**Concerning Blahblahs**

**Exercise 1:** Let A = {p, q} and B = {x, y, z}. Write out the set of blahblahs from A to B.

**Exercise 2:** Let X = {0, 1} and Y= {a, b, c}. Write out the set of blahblahs from X to Y.

I hope you’ll have noticed that Exercises 1 and 2 are more than a little bit alike. Specifically, both can be considered as versions of a certain “higher-level” exercise: “Write out the set of blahblahs from a two-element set to a three-element set” (Exercise 0).

Note that I haven’t told you what a blahblah is; part of the point here is that you don’t *need* to know. (We’ll call them “functions” eventually; forget this for now if you like.) If *I* type out a solution to Exercise 1 and include it here… as I fully intend to do momentarily… then *you* can write out a solution to Exercise 2, simply by substituting a, b, and c (respectively) for my x, y, and z, and simultaneously substituting 0 and 1 for my p and q. Then just leave every other symbol in my solution alone. The result you get will be just as good a “blahblah” as mine *whatever* it is.

**Solution to Exercise 1:**

{

{ (p, x), (q, x)},

{ (p, y), (q, y)},

{ (p, z), (q, z)},

{(p,x),(q,y)},

{(p,x),(q,z)},

{(p,y),(q,x)},

{(p,y),(q,z)},

{(p,z),(q,x)},

{(p,z),(q,y)}

}

Now, there’s real value in such exercises. For example, “substitution”… copying a line of “code” from a source document (typically a textbook or an earlier line of code, say) to a newly-handwritten one, while replacing certain of its symbols with certain others… is one of the most basic tricks in all of Algebra and every literate person will have to perform some variant of this process at least from time to time.

To digress only slightly, and to introduce what I expect to turn into something of a theme in these notes. I speak of *handwriting* though of course other media may be used. Handwriting is much the easiest, and so also the commonest, way to produce what usually call “the code” for a given discussion in my experience. I’ve found very few students willing to join me in typing math and I can’t say I blame the others much: it’s way harder. Calculator code I prefer not to go into just now. For drawings of course handwriting wins hands down. Etch-a-sketch and cel phones notwithstanding.

Note that I don’t consider there even to *be* a discussion until the student actually produces some code (or drawing, or table… *something*) for us to talk about.

Again because it’s commonest and easiest, I’ll generally “speak” here in terms of *oral* discussions, as if one were always already working together with a handful of “students”… usually in dialogue with a particular *one* of them… at a blackboard with plenty of chalk or a tabletop with plenty of (unlined) paper and pencils.

Returning to our Exercises. To produce a solution to Exercise 2 with a copy of Exercise 1 at hand is possible with no understanding at all of functions or even of blahblahs; it requires only what I’ll here call “scribal” skill. You could teach it to a literate foreigner knowing nothing of each other’s languages.

Before we glorify it with the name of Mathematics, though, we’ll need to have… something else. Welcome to the Math Wars. How much “rigor”? How much “understanding”? (And how much homework with how much calculation… and who gets paid and who cleans up the messes… politics.)

At what I’ll here call “University” level, there’s not much dispute: one seeks students that can “work” Exercise 0 (“write out the set of blahblahs from a two-element set to a three-element set”) given only the following definitions.

**Definition 1:** A *blahblah*, *b*, from a set D to a set R, is a set of ordered pairs with the following properties. The first entry of each ordered pair of *b* is an element of D (the “domain” of *b*) and each second entry is an element of R (the “range” of *b*). Each element of D occurs as the first entry of exactly *one* ordered pair of *b*.

**Definition 1′:** A *function* is a blahblah.

**Notation 1:** When *f* is a function from D to R, we can (and *should*, if we want to be clear about it, though few enough textbooks say so) write Usually this is pronounced “eff maps dee to arr”; let this be understood as meaning exactly the same thing as “eff *is a function from* dee to arr”.

But I myself have seldom worked at this level as a teacher. Calculus students, for example, can be counted on to run screaming from anything resembling Exercise 0. Also to crank out Exercise 2’s all day long (given appropriate Exercise 1’s) and beg for more.

I speak here of course not of Exercises 1 and 2 themselves but their moral equivalents. The lazy students I lovingly refer to… as I had *better*, since I’ve followed their pattern myself all my life… prefer exercises involving, if not symbol-for-symbol substitutions merely, still little more than their equivalent at the level of result-of-calculation: rather than “all the *p*‘s get replaced with 0’s, one has something like “differentiate twice and `plug in’ the previous answer”. The trick is to find some exercise that shows you how to do what the author wants without understanding the terminology used in the actual sentences.

You can get through a lot of math courses this way believe you me. Freshman Calculus classes are notoriously often examples of this fact, and so students can even get to consider themselves math *majors* with scarcely any of the down-to-the-ground, from-the-definitions, quote-only-what-you-can-prove this-I-know-for-sure quality that characterizes “real” mathematics.

Real Mathematics occurs at every level of The Art, of course. Such “ostensive” definitions as “Two is *this*, *many*” provide all the formalism needed for *very precise understandings* of the Theorems (if we choose to think of them as such) that are rediscovered whenever anybody anywhere does some Basic Arithmetic.

Which is as far as most people get. Geometry classes are sometimes found in our student’s (typically dimly-remembered) backgrounds. In such cases one sometimes will have had anyway some exposure to “real” math: here if anywhere one is typically introduced to *proofs* that depend on *definitions*.

The importance of definitions to our discussions cannot be overstated.