Why is mathematics useful?

I guess that there are many unrelated reasons why mathematics is useful. I will discuss only one of them in relation to the frequency philosophy of probability.

Consider a scientific law that involves a mathematical formula, such as F = ma, one of Newton's laws of motion. Here F is the force, m is the mass and a is the acceleration. Suppose that we verified that the law is correct with sufficient certainty. What practical benefit is there from knowing that F = ma? If we can measure two of these quantities, say, F and a, then we can calculate the third one using the formula, that is, m = F/a. And then we do not have to measure the third quantity, that is m. We see that mathematics can eliminate some measurements. If we could measure accurately and easily every physical quantity then the need for mathematics would be greatly diminished.

An elementary probabilistic formula says that

(*) P(A or B) = P(A) + P(B) - P(A and B).

According to the frequency philosophy of probability, this formula can be applied only to observed or observable frequencies. The same remark applies to all other formulas in the mathematical theory of probability. In other words, according to the frequency theory of probability, the only useful application of mathematics to probability is to calculate frequencies of some events from the information about frequencies of some other events.

Knowing frequencies is useful but in some of the best known practical examples only one frequency matters. For instance, frequency statisticians claim that the null hypothesis is incorrectly rejected ("Type I error") with a small frequency, given appropriate circumstances. Knowing this frequency is useful but the utility of this information does not have anything to do with the relationship between frequencies expressed in (*) or any other mathematical formula. The probability of Type I error is mathematically related to other probabilities because it can be derived from the statistical model. The mathematical relationships used in the derivation of the probability of Type I error do not correspond to any useful relationships between observed or observable frequencies. This is especially clear when we deal with long sequences of non-isomorphic hypothesis tests, a situation common in academic setting. The frequency of Type I error is the only observable and useful frequency in this context. The frequency theory of probability fails to explain why it is beneficial to use the mathematical theory of probability to derive the probability (frequency) of Type I errors.