The S-K modeling math is in an area that’s been thought “mysterious”:
But we should be able to ask: “What are the arithmetical rules that connect measurable circumstances to abstract math?”
Here’s a question that is practical, but outside pure mathematics:
Here’s a case where the answer matters:
The question of which is right is NOT A QUESTION OF PURE MATHEMATICS.
Analogies between the measurable world and pure math are wonderfully useful, but the abstract “game” of mathematics is not the same as the measurable world.
What’s pure math?
numbers from axiom systems. Here are the Dedikind-Peano axioms commonly used to
the numbers mathematics use:
Pure math is so practically useful because it can form powerful, flexible analogies to physical things.
Math modeling is reasoning by analogy. We need good analogies.
People find physical laws by a process of exploration and refinement on the basis of experience. That’s been going on for thousands of years, and especially over the last three centuries.
Steve Kline and I decided to call these constants “natural law operators.”All are defined according to the same pattern:
Example: A resistance per
unit length determined for a specific wire for ONE specific length increment
and ONE specific current works for an INFINITE SET of other length increments
and currents on that
wire (holding temperature the same.)
Natural law operators work just like dimensional numbers when they are used in exact correspondence with the equation that defines them.
But we have practically no experience with compound natural law operators that contain spatial increments, and the experience we have has often been treacherous.
Wholes should equal the sums of which they consist.
To derive a differential equation from a physical model:
In such a case, two equations are implicitly defined in terms of each other.
When we go from our physical representation, set out as a measurable thing, to an abstract mathematical representation of it, we’re making a jump. Here we are, facing the question:
Our notation isn’t clear about when a symbol stands for a physical model in real detail, and when the notation is a system of “meaningless signs.”
When you continue the expansion, you get relations that set out PHYSICAL interactions, but are not yet abstract, in an equation representation like this:
Here’s one of the crosseffects. It stands for a physical interaction, and is notated so that the arithmetic within the curly bracket is left to be determined.
We’re perfecting an analogy-making procedure, beyond pure mathematics, so that we can represent things within pure math.
Here’s the new rule that passes loop tests:
You can look at the case in a more general way that turns out to rule out terms now called “infinities” that come from some modeling, as well as infinitesimals.
Are these statements true?
Here, “true” must mean:
I believe, and Steve Kline believed, that the logical consistency was there ACCORDING TO THE PATTERN MATCHING STANDARDS RELEVANT TO THE CASE.
Is this new concrete domain to abstract domain mapping procedure consistent with all the evidence?
Direct tests are fundamentally important, and I’m hoping to interest some of you in helping with such tests.
For now, I hope I’ve made you open-minded about this question: