### Eliminating The Middle, Man

I’ve spotted several textbooks in recent years that go out of their way to avoid using certain notations.  The commonest such abuse is in the area of set theory:  the industry has evidently decided that they have to present certain set-theoretical notions, but they’re determined to pretend that they’re really just speaking plain English (and not that scary “math” stuff).  I suppose this was originally motivated by some perfectly reasonable idea of communicating the material in language that readers would understand.  In fact, “perfectly reasonable” isn’t even strong enough here– this is our whole project after all!  So I hope you’ll understand:  I’m not objecting here to some “dumbing down” of the material that offends me because I believe students have to learn to take their mathematics straight, the way we did in the pre-TI, walking-20-miles-to-school-in-our-snowshoes days.

Quite the opposite:  the thing is, once you’ve decided to talk about intersecting sets in any systematic way, it’s just flat-out easier to write $A \cap B''$ than $A$ and $B''$.   If you haven’t looked at a recent Statistics book, you’ll probably find it hard to believe that the phenomenon I’m complaining of exists at all … I can hardly believe it myself.  So I’ve just looked at the first two that fell in my hands:  Johnson/Kuby (10th edition) and Newmark (6th).    The first of these even goes so far as to claim that

probability of A and B = probability of A$\times$ probablity of B, knowing A

is “in words”, whereas

$P(A\, {\rm and}\, B) = P(A)\cdot P(B|A)$

is “in algebra”.  I know for sure that whatever book is used on my home campus matches Newmark in defining conditional probabilty by

$p(A|B) = {{p(A\, {\rm and}\, B)}\over{p(B)}}$

– it just happens not to be one of the books left lying around the big group office I work from (for very good reasons that you can probably guess).  So that’s three out of three books I know about;  I’m not exactly going out on a limb here in calling this contemptable behavior an industry standard.

Hey, why not go all the way?  Shouldn’t we also be writing “four and five” instead of the oh-so-technical “4 + 5″ that replaced it (in certain contexts) several hundred years ago? Since we’re obviously not supposed to be making calculations easier or seeking clarity or any of that stuff, I mean.  Hell, let’s just point and grunt.

I’ll probably have much more to say about this eventually, but I’m tired and hungry and this is harder than it looks.  Yours in the struggle; yours in the faith.  V.

#### 6 Replies

1. Maya Incaand

P means “Proposition” where I come from.

As for “and” I quite like “&”.

Just kidding.

2. Shouldn’t that be “4 plus 5″, which is not exactly the same concept as “4 and 5″ ? :-) OK, I’ll just go point and grunt…

My objection is to inconsistent algebra notation, not algebra per se.

I don’t have any problem with spelling out the meaning of the awful notation every now and then. It’s like acronyms – those who use them all the time are comfortable, those who are new get very lost.

3. JK

What’s wrong with writing P(A and B) when A and B are events? For what it’s worth, you should know that this is the standard in theoretical computer science (my field) — actually, the standard is more like
Pr[A ^ B]
where ^ is supposed to represent logical and (\wedge in latex).

4. but, zac — it’s not just now and then;
they do it throughout the entire presentation!
presumably, students are expected to work
this way, too … which is a big part of my objection.

and of course i have no objection to
“wedge”s (or “ampersands”)! just don’t ask me
to write out the words over and over!
avoiding this is what notations are for!

5. It’s because, you know, natural language is so much more precise than all those symbols.

And when natural language is too hard, you can always refer to a picture.

6. Ah, I remember you giving me a hard time about similar abuse in a probability quiz… I must have missed this post originally… My goal is and remains to get kids to use the precise symbols, but I add, eventually. Oftentimes some transition is necessary.

My little defn of subset, how can I not write “for all” in English (without losing my students)?

That being said, they eat up the new symbols, but in measured doses.

• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)