To him it was indeed a ratio, i.e. the slope, of two infinitely small numbers.
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does the slope toward zero have the same curve as the slope toward infinity. or is "slope" in this instance merely an attempt to map a two dimensional figure (slope) onto, in turn, a one dimensional (zero) and four dimensions (infinity/eternity)
If they did have the same "slope" would that suggest that, say, a black hole is the inverse in "relationship" to whatever is "outside" the universe. What would inverse mean in this case.
I am no mathematician/physicist but as far as I can tell there is a very slushy boundary between what is countable--using the definitions for "countable" as set down in this thread--on the very large scale and what is countable on the very small scale. Things don't just cut off. Now your countable and now you're not.
Rather, things become less and less countable until gradually...infinity--or zero as the case may be.
But I'm not so sure that they become less countable in the same way. (But if they became less countable in the same way they might have the same slope:-)
You ask very intersting questions. Let me try to address the mathematical side of your questions as briefly as I can, but certainly not briefly enough.
Much of mathematics today uses the real number system in which all numbers are finite. This includes zero. However, there is a restiction on zero that the other real numbers don't have: division by zero is undefined. The reason is that division by zero is an infinite process.
Mathematicians who confine their considerations to the real number system have a symbol for infinity, but it is NOT a number. It is a process of applying operations like addition or multiplication or substraction or division forever. And since most mathematicians die before they reach forever, they just call this infinity and or say this is undefined and move on to the next equation.
Slope is a ratio of two numbers. Consider a right triangle. The hypotenuse which is the slanted side obviously has a slope. To calculate it divide the length of the vertical side of the triangle by the length of the horizontal side. The result of that division is the slope of the hypotenuse and it is constant for every point on the hypotenuse.
Now keep the vertical and horizontal legs of the triangle the same, but replace the slanted line for the hypotenuse with a piece of cooked speghetti with lots of curves. What is the slope of this piece of curvey spaghetti? It obviously changes for every point on the curved line that represents the spaghetti.
Back 350 years ago when Liebniz helped to invent the calculus, mathematicians were not resticted to the real number system. They also used numbers called infinitesimals which are so small that you can think of them as zero. To find the slope of the spaghetti curve, Leibniz shrank the vertical and horizontal sides left over from our original triangle until those sides became infinitesimals. So you can think of this as dividing zero by zero, i.e. 0/0. The amazing result was an equation that gave him a finite slope that varied for every point on the curved spaghetti. Wow!!!
About 100 years ago mathematicians became very uneasy about infinitesimals and dividing 0 by 0, so they threw out infinitesimals, and redefined calculus in terms of the concept of limits and real numbers only.
Now, when I first took calculus, I was lost. I didn't understand limits at all and found them confusing and cumbersome. I started studying the history of the development of calculus. Since Leibniz used infinitesimals, I began to think of calculus in terms of infinitesimals. A light bulb went on, and I began to solve calculus problems with ease. But I didn't dare to tell my instructor that I was secretly thinking about dividing infinitesimals instead of taking limits of real numbers.
My purpose for bring all this up was that betty was thinking about writing an essay about numbers and eternity. I thought I would offer more food for thought if she cares to chew on any of it. I will let the philosophers on this thread address your philisophical questions.