A Gentle Introduction to the Art of Mathematics
Joseph Fields, Southern Connecticut State University
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Given that this course is not well established in the mathematics curriculum generally, it is not appropriate for this reviewer to comment read more
Given that this course is not well established in the mathematics curriculum generally, it is not appropriate for this reviewer to comment comprehensiveness. However, the topics that were selected by the author were covered in a comprehensive fashion.
Overall, the author is very accurate. There is one minor item that has to be corrected. It has to do with Exercise 9 of Section !.2. The Twin Prime Conjecture is no longer. Yitang Zhang, lecturer in mathematics at the University of New Hampshire, has proved it recently. The author may have written the textbook before such an event and I am certain he is aware of this but perhaps has not had the opportunity to make the correction.
The author made a strong case for content of this nature to be part of the main stream mathematics curriculum. The idea is indeed very relevant and the idea will prevail as all mathematics programs look for ways to close the wide gap that exists between the Calculus type courses and the proof based courses. As to the specific content that will decide the longevity of such an idea, the question remains to be decided. The author chose excellent topics, but those topics may not be the prime choice of most mathematics curricula.
The author writing style is excellent and goes in great lengths to provide adequate context for terminology. The historical annecdotes the author used are indeed very relevant, useful, and above all, very motivating.
I have to admit that I was very pleasantly surprised when I started reading this text. In the beginning, the author is very eloquent, precise, and yet somewhat informal. Being informal in the early stages is an excellent idea and of course justifies the gentle nature of its title. However, I continued my review and got further in this review process, I developed the sense that the author left that behind and adopted a different style in which he is in a constant dialog with the reader (the learner). This, in itself is a great idea, but there is somewhat of a break, intended and gradual perhaps, but nonetheless a break.
Within the framework of the author's vision for what he wanted to accomplish with students, and particularly the direct dialog, the result is often very long paragraphs. While the content of such long paragraphs is indeed very relevant and often fascinating, the concern is that students are typically not good readers of mathematics textbooks and this may be even more discouraging to them. It should also be mentionned that the text is in PDF format and perhaps, there are limitations for breaking those paragraphs and preserve the integrity of the discussion. Having said that, the text is certainly amenable to changes that would make the student more interested if not excited about reading it.
once again, given that a course of this nature is not established as part of the mathematics curriculum, the author selected great topics as another author with the same intentions and objective would have chosed others. The selected topics by the author are excellent in my view and are presented in a logical fashion.
This reviewer used a PC running Windows 7 and reviewed the text as a PDF file. Within these tools, this reviewer did not notice and problems pertaining to navigation or distortion.
This reviewer did not discern any grammatical problems. I believe the author is very thourough.
The author used historical annecdotes throughout the text. In doing so, I believe he approached with accuracy, integrity, and a sometimes with a sense of humor. This reviewer applaudes the author for including culturally relevant historical notes.
Overall, the author did an excellent job. One of his major objectives is for students to actually read the textbook. The good mathematics students will enjoy such a text. I believe however, that the average student will have difficulty using it and its valuable content. The number of exercises per section is too small. Of course this can remedied by adding more. No resources are being made available to students nor for instructors beyond the textbook.
Table of Contents
Chapter 1: Introduction and notation
Chapter 2: Logic and quantifiers
Chapter 3: Proof techniques I
Chapter 4: Sets
Chapter 5: Proof techniques II - Induction
Chapter 6: Relations and functions
Chapter 7: Proof techniques III - Combinatorics
Chapter 8: Cardinality
Chapter 9: Proof techniques IV - Magic
About the Book
This book is designed for the transition course between calculus and differential equations and the upper division mathematics courses with an emphasis on proof and abstraction. The book has been used by the author and several other faculty at Southern Connecticut State University. There are nine chapters and more than enough material for a semester course. Student reviews are favorable.
It is written in an informal, conversational style with a large number of interesting examples and exercises, so that a student learns to write proofs while working on engaging problems.
About the Contributors
Joseph E. Fields is a professor at Southern Connecticut State University. Research Interests: Combinatorics, coding theory, and discrete mathematics. Ph.D., University of Illinois at Chicago.