented in the U.S.—does moving the bracket have a continuous effect or a discrete one? I need to select vendor A or vendor B, but does the price they charge have a continuous effect on the probability of choosing A or B?

        • Rahul says:

          My point is that it’s silly to criticize NHST as an enabler of binary thinking. Sure, NHST might be a crappy way to do it, but not *because* it leads to a binary outcome.

          The binary decisions are already deeply embedded in the world. Even if you skip the dichotomy of NHST, whatever continuous model you use must at the end remap to a binary outcome (for this class of decision problems).

          The trick is always going to lie in how well you do that. You can’t escape the binary decision because often the problem itself demands it.

          • Sure, there are plenty of binary decisions, but there are lots of continuous decisions as well. How much to spend on cleaning up a toxic waste site, how long to wait before replacing a water pump, what fraction of a shed to paint…

            • Rahul says:

              So I don’t think anyone is advocating that we model the continuous decisions as binary ones. That’d be stupid. So that’s a strawman.

              OTOH, I do think Andrew, Raghu etc. are saying that it’s worth modeling even the ostensibly binary decisions as continuous ones.

    • Nuri says:

      From an occidental perspective, you are right; oriental philosophy sees it the way around. I recommend you read some of the books in which Allan Watts “translates” oriental ideas. My top 3 recommendations are: Psychotherapy East and West, The Book; The Wisdom of Insecurity.

    • anon says:

      If you want to be reductive about it, consider quantum scale physics. Here the position and continued existance of particles are probabilistic rather than discrete. Rather than true or false / 1 or 0, quantum states are superpositions of contradictory discrete states.

      As far as we know, all seemingly concrete / discrete aspects of our universe are composed of these probabilistic quantum elements, so could be considered probabilistic superpositions of truth and false themselves. Obviously, it’s convenient (to say the least) to consider our world in discrete terms, rather than as approximations of probabilistic things that only exist with near certainty.

      In fact, waves and fields can best be understood as continuous — so to me there’s more truly continuous rather than truly discrete aspects of the real world. But, really, if approximations are indistinguishable from what they “actually” are (or can be treated as indistinguishable,) they should be.

  3. Jacob says:

    >I replied that I do think it’s a human tendency to understand things as Boolean, maybe because such rules are simpler to remember and compute.

    I think this is part of it, another big part is that decisions people make tend to be discrete rather than continuous. A paper either gets published or it doesn’t. When a hurricane is predicted to hit my town, I either evacuate or I don’t. My state had 4 ballot initiatives this year, some of which were rather complicated and surely involved negotiations of parties involved, but I only got to vote yes or no. Those were my options. Being 51% in favor of question 3 and being 99% in favor both boil down to the same “yes” vote, so what difference does a preference strength make?

    In the examples I listed above I’m sure people can think of other options besides those I listed, but there are still going to be a small discrete set. Maybe the publisher asks the author for revisions, then publishes the paper. So it’s more like a decision tree, but still a discrete set of decisions at each point. People care more about concrete decisions and actions than abstractions, so the preference for discrete thinking (of which boolean is the simplest) makes sense in many contexts.

  4. BenK says:

    Take a look at Arnold Kling’s economics; and it actually suits much of biology and medicine pretty well. When we ignore the underlying discrete variables and try to construct models based on aggregate variables, we naturally have trouble overlaying a discrete interpretation. However, our instincts correctly tell us that there is a discrete phenomena under there, so we try. Ultimately, instead, we need to skip the aggregation and figure out how to collect and analyze high-dimensional data.

  5. Marcel van Assen says:

    I believe we are hard-wired to perceive and think binary/categorically, following Kant. Our language is full with it too… People are called tall and short, smart and stupid, but all these traits are clearly continuous.
    Scientists are human.
    Some statistical techniques are even used to find classes and clusters (latent class analysis, mixture models) when imo they do not exist in the real world but just in the scientist’s mind.

    • jrc says:

      If this should be read “Unfortunately, some statistical techniques…” then I totally agree.

      If the should be read “Scientists are human, unfortunately” then I totally disagree.

    • Our language is also full of numerical measurements for height (5′ 6″ tall) and direct comparatives (A is taller than B). And with something like “smart”, the scale’s not even unidimensional.

      I think what everyone’s missing is that language needs to balance precision with conciseness. So you indeed get shorthands like “tall”, which convey strictly less information than “6 feet tall” but are more concise. And we do indeed work with stereotypes and other generalizations both true and false (and approximate in most cases so the whole notion of truth and falsity is difficult). But even our stereotypes aren’t necessarily discrete and don’t have sharp boundaries because language doesn’t provide sharp boundaries for most of its concepts.

      I think Andrew’s getting at what you bring up in the last line—that we often use discrete characterizations of phenomena that are continous. Arguably, that’s what “tall” and “blue” are doing to height and frequency/perception. Or what making a three compartment model of soil carbon dynamics does to what’s arguably a continuous process.

      • jrc says:

        For a concrete example but staying with height: we have age/gender standardized growth charts that give each child a height-for-age z-score. This is a continuos measure of child growth. But very often in international nutrition work, people will dichotomize this HAZ variable into “Stunted” (HAZ less than -2) and “Not Stunted”. This is turning a perfectly good continuous measure into a less useful discrete measure, to paraphrase one of our biostatisticians.

        I actually think this happens a lot in clinical work. Many clinical definitions of disability or condition are based on dichotomizing an underlying continuous measure: stunting (HAZ), anemia (hemoglobin levels), hypertension (blood pressure)… Sometimes this dichotomizing makes sense, such as in determining when to intervene with costly or potentially dangerous medicines or procedures. But often times we’d do better to stick with the underlying continuous measure.

        Since HAZ is one I know well, I’ll give an example, which is also a lovely demonstration of Forking Paths. In one of my projects, we experimentally generated a small improvement in child HAZ. That’s nice. But that led to a very large improvement in stunting rates. Why? Because much of the mass of the HAZ distribution was just to the left of -2, so a very small effect could pull a lot of children out of the “stunted” category. In this case, the smallish HAZ effect is probably a much better measure of how much we improved child health with the intervention, because the -2 cutoff is an arbitrarily-set clinical definition without real biological meaning. But of course, a large improvement in stunting rates is more “exciting”.

        So obviously the lesson is – always run your Science/Nature/PNAS-aiming interventions at populations where a lot of people are just on the bad side of a continuous measure’s clinical cutoff, and then only report the dichotomous outcomes. You know, that, or use the continuous measure and learn about how much you actually helped people.

  6. Sean Mackinnon says:

    I think there is something very fundamentally human about thinking in binary terms. This conversation reminds me of theorizing by George Kelly on personal constructs. Of particular interest here is his Dichotomy Corollary: “A person’s construction system is composed of a finite number of dichotomous constructs.” He thought everyone develops a certain sort of ideosyncratic system of binary constructs we apply to the world. So, lots of people look at statistics and think of them as significant vs. non-significant (or maybe in the middle “marginally significant”). Wrong, of course, but I suppose in his theory, he’d say these views persist because they allow people to anticipate future events. Not replication of studies (which people aren’t really doing anyway), but rather this dichotomous thinking lets researchers know if they are going to get published or not, so it does predict the events they care about (getting published).

    A brief summary I found online of what I’m talking about:

    • I don’t know what a “construction system” is, but it sounds awfully limited compared how people actually perceive and act in the world. Sure, we can code up a three-way decision as two binary decisions (and one left over illegal decision), but why would we? I used to have this argument with phoneticians and phonologists in linguistics, some of whom are (or at least used to be) committed to explaining everything in binary terms.

      Yes, we can break things down into discrete bins (like colors), which can make it easier to communicate in broad terms. But then we start breaking them down further and adding more and more color words. And these categories aren’t precise — is that shellacked box a chair? Is that mason jar a cup? Is that yellow-green wall yellow or green or neither? But if we want to get more scientific about color, we tend think of frequency and a continuous scale, even naively when we mix paint on our palette to create our masterpiece.

  7. Jonathan (another one) says:

    There are 10 kinds of people in the world — those who think in binary and those that don’t.

    But seriously, folks: even where we have continuous measures, we unfortunately have discrete comprehension. We use the point value of a regression coefficient and ignore the standard error except in the benighted use of it in NHST. If the effect of an additional year of schooling on the logarithm of income is measured as 0.0327 then by Jove we’ll use every decimal point whether the standard error is 0.01 (still significant!) or 0.00001.

  8. This is a great topic. Yes, from childhood (and probably from the beginning of recorded human history) we learn to think in binary terms. Good to eat, bad to eat. Good to say, bad to say. Right direction, wrong direction. Some of these distinctions hold up over time; others become fuzzy. The problem lies not with binary thinking when it’s appropriate, but with insistence on binary thinking when it does not suit the situation.

    This plays out again and again in education reform. Some new method comes along that supposedly “works.” People try it out, decide a few years later that it doesn’t “work,” and move on to the next one. The problem is that few of these things “work” in every context. You have to ask: what are you trying to accomplish in the first place? What are the assumptions, intentions, and purposes behind this reform or mandate? Where do they coincide with your own and those of the school, and where do they diverge? How might this reform or mandate play out in your particular setting? How and where might you be able to adjust it when it doesn’t fit? When these questions don’t get asked, when the only question is whether it “works,” the answer will at some point be “no,” and resources and effort will go to waste.

    • We also learn to think continuously. How to move our hand to get food into our mouth. How much food to eat. How to place our lips and tongues so that the words we make are understood. How loud to talk so that someone can hear us (or comes running!). And we learn any number of other continuous behaviors.

      • Martha (Smith) says:

        I’d say that “How to move our hand to get food into our mouth,” and “How to place our lips and tongues so that the words we make are understood” are not learning to *think* continuously, but learning to *move* in something that at least approximates continuous motion. (I think I found thinking continuously easier than learning to move continuously.

      • What’s fascinating in all of this is the proximity of statistics and poetry–their common grasp at complexities and continuities, within strict discipline and structure. I don’t mean anything so silly as the proposition that statistics and poetry are the same. They aren’t. But they sure aren’t enemies, either.

  9. Roger H says:

    This reminds me:

    (a) why I dislike classification and regression trees;

    (b) of Stephen Senn’s (or rather his alter ego Guernsey McPearson’s) diagnosis of ‘dichotomania’ among medics: He first cautions us, however, that one of the founders of our discipline of statistics appears to have displayed symptoms of the opposite malady, ‘continuitis’.

  10. Alex Gamma says:

    In Andrew’s original context of causality (and prediction) I find another kind of discreteness. I’ve seen this in my own case, but also with friends, scientists, the media: we seem to be almost unable to consider simultaneously more than one cause for a phenomenon. It seems hard to resist the urge to ask “but what’s the main reason?” or “but what is more important?” and then focus on that.

    In my own case, I found a remedy in an unexpected place and it’s an intriguing connection. As I’ve gotten better with creative artistic tools such as photo editing (e.g. Photoshop) and software for making (electronic) music, I started to *experience* – not just know intellectually – the phenomenon of how many small effects add up to a total. Many tweaks whose effects are tiny in isolation interact almost magically to form the end result. I could start to see, or *feel*, that all of the little steps along the way made a difference. I’m sure that must be a universal experience for people in the arts or in any kind of field where a single person is in touch with a complete, multi-causal process that leads to an outcome (cooking, for example). I’m thinking that if we could harness the power of such experiences for teaching students, that could be one (important!) ingredient to a better breed of scientist.

    • Rahul says:

      Excellent point. Another question people can’t seem to resist is: “Does X increase the effect or decrease it?”

      e.g. Will adding a faster agitator speed up our process?” It depends.

      They cannot seem to reconcile the fact that in multivariate models X can “increase” the effect in some regions of state-space , decrease it in others & have little effect in still other parts.

      Getting a one line plain English summary that describes all the behavior of a complex model is the holy grail everyone is searching for.

  11. There are a lot of excellent points here. Though it is certainly the case that it is “natural” to think in binary terms, and that often we make decisions among discrete choices, I nonetheless think that a key reason science has been successful is that it has (in a lot of cases) moved beyond this to make quantitative measures and continuous models of phenomena. This isn’t a “natural” thing to do, as I am constantly reminded whenever I see a scatter plot with a linear regression and a statement of its p-value. “What could this possibly mean?” I wonder. “How could anyone plot a line and ask ‘is there an effect?’ rather than asking ‘what’s the magnitude and uncertainty of the slope?'”

    I was thinking, when writing a post today that includes a graph of distance-versus-time for muskrats taking over a continent, that some time, long ago, someone made the first graphs of voltage vs. current in a circuit, and rather than writing that current has an effect on voltage (p < 0.05!), they realized that this relationship could be modeled as a continuous, linear function, and that the slope of that function was a parameter worth quantifying. This seems like such an obvious thing to do, but increasingly encountering fields of science where it's not done, I realize it's not obvious at all.

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