The Probability of Being Shaped

I've been thinking recently about how to kind of formalize the idea of probability as something other than a number. And so this has gotten me into an interesting area of math, which is called homotopy type theory. Essentially what these homotopy types are abstract shapes in some sense,. or structures, if you want to think of it that way. I very much see kind of probabilistic judgments as judgments about structure and of trying to fit things together in a way that makes the most sense.