Proofs and Concepts: The Fundamentals of Abstract Mathematics
Dave Morris, University of Lethbridge
Joy Morris, University of Lethbridge
Pub Date: 2013
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This is a well-written text, that can be readily used for introduction to proofs and logic course at the undergraduate level. The text covers topics read more
This is a well-written text, that can be readily used for introduction to proofs and logic course at the undergraduate level. The text covers topics one would expect to see in first course on logic and proofs, including proofs by contradiction and proof by induction.
The content is accurate, error-free, and unbiased.
The examples used range from mathematical and non-mathematical examples. The text makes use of very few "modern" examples that would need to be updated in terms of its cultural significance. It should be able to be used for a long period of time.
The book is very clear. Within each chapter, important ideas are highlighted for the reader, and diagrams support the text throughout. There are sufficient examples (without overdoing the points the authors are trying to make), and interesting games / activities to support the reader with coming to their own conclusions about ideas prior to being introduced formal notions.
The text is internally consistent in terms of its structure and in building a logical system from its foundations.
The first few parts (1 and 2) seem mostly necessary to cover in order to get into the third part, which covers functions, equivalence relations, proof by induction, and cardinality. This last grouping of topics (chapters 6-9) seem very modular if the first few chapters are well understood.
The organization was excellent.
There were no issues related to interface. The text made use of several diagrams that supported the examples effectively.
There were no grammatical errors that I noticed in the text.
I am not sure how to comment on the cultural relevance of the text. I did not find anything that was potentially problematic for implementing the text.
The only area that I felt may have needed further explanation was when the empty set was introduced. The idea of the necessity of the empty set could have been better motivated as this mathematical object is not always seen as intuitive for students. Besides that one very minor topic, I felt that overall the text was well-written and would be easy to use for a course on logic and proofs for students. As many students may find a course on logic new and challenging, the introductory chapters eases students in to the course motivating the need for proof and deduction through engaging problems and interesting examples.
The book gives a beautiful, complete, and careful exposition of its central material -- logic and proofs -- and of several beautiful and powerful read more
The book gives a beautiful, complete, and careful exposition of its central material -- logic and proofs -- and of several beautiful and powerful applications in different areas of mathematics. The methods of logic taught here are so central to how mathematics is done, it would be easy to add either many, many more applications (pretty much all of mathematics!) or to go deeper into more advanced topics in logic. However, for a one semester course, this is a solid introduction to the core material and a nice set of applications thereof. It could add to the students experience if there were more complete and informative indices and glossary. However the book does have a nice Summary at the end of each chapter naming all of the topics covered in the chapter -- students could use this by, for example, making review sheets which simply explain in their own words each of the topics mentioned in the Summary.
There do not seem to be any errors or typos in this book, even in the most intricate of formal logical manipulation it contains.
Obsolescence is not really an issue with this material. The approach to the foundations of mathematics as presented in this book have been the mainstream within the mathematical community for around 100 years -- although some of the beautiful applications it presents are older or younger -- and that does not seem likely to change in the foreseeable future.
This is a brilliantly clear and lucid text. It has clear, precise, and complete explanations of every idea, example, and technical detail. Where appropriate (which is: in parallel to the introduction of some basic ideas of logic, proof, sets, functions, etc.), it gives some non-mathematical ("real world") examples of objects and reasoning to help make the material more intuitive and less intimidating for beginners. Yet the book is clearly written from the perspective of the practicing mathematician. This is its greatest strength, a truly rare and precious thing in today's undergraduate mathematics textbooks. If you want to bring your students many steps towards thinking the way actual mathematicians think, then every definition, example, and proof in this book will help; if you want merely to check off some box "students can mechanically produce blocks of text which are something like 'proofs'," then this is the wrong book. Unlike the humanities, where often students learn to produce their own work while looking at great examples of similar works, mathematics textbooks often are written in some strange hyped-up dialect which is neither common English nor is it at all well written mathematical text that the students can use as a model for their own work. This book is a wonderful exception: students who learn from it, will constantly have before their eyes a solid example of good mathematical writing that they can try to emulate.
Absolutely consistent and clear in terminology, notation, and presentation. This is not to say that everything is explained in exactly one way: there is a great feature in this book, particularly in the first half heavy on logic and proofs, of explaining many ideas both in quite formal, mathematical ways and also in clear and precise but more natural-language sounding ways. That pedagogical strategy is one of the strengths of this book.
The text is about as modular as it is possible to imagine for this material -- so that, e.g., it is hard to imagine doing first order logic without doing propositional logic first, so some version of Chapter 1 has to be done *before* Chapter 4. Other subjects could possible be taken out just for independent use, if an instructor wanted a clean and basic introduction to set theory, for example, or functions, or cardinality -- each of those could stand fairly well on their own. Reorganization within chapters is probably also hard to do, because of the material, but this text would be amenable to such use as much the material permits.
The book is beautifully organized, with a very natural progression of elementary topics leading naturally into greater and greater sophistication and more and more engaging and powerful applications. It is always crystal clear why each idea, term, and technique was presented in the place where it was.
The hyperlinked PDF seems to have the occasional problem with a link (such as, in a proof, to an earlier result or definition) going to the wrong place. This issue is completely invisible if one reads the book in static form, say on paper, but means that it doesn't quite take as much advantage of the potentials for hyperlinking as would be possible.
No grammatical errors at all. Many sentences are written in beautiful mathematical style, which is a dialect all of its own, but they are (as just mentioned) beautiful, and many kind translations into a more colloquial style are also given.
There is a very fine thread of Canadian influence through the book, such as by naming Canadian provinces in examples. However, this is not done in an inappropriate way nor is prior knowledge of anything about Canada required at any point in any way.
This book is absolutely superb for a one-semester bridge course to the advanced mathematics curriculum. As such, it would do students an enormous service by giving them a beautiful and clear introduction to how mathematicians really think and write about mathematics. It also has much material which could be excerpted and used in a course which uses a specific topic (be it abstract algebra, number theory, or baby real analysis) to help students transition to advanced mathematics.
Table of Contents
Part I. Introduction to Logic and Proofs
1. What is Logic?
2. Propositional Logic
3. Two-Column Proofs
Part II. Sets and First-Order Logic
4. Sets, Subsets, and Predicates
5. Operations on Sets
6. First-Order Logic
7. Quantifier Proofs
8. Divisibility and Congruence
Part III. Other Fundamental Concepts
11. Proof by Induction
12. Equivalence Relations
About the Book
This free undergraduate textbook provides an introduction to proofs, logic, sets, functions, and other fundamental topics of abstract mathematics. It is designed to be the textbook for a bridge course that introduces undergraduates to abstract mathematics, but it is also suitable for independent study by undergraduates (or mathematically mature high-school students), or for use as a very inexpensive supplement to undergraduate courses in any field of abstract mathematics.
About the Contributors
Dave Witte Morris is professor of mathematics at the University of Lethbridge.
Joy Morris is associate professor of mathematics at the University of Lethbridge.