I was introduced to transfinite theory as the logical basis of the Calculus. The whole reason that I started this mind numbingly dull task of studying transfinite theory was that I was interested in alternative methods for teaching calculus. My professors demanded that I address transfinite theory before doing my work in Calculus. What a waste of time?
Anyway, back to the subject.
Calculus is a logical model that is independent of the metaphysical beliefs of the person using the calculus. Transfinite theory is not the logical foundations of the calculus. The two primary issues in calculus are the slope of a line tangent to a curve and the area under the curve. The slope of the tangent line exists independent of the mechanism used to find it. Likewise, the area under the curve exists independent of the mechanism used to find it. I could use limits, tangents, fluxions, a good guess, or simple algebra to find these equations. The mechanism used to find the equation is not the same thing as the equation. The logical model itself stands on its own.
As transfinite theory includes a large number of suppositions about the nature of extremely large and extremely small spaces. Physics student's should avoid the theory like the plague. Were physicists holding to Cantor's view of the continuum or to Bolzano's dialectics, they would have never discovered Relativity Theory, nor would they have discovered quantum mechanics. Relativity Theory has its roots in the work of Reimman. Quantum Mechanics has its roots in discrete mathematics and probabilities theories. Even the new String Theory that might provide a method to tie in quantum mechanics and gravity has its ultimate root in equations discovered by Euler that preceed Cantor's transfinite theory.
The logical model of Calculus exists independently of our metaphysical ponderings. This does not eliminate all value from metaphysical ponderings, but I believe strongly that it is better for thinkers to understand the independence of ideas, than to have a metaphysics that tries to explain all ideas.
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