Despite my hesitancy to write a hasty review lest I misrepresent this book, the fact that no one has come forward to place a review on Amazon makes it clear to me that an inadequate review is better than no review at all. What makes the task of reviewing this book difficult, certainly not its clarity of style and organization nor the thoroughness with which the subject is examined, which are all impeccable, but rather the fact that it compares two major works of philosophy each of which demands careful and deep study. Indeed, even the translator of Jacob Klein's Greek Mathematical Thought and the Origin of Algebra says in the preface to Hopkins' study that she did not understand Klein's work until she read Hopkins' text. I have done at least a half dozen readings of Klein's book. I have yet to complete a single reading of Husserl's The Crisis. I have been led to Klein's work as a result of my own reading of the primary sources that Klein discusses in his analysis of what happened in mathematical history.
It has been said that if two people agree on everything, one becomes unnecessary. In my own study of Klein, I don't always agree but I always learn. Hopkins believes, as I do, that Klein's book is one of the most important philosophic works of the 20th century. I find Hopkins discussion of Klein's work completely engaging and thought provoking. His review of Husserl's work and his comparison of Klein and Husserl's understanding of the emergence of modern symbolic mathematics is equally engaging. I am not yet in a position to appreciate fully the part of Husserl's work in the entire argument since I have not devoted the necessary study.
What makes all these studies so important is the notion of desedimentation, which Hopkins discusses at length. There are far too many histories of mathematics and philosophies of mathematics that simply reiterate misleading cliches about Greek mathematics and its relationship to modern mathematics. Klein traces the Greek concept of each number having an eidos and how this concept changes through Greek mathematical history and how the eidetic concept was reborn and remolded in the writings of Vieta, Stevins, Descartes, and Wallis. Hopkins goes through Klein's book chapter by chapter in a way that will only help the reader of Klein's great work.
In my own research, I explore the idea of eidos from the Pythagoareans through Plato and Euclid and the Neo-Platonists, From my point of view and as a result of the research and creative work I am doing, I feel that Klein has overlooked what the full doctrine of the number system might have been. The way they discussed eidos differed from Aristotle's and those who came after is, I beleive, because they did mathematics differently and understood the number field differently from the modern conception. It is only by imagining a coherent theory of how eidetic math worked that the entire Greek mathematical enterprise can be fiarly evaluated and the differences between the Greek approach and the modern approach can be more completely appreciated. What is not true is that modern mathematics simply supplanted ancient mathematics.
I hope this review is sufficient enough to draw attention to Hopkins' book and give impetus to a new study of Greek mathematics as possessing lost wisdom that expands the entire vision of what mathematics is about.