Proofs and Fundamentals: A First Course in Abstract Mathematics

I found this book to be long and drawn out. The chapters are very wordy and dont really get to the point.
Rating: 2 / 5

I picked this book up one day in the bookstore, started reading and left with it. I was a math major at one point in college, but I moved on for other reasons. These days most math programs offer courses such as “Introduction to Proofs.” This book serves that purpose. Now that I am back in college taking math courses again, I feel well prepared in my proof-oriented classes after going through about half of this book. The author does an excellent job of stressing the importance of proofs through language that is easy to understand. There is also a lot to read in the sense that concepts are explained and discussed in English. The exercises are adequate, although it would be better if more solutions were offered in the book (only select ones are). I did most of the exercises and that is where the real learning took place.

This book would not be useful for someone who has any kind of a background in proofs/fundamentals. It’s for those who have only taken computation-oriented classes who want to learn how to do basic proofs. I would recommend finding a copy and opening it up to see if it fits your style. Anyone could learn from the beginning of this text, but it takes some mathematical sophistication to get through more advanced parts. The text seems technically precise (clarity on the difference between range and codomain for instance) and remains reader-friendly throughout. The author also does not hesitate on giving his opinions on how math proofs should be written. I liked the way that he emphasized writing proofs in complete sentences, and in keeping the scratch work and attempts at a proof separate from the proof itself. Also, despite the language, rigor and mathematical symbolism are not sacrificed.

*In response to critical reviews about exercises: At first glance I did not always see how the exercises related to the examples. After rereading the chapter, or trying out some different strategies, I usually was able to make sense of them. The exercises can sometimes be challenging in this manner, but that is not necessarily a bad thing. They emphasize understanding rather than modeling. I found some satisfaction in struggling through some of the exercises upon figuring them out on my own.

**Also, I agree with the critical review that this book will not make you better at doing proofs. If you want to improve upon proof technique, I would look elsewhere. This book is definitely for those entering the world of mathematical proofs for the very first time. Unfortunately, there are not enough solutions of proofs in the back that you can compare to your own.
Rating: 4 / 5

“Proofs and Fundamentals” by Ethan Bloch is undoubtly one of the few books I’d say would be required when beginning (and a light refrence as you progress) the undergraduate mathematics curriculum in most US schools.

There is some bit of controversay about “Foundations” courses in general. It use to be the case that there was no ‘bridge’ course that linked the more applied Calculus classes (something like your average ‘multi-variable calculus’ withi something more theory based such as Abtract Algebra)

The basics of proof writing were usually taught in your first theory course. In some institutions (I hear Cornell as one of them) still teach your basic proof writing in their first theory course. However, much of the large research state flagship schools in the country and even some of the older ivies such as Princeton (which I hear offers 2 – 3 foundations courses depending on your mathematical taste) now have come around to the concept of a bridge course in mathematics.

To be sure, the refinement of the curriculum to include some kind of bridge course in most departments in the US has led (in my opinion) to the ability for more students to study mathematics.

With that in mind, this book is wonderful for that purpose for the following reasons:

1. Its very readable

2. It gives an introduction to propositional logic and naive

set theory.

3. It covers numerous fundemental topics that is the start to

some very interesting mathematics.

To comment on the first point briefly, the author does utilize the standard definiton-theorem-proof-excercise template, however he addds alot of motivating exposition in between. What is motivating exposition? Well, Bloch writes (throughout the text) why we (the student) are learnign these concepts and how they fit into the various structures in mathematics.

He dosn’t only talk about how concepts/theorems are relevent to theory, but mentions why a certain style maybe relevent to communication. One of Bloch’s emphasis is the “communication” aspect of mathematics. That is, it is good to get good notation and adequate writing habits in the beginning (the later is something I have sort slagged on, as I’m more of a concise symbol man myself).

On reason 2, he gives a very good introduction to propositional logic. He even lists most of the rules of inferences early on. This section helped me tremendously. I was always curious on what was the final justification of many mathematical statements (which of course in the naive sense is just proof table verification). Learning this fact allowed me to enjoy mathematics more as it wasn’t so much hand waving as a system of verifable statements (which makes math unique of all the subjects you could study).

After this chapter Bloch has a section for “strategy” in proofs. This section is more useful for the basic courses you will take immediatly after you take your foundations, but I honestly don’t remember getting much of an impression while I was reading it. Stick to learning the naive foundational material and to doing lots of problems and you’ll learn your own strategies. If you want a more concrete book on problem solving a good introduction to that is the “The Art and Craft to Problem Solving.”

Of course the next section is naive set theory. Bloch does alot of work for you (and rightly so) in proving most of the results he mentions. Thus it makes this section very smooth reading. In fact, I do not think I did more then a handful of excercises and yet was able to grasp and use the material taught in the book (for most mathematics textbooks this is not possible).

Finally, the author provides the basics of topics that can lead to interesting studies into various differnt mathematics. A chapter of counting principles is included (Combinatorics), Number Systems (Real Analysis), Groups and Lattices (Algebra) and some small filler chapters in between like a chapter on binary operations, Recursion, Cardinality, Induction, and Fuzzy logic to name some.

I got a very good apprecaition for the Foundations from this book and I blame it for giving me my enthusiasm for Mathematical Logic. There are others which are better or worst on some topics, for instance, those who are interested mostly in Analysis may not find this book appealing (A good pure Analysis undergraduate handbook is ‘Fundamentals of Analysis’) but on balance, this book is good (I’d say excellent). I’d think the best way to utilize it would be to read it the summer before you start rigorious mathematics. Try some of the excercises and then attempt to correct your excercises (determine if they are right or wrong yourself). Then when you take your foundations course (or your first theory course), consult your instructor on your questions. I feel if one sticks to this regiment, they will be well on their way to a succesful curriculum in mathematics.
Rating: 5 / 5