Kimbune's theorem (on The Tyrant Baru Cormorant, part 6)

Kimbune's theorem (on The Tyrant Baru Cormorant, part 6)

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This is the sixth part of a series of articles on The Tyrant Baru Cormorant—part review, part meta, part commentary. For intro and links to the others, go here!

Depending on how much you’d like to read about Euler’s formula, this is either a short article or a long one.

The number of interest

In taking this ecological reading of culture, I am reminded of something Seth once wrote on their Twitter—unfortunately I can’t get the direct quote—about the parallels between cancer and colonialism: a part of an organism or community ceasing to participate in symbiosis, but attempting to gather all resources to itself. (Seth said it better.)

So let’s look at the thing that unites all the powers, and evils, explored by Baru Cormorant books. Powers such as…

• economic growth
• disease epidemics
• cancer
• colonialism
• radioactive materials

What do these have in common? In mathematical terms, there’s one striking point of unity: exponential growth and decay.

What’s that, then? Exponential growth is a pattern which manifests itself whenever the rate of growth of something is proportionate to how much is already there.

That may sound a bit confusing, so let’s consider an example. Imagine a population of bacteria (with plenty of tasty sugars to eat). Every hour, each bacterium splits into two bacteria. The more bacteria there are, the more will be born each generation. When you only have two bacteria, the population grows very slowly… but as the colony gets bigger, the bacteria appear faster and faster. If we count the bacteria every hour, over a number of hours we’ll see a curve like this:

This is the the exponential function, exp⁡(x), or ex. Though there are many exponential functions, like 2x or 10x, they can all be related back to this one.

Exponential functions are of deep importance to capitalist economics, because the terrifyingly powerful engine at the heart of capitalism—at least, once the brutal primitive accumulation phase has seized control of land and materials through direct violence—is the cycling of profits back into making more stuff than the last time around (‘expanded reproduction’ in the language of Marx).

All societies must reproduce themselves, and all growing societies produce more than they need to just continue to exist—but capitalism made this growth the core of what a society is. Everything in capitalism is keyed to an assumption of exponential economic growth, from the interest on a loan to the expectation of annual profits from a company. So if any part of the system—a company, for example—can’t keep up with the pace of growth, it will be crowded out by its competitors, starved of funds or bought up and remade.

But it’s not just capitalism. Diseases, too, spread exponentially at first through a large population—something we’re all too aware of in the age of COVID-19. Cancer cells, not subject to programmed cell death, reproduce themselves on a similar trajectory (although, pedantically) the specifics vary for different cancers, and some only grow at the surface of a tumour).

And radiation? Radioactive decay is something of the opposite: every atom has a random chance to decay in every instant, so the more atoms there are, the more quickly they disappear. In the specific case of uranium, so beloved of the Cancrioth, the most common isotope (by far) has a half life in the billions of years—slow enough that it hasn’t all decayed already, fast enough that, in abundance, it creates some serious activity.

(While we’re at it… in a nuclear weapon, which even the Cancrioth have yet to imagine, the nuclear chain reaction exploits exponential growth in the other direction: one neutron becomes two, becomes four, each time releasing more energy until, in an instant, most of the fissile material has transformed and all that energy is ready to incinerate a city.)

Perhaps alone among the cast, the Brain is aware of the terror of exponential growth on the scale of societies:

“They understand the secret of power, Baru.”

“Which one?”

“The ability to improve one’s own power, no matter how slowly, triumphs in the long run over any other power. Time magnifies small gains into great advantages. If you are hungry, then it is better, in the long run, to plant one seed than to steal a pound of fruit. Falcrest applies this logic in all their work. They do not conquer. They make themselves irresistible as trading partners. They do not keep their wealth in a royal hoard. They send it out among their people, stored in banks and concerns, where it helps the whole empire grow. They do not wait to treat the sick. They inoculate against the disease before it spreads. All their power sacrifices brute strength in the present for the ability to capture a piece of the future.”

The thing about exponential growth is that, though it starts out apparently slow, once it gets into motion it is the fastest-growing mathematical function we routinely encounter. This is what makes disease outbreaks so scary—and it’s what makes capitalism have such force.

Exponential economic growth is Falcrest’s meta-weapon, but—Barhu eventually comes to believe—it is the weapon that can be turned against them too.

Kimbune’s Theorem

But the interest in exponentials comes at a different angle, too. While visiting the Cancrioth, Baru runs into a mathematician who is determined to track down Abdumasi Abd for a different reason than most: in Abdumasi’s tumour is, she believes, the soul of her husband, who died before she could win an argument. And what’s this argument about?

It’s about Euler’s formula. You know:

eiπ+1=0

When I saw that formula on the page, I was like… Seth you absolute dork. Oh, sure, she’s invented the “most beautiful theorem in mathematics” (as decided by vote)…

Then, I ended up spending a very fun afternoon introducing some friends to the significance of this formula, and started thinking about why it would be here.

In the book, Kimbune’s formula comes across to Baru (who can’t understand the proof) as a bizarre connection between unrelated, but important numbers: an indication of the numerical structure of the universe, that Falcrest can’t perceive. But Baru, not a pure mathematician, does not grasp the proof, nor the heart of the formula, which is better rendered in the more general form:

eiθ=cos⁡θ+isin⁡θ

Naturally I came up with a reading of the book’s broader themes in relation to Euler’s formula. But first, to get everyone on the same page, I need to explain the recipe. Since this is a long aside not exactly about Baru Cormorant, it gets the box.

A recipe for Kimbune's Theorem£150×1.5=£225£100×(1+1N)iInterest payments e=2.71281828459…money(t)=£100×et1yearf′(t)dfdtddtet=etx′(t)=kx(t)1+x+x22+x36+x424…ddxx5=5x4ddxxk=kxk−1f(x)=k0+k1x+k2x2+k3x3+…f′(x)=k1+2k2x+3k3x2+4k4x3+…f″(x)=2k2+(3×2)k3x+(4×3)k4x2+(5×4)k5x5…f(0)=k0f′(0)=k1f″(0)=2k2f(3)(0)=6k2f(n)(0)=n!knf(x)=f(0)+f′(0)x+f″(0)x22+⋯+f(n)(0)xnn!+…f(n)(0)=1π=3.141592653589793238...(x(π6),y(π6))=(32,12)distance (x,y)=(cos⁡θ,sin⁡θ)ddtcos⁡t=−sin⁡tddtsin⁡t=cos⁡td2dt2cos⁡t=−cos⁡td2dt2sin⁡t=−sin⁡td2xdt2=−kxcos⁡x=1−x22+x424…sin⁡x=x−x36+x5120…N={0,1,2,…}2+3=53+x=56+x=2n+(−n)=0Z={…,−3,−2,−1,0,1,2,3,…}x×5=x+x+x+x+xx×3=6x×3=2x×1x=1x2=4x2=94x2=22=1.41421356…910+9100+91000+⋯=1x2=−9eiθ=1+iθ−θ22−iθ36+θ424+iθ5120…eiθ=(1−θ22+θ424+…)+i(θ−θ36+θ5120+…)eiθ=cos⁡θ+isin⁡θ

The consequence of Kimbune’s thoerem

eiθ=cos⁡θ+isin⁡θ

This is Kimbune’s formula (or Euler’s, in the boring world). It does a few things. It tells us how to handle complex numbers in exponents, which was not at all obvious. It gives a fascinating geometric interpretation of complex numbers, as a kind of polar coordinates. It leads into the whole terrifying world of complex functions, which add some fascinating headaches. But we’re getting away from the main point…

That relentless exponential growth and decay? Kimbune’s theorem turns it ‘sideways’, transforming real numbers into imaginary numbers and back, in an endless circle.

This short video does a good job of illustrating this concept:

Now, let’s imagine a point roaming the complex plane. At each instant, we can draw a little arrow to see which way it’s going (its rate of change). The bigger the arrow, the faster the change.

direction

Exponential growth means that arrow points away from the origin. Exponential decay, back towards the origin.

And now we have a way to circle the origin, neither growing nor decaying, but always changing, ever faster the larger it is.

In Barhu’s world, the number e is called the ‘number of interest’. Interest is the result of a loan: it is money turned to make more of itself. But through Kimbune’s theorem, the process of interest is turned aside: not to expand but to transform…

Perhaps this metaphor is kind of a stretch? I think it’s a fun reading though, given what Barhu plans to do.

From Kimbune’s theorem to ecology

Although we’ve talked about how bacterial growth gives rise to economic growth, in the real world, very few lifeforms get to happily grow their population without limit… as often as not, there’s a shortage of food… or something out there to eat them.

One of the simplest models involves a population of predators, and a population of prey animals. We can model this as a coupled system of differential equations: the population of predators rises as their food source increases, but predation wipes out the prey, and the starving predators die out. This video discusses a stripped down version of the problem:

Mathematicians have a tool for analysing this kind of problem: they move to a ‘phase space’, not so different from the complex numbers we’ve explored so far. One axis represents the number of prey, the other the number of predators, and the system ‘moves’ through phase space as the two populations vary.

Many choices of parameter result in orbits around a ‘fixed point’. At the ‘fixed point’, the predators devour the prey as quickly as they are born, and themselves die off as quickly as they breed, and the two populations remain stable. Everywhere else, waves of prey breed without fear of predators, only to precipitate an explosion in the predator population which slaughters most of the prey… and then the predators starve, and the cycle begins again.

When we humans want to control a population of animals—for example, of deer—one of the most effective ways we’ve found is to introduce a population of predators to the region. And when we humans disrupt these cycles by, for example, killing off wolves for the sake of agriculture, the cycles of the ecosystem are disrupted.

Unfortunately, it’s rarely as simple as this simple, two-component system. The explosion of prey populations will affect all the things the prey eat. One fascinating example is the reintroduction of wolves into Yellowstone. Prior to the reintroduction of wolves, elk populations had grown rapidly, threatening the reproduction of the things the elks ate, such as aspen trees. The failure of the aspen crashed the beaver population, and the lack of beaver dams caused further effects, increasing the variability of water runoff and removing places for fish to breed.

Reintroducing the wolves, on the other hand, made the elk start moving around a lot more—to avoid getting eaten! The aspen could bounce back, the beavers could bounce back, and so on; the effect was termed a ‘trophic cascade’. The ecosystem started to edge back towards its rhythm of self-reproduction.

So, then, in this rather strained metaphor, the Masquerade’s economic engine is like the population of elk. But there is no predator to keep it in check… not yet, anyway.

The danger of over-abstraction

Yet at the same time, we don’t want to disappear up our own asshole with this. The masquerade’s problems aren’t merely that its number (money) is growing faster than, say, the Mbo’s number: it is that, on the strength of its ever-growing production, it can impose a powerfully self-replicating, horrifying eugenic terror regime on the people living in its shadow. Treating everything on the level of an abstract phase space is to ignore what is actually happening to the living beings inside that system.

Consider: a wolf chases a terrified elk, tears it limb from limb, and drags its corpse back to its cubs. The death of a deer, as Disney demonstrated with Bambi, can inspire a lot of empathy even in us humans. We’ve just explained how that is a really good thing for the perspective of a stable ecosystem, but it’s hardly a good thing for the elk! The humans ruling over Yellowstone made the decision that they valued ‘restoring’ the ecosystem to a particular function enough that the painful death of a certain number of elk was an acceptable price to pay.

The same decision is made by those who set hunt quotas: the course of evolution has produced a world whose stability deeply relies on most lives teetering on the edge of sudden annihilation. Thinking too hard about this is what leads people to making grand declarations that the long term mission of humanity is to ascend to some kind of transhumanist omnipotence, and then reengineer death out of nature… or else to a radical break from the values of capitalist ‘civilisation’. (Tain Shir says hi.)

What of Baru? Her plan is a little less abstract, but as we’ll shortly see, the end she’s pursuing is, for now, merely the economic destruction of Falcrest.

Will it be enough? Perhaps history can help tell us…