Partial Derivatives and Partial Narratives

[Note: The idea needs a lot more work, I’m just throwing this half-cooked metaphor on the wall to see if it sticks]

Hold on to your hats, we’re going to talk about calculus! Or rather, we’re going to talk about ideologies and worldviews and how they’re very vaguely like calculus.

In math, a function describes how a variable depends on another. If we have y = 3x, that means that we can get the value of y by multiplying x by 3. Easy.

Taking the derivative of a function gives us another function that, when evaluated, grants not the value of y but how y changes when x changes. The derivative of y = 3x, for instance is dy/dx = 3. When x increases, y increases three times as much. It doesn’t depict how y is a result of x, but how y:s rate and direction of change is a result of x. In a way, what was a static, neutral way of describing two entities has turned into a description of one dynamic and causal relationship: x affects y this way.

It’s tricky to talk about math metaphorically, since when you use math people reasonably expect rigor. I don’t intend to offer any, so I’ll just say that I see the derivative of a function a little bit like a narrative explaining a set of events or entities. The similarity is poetic, not formal.

The normal way to deploy analogies is to use something simple and intuitive to explain something abstract and difficult. In this post I’m trying to use something difficult and abstract to explain something relatively intuitive. This might be idiotic, but the thought just stuck with me and I want to get it out. So just bear with me, I do have a point and it’s not about math.

Narratives are stories. Stories describing set of events or entities, more or less explicitly putting forth an account of the causal relationship between those events or entities. In that way they have something in common with derivatives.

The narrative version of a simple dy/dx = 3 derivative is a simple story of how something causes something else, like “you dropped the plate on the floor and it broke” or “you pushed your little sister so she started crying”. Yes, I have kids. These are the kind of events my narratives are about.

I had to try hard to find a few examples of such basic, one dimensional narratives. Most are much more complicated, because causation is complicated. Because causation is complicated, derivatives as simple as dy/dx = 3 rarely if ever mirror even the simplest narratives.

The thing with derivatives is that they destroy information. They force us to make choices and put something front and center. Our dy/dx = 3 is the derivative not of one function but an infinite number of them, since when you derive with respect to x any term that doesn’t contain x disappears. The x-derivative of 2x + 5 is 2. The x-derivative of 2x – 367 is also 2. And crucially, the x-derivative of 2x + 100y – 3z^2 is also 2. When a function contains more than one variable, it has what’s called partial derivatives-one for each variable. It becomes impossible to describe its dynamics correctly with a single derivative, just like it becomes impossible to see all of an object at once when it has more than two dimensions.

That last function, f(x,y,z) = 2x + 100y – 3z^2, has three partial derivatives: df/dx is 2, df/dy is 100 and df/dz is -6z. Three partial derivatives from the same function, three narratives describing the same things-in-the-world.

Other variables don’t need to disappear. The partial derivatives of, say, f(x,y,z) = 4x^2 * y – y^z are 8xy, 4x^2 – (z-1)y and y*ln z*y^z. All correct depictions of the same underlying function, all different and on the surface contradictory. What they do is put a different variable into focus, making the derivative “about” that variable and thereby selecting one relationship as the backbone against which everything else becomes an extra, a supporting character.

If we’re not scientists dealing exclusively with formal models, we come across representations of reality in the form of narratives. More difficult, ambiguous narratives  try to balance several partial derivatives, jumping from backbone to backbone avoiding a single point of view. This is difficult and not the norm, as most stories—certainly the most powerful, human-resonant and convincing ones (ask any journalist, propagandist or PR person)—are built by pieces all pointing in the same direction singing in harmony.

I’ve mentioned that I work in consulting. Part of my job is writing reports about the state of some industry or the nature of societal trends. These reports need to be clear and consistent (because people don’t read them for subtlety if they read them at all), so we need to take a bag of individual facts and examples and stitch (a subset of) them together into a narrative with a clear conclusion. That clarity is what we get paid for, regardless of how clear and convergent the reality actually is. Narratives explaining reality are like laws and sausages; when you work with them and become aware of what goes into them you don’t look at them the same way any more.

Reconstructing a function by reverse-engineering its derivative is called integrating, and when you learn to integrate you’re taught, on pain of point deduction on your exam, that you must add an unspecified constant to represent any terms that would have disappeared during derivation. It’s easy to forget, because any such terms are invisible to you and you have no idea of how significant they were or how complicated the function that generated them may have been.

If our narratives are partial derivatives and the reality they describe are the functions, then the way we get our ideas about reality is by integrating our narratives. A complicated, multivariate function has many partial derivatives, leading to disparate results when those derivatives are integrated back into functions (beliefs about the world).

I’ve been having this thought for a while, but what finally made me write it out was Ayn Rand’s Atlas Shrugged. I read a summary of it on Spark Notes because I was curious—curious about how this particular book could be so divisive. Some people love it, cite it as their favorite book and a guide to life. Other people say they can’t get through it because the writing is terrible and its philosophy reprehensible and deluded. To an erisology enthusiast, books like this with the power to cleave readers into lovers and haters are a special treat. It gives us clues as to how people differ, deep down. It’s the psychological equivalent of a chemical indicator.

I’m not going to read a thousand page book to get to the bottom of this particular issue, though. The summary had to do, and it was enough. Things were pretty clear. The book has a view of how the world works and a set of values to go with it. I think people who find its values off-putting probably does so largely because they think its ideas about how the world works are wrong. If the world really did work as envisioned by Rand, her values would make more sense and if you do think they make sense, you probably share her beliefs about how things work. Her narrative is highly coherent and harmonizes strongly with itself, as any piece of propaganda does. I don’t mean “propaganda” as an insult here, I just mean any purposefully argumentative text.

Based on whether you accept or reject the narrative Rand puts forth as accurately describing the world, you’ll hate or love the book. What about me? Did it came across as true or false? Neither, I chose this example because it’s such a clear case of what I’ve been talking about. What stands out to me is not its rightness or wrongness but its vivid, scorchingly powerful partiality. It’s description is not wrong, it’s just really really ridiculously partial. Rand takes the derivative of a single variable, discards all other terms and dimensions, and recreates a reality based on the integration of this particular derivative.

That prosperity depends disproportionally on “heroic” inventors and entrepreneurs isn’t wrong, it’s just not the whole story.

If this variable, this dimension, is central to your own life and experience then recreating the world around it is like removing all annoying noise from a model to make it crisp and perfect. Rounding off into nice, round numbers. It overwhelms you with a sense of absolute clarity (the same thing happens to the political polar opposites when they read Marx). This sensation is powerful and beautiful. It is also how ideologues and fanatics are born.

If this dimension is not central to you the whole thing feels like a violation. A perversion, gross and twisted. An insult to good taste and decency, theology and geometry. Instead of removing the noise to get a clear signal, you remove the signal and warp the remaining noise into something grotesque. A franken-narrative that can only be the result of blatantly ignoring the real signal staring you in the face! Self-serving bullshit anyone with half a mind can see through.

It goes without saying that I don’t think either the lovers or the haters are deluded or evil. They just derive with respect to different variables, get two different derivatives and two different reintegrated functions. Not strictly incompatible—that’s the point, they do live in the same world—but they disagree on what’s signal and what’s noise, what’s central and what’s marginal. What’s the essence of [thing] and what’s its nonessential caveats we discard when rounding off and saving the stem in our gray matter archives?

The nature of capitalism* is one of the clearest examples of partial narratives. What is the true story of capitalism? Well, what is the true derivative of f(x,y,z) = 4x^2 y – y^z? You have “capitalism is when people can trade freely in voluntary agreements and create wealth through their own work and ingenuity” and “capitalism is when the rich can use wealth to assert power over the poor in order to extract surplus wealth from their labor”. They are both partial truths, like a cylinder is a circle from one angle and a square from another. With partial narratives we square the circle, but it remains difficult to keep them both in your head at once.

Imagine that the world was just a set of dots like this picture:

What happens then? Say Alice is told (or, because of psychological predilections, personal experiences or self interest, is more likely to internalize) that the world is a square (left picture), while Bob is told it’s a circle (right picture).

Alice and Bob now have differing ideas about which dots are the important ones, which are expressions of something fundamental (signal) and which are just isolated incidents (noise). They will be interested in and eager to talk about the dots that make up their preferred shape.

When Alice talks about any dot in the square she’s actually taking about the square and other dots are Beside The Point. When Bob talks about a dot that makes up the circle, he’s just ranting about some insignificant dot. If Alice is feeling uncharitable she might think Bob’s just talking about irrelevant dots because he doesn’t want to talk about the square. Bob thinks that Alice taking a great interest in dots in the square and dismisses dots in the circle is hypocritical.

Note that they don’t have to disagree on which dots exist or where they are. Savage political fights can happen without any factual disagreement or fundamental value difference.

There are of course more examples than capitalism. Like nature vs. nurture. “People’s behavior are the result of socialization that works to perpetuate power structures” and “people’s behavior are the result of biological impulses and instincts” are both partial truths. But the full truth is not “in the middle” but on another plane entirely.

“History is determined by the actions of individuals” vs. “history is determined by large scale economic and technological forces.”

“Art subverts the audience’s unexamined preconceptions” vs. “art is the creation of transcendent beauty.”

“Sex is about satisfying basic, impersonal appetites” vs. “sex is an act of intimacy and an expression of love.”

“Ethics is about maximizing happiness” vs. “ethics is about doing your duty”.

“Science works by accumulating knowledge about the world, asymptotically approaching perfect correctness” vs. “science works by replacing one paradigm with another in a series of revolutions.”

“Moral rules and norms are symmetrical, exactly the same for everyone” vs. “Moral rules are there to protect the weak, favoring them over the strong.”

So, it’s narratives all the way down? Just like the “postmodernists” say? We build reality out of narratives and there is no real thing? That’s only (ahem) partially true. That’s the reason I picked partial derivatives as a metaphor. Partial derivatives are seemingly incompatible when you squint, but they are all derived from the same function and describe the same reality.

When people work with different partial derivatives they may not disagree at all about the actual function, which can be revealed if they talk long enough in good faith. Some years ago when I was a student, there was a guy I used to hang out with at parties. He was a Gender Studies student and we had little in common in terms of worldview, me being a bit of a scientismist and all. For some reason I tend to get along well with this cluster of people on a personal level, and we often talked. Despite being too drunk to be eloquent most of the time we did get somewhere. The more we talked the clearer it became that we actually disagreed very little on how the world worked. We had different interests (intellectual interests, not material interests) and because of that our signal-and-noise attributions were often flipped.

In short, reality has a true nature and if we use mental prosthetics (like mathematical models and computer simulations) we can, in theory, get it just right. But our intuitive understanding is severly limited and starts to sweat once we go beyond simple narratives (partial derivatives over a single variable). Any narrative straightforward and compressible enough to comfortable fit in our minds is going to be very partial.**

The only way press on without denial or loss of hope is to gather more partials, keep them on file and train yourself to switch between them so quickly that you can simulate something approaching a complete model. This approach isn’t sexy and you’ll go without that rush of clarity that Atlas Shrugged or Das Kapital offers the precocious teenager with a left- or right wing temperament awaiting ignition, as well as that comfortably detached smugness that comes with being a lazy relativist. But to me it’s the only defensible choice.

There are degrees of partiality. The “circle” and “square” narratives above are about equally true. I could also pick out three arbitrary dots, draw a triangle and declare it a valid signal. The lazy relativist just grants this (in theory), but we all understand on some level that the triangle narrative is more contorted, more partial and less true than the others. How do we make this difference clearer?  I really don’t know. I don’t think anyone knows. It’s really hard. Philosophers seem to have given up (personal impression) on what I think is their most important task.

The partial narrative model based on derivatives is attractive to me because it allows a kind of relativism that is clearly necessary and true; all narrative accounts are partial and just because something has flaws and exceptions it isn’t necessarily false—narratives just have values between 0 and 1 on the truth scale (though partiality is not the same as probability or confidence). And more importantly, just because narratives contradict each other it doesn’t mean any of them are wrong. If I could forcibly inject just one insight into political discourse…

But the partial narrative model accomplishes this without throwing out truth altogether in favor of a nightmarish might-(or social capital)-makes-right relativism, by moving objective truth and a requirement for consistency onto a different plane. Merging science and objective reality with the postmodern ambiguity of language and narratives is the most urgent (by philosophical standards) philosophical project of our era, but I barely hear anything about it. Maybe I’m the crazy one.

*I really truly hate the word “capitalism”, not because I hate capitalism or love it, but because it’s a terrible word that confuses more than it illuminates. If there is one word in desperate need of tabooing and replaced with its substance, it’s “capitalism”.

**This whole model only applies to the factual aspects of explanatory narratives. Words don’t just carve up reality in chunks of different shapes; they make several kinds of value judgments about them too and that complicates things. I’m leaving that dimension out for now.

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18 thoughts on “Partial Derivatives and Partial Narratives

  1. This post feels like a somewhat more poetic variation on a strikingly similar theme to my latest post about “multivariate utilitarianism“.

    When people work with different partial derivatives they may not disagree at all about the actual function, which can be revealed if they talk long enough in good faith.

    Yes, so much this (hope you’ll excuse me for slipping into the dialect of online culture)! I had been hoping to organize and write down my thoughts on this phenomenon one of these days (my favorite example is a certain dispute between Jon Stewart and Bill O’Reilly, not getting into that here). I had never fully noticed the direct connection to the partial derivative metaphor, though.


    1. I suspect Stewart and O’Reilly have more in common than it appears, they just tend to display different parts of themselves. Good faith conversations are rare in public debate, unfortunately, because the format tends to encourage confrontation. I doesn’t have to be like this, but I’ve somewhat accepted that I’m the odd one out here since this format seems to be what most people want.


      1. Yes, good-faith conversations are rare, but Stewart and O’Reilly at least make a little effort towards them, which is perhaps the reason why their disputes provide some of the more obvious examples of agreeing on all the facts but emphasizing different narratives.


  2. Hope you don’t mind if I give this post a mention and a link on my blog. It happens to tie in conveniently well with an idea I’m trying to express at the moment.


      1. I guess I mainly worried that I’d somehow inadvertently twist your intended meaning. But of course if I did, you’d be free to correct me in a comment 😉

        By the way, I was a little hesitant to refer to you by name, as you had mentioned in your first comment something about preferring to be anonymous. My reticence didn’t quite make logical sense, I know, but I am a little confused: is John Nerst not your real name?


  3. This is such a great piece of writing and thinking. The first time I started to read it a few months ago, I stopped when it started getting into the calculus and clicked over to some of your other posts. Big mistake.

    This gets at the CORE of what I’ve been wrestling with over the last few years and what led me to your blog.

    I’m tempted to suggest that the part that reaches me so deeply doesn’t need the derivative analogy.

    I feel like if I could get certain people in my life to read it, they’d understand for the first time what I’ve been trying to understand and communicate. They’re already turned off when I start talking about these sorts of ideas and would surely tune out on the calculus metaphor.

    But who knows… ? It definitely adds a lot to the big picture you’re painting. And it’s a great example of a useful partial narrative. So maybe creating a more digestible version for people who are still high on their Rand or Marx or Resistance would be a waste of time.

    Thanks for the best read I’ve had in a while.

    Liked by 1 person

    1. Thank you. I’ve heard similar things from others — that the message is powerful but the calculus metaphor doesn’t particularly help. I guess that’s true and I might want to make a different version at some point, and probably lean more on how I wrestle with narrative crafting in my job.

      It’s just that partial derivatives is such a perfect metaphor for me, and it’s what gave me the idea for the article in the first place (especially since I could fit in the integration bit as well). It’s a kill-your-darling situation.


      1. This was beautiful post! So succinct.

        A beautiful thing is that while it is convenient to communicate to STEM-familiar people with a succinct phrase “partial derivative,” the concept is SUPER easy to understand: just a picture of any terrain, the “derivative” depends on the direction.

        Of course, non-mathy people might object, they might say, “but there clearly is a more meaningful direction, the one that most directly points at the top!” which then gets at concepts like the gradient as a “natural” direction, and other issues like if you want to define the actual vector (v_local_max – v_current), you need to know that (v_local_max) is.

        Anyway, thanks for the thought provoking post!

        Liked by 1 person

  4. I finally read this – sadly, it had remained in a mostly ignored ‘to read queue’ that I’m finally ‘processing’.

    Interestingly, I think Ayn Rand would (mostly) agree with your own assessment of “Atlas Shrugged”! She even described, explicitly, her own aesthetic philosophy and ‘stylization’ is a big component of it. I think that maps pretty naturally to ‘partial narratives’.

    This also very much reminds me of the idea that _any_ advice is wrong for some people. Scott Alexander has a great post about that: Should You Reverse Any Advice You Hear?.

    I think the analogy to calculus is pretty central tho. Maybe one way to make it more accessible to people intimidated by that might be to use some more graphs or other visualizations. (I’d recommend maybe using a function of two variables for that as visualizing functions with more variables is tricky! The only event partially (ha) workaround is to visualize a third variable as temporal.)

    Liked by 1 person

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