On Orthogonality

The Geometry of Independence

Orthogonality, at its core, is about independence. Two vectors are orthogonal when they point in completely different directions—when one tells you nothing about the other. In mathematics, this is beautifully precise: the dot product is zero, the angle is ninety degrees, the relationship is clean and absolute.

But what happens when we extend this concept beyond geometry? When we think about orthogonality in the space of values, goals, or possibilities?

Orthogonal Values

Consider the idea that certain values might be orthogonal—that is, independent of each other. You can maximize one without necessarily affecting the other. This isn't about trade-offs or compromises; it's about dimensions that simply don't interact.

In practice, we often assume everything is connected. That optimizing for one thing must come at the cost of another. But what if some of our goals exist in orthogonal spaces? What if we've been constraining ourselves unnecessarily?

The Orthogonality Thesis

The orthogonality thesis, particularly in discussions of AI alignment, suggests that intelligence and final goals are orthogonal. An AI could be highly intelligent and pursue any set of terminal values. There's no necessary connection between how smart something is and what it wants.

This is both liberating and terrifying. It means intelligence doesn't automatically lead to "good" values. It means we can't assume that smarter systems will naturally converge on what we consider desirable.

Orthogonal Possibilities

In our own lives, we might think about orthogonality differently. What if some of the things we want are actually independent? What if we don't have to choose between them?

The challenge is recognizing when things are truly orthogonal versus when they're just not obviously connected. Our intuitions about relationships are often wrong. We see connections where there are none, and miss independence where it exists.

The Structure of Possibility

Orthogonality gives us a way to think about the structure of possibility spaces. Not everything is a zero-sum game. Not everything requires trade-offs. Some dimensions simply don't interact.

Understanding this can change how we approach problems. Instead of assuming constraints, we might first ask: are these things actually related? Could they be orthogonal?

The Limits of Orthogonality

Of course, true orthogonality is rare. Most things in the real world are connected in complex ways. But the concept remains useful as a tool for thinking—a way to question our assumptions about relationships and constraints.

Sometimes, the most useful insight is recognizing when things are independent. When we don't have to choose. When we can have both, or neither, without one affecting the other.

The space of possibilities is vast, and not all dimensions interact. Understanding orthogonality is understanding the structure of that space—and recognizing when we're free to move in any direction.