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Mathematics

 

Review Essays of Academic, Professional & Technical Books in the Humanities & Sciences

 

Hierarchical and Geometrical Methods in Scientific Visualization edited by Bernd Hamann, Hans Hagen, Gerald E. Farin (Springer Verlag) This book emerged from a Department of Energy/National Science Foundation-sponsored workshop, held in Tahoe City, California, October 2000. About fifty invited participants presented state-of-the-art research on topics such as:

  • terrain modeling

  • multiresolution subdivision

  • wavelet-based scientific data compression

  • topology-based visualization

  • data structures, data organization and indexing schemes for scientific data visualization.

All invited papers were carefully refereed, resulting in this collection. The book will be of great interest to researchers, graduate students and professionals dealing with scientific visualization and its applications.

Abstract. Currently, large physics simulations produce 3D discretized field data whose individual isosurfaces, after conventional extraction processes, contain upwards of hundreds of millions of triangles. Detailed interactive viewing of these surfaces requires (a) powerful compression to minimize storage, and (b) fast viewdependent optimization of display triangulations to most effectively utilize highperformance graphics hardware. In this work, we introduce the first end-to-end multiresolution dataflow strategy that can effectively combine the top performing subdivision-surface wavelet compression and view-dependent optimization methods, thus increasing efficiency by several orders of magnitude over conventional processing pipelines. In addition to the general development and analysis of the dataflow, we present new algorithms at two steps in the pipeline that provide the "glue" that makes an integrated large-scale data visualization approach possible. A shrink-wrapping step converts highly detailed unstructured surfaces of arbitrary topology to the semi-structured meshes needed for wavelet compression. Remapping to triangle bintrees minimizes disturbing "pops" during realtime displaytriangulation optimization and provides effective selective-transmission compression for out-of-core and remote access to extremely large surfaces. Overall, this is the first effort to exploit semi-structured surface representations for a complete large-data visualization pipeline.

Beginning Functional Analysis by Karen Saxe (Undergraduate Texts in Mathematics: Springer) The unifying approach of functional analysis is to view functions as points in some abstract vector space and the differential and integral operators relating these points as linear transformations on these spaces. The author presents the basics of functional analysis with attention paid to both expository style and technical detail, while getting to interesting results as quickly as possible. The book is accessible to students who have completed first courses in linear algebra and real analysis. Topics are developed in their historical context, with accounts of the past‑including biographies‑appearing throughout the text. The book offers suggestions and references for further study, and many exercises.

Beginning Functional Analysis is designed as a text for a first course on functional analysis for advanced undergraduates or for beginning graduate students. It can be used in the undergraduate curriculum for an honors seminar, or for a "capstone" course. It can also be used for self‑study or independent study. The course prerequisites are few, but a certain degree of mathematical sophistication is required.

A reader must have had the equivalent of a first real analysis course, as might be taught using David Bressouds A Radical Approach to Real Analysis (Mathematical Association of America, 1994) or Walter Rudins Principles of Mathematical Analysis (McGraw‑Hill, 1953); and a first linear algebra course. Knowledge of the Lebesgue integral is not a prerequisite. Throughout the book we use elementary facts about the complex numbers; these are gathered in Appendix A. In one specific place (Section 5.3) we require a few properties of analytic functions. These are usually taught in the first half of an undergraduate complex analysis course. Because we want this book to be accessible to students who have not taken a course on complex function theory, a complete description of the needed results is given. However, we do not prove these results.

My primary goal was to write a book for students that would introduce them to the beautiful field of functional analysis. I wanted to write a succinct book that gets to interesting results in a minimal amount of time. I also wanted it to have the following features:

It can be read by students who have had only first courses in linear algebra and real analysis, and it ties together material from these two courses. In particular, it can be used to introduce material to undergraduates normally first seen in graduate courses.

Reading the book does not require familiarity with Lebesgue integration.

It contains information about the historical development of the material and biographical information of key developers of the theories. It contains many exercises, of varying difficulty. It includes ideas for individual student projects and presentations. What really makes this book different from many other excellent books on the subject are: The choice of topics. The level of the target audience. The ideas offered for student projects (as outlined in Chapter 6). The inclusion of biographical and historical information.

The organization of the book offers flexibility. I like to have my students present material in class. The material that they present ranges in difficulty from "short" exercises, to proofs of standard theorems, to introductions to subjects that lie outside the scope of the main body of such a course.

Chapters 1 through 5 serve as the core of the course. The first two chapters introduce metric spaces, normed spaces, and inner product spaces and their topology. The third chapter is on Lebesgue integration, motivated by probability theory. Aside from the material on probability, the Lebesgue theory offered here is only what is deemed necessary for its use in functional analysis. Fourier analysis in Hilbert space is the subject of the fourth chapter, which draws connections between the first two chapters and the third. The final chapter of this main body of the text introduces the reader to bounded linear operators acting on Banach spaces, Banach algebras, and spectral theory. It is my opinion that every course should end with material that truly challenges the students and leaves them asking more questions than perhaps can be answered. The last three sections of Chapter 5, as well as several sections of Chapter 6, are written with this view in mind. I realize the time constraints placed on such a course. In an effort to abbreviate the course, some material of Chapter 3 can be safely omitted. A good course can include only an outline of Chapter 3, and enough proofs and examples to give a flavor for measure theory.

Chapter 6 consists of seven independent sections. Each time that I have taught this course, I have had the students select topics that they will study individually and teach to the rest of the class. These sections serve as resources for these projects. Each section discusses a topic that is nonstandard in some way. For example, one section gives a proof of the classical Weierstrass approximation theorem and then gives a fairly recent (1980s) proof of Marshall Stone's generalization of Weierstrass's theorem. While there are several proofs of the Stone‑Weierstrass theorem, this is the first that does not depend on the classical result. In another section of this chapter, two arguments are given that no function can be continuous at each rational number and discontinuous at each irrational number. One is the usual Baire category argument; the other is a less well known and more elementary argument due to Volterra. Another section discusses the role of Hilbert spaces in quantum mechanics, with a focus on Heisenberg's uncertainty principle.

Appendices A and B are very short. They contain material that most students will know before they arrive in the course. However, occasionally, a student appears who has never worked with complex numbers, seen De Morgan's Laws, etc. I find it convenient to have this material in the book. I usually spend the first day or two on this material.

The biographies are very popular with my students. I assign each student one of these (or other) "key players" in the development of linear analysis. Then, at a subject‑appropriate time in the course, I have that one student give (orally) a short biography in class. They really enjoy this aspect of the course, and some end up reading (completely due to their own enthusiasm) a book like Constance Reid's Hilber

Infinity and the Mind by Rudy Rucker (Princeton University Press) leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Here Rucker acquaints us with Godel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations. Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Godel's incompleteness theorems. His personal encounters with Godel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism. In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Here Rucker acquaints us with Godel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations. Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Godel's incompleteness theorems. His personal encounters with Godel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism.

The Philosophy of Mathematics Today edited by Matthias Schirn ($140.00, Hardcover, Clarendon Press, Oxford University Press; ISBN: 0198236549) Through the twenty essays specifically written for this collection by leading figures in mathematics, this volume provides an overview of the more innovative areas of current research and theory in this dynamic field. The volume documents important approaches, tendencies, and results in current philosophy of mathematics in a way accessible to not only specialists but also intermediate students of mathematics. The topics include indeterminacy, logical consequence, mathematical methodology, abstraction, and both Hilbert's and Frege's foundational programs. The volume also suggests where fruitful impulses for future work are likely to develop. One feature these studies all share is their essential concerned with foundational issues. Historical questions and the practical functions of mathematical knowledge are down played in these essays.

The volume is divided into five parts: I. Ontology, Models, and Indeterminacy; II. Mathematics, Science, and Method; II. Finitism and Intuitionism; IV. Frege and the Foundations of Arithmetic; and V. Sets, Structure, and Abstraction. Classical positions in the philosophy of mathematics are brought into focus and are critically discussed in Parts III and IV and also, though to a lesser extent, in Part V. The philosophical legacy of Frege and Hilbert predominates in these parts. The collection will be an important source for research in the philosophy of mathematics for years to come.

Matthias Schirn is Professor of Philosophy at the University of Munich. He is the author of identitat und Synonymie (1975), and has books on Frege forthcoming in German and Spanish. He has edited Studien zu Frege/Studies on Frege (three vols., 1976).

Frege: Importance and Legacy (1997) (Perspectives in Analytical Philosophy, Bd 13) by Matthias Schirn (Editor) ($180.00, hardcover, Walter De Gruyter; ISBN: 3110150549) Review pending.

THE APPLICABILITY OF MATHEMATICS AS A PHILOSOPHICAL PROBLEM by Mark Steiner
($39.95, hardcover, 256 pages, Harvard University Press; ISBN: 067404097X)

This book analyzes the different ways mathematics is applicable in the physical sciences, and presents a startling thesis the success of mathematical physics appears to assign the human mind a special place in the cosmos.

Mark Steiner distinguishes among the semantic problems that arise from the use of mathematics in logical deduction; the metaphysical problems that arise from the alleged gap between mathematical objects and the physical world; the descriptive problems that arise from the use of mathematics to describe nature; and the epistemological problems that arise from the use of mathematics to discover those very descriptions. The epistemological problems lead to the thesis about the mind. It is frequently claimed that the universe is indifferent to human goals and values, and therefore, Locke and Peirce, for example, doubted science’s ability to discover the laws governing the humanly unobservable. Steiner argues that, on the contrary, these laws were discovered, using manmade mathematical analogies, resulting in an anthropocentric picture of the universe as "user friendly" to human cognition, a challenge to the entrenched dogma of naturalism.

Mark Steiner is Professor of Philosophy at the Hebrew University of Jerusalem.

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