When you specify how much of some objects you need or want to count, when you first have a pure number in mind and only after that you chose the objects to count, based on that number, you bridge the space and connect pure and applied math. The number you had in your mind belongs to pure math domain, while the quantifiable objects you decide to count, together with the chosen number, belong to the applied math domain.

Applying mathematics means, as per the illustration, filtering out the units, objects counted, and dealing with pure sets and counts, numbers only. With these numbers and quantitative relations you enter the world of pure math, obtain the results, by doing new calculations or by using already proven theorems, and then return back the result to the real world, reattaching the units on the way back.

But, it also means, that the logic, reasoning within pure mathematics, similar to the chain of political reasoning and decisions before a certain action is taken, is important when it is required to know exact quantity that will be used in the real world scenario. Accuracy of pure mathematical processing, calculations, proofs, theorem resuse, is a significant, a central factor to obtain a correct number and hence go ahead with some directive how much of some objects need to be counted. Of course, initial conditions, numbers entered the pure math mechanisms, are coming from quantification in the real world, and that link will be the major connection for units reattachment and decision what to count and at which magnitude.

Hence, the importance of theorems, theorem proofs, although they may seem too abstract and distant from real world application at the first sight, has central role in obtaining accurate results that will be used back in decision making in real world domain from which initial conditions originated.

Quantification result, or the obtained number, need not to be used in mathematical domain only or to quantify something else. It can be used, as a true or false fact, or as an information, as a part of any decision making process of any domain, in any hybrid axiomatic system, or any logical system as a part of logical expression and in any logical connective.

Quantification result, or the obtained number, need not to be used in mathematical domain only or to quantify something else. It can be used, as a true or false fact, or as an information, as a part of any decision making process of any domain, in any hybrid axiomatic system, or any logical system as a part of logical expression and in any logical connective.