The Continuity Debate: Dedekind, Cantor, du Bois-Reymond, and Peirce on Continuity and Infinitesimals

This is not a "lay person" text. True, it is short and very readable, but the topics will be most relevant to undergraduate / first year graduate math students and especially those about to take real analysis. Having already taken a semester of real analysis, I think this text should be required reading, both in terms of providing historical context for the evolution of the number line and in terms of different ways prominent mathematicians have thought about the subject. It is thought provoking and recommendable. This book doesn't purport to apply modern analysis to historical perspectives and if you are already convinced of the invincibility of Dedekind before you read the first page, you'll probably give this book a 1-star review. IMO the author does a good job of counter-balancing the prevailing Dedekind perspective by arguing the merits of counter-perspectives, while providing light criticisms where an inconsistency is obvious. It is true that the author seems to have a partiality for infinitesimals with insufficient focus given to the topic, but such focus seems well outside the scope of this book. If anything, I'd suggest the author swap the "Infinitesimal Interlude" with a chapter on Abraham Robinson.

The Continuity Debate summarizes the subject of continuity in mathematics, in particular its relationship to the set of real numbers and geometry, and by extension its utility to calculus and other applications which utilize infinitesimals. It summarizes and provides commentary on the views of Dedekind, Cantor, du Bois-Reymond, and Peirce after having provided a more accessible foundation in Aristotle. Buckley (the author) is clearly partial to du Bois-Reymond's perspective on the continuity problem, as he explicitly supports it while demonstrating the holes in logic of the other three philosophers. Buckley provides a solid foundation for his support of du Bois-Reymond and makes it hard to disagree once you're done reading the book. Ultimately, infinitesimals in mathematics, and whether or not they actually exist (aside from as a mathematical tool) is still up for debate. We may never know the answer, as closely related problems such as Cantor's continuum hypothesis have been shown to be unsolvable using today's mathematical architecture. With this as a background, The Continuity Debate is an thought-provoking read from an epistemological perspective. I gave this book 4 stars because while it was entertaining and highly readable, it is definitely not for the average lay person. It seems more like supplemental reading for a college course in logic or number theory. I am not a mathematician but have a solid conceptual understanding of the issues involved in continuity. I've read many books that touch upon related issues in mathematics, such as Morris Kline's "The Loss of Certainty", Amir Azcel's "The Mystery of the Aleph" and William Dunham's "Journey Through Genius" - but have no formal training whatsoever. Reading these books prior to "The Continuity Debate" provided enough background to understand the work. Buckley uses terms such as `transfinite numbers' and `the continuum hypothesis' without spending adequate time to explain what these are; he assumes that anyone picking up the book is already familiar with these ideas (which may indeed be the case as this is a highly focused book in terms of subject matter). However, a true layperson wouldn't understand many of the terms without referencing other material. But if you already have a high-level understanding of what continuity means from a mathematical and philosophical/logical perspective, the book is comprehensible albeit challenging at times. All this being said, for me The Continuity Debate was still an excellent book. Buckley has taken upon a difficult task - explaining and providing insight on the issue of continuity in mathematics. He has accomplished this in a manner that is clear and concise, provided you are already familiar with the subject (and related subjects). I would recommend the book to anyone who is interested and has enough prior to knowledge to really appreciate the subject matter.

This is a must for anyone interested in Zeno of Elea, Aristotle's views on Zeno, or the modern development of the real number line.

Arrived very quickly! Completely satisfied!

The author of this book is a philosopher, not a mathematician, and it *shows*. E.g., he repeatedly claims that infinitesimals can be introduced as the multiplicative inverses of Cantor's infinite cardinals, but, in fact, for any infinite cardinal a, a*a = a, so if b were the putative multiplicative inverse of a, we would have 1 = a*b = a*a*b = a*1 = a, a contradiction. In fact, Cantor's infinite cardinals cannot be increased by any act of ordinary arithmetic involving the same or smaller cardinals, which shows the large difference between infinite cardinals and the 'infinities' of non-standard or pre-rigorous analysis.

Buckley's Book is a thorough historical review of the contributions of Dedekind, Cantor, du Bois-Reymond and Pierce on the empiricist/idealist divide — warfare — over continuity, infinitesimals, infinity, existence of the continuity of the set of reals, let alone the set itself… All leading to the "under-the-carpet" conclusion that they are undecidable problems remaining forever outside our possibilities, hence that we should keep on doing our standard calculations without settling the larger foundational issues… Quite valuable, even though Buckley is a philosopher and does not go as far as questioning the very "existence" of the reals… For that sensitive issue, I encourage the reader to follow Norman Wildberger's numerous and wonderful courses on youtube…