Owen Biesel

I’ve been helping put together some materials for a new class we’re tentatively calling “Math and Public Life”, organized around ten or so concepts from higher mathematics and how they relate to the way we think about life and each other. One of the themes I’m hoping to show is that often as our understanding of an issue grows, we move from simpler mental pictures of it to more complex ones. Sometimes that means replacing a yes/no binary with a different structure more suited to the situation, and in this blog post, I’d like to share with you four examples of how that can work.

Before we start, a quick note: usually by “false dichotomy” people mean that there are two alternatives presented (“Would you like apple pie or pumpkin?”) for which there are either more than the two choices offered (additional dessert options) or the two possibilities aren’t mutually exclusive (a slice of each!). In other words, there are extra possibilities that capture either both of the two original ones, or neither. But in the situations that follow, the additional possibilities are somehow both “both” and “neither” of the original two… and also neither. I hope you have some fun with these ideas!

1. The spectrum

This may be the most familiar of the four types: a spectrum of possibilities ranging from one extreme to the other. Think of how there’s no sharp division between “child” and “adult”, but instead a gradual maturing from definitely one to definitely the other. Many pairs we think of as binaries are secretly spectra in disguise — even the archetypal True/False dichotomy is spanned by a range of probabilities, from 0% (i.e. definitely false) to 100% (i.e. definitely true) and everything in between. Whenever we talk about seeing the world in shades of gray instead of black and white, we’re talking about a spectrum.

Spectra tend to come with apparently arbitrary divisions (like, “you’re technically an adult once you turn 18”) and questions of whether the absolute extremes could ever actually occur for real — do the political left and right have endpoints, or is there always “more left” and “more right” going on forever?

2. The hypercube

Like the spectrum, a hypercube is another way allow for possibilities that are “sort of both” of two extremes, but in this case, it’s because each extreme is actually a combination of multiple properties that don’t all have to occur together. For example, people sometimes break down the conservative/liberal binary into two separate binaries: socially conservative or liberal on the one hand, and economically conservative or liberal on the other. Many people will land in either the “all conservative” or “all liberal” camps, but many will not, and it may be more helpful to sort people into a 2×2 grid.

In the picture above, there’s a more complex example with a solid yellow circle as one extreme, and an empty blue square as the other. Since “solid/empty”, “yellow/blue”, and “circle/square” can occur in any combination, there is actually a 2x2x2 grid, or cube, of possibilities, with the edges of the cube connecting the most similar types.

If there are more than three factors, the structure formed will be a 2x2x…x2 grid, or hypercube. For example, there are many different ways people have of thinking about introverts and extraverts:

1. Introverts need time alone to recharge; extraverts need time with other people.
2. Introverts are shy; extraverts are outgoing.
3. Introverts think in private; extraverts think by talking with others.
4. Introverts plan carefully ahead of time; extraverts learn by doing.
5. and so on.

Even if each of these were true dichotomies, they wouldn’t all line up exactly the same way for everyone. Instead, they form a high-dimensional hypercube with most people at or near one of the two ends, but many people with a mixture of introvert and extravert traits.

Hypercube situations tend to come with sleight-of-hand reasoning, in which by establishing one of the related attributes, another is assumed to go with it: “She’s so shy… she must be an introvert. She’s probably fine on her own.” A correlated trait is evidence, but not proof.

3. The plane

Our third better-than-binary structure is when the two alternatives are better represented as coordinate axes, representing a two-dimensional space of possibilities based on how much of each you have. This is the representation I would choose for the scenario in which I’m asked if I want apple or pumpkin pie: the real choice is how much I want of each, and those amounts can be chosen independently in principle: “I’d like a full slice of apple but just half a slice of pumpkin.” The amounts of each option are the coordinates of a point in a plane, and my choice corresponded to the point (1, 0.5) in the apple-pumpkin pie plane.

Plane possibilities usually come with constraints that make the situation look more like a spectrum: I can’t really ask for arbitrary amounts of pie, so there’s actually a tradeoff between asking for more pumpkin or more apple. If I’m thinking about what to read in the coming year, I might ask “should I read more fiction or nonfiction?” but a better question would be “how much do I want to read of each?” — though my limits on how much time I can spend reading mean that if I do read more fiction, I’ll probably have to read less nonfiction. Anything we spend is in principle a plane (or a higher-dimensional plane, called a “hyperplane”), but more often ends up a spectrum because of tradeoffs (or a higher-dimensional spectrum, called a “simplex.” A two-dimensional simplex is a triangle, which is the space of possibilities shown here).

4. The limiting case

Have you ever found yourself thinking along the lines “This activity is either safe or unsafe. If it’s unsafe, I shouldn’t do it at all. If it’s safe, I can do it as much as I want.”? Each sentence sounds logical, but I’ve argued it’s still not the right way to think about safety. Sometimes, there’s a true dichotomy between two possibilties, but there’s an asymmetry in which one possibility is a limiting case of the other. For example, true safety is almost impossible to achieve, and straying from it in any way introduces at least some risk. However, true danger is easy to find, and a slight deviation from one dangerous activity is probably just another danger. Safety is like a tiny dot in a vast sea of danger, and the best you can do is get close enough that the remaining risk is negligible.

Here are some other dichotomies — which would you say is the limiting case of the other?

• Perfect vs. imperfect
• Biased vs. unbiased
• Equal vs. unequal
• Square vs. rectangle
• Flat vs. curved

I would say that perfection is a limiting case of imperfection, absence of bias is a limiting case of bias, equality is a limiting case of inequality, squares are a limiting case of rectangles, and a flat surface is a limiting case of flatter and flatter curved surfaces.

Limiting cases are often marked by disagreements about whether they should “count” as separate or as special cases. (Is a square just a type of rectangle, or does a rectangle have to be non-square to be a rectangle?) Another trademark of limiting cases is that while it’s easy to establish the more general possibility through evidence, it’s hard to establish the limiting case exactly: it’s relatively easy to prove an algorithm is biased: run a statistical test and see the difference in output. It’s much harder to prove that an algorithm is unbiased: at best you can say “we weren’t able to detect bias of such-and-such size.” This is also the mistake that Flat Earthers make when they go out into a big field and say “Yep. Looks flat.” Since “flat” is nearly indistinguishable from “very large and round”, they should really say, “Yep, looks flat or very large and round,” and rely on more sophisticated measurements that actually do establish Earth’s roundness.

But wait, there’s more…

Of course, these aren’t the only mental pictures we can use to get a better sense of the world than through binary thinking: abstract mathematics provides us with many more. (They just tend to have intimidating names like “linear order”, “Boolean algebra”, “vector space”, or “topological space.”) We can even combine different types, like using spectra to fill in a hypercube to make it solid, or viewing coordinate axes as limiting cases in a plane. Do you have another way of thinking about possibilities that doesn’t fit into one of these four types? I’d love to hear about it in the comments — maybe we’ll invent some new math together!