The passage:

"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?

  • Your take on mathematics and its applications seems too narrow to me. Even at the origin, it was not only quantification but also measurement, magnitudes in addition to quantities, that drove it, and modern mathematics (including applied) moved far beyond quantities, variables and equations, see What makes something mathematics? For example, qualitative analysis and complex algebraic and geometric structures find applications from physics and biology to linguistics and AI.
    – Conifold
    Nov 12, 2021 at 21:39
  • Quantification is the expression of measurement. It's the start of mathematics and as we begin to purely deal with them they move beyond just quantities and so they expand. Nov 12, 2021 at 22:38
  • Well, Hardy said that a branch of math was beautiful in direct proportion to its uselessness. And that by that standard, his own specialty of number theory was the most beautiful of all, being completely useless. What would he say if he came back to find that number theory was the basis of Internet commerce and military-grade encryption?
    – user4894
    Nov 13, 2021 at 1:46
  • 1
    Ancient Greeks distinguished quantities (from counting) and magnitudes (from measurement), the two are quite distinct even from modern point of view. But neither of them covers modern mathematics of structures, as expressed in algebra, geometry and topology especially. They do not "quantify" even in your expanded sense, they formalize and relate.
    – Conifold
    Nov 13, 2021 at 11:33
  • 1
    @Rusi-packing-up See Grattan-Guinness, Numbers, Magnitudes, Ratios, and Proportions in Euclid’s Elements on the original conceptions, and Malet, Renaissance notions of number and magnitude on their subsequent reception.
    – Conifold
    Nov 14, 2021 at 6:57


You must log in to answer this question.