Exclusion Principle for Solids

This is an intuitive postulate that provides a basis for comparing solids in terms of volume.

If one solid has a larger volume than another, it is considered dominant; if it has a smaller volume, it is subordinate. If both have the same volume, they are equivalent solids.

At middle or high school level, this understanding is sufficient - you don’t need to delve deeper.

Note. This postulate appears simple, orderly, and clear-cut. Any pair of solids can be compared and arranged without ambiguity. However, this holds true only within the idealized framework of abstract geometry. The moment we step outside that realm, reality disrupts this neatness: not everything is comparable, and not everything can be ranked. The real world is complex, and it cannot always be reduced to a single number. I’ll explore this idea further in the next section.

Challenging the Exclusion Principle

The exclusion principle attempts to rank every pair of solids based solely on volume: either equivalent, dominant, or subordinate.

It’s a neat simplification - but also a potentially misleading one. It works within the confines of elementary geometry, but breaks down in the real world, where objects are multifaceted and evaluated along multiple dimensions.

Put simply, solids can be assessed according to various criteria, and these criteria don’t necessarily yield the same ordering.

Example. Imagine a marble cube and a wooden cube with identical volumes. According to the principle, they are equivalent. But anyone picking them up can tell that’s not the whole story: the marble cube is much heavier, less workable, more expensive, more fragile, and denser. Clearly, volume alone doesn’t capture the object’s full nature.

Once we introduce multidimensional criteria, solids often become incomparable.

Calling one solid “dominant” implies some form of superiority. But superior in what sense? Weight? Surface area? Functionality? Durability? Aesthetic appeal? Without clarification, the comparison is meaningless.

Talking about dominance in absolute terms is like claiming that apples are better than oranges - without specifying whether you're referring to taste, vitamin C content, or price per kilo.

Example. Consider a sphere and a cube with the same volume. They are, by definition, equivalent solids. Yet the sphere has a smaller surface area and is more efficient at minimizing heat loss or moving through air (think of a ball). The cube, on the other hand, is more stable, stackable, and suitable for construction. The sphere rolls easily due to low friction; the cube does not. Which one is better? That depends entirely on the purpose.

The illusion of absolute comparison falls apart when different criteria conflict and no single standard prevails.

This leads us to the concept of incommensurable criteria - when there is no common ground for deciding what is “greater” or “lesser.”

Ultimately, the exclusion principle serves a useful pedagogical role. It’s a helpful tool for introducing the concept of comparing simple, abstract geometric quantities. But insisting it reflects reality in any comprehensive way is a mistake.

In the absence of a clearly defined context, any absolute comparison becomes a deceptive oversimplification. Reality isn’t meant to be ranked - it’s meant to be understood.

And so on.