What can be (un)done?

Okay, bear with me on this (or skip the math examples and shoot down to the break).

Say you want to add the following fractions: 2/3x + 5/y. Doing so is no great difficulty, as any high school student can attest. Multiply the top and bottom of each fraction by the other’s denominator, ensuring they have a common term. Then you can add the numerators. That looks like this:

2/3x + 5/y = 2y/3xy + 15x/3xy = (2y + 15x)/3xy

However, if I’d given you the end result and asked you to reverse engineer the original expression, that would be quite a bit harder. (If you don’t believe me, try to rewrite (8x + 7)/(x2 + 3x + 2) as the sum of two fractions with constants in the numerator.) To do so, you have to use a process called “partial fraction decomposition,” which I bet few high school students or adults are familiar with.

Consider another example. Let’s say we have a function f such that f(x) = 5x2 + 3x + 10, and we want to take its derivative. To do so, decrement each exponent by one and multiply the coefficient of each term by the original exponent:

f'(x) = (2*5)x2-1 + (3*1)x1-1 +(10*0)x0-1 = 10x + 3

Once again, however, if I ask you to do the reverse—to find the anti-derivative of 10x + 3—it’s a different story. You have to undo the same process: divide each coefficient by 1 + the current power of each term of x (in this case, respectively 1 and 0) and reintegrate an x into each term. If you do that, you get this:

10x +3 = (10/2)x1+1 + (3/1)x0+1 = 5x2 +3x + C

But this isn’t what we started with: our original equation was 5x2 + 3x + 10! When we took its derivative, we ended up multiplying the 10 by 0 (because 10 can be written in terms of x as 10x0). In doing so, we irrevocably destroyed the information that would give us the final term of the quadratic. We know it’s something, so we write it as C per convention.


This post isn’t actually about math. I think there’s a worthwhile sociological metaphor to be had: in both cases above, it’s easier to go forward than backward, and in the second example, it’s actually impossible to completely return. There is a strain of thought that major public policy decisions should be taken with little hesitation because “if it doesn’t work, we can always try something else.”

I don’t think that’s necessarily true.

In a particularly inspired Slate Star Codex blog post, Scott Alexander offhandedly rejects the common characterization of the body as a well oiled machine. Instead, he likens it to a careful balancing act that could easily be thrown off kilter.

People always talk about the body as a beautiful well-oiled machine. But sometimes the body communicates with itself by messages written with radioactive ink on asbestos-laced paper, in the hopes that it’s killing itself slightly more slowly than it’s killing anyone who tries to send it fake messages. Honestly it is a miracle anybody manages to stay alive at all.

Scott Alexander, Maybe Your Zoloft Stopped Working Because A Liver Fluke Tried To Turn Your Nth-Great-Grandmother Into A Zombie

I’d argue that what’s true for the organism is true for the super-organism. Complex society is basically a miracle. The stars have to align for it to form, and it can degrade with comparative ease. When we alter it, there’s no guarantee that going back to the status quo is easy or even possible. Sometimes information may be lost permanently in the transition, while unintended consequences can linger for decades.

The decline of marriage rates among low-income households is a great example. The advent of means-tested welfare programs in the 1960s is widely thought to have dissuaded many lower-income women from marriage through the effective imposition of high marginal tax rates.1 (Their benefits would decrease faster than household income would increase if they married, so it made economic sense not to marry.) This has had all kinds of nasty second-order effects,2 which is why welfare reform in the 1990s explicitly attempted to counteract this unintended consequence—but largely to no avail.

Post-industrial America offers another example. The fortunes of former manufacturing towns in the northeast and mid-west have been almost uniformly bleak since de-industrialization. (If you haven’t yet, seriously, read Sam Quiñones’s DREAMLAND.) There have been and continue to be efforts to reverse this downward trajectory: innumerable economic development programs, paid relocation programs, home-buyer subsidies, corporate tax incentives. But no amount of grant funding or wealth transfers alone can replicate the conditions that created prosperity in those areas. The money was only one part of the equation.

As frustration with the societal status quo increases, people will become more pro-action biased. I don’t necessarily think this is bad—it could be great! But it would be a mistake to proceed without caution where sociological elements are concerned. Introducing a universal basic income, for example, could change… a lot. And there’s no guarantee we could ever go back.

  1. This isn’t the only hypothesis put forth, nor are explanations for this phenomenon necessarily mutually exclusive. Pseudonymous blogger Spotted Toad makes the case that the collapse in men’s wages (also) played an important role.
  2. The consequences of the decline of marriage have different valence depending on who’s doing the analysis. There are plenty reasonable-enough takes on why the decline of marriage is good. I think the most reasonable analysis is that it’s a mixed bag generally, but for lower social classes, the results have been less ambiguously destructive. Having been raised in a rather unstable cohabitating household, I’m personally rather attuned to the drawbacks, but that’s just me.

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