Conspiracy Theory

The text for my 148 class says

A relation is a correspondence between sets.
The boldface type (implicitly) means that this is given as a definition. But of course this won’t do: what’s a “correspondence”, after all? To go on to say that “If x and y are two elements in these sets and if a relation exists between x and y, then we say that x corresponds to y or that y depends on x, and we write x\rightarrow y” isn’t helpful. In fact, it proves very clearly that our authors don’t believe their own words: if a “relation” is a correspondence between sets, then how the Heck can a “relation” exist between x and y (which are [again implicitly] numbers, not sets)?

There’s more—a lot more—to complain of even in the two sentences I’ve copied out here (and it stays this bad for at least a few paragraphs), but let me leave my analysis at that for a moment.

Actually, of course, a relation is a set of ordered pairs. Our authors know this of course. They’ve even tried to tell us so: an earlier edition had “A relation is a correspondence between two variables, say, x and y, and can be written as a set of ordered pairs (x, y).” … but this was evidently too clear for somebody at Prentice-Hall, so now we’ve got the handwaving.

“Oh, this is hard to understand”, we can imagine somebody saying … “so let’s make it impossible to understand instead.”. Somebody out there prefers imprecise “definitions” to the real ones. I’m convinced that they don’t want students to know how simple the truth can be: math is supposed to be hard for these villains.


  1. Peter

    I don’t know wat a 148 class should normally do but:

    Maybe the writers didn’t mean to write a book for the formal mathematician. The sentence could as well be used to give some vague idea of what a relation is. The formal definitions, if I’m correct where a rather late addition to mathematics. I don’t think Newton knew them.

    I think you’ll have to start somewhere. Do you define what a variable is, or a set?

    Don’t know the book, don’t know the title, don’t know the intended audience, so I’m probably talking complete nonsense here.

  2. brutus12

    A little off topic but the damn book shouldn’t cost near the $114.99 it is new, no matter if it the best 148 book or not, but then again the cost of most of our (students) books are over priced.

  3. @peter:
    this is *far* from complete nonsense.
    in fact, you’ve chosen an outstanding example:
    newton did, indeed, succeed in “inventing calculus”
    *without* formal definitions of functions and relations.
    and it was great mathematics. but he was newton.

    a few hundred years later,
    *anybody* can learn calculus;
    the “bugs” have been worked out.
    part of the process of debugging
    was arriving at the “right” definitions.
    these make things *easier*, not harder!

    “variable” is notoriously hard to define;
    certainly i’ve never taught a class that
    included a formal definition.
    “set” is usually *undefined*—explicitly.
    (no formal system can define *all* its terms …)

    “late” or “early” doesn’t appear to have
    much to do with what i’ve been thinking about.

    textbook prices are indeed
    a scandal on the industry.
    i’ve been saying this for years.
    (the “letter to the UME trends” i refer to
    in that post isn’t online anymore as of now
    [the pages from my old AOL account died]
    but i’ll probably repost soon; also the link
    by “mobius stripper” is gone [probably forever, alas]).

  4. W1ng5Up

    I am in total agreement with your perspective. Simply put, math is math – rhetoric is rhetoric. Math instructors teach math, textbook authors apply rhetoric to relate math concepts. I perceive a disconnect here. Additionally, if math instructors are forced to use the text in order to meet standardized department test requirements, is it any wonder today’s math students are hopelessly lost in the void? You are the Voice crying in the Wilderness, Vlorbik! If these first few days of the quarter are any indication of what can be expected the next eleven weeks, I may come out the wiser, and learn some *real* math along the way!

    Speaking of standardized tests, and with a nod to an earlier post {(0,0), (0,1), (1,0), (1,1)}, how are standardized tests an accurate measurement of a student’s “demonstrated skill in mathematics?!” In my opinion, standardized tests are no more a demonstration of my comprehension of the material, or of my accumulated math skills, than the designated textbook (that is ‘bigger . . . [with] more user-proof crudware and . . . even less precise”) is an effective learning tool for us remedial students! The Socratic Method has effectively been bastardized!

  5. thanks for the kind words.
    a voice in the wilderness, eh?
    (repent! the kingdom is at hand!
    — yeah, that sounds kinda like me …)

    “standardized” works out to a good first approximation
    to “multiple choice” (i.e. *objective* grades).
    without something like this, one section of a given class
    (never mind students from entirely different *schools*)
    can’t really be compared in any way
    worth keeping track of.
    a large wing of the industry
    (i mean the former academy)
    devotes itself to trying to hide this obvious fact
    (on the principle that if one should instead simply
    furnish the emperor with some visible clothes,
    they’d all be out of work).

    you’ll never prove the riemann hypothesis by studying
    for multiple choice tests … but if i’d’ve been, let’s say,
    in the *top* third of the math GRE takers in my season
    instead of the *bottom* third, it would have been
    because i’d done a lot more homework as an undergrad
    (& i would have been a *whole* lot better prepared
    for my graduate program; maybe i’d’ve gotten good).

    probably the tests are deteriorating …
    somewhere in all the complaints about ETS
    there’s probably plenty of substance:
    money corrupts; lots of money corrupts a lot.
    but the texts (particularly those “below”
    our level in 148) are already a complete loss;
    i sincerely believe they’re doing more harm than good.

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