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The Continuity Debate: Dedekind, Cantor, du Bois-Reymond, and Peirce on Continuity and Infinitesimals, by Benjamin Lee Buckley
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The topic of this book is the historical struggle to define and defend a real number continuum which could do the work limit theory required of it. These definitions drew heavily on philosophical and foundational assumptions, and each raises numerous philosophical questions of its own. As we shall see, attempts to formulate a non-geometrical mathematical continuity raise questions such as: What is a number? What, in particular, is a real number? What is the true nature of continuity itself? Does a philosophically coherent definition of continuity logically commit us to infinitesimally small quantities? Is the concept of an infinitesimally small quantity even logically coherent? What is the relationship between this real number continuum and other well known continua, such as the geometrical straight line? The main question to be addressed, of course, is whether mathematical continuity exists at all.
- Sales Rank: #157653 in Books
- Published on: 2012-12-19
- Original language: English
- Number of items: 1
- Dimensions: 9.00" h x .41" w x 6.00" l, .55 pounds
- Binding: Paperback
- 182 pages
Most helpful customer reviews
23 of 23 people found the following review helpful.
Thought provoking and challenging
By BMean2Puki
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.
1 of 1 people found the following review helpful.
Don't waste your time
By Jonathan C. Cast
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.
1 of 9 people found the following review helpful.
Completely satisfied!
By Driftwin Books
Arrived very quickly! Completely satisfied!
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