When I was in
graduate school, it sometimes seemed as if one couldn't refer to motives
without using the phrase "Grothendieck's conjectural theory of motives." As
Yves André's Une Introduction Aux Motifs shows, things have
changed. Grothendieck's vision, 40 years ago, was that the arithmetic properties of algebraic varieties suggested the existence of some sort of underlying object, which he termed a "motive" (or, perhaps, a "motif", as in music or literature). The study of cohomological properties of varieties suggested common themes, motifs that ran through the theory (for example, the notion of "weight"). At the same time, common properties of very different cohomological theories seemed to call for an explanation, a motive. At first this was only a conjecture, but over the last fifteen years the theory has taken shape and become an important part of Arithmetical Algebraic Geometry.
Grothendieck's punning choice of name seems to have inspired his followers. There are "pure motives" and "mixed motives". I understand that someone once wrote a paper with the sole purpose of defining "ulterior motives".
André's book is an introduction to the theory aimed at "non-specialists" (meaning roughly, I think, specialists in arithmetical algebraic geometry who don't yet know about motives). Its first two sections discuss pure motives and mixed motives, respectively, and the third section puts the theory to work in an interesting way by studying periods of motives and their connection to polyzeta functions. Each of the first two parts is preceded by a motivational chapter that, though still not easy, tries to show why the theory to be developed is necessary. All in all, a valuable contribution to the literature. [Fernando Q. Gouvêa]
For
many years, Green Lion Press has been a reliable source of serious books in
the history of science and mathematics, including many excellent
translations of crucial texts. For example, their edition of Euclid, noted
here some time ago, is beautifully done. The Green Lion now has a separate
imprint, Green Cat Books, that seems to be dedicated to publishing more
accessible versions of the books that interest the Lion.
The two books under review, Selections from Kepler's Astronomia Nova and Faraday's Experimental Researches on Electricity, contain extracts from some of the most important texts in the history of science. Also in the series is a similar book with extracts from Newton's Principia, to be reviewed soon in Read This!.
All three of these books bear the rubric A Science Classics Module for Humanities Studies. The idea, as described in "The Green Lion's Preface", is to present "study modules designed to bring fundamental works of science and mathematics within the grasp of students and other readers without the need for specialized preparation. The series reflects the Green Lion's conviction that scientific and mathematical inquiry, unquestionably human activities, are not to be walled off from humanities studies but are, on the contrary, integral to them."
I agree completely with the Lion on
this, though I suspect either of these books would be hard going for the
typical student at my school. Faraday would probably be the more accessible
of the books, though in my mind Kepler is more interesting. The hardest
part of the job would be to convince students that the work needed to read
and understand these texts will indeed deepen their understanding of the
world and the culture in which they live.
Each book contains a brief biography of the author, an introduction, and then selections from the work in question. In Kepler's case, we get annotated selections from chapters 1, 2, 7, 24, 32, 33, 34, 39, 40, 44, and 57 of the Astronomia Nova. In these chapters one finds Kepler's description of the solar system, including arguments for the theses that "the power that moves the planets is the sun" (chapter 33) and "the orbit is not a circle" (chapter 44). The notes are very helpful.
In the Faraday book, the editor provides a long introduction explaining various experimental devices used to study electricity. Then follows the first series of lectures on electricity presented by Faraday in 1831.
In both cases, students are offered the chance to see how ideas that today are treated as "obvious" came to be established. If one can make students see the drama involved in setting out into the unknown and returning with actual knowledge, it may well be possible to motivate humanities students to work through these texts. I hope so! [Fernando Q. Gouvêa]
Here is
cause for celebration. Steve Russ and Oxford University Press have given us
an English edition of (a selection of) The Mathematical Works of Bernard
Bolzano. What else is there to say but "thank you"?
Despite his fame as one of the first people to really think about the foundations of analysis, much of Bolzano's mathematical works had not appeared in translation until now. Russ's volume contains all of the mathematical works published during Bolzano's lifetime, and also a couple of works that were prepared for publication but not published until much later. It does not include Bolzano's extesive mathematical diary or his work on logic.
Bolzano was a Christian humanist philosopher whose goal, in a sense, was to reorganize and reframe all of human knowledge. His collected works, which are in the process of being published, will include 120 volumes. Bolzano's mathematical work, then, was just a small portion of a much larger project. Russ emphasizes this in his introduction, expressing the hope that his translations may encourage students (and their advisors) to dig into this vast material in order to come to a better understanding of Bolzano's thought.
The works included here are divided into three groups, each of which has a separate introduction by the editor. The first, on "Geometry and Foundations", includes Considerations on Some Objects of Elementary Geometry and Contributions to a Better-Grounded Presentation of Mathematics. These titles already highlight Bolzano's typical concerns: how ideas and objects are to be represented, how knowledge is to be discovered, organized, and presented. The second section, "Early Analysis", contains texts on the binomial theorem, on the intermediate value theorem, and on integration. The final section, "Later Analysis and the Infinite", includes Bolzano's theory of the real numbers, work on functions, and his famous Paradoxes of the Infinite.
The conclusion is clear: no self-respecting library will be without this book. History fanatics may want their own copies. And let's hope that Russ's expectation that his book will stimulate further work will be fulfilled. [Fernando Q. Gouvêa]
What
happens when you publish an author's collected works while he is still
active? Why, you have to publish additional volumes! Thus, here is volume
six of Sir Michael Atiyah's Collected Works.
The original five-volume Collected Works was published by Oxford in 1988. In the sixteen years that followed, Atiyah continued to write and publish. He also received the Abel prize in 2004. That seems to have been the added stymulus needed for the appearance of a sixth volume. (And, of course, there may yet be a seventh!)
In his preface, Atiyah says that
Inevitably, with increasing administrative roles and with advancing years, a greater proportion of my papers have been surveys of one kind or another, rather than technical mathematical papers. Such surveys (...) tend to cover similar ground and are in danger of being repetitive. For this reason not all my papers, for the period 1988-2004, are reprinted here. I have been somewhat selective and only included papers which contain new results or at least a new perspective.
This volume has 1030 pages. One wonders at what might have happened if all the papers had been included!
Given the expository nature of many of these papers, this volume may actually be of more interest to some libraries than the first five were. But most good libraries will already have the first five volumes, and will jump at the chance of getting the sixth. [Fernando Q. Gouvêa]
Since
2001 marked the centennial of the publication of Lebesgue's original note
describing his integral, a celebratory conference was held at the
École Normale Superieure de Lyon. This book is the result, and it's
definitely worth looking at.
The first two essays in the book are historical. Gustave Choquet describes the careers and thought of Borel, Baire, and Lebesgue, who were working on similar topics at about the same time. Jean-Pierre Kahane tells the story of the impact of Lebesgue's integral on 20th century mathematics, providing a link between Choquet's article and those that follow.
The last four essays are expository accounts of more recent work. Pierre de la Harpe writes about the Banach-Tarski paradox and related results, then moves on to finitely additive measures. Bruno Sévennec covers invariant measures on compact groups and results about equidistribution. Thierry de Pauw gives us a tour of conditionally convergent integrals (such as the Henstock integral) and relates them to generalizations of the divergence theorem. Hervé Pajot talks about rectifiability and the "geometric traveling salesman problem".
The first two (maybe even the first three) chapters will be of interest to a wide range of mathematicians, particularly those interested in the history of the calculus. This centennial celebration is one in which I am glad to take part, even if passively. [Fernando Q. Gouvêa]
Talk
about "last but not least"! Judy Grabiner's The Origins of Cauchy's
Rigorous Calculus is back in print, and at a friendly price. This is a
classic study of the rigorization of analysis that is a "must have" for
anyone who is serious about understanding the history of the subject.
Grabiner's book, which was originally published by MIT Press in 1981, is well-described by its title. Though the first chapter argues for Cauchy's central role in creating a rigorous calculus, the focus is really on "origins." In other words, Grabiner's question is about the context of Cauchy's work, the authors from which he learned and with which he interacted, and the early development of Cauchy's thought: "Since no great work arises in a vacuum, what, in the thought of his predecessors, made Cauchy's achievement possible?"
It's an interesting historical question, but it also has implications for how we teach calculus. Here's how Grabiner herself puts it:
The history of the foundations of the calculus provides the real motivation for the basic ideas, and also helps us to see which ideas were — and thus are — really hard.
In order to keep the book accessible to those mathematicians and teachers who might not be specialists in history, Grabiner has been careful to cite her sources in translation. She even provides her own translation of a few key passages in an appendix.
Like most Dover reprints, this is an unchanged photographic reproduction of the original edition. As such, it preserves the original's somewhat strange page layout, with narrow columns of text and section headings in the margins. (But always in the left margin, so sometimes in the inside margin and sometimes in the outside margin, which seems particularly strange.) So ignore the layout, and enjoy the book. [Fernando Q. Gouvêa]
Publication Data
Une
Introduction aux Motifs: Motifs Purs, Motifs Mixtes, Périodes,
by Yves André. Panoramas et Syntèses,
17. Société Mathématique de France, 2004. Distributed
in the U.S. by the American Mathematical Society. Paperback, 261 pp.,
$37.00. ISBN 2-85629-164-3.
Selections from Kepler's Astronomia Nova, selected, translated, and annotated by William H. Donahue. A Science Classics Module for Humanities Studies. Green Cat Books, 2005. Softcover, 110 pp., $9.95. ISBN 1-888009-28-4.
Faraday's Experimental Researches in Electricity: The First Series, by Howard J. Fisher. A Science Classics Module for Humanities Studies. Green Cat Books, 2004. Softcover, 84 pp., $8.95. ISBN 1-888009-27-6.
The Mathematical Works of Bernard Bolzano, by Steve Russ. Oxford University Press, 2005. Hardcover, xxx + 698 pp., $229.50. ISBN 0-19-853930-4.
Michael Atiyah: Collected Works, volume 6, by Michael Atiyah. Oxford University Press, 2004. Hardcover, xxv + 1030 pp., $194.50. ISBN 0-19-853099-4.
Autour du Centenaire Lebesgue, by Gustave Choquet, Thierry De Pauw, Pierre de la Harpe, Jean-Pierre Kahane, Hervé Pajot, and Bruno Sévennec. Panoramas et Syntèses, 18. Société Mathématique de France, 2004. Distributed in the U.S. by the American Mathematical Society. Paperback, 156 pp., U.S. price unknown. ISBN 2-85629-170-8.
The Origins of Cauchy's Rigorous Calculus, by Judith V. Grabiner. Dover Publications, 2005. (Original publication: MIT Press, 1981.) Paperback, 251 pp., $16.95. ISBN 0-486-43815-5.
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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouv&ecric;a, Dept. of Mathematics, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.
MAA Online is edited by Fernando Q. Gouvêa (fqgouvea@colby.edu). Last modified: Thu May 26 14:04:16 Eastern Daylight Time 2005