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The Tangled Origins of the Leibnizian Calculus: A Case Study of a Mathematical Revolution, by Richard C Brown
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This book is a detailed study of Gottfried Wilhelm Leibniz's creation of calculus from 1673 to the 1680s. We examine and analyze the mathematics in several of his early manuscripts as well as various articles published in the Acta Eruditorum. It studies some of the other lesser known calculi Leibniz created such as the Analysis Situs, delves into aspects of his logic, and gives an overview of his efforts to construct a Universal Characteristic, a goal that has its distant origin in the Ars Magna of the 13th century Catalan philosopher Raymond Llull, whose work enjoyed a renewed popularity in the century and a half prior to Leibniz.
This book also touches upon a new look at the priority controversy with Newton and a Kuhnian interpretation of the nature of mathematical change. This book may be the only integrated treatment based on recent research and should be a thought-provoking contribution to the history of mathematics for scholars and students, interested in either Leibniz's mathematical achievement or general issues in the field.
- Sales Rank: #3584104 in Books
- Brand: Brand: World Scientific Publishing Company
- Published on: 2012-05-26
- Original language: English
- Number of items: 1
- Dimensions: 9.00" h x .75" w x 6.00" l, 1.30 pounds
- Binding: Hardcover
- 312 pages
- Used Book in Good Condition
Most helpful customer reviews
7 of 7 people found the following review helpful.
Incoherent
By Viktor Blasjo
This chaotic and rambling book does not rise to the level of coherent scholarship. It reads as if the author simply recorded facts as he came across them and reflections as they occurred to him, and christened their totality a book when they reached sufficient bulk. This means that the book may contain some remark of value here or there but that it is on the whole unreliable, as the author has not conducted systematic research on the topics he is expounding.
For instance, Brown claims that 17th century research "often demonstrated that what some ancients had considered 'line-like' problems like angle trisection ... were really 'solid'" (p. 20), a claim which is backed up by a footnote saying that "Pappus had used the quadratrix ... to trisect ... an angle" (p. 20). This is highly misleading to say the least since Pappus explicitly used the trisection of the angle as an example of a solid problem when motivating his classification in Book IV. In fact, Brown himself gives the demonstration in an appendix: "Here is one of two solutions given by Pappus to trisect an angle using a hyperbola" (p. 271). It is a sign of how disjointed and poorly researched the book is that Brown does not even notice that he is blatantly contradicting himself.
Brown also makes quite a fuss about some relatively harmless ways in which Leibniz's calculus is different from ours, e.g., in being based on curves and their differentials rather than functions and their derivatives. But unfortunately his own analyses are inconsistent with his posturing. For example, when Leibniz states the integral representation of the arcsine, Brown proposes to "verify" it "by what was probably his method" (p. 142). Brown's method is to differentiate sin(arcsin(x)), which is a prototypical instance of the exact opposite of the essence of the Leibnizian calculus in that it is based on functions, inverse functions, and abstract derivatives. Much more likely, and in line with Brown's own characterisation of Leibniz's calculus, is that Leibniz found the result as an arc-length integral, which reflects directly the geometrical meaning of the arcsine and does not involve any mysterious alchemy of functions a la Euler.
This book also proves once again that wherever the revenues of despicably overpriced books such as this are being spent, it is certainly not on proofreading. There are superfluous words ("by from which", p. 144), missing words ("it will always the case", p. 149), double commas (p. 45), and periods at the end of questions (p. 147) but often not at the end of sentences that end with a footnote (e.g. pp. 7, 24, 110, 232).
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