How can we define something and have it translate to the real world?

Question

I'm looking at a definition in my textbook (although my question applies to most definitions)

Let A, B be two events. Define $P(B|A)$ i.e. the probability of B given A, by: $$P(B|A) = \frac{P (A\cap B) }{P(A)}$$

How can we define $P(B|A)$ in this way? Shouldn't it be a theorem? How can I calculate the probability of an event B in the real world given an event A in the real world, with this definition?

How can we define something and then use it to answer quesitons about the real world?

Another example: Where is the proof that $$\iiint_K1dxdydx$$ where $K$ is an abstract sphere with radius $r$, actually calculates the volume of such a sphere in the real world?

1. How can we define something and have it translate to the real world?
2. Why is it that mathematical models translates to the real world at all?

Answer 1

A possible answer to both of your questions is "trial and error". To understand this viewpoint it is helpful to think of mathematics as one of the exact sciences rather than being a field of investigation apart from the sciences. In the exact sciences one builds models and then tests them against the "real world" to see if they predict meaningful results. Similarly in mathematics we have no apriori reason to expect that this or that type of calculation will necessarily have this or that meaning in the "real world"; rather, trial and error have shown that certain mathematical theories are indeed applicable in the sense of producing seemingly meaningful results.

To answer your second question: in the framework just outlined, mathematical models indeed do not automatically translate to the real world. Through trial and error, some of them have been found to be useful. Others have not (take for example the mathematical entity called a nonmeasurable set and the theorem asserting that a unit ball can be broken up into 5 pieces which, when rearranged, produce two unit balls).

In fact, Bayesian statistics is an excellent case in point: half the experts think it is applicable to describing things in the "real world" and the other half think it is useless. This indicates that the existence itself of a coherent mathematical theory is no guarantee of its "translatability to the real world".

Answer 2

First of all, it should be $$P(B|A) = \frac{P (A\cap B) }{P(A)},$$ and then, that wasn't some arbitrary definition, but a generalization of simple models in finite probability spaces. And then, it's a somewhat unfortunate example: nothing is more controversial than the real world interpretation of probabilities, especially conditional probabilities (and Bayesian statistics is yet another can of worms). Concerning that sphere: it's an abstract sphere. Does it exist in the real world? No, not exactly, look at some almost ideally polished metallic sphere through an electron microscope. It looks a bit rough and cubistic, almost like the integral sums we use to define integrals, doesn't it?