“The present book proposes three abstraction principles, one for natural numbers, one for the rational numbers, and one for the real numbers”
“The book adopts a unique approach to logicism, which it calls “natural logicism” due to its use of Gentzen-style natural deduction rules”
“The method of the book is described as logico-genetic theorizing because it aims to show why a rational being is entitled to their conception of the numbers and the assumption that such things exist without presuming a prior understanding of the topic”
“Neil Tennant is, to this reviewer’s knowledge, the only person who has articulated a very natural idea: that one might combine proof-theoretic semantics and logicism. The section on the natural numbers attempts to do so within the background of core logic, which should satisfy those committed to stronger logics as well”
“Tennant aims to present the rationals using pre-mathematical resources and the natural numbers constructed in the earlier section”
“The ingenious approach taken is to adopt mereology (a theory of part whole relations) as a background”
“division of objects is not primarily a mathematical issue but rather a question of logical possibility”
“The rationals are taken to be the abstracts of divisions of one group between another, which may require breaking members of the first group into their parts (hence the mereology). This approach captures the applicability of the rationals and their difference from the reals (dividing vs measuring).”
“The largest section of the book is devoted to developing and defending an approach to the reals”
“At the heart of Tennant’s strategy is a rejection of the arithmetization of the reals and a return to a more geometric approach”
“It is argued that the correct view of the reals is to see it as the measure of a dimension by some unit (think length, duration, weight, etc.)”
“a real number can be reached by an infinite sequence of divisions of the dimension measured”
“Tennant uses length as a paradigm”
“if we have a point on a line, we can find the real corresponding to it by splitting the line in half and giving the left half the label 0 and the right 1. By continuing this process, we end up with a binary expansion of the real number (e.g., 0.1001001010. . .). By considering all such infinite sequences, Tennant ensures we are talking about the uncountable entirety of the reals”
“The book emphasises the fact that the natural numbers are embedded in the rationals and the rationals in the reals”