Hence, the limitations of Norton's logic are not entirely due to the failure of additivity, nor to the fact that all infinite, co-infinite sets of outcomes have the same chance, but to a more fundamental problem: We have no well-motivated way of comparing disjoint countably infinite sets. Consequently, it yields better advice about choosing between sets of lottery tickets than Norton's, but it does not appear to be any more helpful for evaluating multiverse models. This logic satisfies both label independence and a comparative version of additivity as well as several other desirable properties, and it draws finer distinctions between events than Norton's. Here we define a purely comparative infinite lottery logic, where there are no primitive chances but only a relation of ‘at most as likely’ and its derivatives. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. However, his negative results depend on a certain reification of chance, consisting in the treatment of inductive support as the value of a function, a value not itself affected by relabeling. Download PDF Abstract:Abraham Robinsons framework for modern infinitesimals was developed half a century ago. This makes the logic useless for evaluating multiverse models based on self-locating chances, so Norton claims that we should despair of such attempts. It follows, Norton argues, that finite additivity fails, and any two sets of outcomes with the same cardinality and co-cardinality have the same chance. This is due to a requirement of label independence. The problem of the nature of infinitary notions is still of central importance in the philosophy of mathematics.Īs an application of his Material Theory of Induction, Norton (2018 manuscript) argues that the correct inductive logic for a fair infinite lottery, and also for evaluating eternal inflation multiverse models, is radically different from standard probability theory. It compares the infinitely small differences of finite quantities discovers the relations among these differences and introduces the relations among finite quantities, which are infinite as compared with the infinitely small quantities. Ordinary analysis deals with finite quantities and penetrates as far as infinity itself. At that point, infinitesimals were all but banished from the mathematical world. Some examples that suffice to give a hint on the way the differential and integral calculus can be developed within the framework of nonstandard analysis are also provided in the chapter. century mathematician Karl Weierstrass did Calculus receive a rigorous treatment, removing the references to infinitesimals. The basic ideas of nonstandard analysis are sketched in two sections––a comprehensive development of the theory and a more detailed discussion of the historical issues. The background of nonstandard analysis as a viable calculus of infinitesimals that provides a more precise assessment of certain historical theories is discussed in the chapter. This chapter describes and analyzes the interplay of philosophical and technical ideas during several significant phases in the development of the calculus.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |