How to develop a proper theory of infinitesimals I | Famous Math Problems 22a | N J Wildberger

Infinitesimals have been contentious ingredients in quadrature and calculus for thousands of years. Our definition of the term starts with the Wikipedia entry, modified a bit to reduce the dependence on "real numbers", which is actually quite unnecessary--- but as a logical definition it is still clearly unsatisfactory. A quantity which is positive and non-zero but smaller than any other strictly positive rational number: does this make any sense??

Is there a modern way to establish these mysterious quantities without resorting to philosophical or logical hand-waving? Yes there is, and it involves yet another application of the remarkable Dihedron algebra that we introduced in the previous Famous Math Problem 21 on the true complex numbers.

This first video sets the stage, reviewing in some details Archimedes' approach to the quadrature of the parabola using The Method of infinitesimal balancing based on his Principle of the Lever. Then we move to the 16th century with the work of Cavalieri and the Leibniz with the foundations of Calculus. And then to the 1960's with the introduction of non-standard analysis of Laugwitz and Robinson.

Our approach is based on the dual complex numbers, originally introduced by Clifford in the 1870's.

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