Epistemology of Fluxions: Volume 2

Higher mathematics does not exist without infinitesimals and the limit theorem. It is the fundamental for real and complex analysis. Clearly, integral and differential calculus would remain dormant without them. In an odd twist, we can think of integral and differential calculus as a “glorified algebra” for removing the use of infinitesimals and the limit theorem we are left with high school algebra. So, how does one work with numbers that cannot be measured (with current technology, or, for that matter any finite technology, touch, see, feel, hear let alone imagine such numbers. Conjure in your mind the smallest number that you think is possible. Make no mistake we can find a number even smaller. On the computational side current computer technology is able to work with only a subset of the reals R. There is no computer that can compute the last digit of p or 2. Obviously, it would take to infinity to find the last digits of these numbers. Unfortunately, the “infinite computer” is only available to God who is the only One that can see the last digits of both numbers. The fact is, there will always be numbers that are too small to be managed and, therefore, accessible only in our mind. Then, how is a structure derived that uses the “invisible” yet providing correct results? This is the very essence of the study of infinitesimals. A close look at the “derivative at a point” is based on a contradiction. It takes two distinct points to define slope yet with infinitesimals we can let the two point infinitely approach each other so closely that they are indistinguishable and, therefore, define the derivative at a single point. Hence, the distance between the two infinitely close points cannot be measured and will never affect an outcome resulting from the derivative computation.