"To introduce rigorous mathematics, I believe it's essential to discuss the whys and establish a core relation between mathematics and application. Mathematics begins with quantification, a necessary and humanely natural way to organize what we experience into numerical abstractions. We expand from just pure quantities to correlate to the higher complexities we experience. What we derive and conceptualize mathematically is how our mind understands the universe from the expansion of quantities. If we try to correlate quantities to the events (cause and effects), then it begs the question, "is the quantitive outcome of an event subject to a universal governance?" Most outcomes seem to follow a governing system, and others don't. It distinguishes two possible quantitative outcomes; the random and the controlled. Events with apparent factors that dictate their outcome are controlled. The proper way to look at it is as a description and a relationship. Mathematics has granted us the ability to describe these outcomes and reveal the governing system of these quantities. Accordingly, we establish a system of laws, quantities, relations, operations, etc. We derive all these tools from universal relationships and events that surround quantification. We begin with relationships; the most common and functional is the relation of equality "=" which relates to equivalence. Equations will serve as a whole tie between outcome and description by equivalence. Quick note, it's conventional to format an equation, so the outcome is secluded in the equation.
If we want to be general and incorporate the lack of quantitative information, we must rely on variables. We then encounter two types of equations; an equation with a fixed outcome or one with a range of outcomes. If an outcome is a constant, then it's fixed in all cases. This constrains variables. If an outcome is a variable, then its quantity ranges with respect to inputs in the description. This creates dependency. Take this example, the daily salary of an average person is dependent on the number of hours times a rate of pay. If the outcome is the daily salary, then it's dependent on a quantity of time. If the salary is the outcome then the example is this simple equation $S = pt$; if not then various other outcomes can be considered. The point is that variables are simply unknowns unless related information constrains their possibility to a definite value or set."
Is this take good? Does it set the basic relationship between mathematics and the world of application?