The positive integers that are referred to in these examples are a set. Well, a particular set we call the Real numbers, mostly out of deference to their direct physical analogues (any counted object).

In math, as in the physical realm, Real numbers have an identity (a + i = a), and that identity is zero. Any Real number(a) plus it’s identity(0), equals itself(a).

In a set of numbers defined by [mod 4], the identity is 4. Given 2, and you add four (then you get 6, and then you mod 4 to stay in the set, and finally) the result is 2.
In this set, 2 + 4 = 2.
1 + 4 = 1.
3 + 4 = 3.
0 + 4 = 0.
Any mod x set contains the numbers 0 through x (by definition).

Given a set of numbers you can infer certain properties from them. Based on the properties, they may be a formal group, ring or module. If you take any two numbers and perform some operation, it’s called a binary operation. And if you use three numbers it’s a ternary operation. But guess what? A ternary operation is just two sequential binary operations. So most all operations reduce down to binary.

All subtraction is addition in reverse. All multiplication is addition in fast forward. All division is multiplication in reverse.
Given two numbers, you have only one operation on them: addition.

And one more biggie: commutativity!
When A operating on B (AB) yields the same result as B operating on A (AB = BA), then you have the commutative property. In your Real numbers examples the operations of addition and multiplication are perfect examples of the commutative property.

In binary terms:
a * b = ab
b * a = ba
ab = ba for Real number multiplication

a + b = ab
b + a = ba
ab = ba for Real number addition.

This is very convenient for us to have Real number commutative addition. It makes learning to count WAAAYY easier. 1 + 1 always equals 2.

Now outside of the Real numbers set, you can have sets of numbers where 1 + 1 = 0 (inverse) or 1 + 1 = 1 (identity) and/or 1 + 1 = 11 (depends on how you write it: A operating on A equals AA, literally) and 1 + 1 = 3456 (make your own rules).

Again, depending on the relationships between the numbers, you could have a ring, a group, an albelian group or a module, or an algebra or lots of made up stuff that creates abstractions from abstractions. Math is fun, but let’s take off…

“Complete abstraction is the natural state of the mind.”

I did try not to accuse the list of providing a list of the greatest or best books, though in my recommendations I have applied a more stringent triage, which perhaps implies it.

I actually don’t dislike fiction, but I do have one great difficulty with it which might sound just a bit ridiculous:
My lifespan is finite, and the number of great books out there is certainly beyond what I can take in. It seems to me that a great number of fiction books are primarily geared towards entertainment, rather than teaching something. While not true of all fiction, it seems to me a safe assumption that nonfiction books are more likely to be intended to edify than fiction books. If I get the same enjoyment out of fiction and nonfiction*, but I am more likely to learn from the nonfiction (because it is intended to teach or because I more clearly discern the lesson, perhaps), then the nonfiction seems to me to be the better choice. In some cases, I fiction can teach as well or better, than nonfiction. However, in most cases, nonfiction is a better, or at least more efficient teacher… and I am very much in favor of efficient self-improvement!

I want to thank you for compiling the list; I love both it and the idea of it! I actually prefer it as a list of recommendations rather than an official “best” list. Generating something for to discuss is key, if you ask me. If I could, I would love to take the time to catalog all of the faculty books and gather statistics on which books are popular! I am a member of a website called librarything, which contains personal and library catalogs with which I can compare my own book collection. They even have a large project called Legacy Libraries, in which historic figures’s libraries are re-created, often with details right down to the proper edition of each book that each figure owned. I see a number of the books from those libraries featured here (I think among all the legacy library members, Don Quixote narrowly beat the Bible for most-owned). It’s quite fascinating.

Thanks again!