Unsolicited Testimonial

James Allen’s ScribTeX just… works. You shove in TeX code and click and out pops a webpage with your typeset pages.

Probably I’ll be moving the math stuff back to Vlorbik On Math Ed. Now that I know how to… you know… post math on the web and all. Geez.

It’s Been Too Long Since We Took The Time

This just uploaded PDF is the end of my decade-spanning quest for a procedure for putting pages, written and typeset by me on my own computers, onto the web with my own computer. The tables are very ugly. This is because I had to kluge ’em together by hand with no prior knowledge of how it’s really done, just to get something on the page. I was using plain TeX. It appears that LaTeX will have solved many of my problems. So I’ll have to learn the new ways and redo all my tables to reuse the files for my lecture notes from Dominican.

Meanwhile, I’ll experiment with TeX right here in the blog.
.

all roads lead to shakespearefest

recently i’ve posted (what i’ve been calling)
rambles on foundational topics like
plugging without even chugging,
how to write cross-products,
and symbolic logic via ordered pairs.

sketches of lectures never to be given
as i now choose to think of ’em…
not that there’s anything wrong with
“rambles”. i’ll probably go right on
*calling* ’em that. just please don’t
think my actual *lectures* have ever
been so rambly. necessarily.

okay. what’s on my mind today.
about sets and algebra natch.
big picture. how big.
my life story. too big.
publishing and algebra.
and me. just right.
here goes. publishing first.
annus mirabilis:
1968
(actually 68-69; one thinks
in academic years since before
the beginning.)
cops and kids fighting in the streets
and martin and bobby and all that, sure.
but for my purposes:
my beloved 6th-grade.
both my main teachers i loved.
pretty doggone undyingly as it turns out
though of mister ratts i think but seldom.
mary ann di baggio taught us math
and taught us well. *i* got a model
of “math teaching done right”.
also, though this had little to do
with miss di baggio directly…
her role was to provide access to equipment
and a receptive environment for us to
share our results in (no small thing
and indeed a very big thing and if
there were a heck of lot more of this
instead of a heck of lot less we wouldn’t
now be in a position of having to make
the events of 68 look like a party…
but i digress)… zines.
i did my first self-publishing
in di baggio’s class (with
andrew mcgarrell,
peter strickholm, and
tom hoffa (who appears
not to have a webpage):
twenty copies or so (at a guess)
of each of about four issues of GlOAT
(the lowercase L
is not a misprint). purple “spirit
master” one-side-to-a pagers. (dittos!)
i soon went on to do
(in “printings” of one;
i circulated these kid-by-kid
myself at school after drawing
’em at home) the _ten_page_news_.
i revived the title years later,
beginning just before the zine boom
(before hypertext… so called e-zines
[and blogs and such]… captured those
easily lured by the easier softer way
of the dark side [including me alas]).
anyhow you get the idea: self-publishing
goes way back for me. i soon began doing
a zine *about* zines: _indy_unleashed_.
in this TPN piece from 1998, i claimed
that “the ten page news is most
of my social life”… showing that
my interest in self-publishing
also goes way deep.
hypertext was a natural for me;
indeed mike cagle (linked-in profile) had given me
a copy of ted nelson’s
(seminal, self-published)
_dream_machines_
almost twenty years before the web
broke big (with graphical browsers).
i’ve recently rambled on this part
of my publishing autobiography already
(in my “about” page; not
necessarily a good place for it).
so. turning to algebra.
i majored in algebra in the sense
that i wrote my doctoral dissertation on it. so i trained
as a professor. and was one briefly.
and felt (and still feel) that it was
work i was “born to do”. but i soon
lost my professional rank and title.
after a year of death throes in the form
of never-even-an-interview applications
all over the country i quit trying and
worked freelance from then—1996—
till the day before yesterday. spring quarter.
as a teacher and tutor. and… ta-dah!
published _vlorbik_on_math_ed_
until the efforts to write up algebra
in blog format finally snapped
my patience (and i quit
blogging for five hot minutes).
part of the point of the turn-of-century
_ten_page_news_ was to have some extra-academic
use for the typesetting system…
…
i’d learned for writing up my thesis
and gone on to write exams and quizzes in.
as well as a set of lecture notes.
source files now lost. also for that
matter part of the point of _VME_ itself
was *also* to work with TeX.
it was harder than i thought it would be.
anyhow now i’m at it again.
wordpress i’ve only learned to do
little things on. and those badly.
but i’ve got a copy of the real now
on Legion (and it isn’t even bootlegged!).
leslie lamport claims that things
have stabilized recently qnd i’m
inclined to trust him even though
he evidently works for microsloth.
so far so good. i’m producing
pretty pages with unfamiliar ease. but i
can’t yet put .pdf’s online for all to see just now.
which is sort of strange. looks like google
had a page you could do it with easily.
for about a day. but this is conjecture.
by the time i got there they’d quit
taking on new users. some experimental deal.
all for this morning. what’s to eat?

notes for chapter zero, modified by hand with considerable grumbling from the TeX code but we won’t be doing this again soon

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.

t. tao on displaying math on WWW.

Too Late To Do *Me* Any Good Though I Imagine

Randomly-walking Mike C. on Detexify. Amazing.