it said there’d be some thunder at the well

i drew the i ching yesterday.
there are sixty-four “hexagrams” here.
each one consists of six “lines”;
each “line” is a yin (“broken”)
or a yang (“unbroken”). one counts
from bottom to top so that for example
_ _
_ _
_ _
___
___
___
has yang lines in its first,
second and third positions
(the “lower trigram”).

the arrangement of the hexagrams here
is my own; i copied it from a similar
arrangement of sixty-four objects
having exactly the same
“six things, with each ‘thing’
taking one of *two* possible
values” form… namely an
arrangement of “subgraphs”
of a certain “graph” having
six “edges”. (i don’t know why
i’m getting all “quotemark happy”
here exactly… maybe it’s the
*opposite* of “scarequotes” i’m
aiming at: by marking technical
terms in this way, i mean to indicate
that their technical sense is
here *recognized as such* [but
without actually wishing to
bother ourselves with their
technical *meanings*]…
“don’t worry about this;
it’s from another part of
a different course [but
worth mentioning here
just the same]”. oh never
mind.)

so much of the “work” of this “exercise”
i set myself will have been, if we want
to think of it this way, *calculation*:
translating each *graph* of the original
drawing i was working from into a
*number* (using binary arithmetic;
the zine i took the original drawing
from [K_4, MathEdZine #0.8] is
essentially *about* binary arithmetic)
and then translating this number
(from 0 to 63… starting at zero
makes things much easier) into the
“hexagram” notation.

of course i could also have just learned
directly which lines on the hexes
corresponds to which line of the graphs,
and spared myself lots of calculation
(by paying the price of having to briefly
memorize subsets of {1,2,3,4,5,6}
a whole bunch of times instead…
or maybe “wordstrings” like
“yes-no-no-yes-yes-no”).
but my way looked easier for me.

then i went ahead and drew ’em all
over again on lined paper. and in
a much more natural order: eight
rows of eight, 0–63. the book i then
turned to *also* had eight rows
of eight but in a different order…
and so i went ahead and looked
up all the correspondences so i’d
have the names… and the numbers
(in the “standard” order for the
writings-of-the-sages associated
to the various hexagrams). and
here i used a “mixed” strategy…
the trigrams have chinese *names*
so as i checked back and forth
i’d subvocalize along the lines
“chi’en over chen” more often
than “seven over zero” since
(as it turns out) i can recognize
even nonsense syllables (in
latin letters) with
less effort than it takes to
“read” a trigram as a number
from 0 to 7.

and i got to thinking about
drilling and killing. now
obviously if i were doing
this kind of thing all day
on a deadline or something,
i’d want a quick-and-easy
*routine* and soon would
arrive at one, too. i’d
slip into a bit of a zone
where i’d devote most of
my conscious effort to
controlling the movements
of my hand to make better
lines and let my inner robot
do the names-and-numbers bits.
i even did some of this yesterday.

but only for a little bit at a time;
this takes *concentration* and
is, pretty specifically, no fun. no.
i haven’t *got* a deadline (alas)
and sure as heck ain’t gonna
do this all day, anyhow not
yesterday i wasn’t. not a bit.
i was also looking up some of
the hexagrams… the one i
typed out a moment ago shows
the “female” sign over the “male”
and denotes “peace” (and when
the positions are reversed,
“stagnation”)… for example,
and thinking back over certain
classroom work in my past and
fantasizing forward about working
with classes of students again.
it’ll never replace DEAD BEEF
but i’d love to give an “i ching
as intro to binary logic” lecture
before they finally haul me away.
to everything there is a season.
a time to gather stones for casting.

In Media’s Rays

Finite Sets are most easily displayed
by listing their elements (and, when it’s
convenient so to do, also naming them):

A = {a, b} and T = {a, e, i, o, u}

for example.

We note that A\=a here;
this means that

and that typing math
is here considered as
part of the problem
instead of as part
of the solution.

[
The point of having so noted
is that we are, as we so often do,
pretending to begin again.
This is in part the effect of
a lifetime’s classroom work
but in this case is also something
of a conscious choice. Anyhow,
I’ve been going on about handwriting
for a while now and intend to
continue. Hypertext is all well
and good and I’ll drop links
as usual according to my whim
(or careful design… you never
can tell… [until you can]… but
there’s nothing like the body
of the post when you want
to call your reader’s attention
to something).
]

Make sure that whatever symbols
you use to represent my “A” and my “a”
are easily distinguished. This kind of thing
can sometimes be more trouble than you’d
think. For example,
something of a problem for me,
not in writing down credible versions
of
a\=A
b\=b
or
d\=D
but in… but you’ll have guessed…
c\=C
o\=O
and suchlike U&lc
(upper-and-lower-case [RIP])
pairs distinguishable,
as i tend to write them
(unless i’m being very careful),
only by their relative sizes.

Gedanken Experiment (I).
Think through which letterpairs
will give you trouble
(when you’re not careful).

[
in “script” versions of handwritten
letters there are common flourishes
to distinguish, say, script-c
from script-C; “print” letters
aren’t always so easy.

story making very little sense here
(you have to see the letters as i draw them).
my script-y’s and script-z’s
looked too much alike…
and, as you can imagine,
$x$, $y$, and $z$ come up
*a lot* when you stand up
in front of basic-algebra classes…
so i started writing my z’s
differently. had to start
*crossing* ’em to tell ’em
from 2’s.
no wait. actually this story
makes *perfect* sense.
it gives me an opportunity
to report that students
won’t believe that this
is a good idea when told
or shown or even when
they themselves make
a the mistake you’ve
warned ’em about on
work submitted to you
(yourself; that would
be me in this case).
i’m only *vividly* aware
of this having happened
*once* but i can’t swear
it didn’t happen before
or since then. nobody
believes anything you
tell ’em in this sorry
racket and it’s heart-
breaking. heartbreaking
i tell you.
]

Recall that
A = {a, b} and T = {a, e, i, o, u}
.

The set product (or cross product)
of a given pair of sets is the set of all possible
ordered pairs consisting
of a first (or left-hand) entry
taken from the first set and
a second (right-hand) entry
from the second set. Thus

A x T=
{
(a, a), (a, e), (a, i), (a, o), (a, u),
(b, a), (b, e), (b, i), (b,o), (b, u)
},

and

AxA={(a,a),(a,b),(b,a),(b,b)}.

The “carriage returns” in our display
of AxT are here as a convenience;
it’s just as correct… and in some circum-
stances correcter… to smash ’em.

AxT={(a,a),(a,e),(a,i),(a,o),(a,u),(b,a),(b,e),(b,i),(b,o),(b,u)}.

[
i’ve smashed space too.
this can be well worth it.
a *lot* of trouble lies in those
invisible characters.
and if you can learn to read
code that tight–and prove
it by *writing* code that tight–
you’re way beyond this lesson
and it remains only to endure
my plea that you comment on it.
]

Anyhow. This construction
can be taught to beginners
knowing nothing of set theory
in a single lecture obviously.

And without it, the whole
god-damn “functions as sets
of ordered pairs” thing…
the “let’s cram hundreds of years
of post-scientific-revolution
philosophy-of-mathematics
down their tender little throats
in one big enormous blob
of sticky incomprehension” thing…
is doomed. Because what the
hell is any of this going to have
to do with so-called “graphs”
in the student’s imagination
if the instructors haven’t themselves
ever connected these dots?

Well, quite a bit actually when the student
isn’t already badly damaged when they
encounter the nonetheless-incredibly-mangled
presentations we get in the “standard” treatments…
I was ahead of the instructor sometimes myself
in early days and it did me a world of good.

But the real point.
This should be presented by about 7th grade
and then again and again until everybody can teach it
to a 7th grader easily.
Because it’s the kind of down-to-the-ground
nobody-will-ever-be-righter-than-*i*-am
this-is-the-*one*-thing-i-know-for-sure-so-far
moral fucking certainty
without which math
is just more mental masturbation.

This is sadly lacking in our culture at large;
obviously there’s only so much we can do
in our classrooms about it. And creating
*programs* for the culture at large is a
fool’s errand or showbusiness or politics
not math. But dammit. Give the poor devils
a chance to be right when the teacher’s wrong
and know it in this one arena I beg of you.
It won’t make ’em into little
subversives-like-Kibrolv
and even if it did, they’ll
be just as easily marginalized
as I’ve been fear not of that.

The trick has been to get ’em into “statistics”
with no probability. But just to talk about rolling
two dice, any math-head worth their salt
requires a cross-product (DxD
where D= {1, 2, 3, 4, 5, 6}).

The display alone
11 12 13 14 15 16
21 22 23 24 25 26
31 32 33 34 35 36
41 42 43 44 45 45
51 52 53 54 55 56
61 62 63 64 65 66
and a few hours with
a competent fellow-student
would be more helpful
to a beginning student
than a wilderness of stix classes
as I see it. So. How to keep
students apart? Online classes baby.

Exercise in case you missed the whole rant.
Let and .
Write out
(including every bracket, brace, paren,
and comma [and no extras]…
but with space on the page
left to your taste and discretion)
copies of and .