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The paradox of tolerance examines an apparent self-conflict in cultural values of tolerance and inclusivity: it raises a question of whether it's just for a community to include or exclude people who strive to exclude certain others. For example, does it align with the principle of tolerance to exclude racist people from a community because their inclusion would endanger and eventually eject people of targeted skin colors?

I propose this apparent paradox can be resolved by reframing the implicit problem statement of tolerance as a self-consistent spatial metaphor.

The bin packing problem is an optimization problem where the goal is to most efficiently pack objects into a space.

The problem in the general case is NP-hard, which, without getting too deep into a tangent, more-or-less means we're pretty sure there's no possible approach to solving it that both works in all cases and can be worked out quickly. A universal general-case solution may exist, but it will be agonizingly slow to carry out. Fast solutions may exist, but they will all be special-case. At least we're pretty sure this is true.

Here, though, we won't be looking at the NP-hard general-case bin packing problem. We'll be looking at a really trivial special-case variation on the problem: fitting equally-sized perfect squares into a bigger perfect square whose side length is a multiple of that of the smaller squares.

The optimal packing here is plainly self-evident just by looking: simply align the smaller squares strictly on a grid within the larger square, so that every row and every column is a straight unbroken line, and we can fit exactly the maximum number of squares into the bin, with exactly zero space wasted.

The squares also have different colors. Intuitively, this should have no bearing on the optimal packing. However...

Suppose the squares attract and repel one another. How much a square attracts or repels other squares varies by individual square (not by color). Crucially, so does which colors the square attracts or repels. Some squares may attract all colors, some may repel all colors, some may attract specific colors, and some may repel specific colors.

Assume a square has a sort of agency. It has no control over what color it is, but it can control whether it attracts or repels, how much, and what other colors. It can control these things, but it won't. It's set in its ways.

Now, as we look for the optimal packing of the squares, let's change our definition of "optimal" to reflect these extra attributes the squares can have. We're optimizing by the following rules:

1. Maximize how many small squares can fit in the big square. Equivalently, minimize wasted space. (This was the rule we had already.)

2. Minimize total repulsion exerted on all small squares.

3. Minimize "unfairness:" total squares excluded for reasons they cannot control. In other words, minimize cases where a square is excluded because of its color.

Assume the forces exerted by the squares on one another according to color follow an inverse-square law: the force is directly proportional to the magnitude intrinsic to the exerting square (the length of the arrow) and inversely proportional to the square of the distance between the squares. (For our purposes, the exponent here isn't actually important, just the principle that the force is stronger with proximity.)

We could just throw all the squares that repel each other into the bin together. As it turns out, this would be a really bad packing strategy.

If we do that, then the squares want to keep their distance. This approach would have to compromise in one of two ways: it would inevitably either minimize the number of squares packed / maximize space wasted (as illustrated) or maximize repulsion exerted between the squares (as a side effect of minimizing the distance between them).

Alternatively, we could throw squares together that are all the same color and all repel the same colors, or, more broadly, a configuration of squares such that no square repels any other square included even though every square included repels some non-included squares.

Going only by rules 1 and 2, this packing strategy is optimal. No space is wasted, and no repulsion is exerted on any square included in the packing. However, this packing is highly suboptimal according to rule 3: there's an obvious pattern here where squares are included or excluded entirely on the basis of color.

Some people think the squares excluded on the basis of their color should just go pack themselves somewhere else.

There's a huge problem with that way of thinking: there aren't enough bins. The optimal shape of the packing won't hold without the underlying framework of a bin to enclose it on all sides.

In the usual formulation of the bin packing problem, the optimization condition is expressed as minimizing the number of bins that must be used to pack all of the shapes. We are instead formulating the problem as maximizing the number of shapes that can fit into the bin for two reasons:

• We do not necessarily have a finite number of shapes to pack.

• We only have one bin.

Now let's try including only squares that do not repel other squares.

Under our modified rules, this is the actual optimal packing:

1. The number of squares that fit in the bin is maximized and no space is wasted.

2. No repulsion is exerted on any included square.

3. The number of squares excluded for reasons beyond their control is minimized.

Since we have limited space, some squares will always be excluded arbitrarily even in an optimal packing, simply because no more squares can fit anymore. We could say these squares are excluded for reasons beyond their control, but they are not specifically chosen to be excluded. If we start with that inevitable baseline unfairness and then add rules that exclude squares based on their color, which they cannot control, then we are specifically choosing squares and guaranteeing they will be excluded for reasons beyond their control, and so, although the number of squares excluded for reasons beyond their control could never have been reduced to zero anyway, we raise it beyond its baseline.

With the strategy we're trying here, however, where squares are excluded based on what colors they repel, which they can control, the number of squares inevitably excluded for reasons beyond their control is not increased beyond its baseline. In fact, it is necessarily decreased, because if the total number of squares included in every optimal packing is always the same, then it follows that the total number of squares excluded is always the same, which means any number of squares excluded for reasons within their control subtracts from how many are not.

In case it's not painfully obvious by now: the large square, or bin, is meant to represent a community; the smaller squares are human beings; their colors are qualities a person cannot change voluntarily and healthily, such as skin color, gender identity, sexuality, economic class, ability, etc; attraction, denoted by inward arrows, represents sympathy for a common cause; and repulsion, denoted by outward arrows, represents bigotry. The thought experiment demonstrates unambiguously that the optimal packing strategy is to refuse to tolerate intolerance.

Where's the paradox? There is none. We've elided it. Excised it. The paradox appears when we frame the problem of tolerance as some grand universal principle that can be cleverly pitted against itself, but when we use a practical spatial metaphor, there's no room for self-contradiction. Where would we find it? Somewhere in the bin, among the squares? There are no self-referential statements at work here, no semantics, no propositions and predicates—just colored shapes in a box.

What is represented by the requirement to only use a single finite bin for packing, you might reasonably wonder? After all, more than one community exists in the world.

That's definitely the weak link in my metaphor here, but even it is not without meaning. The intended meaning is hinted at in how, in the diagram where the squares of the excluded color try and fail to form their own packing, arrows signifying repulsion of that color are still extending from the bigoted squares in the successful packing, even extending out of the bin itself.

The meaning is this: In practice, the requirement to use only a single bin reflects the unfortunate reality that when communities are fragmented, there will be conflict between them, and one community will dominate the conflict space and make life unlivable for everyone else. I'd like to believe a world can be attained where things don't work out like this, but I've never seen it. I've never seen a world where it is not the case that one community controls all the land, all the food, and all the firepower, among other things, while all the other communities in effect have little to no say in supercommunal agreements.

Moreover, even with adequate resources, I've never seen a world where the dominant community won't invade all the other communities and try to harm and control and drive out the colors of squares they repel. Even if the squares excluded from the dominant community went and formed their own community, the dominant community would just conquer it. This happens not just at the level of nations, as I was implying in the previous paragraph, but also e.g. among cliques, businesses, gangs—everywhere there are people. Everywhere there are people, there is senseless conquest. The squares that repel other squares on the basis of color say they just want those they exclude to go form their own packings, but what they really want is to wipe out their existence entirely, and they prove it at every opportunity they're given. One would hope a community not united by intolerance would thus not be driven by intolerance, and so would not exhibit this same destructive behavior toward competing communities.

Thus, to bring things back around to the metaphor: the reason we have only one bin to work with is because the force of repulsion emanating from within that bin will prevent the formation of additional packings.