Hossein Hosseini Giv is an Assistant Professor of Mathematics at the University of Sistan and Baluchestan in Zahedan, Iran. The AMS Bookstore’s description of his book Mathematical Analysis and its Inherent Nature begins, “Mathematical analysis is often referred to as generalized calculus. But it is much more than that. This book has been written in the belief that emphasizing the inherent nature of a mathematical discipline helps students to understand it better. ” This point of view sets the stage for a friendly and engaging textbook that is sure to appeal to students who are curious about the underlying mathematics behind single-variable calculus.
What made
you decide to write the book? Teaching is my favorite mathematical activity, and I consider writing books as some kind of teaching. This has a simple reason: When you teach at a class, you are working with a limited number of students, but when you write a book, you have the opportunity to train a large number of students, some of whom may never meet you.
When I was asked to teach the foundations of mathematical analysis for the first time, I thought of the ways I could make the material accessible to my students. Mathematical analysis is one of the first courses in which students deal with abstract ideas seriously, and it is therefore absolutely essential to h
elp students in finding some intuition. When I was thinking about the abstract parts of the course, I noticed that some parts are not of an abstract flavor, and they aim instead to complete the students’ calculus-based knowledge theoretically. This was against what I heard previously about undergraduate mathematical analysis, namely, its description as generalized calculus.
For this reason, I first tried to talk about the essence of any issue I was teaching. After I used this approach successfully several times in my classes, I tried to popularize it by presenting it within a book. In writing the book I did my best not to present the material in the usual definition-example-theorem style of mathematics books. I did this by spending more time on the interpretation of results and talking about the essence of the issues.
How did you decide on the format and style of the book? The format and style of the book was determined by the essence of undergraduate mathematical analysis, that is, what it has to do to the calculus-based knowledge. In fact, since analysis generalizes some aspects of calculus to wider frameworks and completes some others theoretically, I decided to present the material in two parts. These parts, which concentrate on the completion and abstraction of our calculus-based knowledge, respectively, allow students to go from concrete arguments to abstract ones. For example, Chapter 3 of the book presents some important aspects of the metric space theory within the classical space $\mathbb{R}$ equipped with the Euclidean metric. Most of the concepts and results of this chapter are then generalized to the abstract context of metric spaces in the second part of the book.
One reason I considered this style of presentation was to help students to understand which concepts and results are generalizable to the abstract setting and which ones are not. This approach helps students to understand the way abstract theories are developed.
What did you focus on the most when writing? What was the most challenging aspect? What came easily? My main focus was on the clarity of exposition. The most challenging aspect of this work was to develop mathematically rigorous ideas within an expository framework. A further challenge was to present enough material with sufficient interpretations and justifications in a book of reasonable size. All that said about my challenges, I should say that the writing itself was very easy for me. More precisely, although I was concerned with the choice of material and its volume, I have never had difficulty with how to write something. I think writing is my best ability and I enjoy writing expository texts.
What was the writing process like? I wrote more than one half of the book within the summer vacations of 2014 and 2015, when I had enough time for both writing and recreation. The remaining parts of the book were written when I was busy with my classes, and I sometimes had to write on weekends.
Generally speaking, I write whenever I can. I think writing is like playing a musical instrument, and one is able to write an influential text when he/she is mentally and physically at a good situation. For this reason I have never had a daily schedule for writing. Sometimes I even write on midnights. This is because on midnights there are no disturbing sounds and thoughts. Sometimes I stop writing one chapter to start or continue some other. This happens when I feel that working on some argument is easier than some other in a particular situation. Of course, I never work on more than two chapters simultaneously. Also, to remain focused and motivated, I never stop writing for more than one week.
What were the positives and negatives of the experience? Did anything about the experience surprise you? I think that writing a book for the AMS cannot involve any negative aspects. It is a great opportunity which has many positive aspects. One important positive point of publishing with the AMS is that you work with a society of mathematicians. This is very helpful, because they are able to evaluate your book idea quickly. Moreover, authors receive unfailing support from the AMS personnel. They are all strongly committed to work with authors and to help them effectively. I was really surprised when I found that professor Gerald Folland is the chair of the editorial committee of the AMSTEXT. I always admired him as a great author, and I learned measure theory and harmonic analysis from his excellent books. I have taught measure theory several times using his Real Analysis book.
Did you have a special place where you liked to write? How did you stay motivated and focused? I write in both my university office and my home. More generally, I can write in any quiet place where I have access to a pen and enough sheets of paper. Of course, access to appropriate books and an internet connection is sometimes necessary for me to write. When you sign a contract with a great publisher like the AMS, motivation and focus come automatically to your assistance!
Did time pressure or other responsibilities help or hurt your writing? Deadlines allow us to work more efficiently. I always consider deadlines for my projects, even if I am not asked to do so. No matter a deadline is proposed to me or it is considered by myself, I do my best to finish my task on or before that date.
What advice would you give to new authors? Writing books is a nice experience in which one is able to help students worldwide by making some parts of mathematics accessible to them. It is also a great opportunity for those who wish to share their teaching plans and approaches with other instructors. So, if you think you have hot ideas for book writing, please try to work hard on your ideas and help the mathematics community by writing influential books!
What made you decide to write the book? Was there a gap in the literature you were trying to fill? Did you use existing notes from teaching?
d a window on the other. Perhaps I stayed focused because there wasn’t much else to do there in the winter, other than the eternal problem of collecting firewood dry enough to burn. I was not well-prepared! Being in an isolated place without much distraction helped me concentrate and devote daydreams to issues concerning the book, rather than other responsibilities.
I wrote later versions of that first book because I thought that other children would like them as well. I wrote my second book, “Really Big Numbers” because I wanted to share my fascination with really huge numbers. Even though there are many counting books in the literature, there are none which explore the kind of insanely huge numbers that mathematicians occasionally think about. I thought that kids might like to see this.
you decide on the format and style of the book?
The most negative part of the experience is when I find out about a mistake in the published version. Like many mathematicians, I hate making mistakes. This doesn’t happen too often in the picture books, but I have found many little mistakes — mostly typos or notation errors — in my published works and this really sickens me. Another negative experience is getting bad or luke-warm reviews of my books online, especially if they complain about the artwork. One of my unpublished books, “Life on the Infinite Farm” somehow got onto tumblr, and about 20,000 people looked at it. There were lots of comments to the effect that the book was horrifying or monstrous or otherwise terrible, and these made me pretty unhappy.
for comments. I am generally very grateful for criticism and happy to implement suggestions I get.
On the other hand, do solicit a lot of feedback eventually. Once you have written a large part of your book, by all means solicit lots of feedback. I have found that other people’s suggestions have made my books much better. A lot of times you develop blind spots in isolation, and think that something will be clear to other people just because it is clear in your mind. By finding out what people actually experience when reading your book, you can adjust things so that they make sense to the outside world.
