Math took off as a separate discipline when thinkers realized that 2 + 3 = 5 no matter what are you counting. Then, axioms came into light, setting math on a firm footing of logical thinking. But, when you use math in real life situations or when you apply math in various disciplines, you do not go through axioms and theorems, not even proofs, at least until later. You state something, like Coulomb's Electric Force formula or Newton's Law of Gravitation, but, what are you stating, what is it, from the mathematical point of view?

You stated theorems, or premises, or postulates, directly provable from ZFC axioms! From inside math it looks trivial. But, the actual selection, what postulate, assumption, premise, you chose, as your starting point, really matters a lot from the point of view of discipline that stipulated that formula. Axioms can not tell you which postulate will be of special interest to you. Axioms just serves to tell you what operations and concepts you have at your disposal. And it's not much. You have a concept of a set, and few operations on sets, and that's, essentially, it!

Once you see, note, distinguish, things that are common to many other concepts,or objects, perhaps some common property, there is a big chance that that property and relationships between those properties, can take off as a separate discipline, with its own axioms and postulates, premises, theorems (and I am not talking only about math here!). World#1, is the world that has those objects with common properties between them. These common properties and their relationships, can be abstracted from World#1 into separate (possibly and desirably axiomatic) system. Then, World#1 will dictate the genesis of postulates and premises in that abstracted (axiomatic) system, let's call it World#2.

The postulates in World#2 can be generated from two sources. From World#1, with all descriptions and explanations using World#1's language, or, from World#2 axioms! Example in math and physics. Math expression y = ax^2 can be obtained directly from fundamental axioms of mathematics, which just tell you that you can, you are allowed, to do that. Hence, you can just chose to generate a function y = ax^2, without any further explanation. But, then, in physics, y = ax^2 can be E = mc^2 which required Nobel Prize way of thinking which objects to put in place of "a" and "x" in the formula y = ax^2. Here in physics, it matters what is counted and not only the mathematical form.

Hence, y = ax^2 can come from physics, engineering, economics, finance, and, while in math it is only counts what matters, in all those disciplines matters what you actually counted, or measured. The subject of kind or nature of countable objects, i.e. what is counted and reasons they are counted doesn't enter mathematics. Math only sees the count and what you require to do with the counts, namely you squared x, then multiplied it by a. Was it money (finance), was it production output (economics) or speed of light and mass (physics) math does not care. It will just deal with counts and give you result back after the multiplication.

You can read more "How math can be applied to so many different fields and how we can use math in real life".

Here are more links you might like as well:

You stated theorems, or premises, or postulates, directly provable from ZFC axioms! From inside math it looks trivial. But, the actual selection, what postulate, assumption, premise, you chose, as your starting point, really matters a lot from the point of view of discipline that stipulated that formula. Axioms can not tell you which postulate will be of special interest to you. Axioms just serves to tell you what operations and concepts you have at your disposal. And it's not much. You have a concept of a set, and few operations on sets, and that's, essentially, it!

Once you see, note, distinguish, things that are common to many other concepts,or objects, perhaps some common property, there is a big chance that that property and relationships between those properties, can take off as a separate discipline, with its own axioms and postulates, premises, theorems (and I am not talking only about math here!). World#1, is the world that has those objects with common properties between them. These common properties and their relationships, can be abstracted from World#1 into separate (possibly and desirably axiomatic) system. Then, World#1 will dictate the genesis of postulates and premises in that abstracted (axiomatic) system, let's call it World#2.

The postulates in World#2 can be generated from two sources. From World#1, with all descriptions and explanations using World#1's language, or, from World#2 axioms! Example in math and physics. Math expression y = ax^2 can be obtained directly from fundamental axioms of mathematics, which just tell you that you can, you are allowed, to do that. Hence, you can just chose to generate a function y = ax^2, without any further explanation. But, then, in physics, y = ax^2 can be E = mc^2 which required Nobel Prize way of thinking which objects to put in place of "a" and "x" in the formula y = ax^2. Here in physics, it matters what is counted and not only the mathematical form.

Hence, y = ax^2 can come from physics, engineering, economics, finance, and, while in math it is only counts what matters, in all those disciplines matters what you actually counted, or measured. The subject of kind or nature of countable objects, i.e. what is counted and reasons they are counted doesn't enter mathematics. Math only sees the count and what you require to do with the counts, namely you squared x, then multiplied it by a. Was it money (finance), was it production output (economics) or speed of light and mass (physics) math does not care. It will just deal with counts and give you result back after the multiplication.

You can read more "How math can be applied to so many different fields and how we can use math in real life".

Here are more links you might like as well:

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