I addressed the OP in my post # 16 ... it seems this paper is beyond your ability to understand ... no, I'm not going to explain it to you, you'll need couple years in a college math classes before I'll even try ...
What you want is some Middle School explanation ... and such doesn't exist ... must suck to be you ...
I saw post #16. My previous response was sarcasm. Let me be more responsive this time.
It's a mathematical treatment on selection. You seem to be under the false impression that creationists and/or ID theorists deny selection and the speciation thereof. Wherever did you get that silly notion?
Precisely how does the treatment prove that evolution is necessarily true, let alone that naturalism is true?
It. Doesn't. Do. That. Does. It?
And don't give me any crap about my alleged lack of understanding when you can't/won't explain the thrust of the treatment or its relevance to my observation, most especially given the fact that Haldane's calculi are inconclusively mixed regarding the transmutation rate in the past.
As for math, do you mean like the college courses in mathematics you need to take, apparently, in order to grasp the real world ramifications of infinitesimals and the concept of infinity per
my mathematical treatment in the other thread? That was significantly less complex than the math in your citation. Hell, I even explained it to you in detail, in the simplest terms possible, and you still didn't get the thrust of it.
Maybe you were ill that day. Give it another try.
The most straightforward mathematical illustration of the existential impossibility of an actually infinite regress in nature would entail a limit function of systematic division. Excerpt from my article:
But, once again, what do we do with any given integer divided by Infinity? The quotient would obviously not equal ±∞. Nor would it equal 0. If we were to divide ±1 by ∞, for example, and say that the quotient were 0, then what happened to ±1? Calculus entails the analysis of algebraic expressions in terms of limits, so in calculus the expression n ÷ ∞ = 0 doesn't mean the quotient literally equals 0. Rather, 0 is the value to which the quotient converges (or approaches). Again, Infinity is a concept, not a number. We can approach Infinity if we count higher and higher, but we can't ever actually reach it. Though not an indeterminate form proper, n ÷ ∞, like any other calculation with Infinity, is technically undefined. Notwithstanding, we intuitively understand that ±1 ÷ ∞ equals an infinitesimally small positive or negative number. Hence, we could intuitively say that ±1 ÷ ∞ = ±0.000 . . . 1, and we would be correct.
For the proof, let the input variable = x, and let the integer = 1:
x | 1 ÷ x |
1 | 1 |
2 | 0.5 |
4 | 0.25 |
10 | 0.1 |
100 | 0.01 |
1,000 | 0.001 |
10,000 | 0.0001 |
100,000 | 0.00001 |
1,000,000 . . . | 0.000001 . . . |
Note that as x gets larger and larger, approaching Infinity, 1 ÷ x gets smaller and smaller, approaching 0. The latter is the limit, and because we can't get a final value for 1 ÷ ∞, the limit of 1 ÷ x as x approaches Infinity is as close to any definitive value as we're going to get. The limit of a function in calculus tells us what value the function approaches as the x of the function (or, in shorthand, the x of the f ) approaches a certain value:
lim f(x)
x→a
We know that we're proving the limit for 1 ÷ ∞; hence, the following reads "the limit of the function f(x) is 1 ÷ x as x approaches Infinity":
f(x) = lim 1 ÷ x
x→∞
Additionally, the output values of function f depend on the input values for the variable x. In the expression f(x), f is the name of the function and (x) denotes that x is the variable of the function. The function itself is "the limit of 1 ÷ x as the inputs for x approach Infinity." When we solve for the limit of more than one function in an algebraic combination, we typically call the first of the functions f for "function." It really doesn't matter what we call any of them as long as we distinguish them from one another. The names given to the others typically follow f in alphabetical order merely as a matter of aesthetics: g, h, i, j and so on.
Hence, as we can see from the table above, the function proves out that the limit of 1 ÷ x as x approaches Infinity equals 0. That is to say, as x approaches Infinity, 1 ÷ x approaches 0:
lim 1 ÷ x = 0
x→∞
Altogether then:
lim f(x) =
x→a
lim 1 ÷ x = 0 (i.e., 0.000 . . . 1)
x→∞
x | 1 ÷ x |
1 | 1 |
2 | 0.5 |
4 | 0.25 |
10 | 0.1 |
100 | 0.01 |
1,000 | 0.001 |
10,000 | 0.0001 |
100,000 | 0.00001 |
1,000,000 . . . | 0.000001 . . . |
In nature
t = 0 is never reached via an infinite regress into the past. Hence, an infinite regression can never be traversed to the present.