Proofs and Concepts: The Fundamentals of Abstract Mathematics

Reviewed by Swati Kelkar, Professor, Bunker Hill Community College on 3/25/23

Comprehensiveness rating: 5 see less

This textbook covers a comprehensive list of topics in a typical undergraduate advanced mathematics curriculum. Topics include logic, set theory, functions, relations, and mathematical induction. Proof techniques form a foundation for mathematical reasoning. Direct proof, proof by contrapositive, proof by contradiction, and mathematical induction are covered in detail. A discussion of advanced topics such as cardinality, countability, and the Axiom of Choice is also found in this book. Perhaps including a discussion about computer assisted proofs can spark student curiosity.

Content Accuracy rating: 5

All proofs, definitions, techniques, and terminologies found in this textbook are accurate.

Relevance/Longevity rating: 5

The textbook covers timeless logical concepts and will not be affected in a material way by changes in world order or mere passage of time.

Clarity rating: 5

The book introduces mathematical concepts and proof techniques in a clear and concise manner. Examples provided are to the point.

Consistency rating: 5

All symbols and terminology used in the textbook is consistent. It also includes an index of terms and a list of notations used.

Modularity rating: 5

The textbook is divided into 3 parts: proof logic, set logic, and advanced topics, that build upon each other. The parts however can also serve as standalone units as they require little prior knowledge. The focus is not on finding the right answer but on being able to explain the answer.

Organization/Structure/Flow rating: 5

The textbook is well-ordered and topics build upon each other in a logical progression. Examples provided and exercises assigned are pertinent. This textbook will lend itself very well to classroom reading as well as self-learning.

Interface rating: 5

The textbook has a user-friendly interface. There are no visual or navigational issues.

Grammatical Errors rating: 5

There are no grammatical errors that I found.

Cultural Relevance rating: 3

Although the textbook is well-written and informative, it exclusively speaks to the western audiences. Examples and exercises predominantly draw upon the culture and experiences of the English speaking world. Though not an issue by itself, it can possibly alienate foreign students. For sake of inclusiveness, the book could use examples from mathematical cultures from around the world.

Reviewed by Davide Fusi, Assistant Professor, USC-Beaufort on 8/11/20

Comprehensiveness rating: 5 see less

The book covers all the topics included in an undergraduate course that aims to be students' first exposure to mathematical proofs: propositional logic, induction, sets, and functions. The book has plenty of good examples and exercises appropriate for the undergraduate level. A summary containing definitions and main concepts is at the end of each chapter to help the students.

Content Accuracy rating: 5

I didn't find any typo or mistake in this book.

Relevance/Longevity rating: 5

I think that the book covers all the building blocks of modern mathematical thinking. Chapter 5 could be implemented offering more sample topics.

Clarity rating: 5

The book is well written and uses appropriate terminology.

Consistency rating: 5

Consistent.

Modularity rating: 5

I think that the first two chapters are a little bit longer than needed.

Organization/Structure/Flow rating: 5

The book is well organized.

Interface rating: 5

The interface is very good.

Grammatical Errors rating: 5

Good grammar.

Cultural Relevance rating: 5

I don't see the book be culturally offensive in any way.

Comments

I think that "Proof by Induction" should be introduced earlier in the book and not at the end. Other than this personal observation, I consider the book a very good adoption for an introductory course to mathematical proofs.

Reviewed by Adriana Salerno, Associate Professor, Bates College on 5/25/20

Comprehensiveness rating: 5 see less

I may even cover too much logic before getting into the more mathematical applications and proof writing.

Content Accuracy rating: 5

It seems accurate.

Relevance/Longevity rating: 5

This is classical material so it's definitely going to remain as relevant as it has been.

Clarity rating: 5

Clearly written.

Consistency rating: 5

Consistent.

Modularity rating: 4

Again, I think it might take too long to get to the "meaty" stuff, and spends too much time on the logic portions.

Organization/Structure/Flow rating: 4

Well organized, but perhaps too much logic at the beginning.

Interface rating: 5

Good interface.

Grammatical Errors rating: 5

Good grammar.

Cultural Relevance rating: 4

Understandably so, but the examples may be too Canada-centric.

Comments

I think it's a bit too much logic at the beginning, and proofs by induction are left until the very end, but that part doesn't bother me much. It covers everything one would want for an introduction to proofs and abstraction, but I think it might take too long to get to the "meaty" subjects. However, it is well written, well organized, and has LOTS of exercises in the middle of the text, instead of at the end, something that allows for a more organic (and better pedagogically) reading of the material.

Reviewed by Chris Hiatt, Associate Professor, University of Texas of the Permian Basin on 6/19/18

Comprehensiveness rating: 5 see less

This book is a very comprehensive look at proof methods. It appropriately covers the subject starting at logic and moving to various topics. One difference from other books of its type is that the text on proof y induction is not with the other proofs methods introductions. Instead proof by induction is at the end of the text. The index and glossary are detailed and very useful.

Content Accuracy rating: 5

This book appears to be accurate.

Relevance/Longevity rating: 5

Introduction to proofs material is fairly fixed so this text should have longevity . Also other topics can be added easily at the end of the text.

Clarity rating: 5

This text if very clearly written and would be easily readable by an undergraduate student. The author mentions other widely used alternate notations when introducing new notation.

Consistency rating: 5

This book is very consistent in definitions and notation used. There is a clear consistency in notation throughout the text.

Modularity rating: 5

This book is easily divided into modules and has been divided into many subsections.

Organization/Structure/Flow rating: 4

This book is structured in a very clear manner. One difference from other introduction to proofs texts is that the proof by induction section is at the end of the text separate from the other proof techniques sections. This text has a natural flow from logic based proofs at the beginning of the text to mathematics based proofs in the middle of the text. The last part of the text introduces important concepts needed for higher level mathematics.

Interface rating: 5

There does no appear to be any interface issues. In fact if you click on references in the text it moves you to the referenced part of the text instantly.

Grammatical Errors rating: 5

There does not appear to be any grammar issues with this text.

Cultural Relevance rating: 4

I did not notice any cultural in-sensitivities in this text.

Comments

This text appears to be a very adequate introduction to proofs text with many good examples. There are many exercises in the text, but there are no solutions in the back of the text for any of the exercises.

Reviewed by Jonathan Poritz, Associate Professor, Colorado State University - Pueblo on 2/1/18

Comprehensiveness rating: 5 see less

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.

Content Accuracy rating: 5

There do not seem to be any errors or typos in this book, even in the most intricate of formal logical manipulation it contains.

Relevance/Longevity rating: 5

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.

Clarity rating: 5

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.

Consistency rating: 5

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.

Modularity rating: 5

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.

Organization/Structure/Flow rating: 5

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.

Interface rating: 4

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.

Grammatical Errors rating: 5

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.

Cultural Relevance rating: 5

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.

Comments

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.

Reviewed by Nic Gilbertson, Assistant Professor, Winona State University on 2/1/18

Comprehensiveness rating: 5 see less

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.

Content Accuracy rating: 5

The content is accurate, error-free, and unbiased.

Relevance/Longevity rating: 5

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.

Clarity rating: 5

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.

Consistency rating: 5

The text is internally consistent in terms of its structure and in building a logical system from its foundations.

Modularity rating: 4

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.

Organization/Structure/Flow rating: 5

The organization was excellent.

Interface rating: 5

There were no issues related to interface. The text made use of several diagrams that supported the examples effectively.

Grammatical Errors rating: 5

There were no grammatical errors that I noticed in the text.

Cultural Relevance rating: 3

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.

Comments

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.