Is there better alternative to Princeton Companion to Mathematics, because I can't make sense of it?

Question

I am currently reading first chapter of this book, and here are few quotes from the book and I can't make sense of these,they are -

1.) Historically, the abstract structures emerged as generalizations from concrete instances. For instance, there are important analogies between the set of all integers and the set of all polynomials with rational (for example) coefficients, which are brought out by the fact that they are both examples of algebraic structures known as Euclidean domains. If one has a good understanding of Euclidean domains, one can apply this understanding to integers and polynomials.

2.) This highlights a contrast that appears in many branches of mathematics, namely the distinction between general, abstract statements and particular, concrete ones. One algebraist might be thinking about groups, say, in order to understand a particular rather complicated group of symmetries, while another might be interested in the general theory of groups on the grounds that they are a fundamental class of mathematical objects.

3.)A supreme example of a theorem of the first kind is the insolubility of the quintic [V.24]—the result that there is no formula for the roots of a quintic polynomial in terms of its coefficients. One proves this theorem by analyzing symmetries associated with the roots of a polynomial, and understanding the group that is formed by them. This concrete example of a group (or rather, class of groups, one for each polynomial) played a very important part in the development of the abstract theory of groups.

What I mean by "I can't make sense of it" is I can't comprehend what author wants reader to understand,I have read these few times and I don't know how to link symmetries with roots of polynomial or how to apply Euclidean domains to integers and polynomials and I don't know what are groups of symmetries? So the problem is I can't understand what author wants its readers to grasp. So what should I do?

Do you know a better alternative to this book? or Should I read this book differently? If yes,then how? Basically, I want a book for broad overview of mathematics.

Answer 1

A brief word from Timothy Gowers (Preface, 'Who is The Companion aimed at?'):

If you have embarked on a university-level mathematics course, you may find that you are presented with a great deal of difficult and unfamiliar material without having much idea why it is important and where it is all going. Then you can use The Companion to provide yourself with some perspective on the subject. (For example, many more people know what a ring is than can give a good reason for caring about rings. But there are very good reasons, which you can read about in RINGS, IDEALS AND MODULES [III.81] and ALGEBRAIC NUMBERS [IV.1].)

How I use the book: It may be the case that you are approaching the book as if you want to study it, or read it straight through, from front to back. I don't believe this is how the book was intended to be used; at least, it's not how I use it. Rather, when I feel like looking at the book again:

• I go to the contents page first;
• then I choose a topic that sounds interesting at the time;
• then, I try to read those pages.

I'll start at the beginning of that section, and continue reading; if I run into difficulties understanding some paragraph, the first thing I check is "if I skip this paragraph, can I still make some sense of most of the rest of the exposition?" I definitely do not read the book like a textbook. If a section concerns mathematics that I don't understand, then I have two options:

• learn about that mathematics (I very rarely go for this);
• read some other section of the book (this is much easier, and if I wanted my reading time to be difficult then I would have chosen a book by Serge Lang rather than The Companion).

The Princeton Companion to Mathematics is just that: a companion. The point of the book is to be something to "dip into" every now and then during your studies. The point is not to actually learn a whole lot of new mathematics, but rather to come to understand the motivations of a subject, or to experience the general flavour of something.

Why use the book this way?

The Companion is not really designed to teach mathematics in any kind of formal sense; yes, there are sections introducing some very technical concepts, and there is a section at the beginning of the book devoted solely to explaining the mathematical definitions that you'll definitely need to know to understand all of the rest of the book. However, there are two important things to take note of:

• those "Fundamental Mathematical Definitions" cannot be learned all at once, and are best learned from other sources. A key thing to note is that there are no exercises, and mathematics is learned by doing!
• there is no reason to expect to come to understand "the whole of the rest of the book" all at once. Rather, it is probably much better to "dip in" every now and then, and come to understand as much as you can section-by-section.

What does this mean for you? By the sound of it, you aren't acquainted with a lot of higher mathematics (you mention that you don't know about groups of symmetries). That's perfectly fine, and it's also fine that you're reading The Companion. However, you cannot expect a book like The Companion to do everything; it has good explanations on a lot of complicated topics, but understanding any particular section requires a certain level of prior understanding of mathematics. With that in mind, I recommend that you read the book just as I do (there are large swaths of the "Fundamental Definitions" section that I haven't properly looked into yet), but with the understanding that it's okay if you have to stop reading at some point, or skip a few paragraphs that involve lots of strange symbols.

If you're after a book that has a different intention, i.e., if you want a book that, for example, has exercises, and that you can really learn mathematics from, then that's another matter.

Answer 2

Well, you can't expect to learn Algebra from a few paragraphs talking about the general philosophy of mathematics. If you knew "how to link symmetries with roots of polynomial" or "how to apply euclidean domains to integers and polynomials" and "what are groups of symmetries" without studying Galois Theory, Ring Theory and Group Theory first, then you'd be a prodigy!

I have never read this book, but from those paragraphs alone, it seems that the author is giving a broad overview of what mathematics is, they are not actually trying to teach you mathematics.

Answer 3

For me, the Companion serves the purpose of increasing the range of math I'm interested in. There's a ton of really cool math out there, but only a very small amount of it is directly related to the kind of math I'm doing research in, and this is the case for the vast majority of mathematicians. That means it's easy for me to lose sight of all the really cool math happening outside my own field - especially since there are very few good descriptions of advanced topics which are readable by outsiders (even mathematically trained outsiders).

The Companion helps me find awesome things in math that I would otherwise not run across. The way I use it is:

• Flip to a random article.

• If I can't understand it, or I just don't find it interesting: flip to another article.

• If I hit on an article that really grabs me: wait, where did the afternoon go? I had things I was supposed to do, darn it! On the plus side, generating functions are pretty sweet.

Now, like any book, the Companion isn't for everyone. It sounds like right now, you won't find the Companion super useful, because it assumes some background you don't have yet. One book that fills a similar role (or at least, did for me) is Ian Stewarts From Here to Infinity; this was one of the books that got me seriously interested in mathematics. Unlike the Companion, by the way, it can be read front-to-back.

Answer 4

The Princeton Companion is a quite wonderful book: but it is for those already embarked on a university level maths course (as is made clear from the quotation given by @WillR). And perhaps we should remember that the editor is a professor at Cambridge, and the university level maths courses that he and a number of the contributors are most familiar are the Cambridge tripos, or equivalents -- among the tougher courses on planet earth. So in fact, by many standards, the book is not really for those recently embarked on their courses but rather for those already a year or two into their university studies. For such readers, however, the book is an unbeatable resource. But you do indeed need to bring something to the party by way of "mathematical maturity" if you are going to get out of it all you could.

To return then to the OP's question, is there a lower-level alternative? Not really, as far as I know. Though there are some fine introductory invitations to university mathematics (one of Ian Stewart's books is mentioned in another answer). And in a different vein, it is worth remarking that actually, Wikipedia entries on topics at entry level (although of course variable in quality and detail) can be very good.