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I might be phrasing this question incorrectly, but my students asked me about it and I did not have a good answer. I am in statistics and I look at all sorts of social data, network data, climate data, etc. There is a very simple algebra of events that works within statistics--$\sigma $-algebra and of course linear algebra. However, I was wondering how I would recognize whether a particular phenomenon that I was studying seemed to behave according to the rules of some sort of algebra--i.e., if the states of the system evolve based upon some particular algebraically defined operations. As I said, I am sorry if I am not phrasing this correctly.

The question was intriguing because at least in stats and the math at my level, we generally learn the algebraic rules and then apply the operations. We never come at it from the other way and say, "look at this phenomenon, there seems to be a set of operations that seem to explain the patterns of behavior and these patterns seem to be algebraic." Is there such a type of analysis. Please correct me if I am phrasing this incorrectly, but hopefully someone will understand the essence of the question. Thanks.

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Interesting question. (This is more like a long comment.)

I think that usually, algebraic structures don't describes phenomena directly, rather they're used as ingredients to build models of those phenomena. For example, the phenomenon of motion can be described using the Newtonian synthesis (a model), which in its modern form makes use of vector spaces (an ingredient in the model); however, vector spaces alone aren't really a "model" for motion.

So really the question is: how can we learn to recognize the phenomena with the property that to build a good model of said phenomenon, we're going to need a particular (perhaps exotic) algebraic structure as an ingredient in our formulation of the model?

Unfortunately, that's a tough question, and I don't have answer.

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