1 Introduction 2 The Limits of Classical Probability Theory 2.1 Classical probability functions 2.2 Limitations 2.3 Infinitesimals to the rescue? 3 NAP Theory 3.1 First four axioms of NAP 3.2 Continuity and conditional probability 3.3 The final axiom of NAP 3.4 Infinite sums 3.5 Definition of NAP functions via infinite sums 3.6 Relation to numerosity theory 4 Objections and Replies 4.1 Cantor and the Archimedean property 4.2 Ticket missing from an infinite lottery 4.3 Williamson's infinite sequence of coin tosses 4.4 Point sets on a circle 4.5 Easwaran and Pruss 5 Dividends 5.1 Measure and utility 5.2 Regularity and uniformity 5.3 Credence and chance 5.4 Conditional probability 6 General Considerations 6.1 Non-uniqueness 6.2 Invariance Appendix. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general. Another way of looking at your formula is. There are several different ways to develop a rigorous theory that includes infinitesimal numbers. However, the basic intuitions of calculus since its beginnings have dealt with infinitely small (and sometimes also infinitely large) numbers.
the whole thing is notation for the derivative. In the ordinary analysis of real numbers, the only infinitesimal number is zero. Today d y d x is not a quotient but is notation for the limit after the limit has been applied, i.e. Non-Archimedean probability functions allow us to combine regularity with perfect additivity. He still thought of d y d x as a quotient with d y & d x being very small.
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