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Essentials of Mathematics: Introduction to Theory, Proof, and the Professional Culture

Margie Hale
Mathematical Association of America
Publication Date: 
Number of Pages: 
Classroom Resource Materials
[Reviewed by
Marion Cohen
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I feel (happily) as though I'm destined to read and review books on "what's really going on with numbers" (natural, rational, negative, real, complex...), meaning the axiomatic development of arithmetic. The last book I chose to review, among the selection that the editor gave me, happened to also be on this wonderful topic; both times that was a surprise to me, and both times I did not complain. "Axiomatized arithmetic" is a pet passion.

I suspect that the author of this book feels the same way! That is, though the stated purpose of her book — which she also accomplishes — is to provide "the knowledge needed [by sophomore math majors] to move onto advanced mathematical work, and a glimpse of what being a mathematician might be like" (from the back cover), I feel as though a large part of what she really wants to do is to take on the work of the likes of Cantor and Dedekind. And I don't blame her!

From the back cover, which summarizes the book as well as anything I might say: "The content is of two types: ... material for a 'Transitions' course at the sophomore level: introductions to logic and set theory, discussions of proof writing and proof discovery, and introductions to the number systems... The second type of content is an introduction to the professional culture of mathematics. There are many things that mathematicians know but weren't exactly taught... the philosophy of mathematics, ethics in mathematical work, professional (including student) organizations, famous theorems, famous unsolved problems, famous mathematicians, discussions of the nature of mathematics research, and more."

Much of this extra material is in the last chapter, titled "And Beyond...", which attracted me enough so that I read it first. It begins by describing math research. (P. 129. "It is difficult for non-mathematicians to imagine anything new in mathematics. "Do you look for really big numbers?" is a question we sometimes hear..." That made me smile.). She breaks math research up into several categories, "the old, open problems whose statements are accessible to a beginning math student, but whose solutions have eluded even the best minds... problems that no one has thought to work on yet, some of which may not require so much background... problems that are new to you. Working on this kind of problem develops your ability just as surely as working on an unsolved problem..."

On the next page (p. 130) she describes the research itself. "It is a cliché: mathematician, feet on desk, gaze directed out the window, the only movement the blinking of the eyes. But this is one of the true clichés... If you are a more active or social personality, do not despair. For many mathematicians, inspiration can strike during any activity, and new ideas arise in collaborating with colleagues. The physicist Per Bak, who does a lot of mathematics in his work on complexity, enjoys sight-seeing, watching magic shows, walking on the beach with a colleague, all the while thinking, joking, playing with ideas... [he says,] 'the harder one tries, the less likely the prospect of success.' There are false starts and periods of no apparent progress at all... persist, read, talk to others, never let go..."

She goes on to talk about math-games such as nim, chomp, and Conway's Game of Life, undergraduate research programs, non-Euclidean geometry, Tom Lehrer, famous theorems, famous unsolved problems, and the professional math-life including a list of the better known organizations with description and contact info, internet activities, contests, and meetings. This part of the chapter could be an excellent easy-to-access-at-a-glance source.

The first chapter has perhaps the same theme as the last chapter, which is acquainting beginners with "the math life". Explored briefly are "What is Math?", "Pure vs. Applied", "What Kind of People are Mathematicians?", and "Mathematics Subject Classification". Possibly much of it (though not all) is known even to beginners, but even for those who know all of it, it's nice to have it written down in one place, and it feels comfortably familiar.

Most of the rest of the book is the "indulgent" stuff I mentioned in the first paragraph, namely axiomatized arithmetic, along with other "advanced math fundamentals". Here is a partial listing: Logic and proofs, set theory and paradoxes, cardinal numbers, induction, well-ordering, rationals as equivalence classes of ordered pairs, filling in the holes (not only the square root of two) à la Dedekind, solving x2 = -1... what's not to like?!

There are many places where she makes very clarifying and enlightening comments. P. 19: "Think of a predicate as a 'potential statement' which becomes a statement when the variables are replaced with elements from the appropriate ranges." P. 27: "Unlike in politics or sociology, a mathematical argument must be irrefutable, not simply persuasive." (I also like that, on that same page, she says something which I often find myself saying — "The best proofs also provide understanding of why the theorem is true, but it is possible to give a proof that provides no such insight.") And on p. 49 it's good that, in demonstrating the correct usage of notation, she distinguishes between the three expressions: x(n+1), x(n+1), and x(n+1). "The size and position of the expression (n+1) indicate it is being used in a particular way." Finally, p. 126, concerning complex analysis: "A reason for the subtlety is due to more stringent requirements for the existence of a limit in the complex domain: on the real line, a limit need only be checked from two sides: the left and the right. But in the complex plane there are many different paths to a given point: lines, curves, spirals, etc. It is possible to have a function and a point where the limits are different along different paths..."

I also thoroughly approve of her format, which is consistent with the Moore Method (which involves hitting students with a bunch of theorems or facts and asking for the proofs). In so doing, she begins each (mathematical) chapter with "warm-up exercises", then moves on to the (in some cases, fill-in-the-blank) statements but not proofs of the theorems which provide the introduction of, for example in Chapter 4, the positive rationals (from the natural numbers). Here, for example, is a "warm-up exercise" from that chapter: "How is 3/5 related to the equation 5x = 3?" and "Given a set of things we call fractions, list some theorems you would expect to hold for this set." And here are some of the fill-in-the-blanks: "The sum of two fractions is defined by x/y + u/v = _______." "The product of two fractions is defined by: ________." Sometimes the blank to be filled in is the "title" of the "proposition". I was slightly puzzled when I noticed that she doesn't leave very many blanks (that is, in the statements of the definitions/propositions/corollaries; the proofs are, of course, entirely blank.). Is that an oversight, or is it that way because she doesn't want to put too much on the student? I'm not necessarily criticizing the book for this; in fact, it might very well be a good thing. But I wonder whether she herself has noticed.

I do have a few other questions. First, each of the six "math chapters" seems to have a first section that is about mathematics in general (and not about the defining/creating/inventing of new types of numbers) — for example, the section on "Mathematical Notation" in the "Natural Numbers" chapter, appearing before actually introducing natural numbers, and the section on "Philosophy of Math" to begin the "Positive Rational Numbers" chapter. My question is: Is there an inherent reason for these particular pairings? And if so, shouldn't that be explained? (Or did I miss something?)

Second question: each of the math chapters seem to me to contain too many propositions! Specifically, there are too many propositions without our being given enough of an idea how these propositions figure in the scheme of things. For example, I'd like to see a distinction made between those which are "fundamental" in the sense that they must be proven using the definitions and those which, on the other hand, are "corollaries" of the "fundamental" propositions (and thus can be proven in the "regular" way, that beginner mathematicians are used to). (For example, in the chapter on "The Real Numbers", a proposition of the first kind is Proposition 5.6 on page 87: "If X in an upper number for α and Y > X, then Y is [also] an upper number for α", whereas a proposition of the second kind is Proposition 5.41 on page 89: "Given cuts α and β, the equation α = β x σ has a unique solution σ." (The proof of this follows from Proposition 5.39, which allows the concept of "reciprocal" to be defined.) Also, the significance of some of the propositions could be explained; for example (p. 90), Propositions 5.44 and 5.45 could be prefaced by the statement that they amount to: X goes to X* is a "good" embedding, in the sense that it preserves order and all four operations of arithmetic. (Also, Part b of Proposition 5.44 says that X goes to X* is in fact an embedding — that is, it is one-one.)

Third, in some ways the author doesn't make us appreciate the problems — in particular, why do we need to be so "rigorous" in defining the various kinds of numbers? (That is, why do we need to "axiomatize arithmetic"?) Why, for example, isn't "grammar school" arithmetic enough to explain natural and rational numbers? To my mind, the reason is that, in grammar school, addition, for example, is defined (effectively) in terms of properties — commutative, associative, what the times table is, how to work with numbers of more than one digit,... But no one in grammar school asks whether there exist entities (called "numbers") with these properties nor, more important, just what these entities (and the operations we do on them) might be, and whether these properties are consistent. Thus grammar school students wind up with unanswered questions such as "Why is the product of two negative numbers positive?" and "Why is any number to the zeroth power equal to one?" This, to me, is one important reason to be axiomatic about numbers.

There are a few small errors, not very numerous and not very serious. On page 87, where real numbers begin to be defined, it would be more clear if she said that X and Y are rational numbers (meaning, the numbers that we already "have"). On page 45, describing Richard's Paradox, here is her version: "Each whole number has one or more descriptions in English. For example, some descriptions of the number 2 are : 'two', 'the first prime', and 'the square root of four'. Because descriptions are finite, every whole number has a shortest description — that is, one using the least number of words. What number has the following description? 'The smallest whole number whose shortest description exceeds eleven words.'" Well, first of all, every whole number has a one-word description, namely the word consisting of the number itself (as she herself pointed out with "two"). So the shortest description of any number cannot exceed eleven words. This should, perhaps, be acknowledged, even though it makes no difference as far as the paradox is concerned. Secondly, who says there is a unique number which has "the following description", or any description? That is, the paradox-question needs to be worded, or paraphrase-able, this way: "What is the smallest whole number whose shortest description exceeds eleven words?" (Or am I being picky? I do see the reason that she worded it that way — in order to have the word "description" appear twice, displaying the paradox-ity more clearly.) Third, shouldn't the number in her description be "ten" (since there are ten words in that description, and in view of her word "exceeds", meaning "is strictly greater than")? Sure, "eleven" works, but "ten" is smaller and somehow seems to convey the essence of the situation better — meaning that it's the smallest possible. ("Nine" won't work.)

And on page 113, she says that Erdös numbers can be defined recursively as follows: "Erdös has Erdös number 0. If you publish jointly with someone with Erdös number n, you have Erdös number n+1." Well, here's a counter-example: Suppose you publish with Erdös and with Ron Graham; then (1) you publish with someone (Ron G.) who has Erdös number 1 so, according to the author's definition, you have Erdös number 2, but (2) also according to the author's definition, your Erdös number is 1, since you publish with Erdös. Contradiction! I think we need to say that n is the least such Erdös number among all the people with whom you publish.

And now this is my chance to say what I think of the Moore Method. As a (former) home-schooling parent, and as a mathprof who tries to make her classes feel like conversations/workshops/support groups, I think a lot of the Moore Method! However, I also believe that, at least in some situations, we could go even further. For example, at least for pure-math majors, we need to encourage students to discover, not only proofs of theorems that we already give them, but actual statements of theorems — and with less info given, thus making it more than a fill-in-the-blanks situation. In my own high school days, instead of taking advanced math courses at local colleges or entering math competitions, I spent time and energy writing what I called my "Treatise", with "theorems" like "The center of a circle is inside the circle." If I had kept that up for all of my years, I might not have written anything worthy of a Ph.D., but keeping that up for several of my youthful years might have just been what enabled me to think creatively. In the case, say, of defining rational numbers, perhaps a teacher might simply say to students, "the number system of just-integers does not allow for the solving of all linear equations, does it? Can we define a system which extends the integers so that we can always solve any linear equation?" If we sent them home with that question (or, perhaps, the less "manipulative" question, "How would you rigorously justify the reason for the rule for multiplying two negative numbers?"), what might they come up with? (Disclaimer: I have never sat in on a Moore method course, nor read extensively about the Moore method, so my comments here might not be well-taken. Perhaps Moore himself did just what I was suggesting.)

All-told, this book is well written and will, I'm sure, be useful. I would love to teach a course from it.








 Marion Cohen ( teaches at the University of the Sciences in Philadelphia

0. Mathematics: What is Mathematics?; Pure vs. Applied: What Kind of People are Mathematicians? Mathematics Subject Classification. 1. Logic: Mathematical Systems: Essentials of Logic. The Axiomatic Method; The Results of Gödel. 2. Set Theory: Proofs; Essentials of Set Theory; Paradoxes; Axiomatic Set Theory: 3. The Natural Numbers: Mathematical Symbols and Notation; Essentials of the Natural Number System; Cantor’s Infinite Arithmetic. 4. The Positive Rational Numbers: Philosophy of Mathematics; Essentials of the Positive Rational Numbers; Ethics. 5. The Real Numbers: Famous Mathematical Objects; Essentials of Positive Real Numbers; Essentials of the Real Number System; Important Properties of the Real Number Line. 6. The Complex Numbers: Famous Mathematicians; Essentials of the Complex Number System; Important Properties of the Complex Numbers; 7. And Beyond: What is Mathematical Research?; Famous Unsolved Problems; Professional Organizations; Extracurricular Activities.