What does a "rating number" actually represent? Reflections from building a toy model

[EthanWalker52]

Hi all — long-time lurker, first post here.

I've been chewing on a question that sits somewhere between philosophy of mathematics and epistemology, and I'd value this forum's take.

Over the last few months I've been tinkering, as a hobby, with a small Elo-style rating model for tennis players in Python — feeding it historical ATP/WTA match results and watching the numbers settle. It's purely a personal data project (no betting involved, before anyone asks), but it keeps raising a conceptual puzzle I can't shake.

When we assign a player a "rating" of, say, 1850, what is that number actually a claim about? A few framings I keep circling between:

1. Dispositional — it's shorthand for a disposition: "would win roughly 75% of the time against a 1650 opponent." But a disposition that is never fully actualised feels metaphysically slippery.

2. Epistemic / Bayesian — it's not about the player at all, but about the state of our knowledge: a compressed summary of the evidence so far, revised as new results arrive. On this reading a rating is a belief, not a property.

3. Instrumentalist — it means nothing beyond its predictive usefulness, and asking "what it really represents" is a category error.

What I find genuinely interesting is the Elo update rule itself: it's a tiny, self-correcting inference engine. Each result nudges the estimate toward consistency with observed outcomes, which feels Bayesian in spirit even though the original 1960s formulation isn't framed that way at all.

So the question I'd put to the logicians and philosophers of mathematics here: is a rating a property of the world, a property of our knowledge, or merely a useful fiction? And does the same analysis carry over to any quantity that is only ever defined through comparison — temperature before absolute scales, utility in decision theory, and so on?

Keen to hear how you'd carve this up.