Why State Spaces Must Be Sets Before They Can Be Anything Else

So lets begin with the Schuller lectures. I watched the whole series a few years ago without understanding anything of what was going on, but this time I return armed with Linear Algebra, Calculus some Real and Complex Analysis, and more Probability knowledge. Let us see how far I am able to go this time.

Introduction/Logic of propositions and predicates- 01 - Frederic Schuller - YouTube

After realizing that physical theories begin by restricting what can be observed, the next question is where those observables are supposed to live. Schuller’s answer is deliberately unglamorous: before a state space can have geometry, algebra, or dynamics, it must first be a set.

At first this feels almost trivial. Sets seem like mathematical background noise, not something worth emphasizing. But that reaction is exactly the point. Sets are invisible because they are doing their job too well.

Schuller begins with logic, and from logic, builds sets. This is not pedagogical excess. Logic fixes what counts as distinction and implication. Sets then fix what counts as identity and collection. Only once these are in place does it make sense to talk about states at all. Without a set, there is no meaningful notion of “this state rather than that one.”

A state space, at its most basic level, is just a collection of possible states. No structure has been imposed yet. There is no notion of distance, addition, probability, or evolution. But something crucial has already been decided: states are distinguishable elements, and statements about them obey ordinary logic. This is the minimal condition for prediction to even be conceivable.

The temptation is to rush past this step and start adding structure immediately. Vector spaces, manifolds, Hilbert spaces. But every added layer presupposes the one before it. Geometry assumes a set with points. Algebra assumes a set with operations. Probability assumes a set with events. If the underlying set is ill-defined, every higher structure becomes unstable.

This mirrors a pattern I have seen repeatedly in mathematics. Before division can be trusted, we need a field. Before linear algebra works, we need scalar inverses. In each case, the theory protects itself by first fixing a minimal structure and only then allowing richer behavior.

In physics, the same logic applies. The first commitment is simply that states form a well-defined set, so that comparison, repetition, and distinction are even possible.

What looks like a mundane starting point turns out to be a strong constraint. By insisting on sets, physics rules out vague or context-dependent notions of state before they can cause trouble. It commits to clarity at the cost of apparent generality. As elsewhere, restriction is what makes reasoning reliable.

Seen this way, starting from sets is not a mathematical convenience. It is a refusal to smuggle in structure prematurely. Only after identity is fixed does it make sense to ask how states relate, evolve, or interfere.

Physics does not begin with spacetime or equations. It begins with the quieter decision that states must be things that can be collected, compared, and spoken about consistently. Everything else is built on top of that choice.