we are not data

consider “yin and yang”.
emphatically *not* “zero and one”
(except… maybe… just a *little* bit like).

is there *ever* an “always”?

for there to BE “always some
black in the white” (as the
standard yin-yang symbol seems…
on the standard interpretation…
to imply) already somehow implies
that we can (somehow) *distinguish*
“black” from “white”.

[
or this.
suppose "everything is shades of grey".
consider *tendencies* then.
"getting darker"... "getting lighter"...:
aren't these "directions" actually
*opposites*... & this in some "absolute" sense?
("diametrical"... "one-hundred-eighty
degrees"... one could go on.)
or are there, after all, no *shades*?
wait... *none*? ever? always?
]

it looks like there can, um, “be”
no “polarity” without “poles”
(& you can’t say “everything
is relative” without saying
“everything”).

all this is perfectly well-known of course
and indeed altogether obvious. i don’t
deny that. you win. i didn’t come here
to argue.
**********************************************
the path of excess leads to the palace of wisdom.

one of the many interesting things
that happens when you overdose
(on any of a pretty wide variety
of psychoactive substances)…
one of the things that *characterises*
“overdosing on a drug”…
is a very-convincing feeling
that now, at last, one has access
to some simple-yet-elusive truth
about the-way-things-are:
the “moment of clarity”.

simultaneously to experience (for example)
“mortal terror” and “full acceptance”…
well, people go an awful long way sometimes
(and i don’t *blame* ‘em). never mind
the swinging-back-and-forth rapidly
(between shadows of suchlike feelings)
that makes ordinary experience
so *ordinary*… forget “philosophy”
and… for sure… forget a fist-fight…
but. dammit. right-here, right-now!
both-at-once-and-neither
forever-for-just-this-instant
at long last (as always)!
i *get* it!

[
this matters a great deal!
this matters not-at-all!
]

losing my religion.
*********************************************
math… my religion…
has, in certain contexts,
sometimes dealt with some part of
the perennial philosophical question
i’ve been gesturing at
(more or less helplessly)
by replacing certain “sets” of “points”
–taken to be well-understood–
with new sets consisting
of (old) “ordinary” points
(taken from the well-understood set)
together with (new) “ideal” points
created on the spot to account for
what i’ve been calling “tendencies”
in the last few paragraphs.

so. what *if* the obviously-impossible
*were* possible? (while, somehow,
“logic” and even “rigor” had passed
through the upending-of-the-world
that would ["obviously"!] follow…
what *then*?)

sometimes whole new systems emerge, is what.
(& sometimes… usually… one soon arrives
at a complete logical dead-end;
back to common-sense then and no harm done.)

“all math-heads are platonists”
or so they say. (as if we
[who take symbolism seriously]
were *more* inclined to confuse
(I) our axioms-taken-without-question,
with (II) the very “laws of nature”,
than are the common-sense majority
[for whom the question simply
cannot even arise; "obviously"
(with or without invoking
"god's will") these (a-t-w-q
and "l's of n") are *one and
the same*]).
**********************************************
suppose you had some idea.

suppose your idea was
(i) important and
(ii) not already well-known.

well, then, my child, wow.
you’d've done what very few people do.

[
of course, *i* always knew you could
do it if that's what you wanted to do.
but "wow" just the same (and i mean it).
]

now. get ready. shut up quick.
or spread it around if you must
but please let somebody else
take the blame.

otherwise.

you’re going to be punished, good and hard.

and not just until you give up. for *ever*.
*********************************************

new guitar track today.

the swirling beachball of doom

my life in schools.

fast forward.
bill carroll’s humanities initiative
at ohio dominican college.

if word-choice were anything like
as influential as many in politics
(and almost all academics)
like to pretend, why then,
my “indoctrination” would have been
*over* when i got my “doctorate”.

but one door closes when
another one opens and
as far as “turning pro”
goes, i was almost completely
unprepared. (that this was
in large part because of choices
i’d made with eyes *at least*
as wide open as the generic
next-guy player in whatever
game it is we’re here considering
[or are *about* to; somebody
say "go"]… well, that’s very likely
true i hope and feel. so what.
[i hope you don't like it
and want to do something
about it].)

“i hope and feel.”
“trust” doesn’t enter into it of course.

personal slogans of the season part n+1.
‘our medium is handwriting.”,
“trust no one.”,
“tools at hand.”,
“no yin without a yang.”,
“look again. look closer.”,
and “where was i?”.

to name but a few.
where was i?

bill carroll and humanities at dominican.
of bill i know but little and will
say only slightly less. young administrators
“going up” some career “path”
(a “graph”? a “*tree*”??) will…
or so conventional wisdom has it…
generally try to distinguish themselves
by *changing things around*. if nothing
too horrible happens, they’ll have proved
that they can get people to *do things
their way* (instead of some tried-and-true
*other* way)… which, on some prominent
world-running models anyway, is pretty
close to the *whole point* of “administration”.
(or, as i’d rather call it, “management”.)

anyhow, when i turned pro in ’92
and moved to columbus to work at
dominican (as a newly minted
math Ph.D.), bill was academic V.P.
there and was energetically recruiting
faculty from all departments
to teach in a newly-redesigned
humanities program.

i loved this idea and signed on right away.
so along with teaching my math classes,
i sat in as an auditor-with-benefits
on (young theologian) leo madden’s
Hum101 (as i’ll here style it;
the actual course number is lost
to me and good riddance).

it was great. just for the regular
direct contact with leo, it’d've been
totally worth it. but the big draw,
as you’ll’ve guessed if you know me,
was the *texts*: the iliad,
the aeneid, luke and
acts, some readings in plato
(“the death of socrates” and,
the intro to the whole course sequence,
“the parable of the cave”),
and (alas) the confessions
of augustine. (i’m pretty far from
catholic [nanny converted to marry
owen thomas ("senior"); their son
owen thomas ("junior") was my--
owen thomas ("the third")...
"owen by the way"...--dad;
dad was raised catholic
but *never* spoke of church
matters (with me) except
in the abstract; some years
after his death i was somewhat
surprised to be reliably informed
that he (dad) had considered
studying for the priesthood
as a young man]; as a lifetime
academic one cannot escape
having a certain respect for
the “catholic school” tradition;
that’s about it.)

anyhow, one semester later i was ready
and led my own classes. and i’m sure
i had a point when i begain this ramble
and i even expect to get *to* it…
at some later date. suffice it to say
here that i taught Hum101 three times
and 102 once (a “team teaching” situation;
we-must-do-our-*ex*ercises…) and
loved it and that i even have some notes
left over (that i’ve had it mind
to port up onto the net for quite
a while now and maybe soon will).

more light!

the day i first loved henry james

quoted in pseudopodium. i didn’t laugh as long or as hard as at that dubbed startrek thing that’s been going around. but i did laugh. who knew. henry’s riotously funny. you just have to be old. enjoy!

the man went to earth and looked again

madeline made bread.

there’s nothing in the world
like fresh bread. mmm-mm!
i was barely in the kitchen at all
for the whole process but from
time to time various thoughts
about fresh bread drifted around
through my considering-apparatus.
dominant among them dorothy stewart’s
brown county loaves. dorothy was
a dear friend to our whole family.
she made much the most homemade
bread strictly-so-called that *i*’ve
ever had and boy it was good.
when i was little i wanted plain
old grocery store air bread but
i got over that. probably around
the time i found out i liked pizza.

anyhow when the loaves are cooled off
i bring one into the TV room along
with knives and butter. but first
of course… never mind the butter…
i break off a nice crusty piece
and have at. boom. i’m back just
like that in zagreb across from
the school where there was a little
shop where usually i’d get a
lollipopish thingie costing, like,
a couple dinars or something…
about a tenth of cent US at that
time… but would sometimes splurge
on a small loaf… or big roll…
tasting *just like this*
wonderful stuff right here.
and i’ve got impressions in here
of both buildings and the street
that separated ‘em and the gravel
of the school courtyard and the night
jasna got knocked out cold on the gravel
somehow (and took quite a while to come around)
and of other more playground-like goings-on
in sight of that store (though nothing
much of its *inside* at all). mainly, though?
the taste of bread. just like back in zagreb.
gee it’s good.

now this is the strange part:
it wasn’t until *after* i’d started
drafting a blog post (still in
my considering-apparatus you
understand) that i thought
of proust. imagine that.

m’aidez

founding meeting of covoc:
preliminary report. the
central ohio volunteer
organizing committee is
in full effect as of now.
yesterday was my best day
as an organizer since i was
actually on a union’s payroll:
i organized one worker twice.

i’m a rank-and-file member
of AAUP & IWW; my prospect
signed on to both outfits
right in front of my eyes
in a coffeehouse on main
before i’d even properly
worked up a head of steam
for the pitch itself… and we’re
talking “click here to pay dues”
not just “sign here for the
abstraction” (authorizing a
bargaining-committee-yet-
-unformed to try to cut deals
in one’s interest… with dues
actually *due* only after
a contract is negotiated
with the bosses; for a year
of my life i tried and mostly
failed to get people to sign on
with cfa/uaw on these terms
[and had a couple of *three*
signature days along the way;
never the same worker twice
till now though obviously]).

so we two are now planning to work
the “high-road, low-road” approach
to recruit new faculty… our targets
are non-tenure-track college faculty
in columbus ohio and environs. this
includes grad students in case that
isn’t obvious.

the aaup is of course a *professional
association* rather than a union.
many of our prospects are likely to
feel that labor unions are “part of
the problem” in the u.s. economy
rather than “part of the solution”.
in academic organizing we’re blessed
with a sort of middle ground here:
even faculty committed to “free market”
economic policies often feel that
the norms for “professional” work like ours
can and should be determined at least
in part by its practitioners. indeed, if
one’s professional indoctrination in grad
school has been at all effective, it’ll
generally be understood that “service
to the profession” is a duty… and that
this is entirely *different* from service
to the department or the college or the
vice-president for academic affairs.
which is enough. tell me about some
of the problems on *your* campus
and *i’ll* tell you where to find some
people who’ve *done something about*
similar problems on similar campuses
or anyway to learn *something* useful
about your problem… while meeting up
with some of the liveliest fellow scholars
in the game and working hard on
worthwhile projects. maybe somebody
will even *read* something you wrote
with real interest. why not ask for
the moon. anyhow that’s the high road.

(one should also belong to a disciplinary
association in my opinion; i’m currently
*not* plugged into ams, maa, or nctm
but am ex- in all of ‘em and’ll probably
pay dues to at least one of ‘em again
soon. one needs a pro peer group
*particularly* when not actually working
as a pro… anyhow insofar as one still
includes “highroad” health-of-the-profession
issues on one’s personal agenda…)

by calling iww the “low” road i intend
to *dignify* it. if bosses are “high class”,
we’re the other thing: we won’t have
anybody’s boss *in* our union but
only fellow workers. that’s just basic.
if it’s “low” to call exploitation by the
name of exploitation … to call violence
by the name of violence… and to invoke
heroes like joe hill at every opportunity
in *doing* it… well sign up here,
because we’ve got lots more good songs
and the strong hearts to keep singing ‘em
when the heads start cracking.

it’s pretty widely understood that there’s
an earthquake on and everything’s changing
all at once all around us. the high-tech god
pretty clearly has feet of clay, too, anyway
for more and more people as it appears to me.

so it’s a great moment. *something* like colleges
will emerge from all this mess with any luck
and it’s not unreasonable to hope to be a part
of whatever that might happen to be. any luck,
there’ll be a general strike soon and one can
be part of that along the way.

meanwhile i’ve got my publishing projects
and haven’t done much about founding
a school or a resource center or any of that
so it’s not like this’ll take over my whole
life or anything. just, you know,
welcome back to the struggle, me.

s,n! is still at it. yay. self-reliance. nothing can bring you peace but the triumph of principles. peace peace when there is no peace.

the web has evidently killed my camera. trust no one. more light!

dan kennedy’s math ed stuff via JackieB.

the chinese room

this moving-around-of-letters
activity of the past couple of
rambles is, or could and (i hope
someday to convince *some*body)
should be, as foundational
in the study of mathematics as
elementary arithmetic (+, -, *, 1/n) or
compass-and-straightedge constructions.

“trust the code” shall be
the whole of the law whenever
*i* set up as math dictator.

this means symbol-by-symbol
every-keystroke-perfect *code*
is, first of all our *subject matter*
when we’re studying algebra
every bit as much as it is for its
johnny-come-lately derivative
“computer programming” (whatever
the proper euphemism is these days).

enforcing this level of attention to
detail *without* a computer turns out
to be quite difficult. one of the great
frustrations of my life is that *with*
a computer you can pretty much get
*any*body to perform rituals of
*arbitrary* complexity as long
as no actual *reasoning* is involved
just by convincing them that there’s
a paying job in it for them somewhere
if only way down the line behind all
those other poor desparate bastards
that already graduated and have nothing
better to do now but spy on *them*.

but computers are are hard.
to pay for. to understand.
and altogether *impossible*
to maintain for long.

whereas the game
is “simple things first”.
(another fine game is
“don’t let machines
tell you how to live”.
this one’s *much* harder.)

*you can do this*.
what’s more, having done it…
and had the right *conversations*…
you’ll be darn *sure* you can.
and when anybody else…
human or robot overlord
or one of the many blends
emerging all around us daily…
has it *wrong*, you’ll *know*.

here is power.
*that*’s what the simplicity is for.

let me go ahead here and admit that
there’s plenty of good math you can do
*without* this almost-machine-code
letter-by-letter detail-oriented
okay-i-admit-even-somewhat-obsessive
*algebra* stuff.

i was an algebra *major*. so i’m biased.
anyway, logicians are worse. but no. really.
this is the stuff that’ll make you *good*.

story-of-the-blog-so-far stuff.
last winter when i was blogging
about my math148 precalculus class
(as i think of it; three classes really),
i devoted quite a bit of attention to
finding and implementing the “right”
notation for, what was one of
the big themes of the course,
transformations of the xy-plane.

here as maybe nowhere else
one has an opportunity to *use*
the “points as ordered pairs”
point-of-view so sloppily
developed throughout math101.

because the centerpiece
in everybody *else’s* imagination
seems to be the xy-plane
itself… the admittedly epoch-making
observation that by laying down
co-ordinates over a euclidean plane
you get a cartesian plane and all
of a sudden equations have *pictures*.

ooo. aaah.

and these pictures are all well and good
and the basis for the scientific revolution
whether *i* like it or not and all that.
but.

the kids don’t get it. and won’t
until they believe they can. and
as to “functions as sets of ordered pairs”,
the examples given typically…
graphs of polynomials and whatnot…
have manymany scary confusing aspects
already known by the audience to be
well beyond their comprehension.

so it’s… well… just *logic*
(not *rocket science*[!]): simple
things first. confused about why
some “transformation” (that doesn’t
even have a proper name, let
alone appropriate symbol)
causes “it” (the graph of…
something… but “it” isn’t usually
any one thing in these discussions)
to *change* in some particular way?
well, how about a bunch of highfalutin
*technical terms* that you know very
well *you* don’t know (and have no
very good reason to be sure about
the teacher)? that’ll sure be useful.
(depending on your goals.)

confused about A, B, and C?
*where*, precisely?
how did *yours* look?

in the *spirit* of “keep it simple”
i now propose to ramble some more
about the “simplest interesting case”
of permuting the elements of a set:
the case of *three* elements.

ABC ACB BAC BCA CAB CBA
XYZ XZY YXZ YZX ZXY ZYX

here are two isomorphic “strings”.

“isomorphic” means “having the same form”.
that the strings… lists of symbols…
*do* have the same form
in some sense is probably obvious to
any reader. heck, six groups of three.
but more than this.

the set isomorphism



“induces” (what i’m here calling)
an isomorphism of lists:
replacing each left-hand object
wherever it appears in
our first string with the
corresponding right-hand object
produces the second string.

note that “isomorphism of sets”
is (and deserves to be) standard language
for the kind of one-to-one (and “onto”)
function we’ve displayed here.
two (finite) sets “are isomorphic”
as soon as they have the same number
of elements.

but there will be many different
isomorphisms between any
pair of isomorphic sets.
indeed… theorem 1!… there’ll
be n! (en-factorial) of ‘em
between any pair of n-element
sets. (you see this, right?…
remember that factorials count
permutations…)

.

now. in the spirit of the introductory
ramble from a couple weeks back.

two exercises are isomorphic
when one can be worked out from the
solution of the other simply by
replacing “letters”.

consider the six isomorphisms
from {A, B, C} to {X, Y, Z}
(as shown above).

for a low pass, write out all six
isomorphisms from {a, b, c} to {x, y, z}.

for a passing grade, write out all
six isomorphisms from {P,D,Q} to {E,I,O}.
let (the particular isomorphism)

be denoted by “elbowgrease”.
write out the result of applying
elbowgrease to the string PDPDQ.

for a high pass write out the
iso’s from {1,2,3,4} to itself.
what happens if you “apply”
an isomorphism to the result
of the application-of-an-iso’ism?

for a pass with distinction learn
“cycle” notation and how to calculate
with isomorphisms-of-sets considered
as members of the so-called
symmetric group on three elements.

essay question for advanced credit.
we’ve “gone meta” twice in “lifting”
correspondences of sets first
to what we called isomorphisms
of strings, and then to
isomporphisms of exercises.

one could continue to “lift” the
concept to even “higher-level”
groups of data… perhaps introducing
some metaphor along the way to
replace strict symbol-for-symbol
sustitution.

find a pair of textbooks covering
transformations of the plane.
display an “isomorphism” between
the bone-headed wrong ways the
relevant sections of your chosen
texts leave out crucial concepts and
fudge important details.

develop a theory of how this state
of affairs came about. for the
love of god and the gratitude
of generations still to come
do something to change it.

there is no frigate like a boat

in recent posts i’ve typed out
rather detailed descriptions
of a fairly simple process:
forming lists of all the
permutations of a given
(small) finite set.
four or five elements, say.

any more than five
would be impractical for in-class work
though i can easily imagine assigning a
“neatness counts” *poster* project along
the lines “create a display showing…
exactly once each, prominently and distinctly…
all 720 permutations of the set
{A,B,C,D,E,F}”. you fail if it’s wrong
of course so check it over several times…
by several different methods if
possible (usually one settles for two
actually… but this is math ed so
the more the better [let's pretend]).

so i now propose to look at methods
of *generating* our lists. thus far
we’ve examined what i hope is much
the commonest: trees.

given the symbols of {@, ^, *}
(“at, hat, and splat”) to permute.
let’s see if i can draw it
with this typer here.

[tries and fails; that would be "no".]

to heck with it then. the point is,
you can draw what’s called a tree.

[
this is as good a time as any to mention
that if there were any doubt in anyone's
mind what the heck i'm going on about
they'd take advantage of the web format
and enter a few well-chosen words into
the search engine of their choice. what
would owen do. quite often i tack "wiki"
onto the end of my search string (i.e.,
my list of words to search *for*) so as
to be taken quickly to the amazingly
useful wikipedia (the pearl of the net).
on "tree diagram" it's *not* so useful
but you'll see a real one quick enough
if you do the search. think "family
tree" by the way if you haven't seen
the connection yet....
]

@
@

^
^

*
*

each pair… the ats, the hats, the splats…
should be imagined… since i’m helpless to
*draw* it… as connected by “branches”
to a “root” to the left of the left margin.

*three* branches at this stage…
one for each of the “objects to be permuted”.

in the next stage, two branches
sprout from each of our current ones:

@^
@*

^@
^*

*@
*^

(the “at” is here imagined as having
sprouted new leaves: one for “hat”
and one for “splat”… and so on down
[*two* leaves for each new set because
there are two "unused" symbols
left to permute]).

there are no more “choices to make”:
we can complete each partial-permutation
at this stage by appending the
only-available-symbol. and of course
we get
@^*, @*^, ^@*, ^*@, *@^, *^@
: i hereby propose to call this
the tree order for the
permutations of the symbols
in the (ordered) list [@, ^, *].

note that this “tree” procedure
vividly illustrates the “factorial”
process. there are six orders
because 6 = 3*2*1…
three choices of first letter
(the first set of branches),
two of the next… so on.

this “tree” algorithm is exactly that
of the previous posts; if this isn’t
obvious i urge you (strongly) to
write out the tree for [A,B,C,D]
and compare your result to
(what i’ve already posted)
ABCD
ABDC
ACBD
ACDB
ADBC
ADCB
BACD
BADC
BCAD
BCDA
BDAC
BDCA
CABD
CADB
CBAD
CBDA
CDAB
CDBA
CDAB
DABC
DACB
DBAC
DBCA
DCAB
DCBA

now it’s no coincidence that this
is not only *tree* order but
*alphabetical* order…
also known as “dictionary”
or “lexicographic” order.
this can be very helpful when
alphabetical symbols are used.

alphabetical order is actually a technique
of astonishing power. this is why we must
endure shrill voices *singing* the damn
thing over and over, and even encourage
them to continue until they’re sure they
have it right, and to teach others, and to
correct mistakes whenever they’re encountered.

once you can find words in a dictionary
you’ve got it made: you don’t need
teachers to learn what words mean
anymore. just tools and motivation.

back to permutations.
where i believe something very similar
is going on.

a hugely important point:
“list the permutations”
calls for full credit when
the permutations are given
in *any* order. hard work
for the grader then. too bad.

my {ABCD, ABDC, …, DCBA},
listed alphabetically, is no better or worse
(unless there were more *rules* imposed)
than any other.

i think… vaguely… of our “tree” method
as *top-down*. we’ve got four letters
so we spread ‘em out onto different
parts of our working space and work
“down” the list.

by contrast, a “bottom-up” method might
examine a *single* permutation… the
“standard” ABCD, say… and work “up”
from there. as follows.

suppose i decide to “move letters”
in producing new permos from old.
start at the end with D. move it
as far as it’ll go.

ABCD
ABDC
ADCB
DABC

that’s it for strings having A, B, C
in their standard order. what’s next?
well suppose we reposition the C
one step leftward and repeat.

ACBD
ACDB
ADCB
DACB

so on (“reposition” C again and slide the D’s)

CABD
CADB
CDAB
DCAB

and so on (we’ve “gone as far as we can”
moving D’s and C’s only… so B is up next.
and so on:

BACD
BADC
BDAC
DBAC
BCAD
BCDA
BDCA
DBCA
CBAD
CBDA
CDBA
DCBA

notice that
once having “repositioned” C
for the first time we then
“go through a D subroutine”.
we then reposition the C again
and go through the D-subroutine
again. until we can’t repostion C.

we then reposition *B* and go through
the C-and-D subroutine
(consisting of running *several* D-routines).

the towers of hanoi should be mentioned around here.

that’s it for now; i hope there’s somebody still following.
exercise. take a hostage and don’t let ‘em go
until they satisfy you that they can write out all 24
permuations of {A, ?, 17, e}.

never get outta the boat

i first encountered the factorial function
at about age ten. in fact, i recently
acquired a copy of the very book i
leaned about factorials from.
i blogged about it here,
mentioning the (classic!) problem—”four fours”—
i learned about ‘em from.
the game is, using only “standard” operations
like powering and rooting and multiplying
and subtracting and whatnot…
and *exactly* four 4′s…
and no other numerals..
to write representations of
small natural numbers.

e.g.
1 = 44/44
2 = 4*4/(4+4)
3 = (4+4+4)/4
4 = 4*4^(4-4)
and so on. a great game for kids.
(you can see it had something of
an influence on *me*…)

kids of all ages lest that go without saying.

anyhow, sooner or later you’ll get stuck.
two things happen. you give up or you
get mad and start looking more carefully.
okay, three. you can *cheat* and allow
“new” symbols… like factorial (!).

the factorial function “counts permutations”.
in the example that should be given
every time the subject come up
until the student indicates that
they’re already doing it “in their
head” every time it come up already
and you can stop again (already):
the permutations of the elements
of {A, L, T} are
ALT, ATL, LAT, LTA, TAL, and TLA.

*any* three letters can be used of course;
the permutations of {X, Y, Z}
(i’m being sloppy) are
XYZ, XZY, YXZ, YZX, ZXY, and YXZ.

the point… *a* point anyway…
is that a set of *three* letters
will always have *six* permutations.

one easily sees that this is “because”
6 = 3*2*1. likewise for {A,B,C,D}
one has 4*3*2*1 permutations.

notation:
4! = 4*3*2*1 = 24
3! = 3*2*1 = 6
2! = 2*1 = 2
1! = 1 = 1

the “factorial of” a (natural) number…
n, say…
is denoted by “postfixing”
(like some… trouble aplenty…
adjective-postpositive)
the symbol “!”
(i pronounce this “bang”
usually in class…
“exclamation point”
has five times the
necessary number
of syllables…).

we now introduce the weird-looking
but not-so-weird-if-you-just-look-closer
convention that
0! = 1
(there’s one way, from anyway
one point of view to “list”
the “elements of” the empty set
[i.e., the set of *no* elements...
the "zero" case of "how many elements?"]:
namely the empty *list*).

we can now (though i consider it highly
optional) define the factorial function
!:N—>N
by

!(0) = 1
!(n) = n*!(n-1) [ n\=0],

a “recursive” definition.
these amuse prepared minds
and horrify the rest.
best not try it on the class as a whole
unless they’ve got some “math maturity”.

really n-factorial is spelled “n!”.
i used !(n)
to be perfectly explicit
about the fact that we
*are* considering a
function on N
(the set of natural numbers
[including zero; rant still
to come unless it’s around
here somewhere.

the point is to know like your own middle name
that when you need to count orderings you’ll
*use* this thing (and to know when you *see* it
what the heck it is).

students that can’t write out all 120
permutations of {E,G,B,D,F} at this point,
and go on to the rest of the course anyway,
are *damaged* thereby
and indeed constitute damage to
their whole class and to society at large.

i don’t like this any better than anybody else.
but what i *really* don’t like is being the only
god-damn doctor of philosophy i know of
saying so on the record at this level of detail.

your philosophy is sick and i’m here to fix it.
oh, cursed spite.

you don’t have to be all tough-guy
this-is-college-kid about it… never mind
the if-you-were-serious-you’d-already-*know*
game that wrecks most math classes
before they even get started…
actually, starry-eyed idealist that i am,
i believe that material much easier
than tying your fucking shoes
can probably be taught even
to the dimmest kid admitted
to your college if that’s what
you actually want to fucking do.

i could be wrong of course.

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 .