math carny #96 at MMW.
here… but not for long… is another
page from the display i’m removing
(the big version from the previous
post is gone already).
you can see a corner of the
car-wreck-into-house shot again;
also one “line”… the line at infinity,
as i think of it, in fact… of a page
showing a duality diagram for
(in my standard notations of a couple
years ago when i drew a lot of those).
but i’ve *really* got to get back to work.
yesterday i put in, not just an honest
day’s *work* (i do *that* all the time)
but an honest day’s *labor* (moving
stuff around [mostly books, a little
furniture, some etceterati])… for the
first time in, oh, probably a couple years.
naturally i’m somewhat stiff and sore
now. no big deal. back to work.
in fact, i *wish* i meant “get back to
moving stuff around”. but that’s not
what i’m blogging just now to avoid.
no… yesterday’s burst of moving-
-stuff-around was *itself*… necessary
as it was… in large part just another
*grading* avoidance ritual.
so. time to take this off the wall
and get to some *paying* work.
… hey, what’s that? dirty dishes!
wow, what luck!
here’s the poster-sized version
of my portrait of 
21 copies of the zine page that itself
displays the 21 “points” of S. each page
has been identified with a point of S
by hand (twice; once by drawing 5
circles and once by placing it properly
on the wall); the seven pages associated
(in the zine the poster’s based on)
with 
also in sight: a shot of madeline’s house
the day after some lunatic drove into it.
also a couple of other stray zine pages.
and a weird little asterisk-lookin thing
with a 
anyhow, it’s all about to come down.
to make room for what but another wall
of books. yay! eventually i should
posterize this idea again.
(first post since the name change.)
things’re going fine all around…
other than some computer-interface
hoo-hah snafu-ing the grade entry…
and it’s about time for the first test.
which the lecturers grade for themselves
according to the usual department practice.
so there’ll be maybe a little better chance
of me actually *posting* something around here.
meanwhile, let me go ahead and report for today
that for quite a while now i’ve actually been
*reading mathbooks* pretty avidly again.
recently it’s been one of my favorite
work-avoidance tricks.
reminding me of nothing so much as myself
in grad school when i’d spend hours in the
math/physics/computo-stuff library (in swain
hall west) avoiding the actual sit-down-and-*try*-
-something work that horrifies almost all beginners.
browsing around in the stacks at IU…
i didn’t neglect the *main* library
or that of the education school…
certainly wasn’t a *waste of time*.
i was one of the best-read amongst
a bunch of *very* smart people and
learned a lot of “math culture” stuff
most grad students never find time for.
because they’re, you know, actually *working*.
certainly it’s a relief to look over well-
-reasoned (& sometimes even well-written!)
work after… or during… a marathon session
of correcting basic-mistakes-stubbornly-
-clung-to created by students who should
have long since known much better. (it all
starts with the “=” sign… but [for now]
i’ve ranted enough on that matter in earlier
posts.)
and now that i’m mostly working with my *own*
books… almost entirely gotten on the cheap,
i assure you… “free” and “dover” are the
biggest categories here… i feel (more) free
to *write in ‘em* (as i should’ve been doing
much more of all along [in *pencil* of course!]).
making my avoidance-technique that much more like
the actual work, you see.
heck, even *crossword puzzles* (usually
a big enough part of any go-to-campus-&-back
day since i almost invariably work ‘em on
the bus) have a certain reading-and-writing
… and even *problem-solving*… feel to ‘em.
so i’m spending a *lot* of time in some
mathy-or-anyhow-*kind*-of-mathy state.
immersed in the world of symbols-and-syntax
(or of concepts-and-code if you prefer…
i’ve got a million of ‘em…).
and i *like* it like that. the trick is
to avoid, not facing-down-unfamiliar-symbols
or even grading, but, you know, those messy
“real world” encounters with other beings.
some math users… famously the great pascal…
have even reported an anesthetic effect
(he used analytic geometry against toothache).
the *dreams* you get from a math OD, though?
usually not so good in my opinion. usually
it’s too much like real life: not “one
damn thing after another” but the *same*
damn thing, over and over.
anyhow. past bedtime now. wish me luck.
grading linear algebra again.
so far this quarter, i’ve (1) returned
the text from last quarter (late) and
i’ve (2) gone back and got a different
copy of the same text a week later.
and that’s it… a week and two days
into the semester.
but starting today there’ll be bigfat
envelopes full of homework piling up
in my mailbox. so very likely i’ll be
posting less over in the cooking show
(where i’ve had a pretty good run going
for about a week).
this i vow. i believe in the power of symbolic-objects-carefully-defined. in the power of “math”, in other words. and the holiest-of-all-math-holies i swear i will keep most sacred.
oh, equal sign! great in perfection!
none are like thee in the power to reveal!
no not “continuity” nor “magnitude”…
nor even “quantity”… nay, *nothingness*!…
not even “infinity”…
how are we to speak
how are we to write
how are we to think…
& who’ll understand *any* parable
that won’t understand that there
even *are* any parables…
it depends what the meaning of “is” is.
and the meaning of “is”, is, dammit, “is”.
otherwise, fuck it. what is *wrong*
with these undergraduates? happy christmas.
or whatever.
and a very new year.
at 12:12 on 12/12/12, i’ll’ve been here in the
math-tower office cranking out final exam grades.
(they’re not looking real good.)
“12:12 12/12/12″ caused google to report
18 billion results (in a fifth of a second).
getting back to work.
so i’m grading linear algebra like i usually do.
where we need to speak… or anyway, to read
and write… frequently about *column vectors*.
much the usual thing (for example) in defining
a linear transformation (called F, say)
on “real three-space” (so F: R^3 —> R^3)
is to *consider R^3 as the space of real-valued
column 3-vectors* and then to supply a matrix
(called [F], say; [F] will be a 3-by-3
matrix in this context) such that
F([a, b, c]^T) = [F][a, b, c]^T.
the “caret-T” here denotes “transpose”…
the idea is that one has something like
[a, b, c]^T=
[
a
b
c
]
;
in laypersons’ language, the transpose
symbol ^T tells us to turn our old rows
into the new columns (which simultaneously
turns our old columns into the new rows).
in the language of the widely-used
TI-* calculator line, one has
[a, b, c]^T = [[a][b][c]]…
and this is starting to look
better and better to me right
in here.
but what one really seems to *want*
here is a quick-and-dirty notation
for expressing (what we will still
continue to *speak* of as)
column-vectors, as rows.
and i’ve noticed student papers using
< x, y, z > = [x, y, z]^T.
this looks like a real useful convention
to me and i’ve adopted it for my own use
until further notice.
angle-bracketed vectors have been useful
to me before. mostly, i think, in the context
of “sequences & series” typically dealt with
in about calc 2 or 3.
LANGLE x_0, x_1, x_2, … RANGLE
(i.e., < x_0, x_1, x_2, … >
… “angle brackets” are special characters
in HTML and so i prefer to avoid ‘em)…
in either notation…
represents sequence of objects
LANGLE x_n RANGLE
(which is of course *not* the same
as the *set* R={ x_n } = {x_0, x_1, x_2, …}
[the set of values taken by the function
f(n) = x_n
on the natural numbers NATS:={0, 1, 2, ...}]).
this usage actually *extends* a usage,
ideally introduced and maintained earlier
in a given presentation (class or text or
who knows maybe someday even both at once)
using angle-brackets for (finite-dimensional)
*vectors*.
LANGLE 3, 4 RANGLE
now represents the vector that, ideally,
we would represent in some other part of
our presentation as [3,4]^T…
a *column vector*.
i remark here that meanwhile
(3,4)
represents a so-called “point”
in “the x-y plane”… an entirely different
(though closely related) object.
we pause here and take a deep breath.
i’ve pointed at two distinctions:
sequences-as-opposed-to-sets and
vectors-as-opposed-to-points.
many textbooks… and many instructors…
are *very sloppy* about keeping these
(and many other suchlike) distinctions clear.
the *notations* used in *distinguishing*
the distinct situations in each case were
“delimiters”: pairs of opening-and-closing
symbols used to mark off pieces of code
meant to be handled as single objects.
delimiting conventions are vital even in
ordinary literacy (“you see? he” sa)i(d.
and i claim they’re all the more so in maths
(since we get fewer and weaker “context clues”
when the code gets munged [as in the example]).
and students’ll just leave ‘em out altogether
if they think… or god help me, know… they
can get away with it. failing that, choose
randomly (itt…oghm,k…).
failing that, “well, you *know*
what i *meant* was”…
too late, too late. here endeth the sermon.
in our next episode of “who stole my infrastructure?”:
the dot product. when they came for the opening-apostrophe,
i pleaded and begged. when they came for the
sign-of-intersection i raved incoherently.
never had a chance, no hope, no hope. doom doom doom.
can somebody pick up the torch, here?
i don’t think i can go on much longer.
vlorbik on punctuation for the twenty-twelve.
more clarity!
at the foundation of (an earlier version of)
this blog i ranted and rambled about
a tendency on the part of (lower-division
college math) textbooks to hamper the work
of the teacher by (deliberately!) suppressing
correct technical language.
of course things have continued to deteriorate.
but, by some miracle, i’m still earning
the random crust of bread by helping students
learn to *read* these ever-more-horribly flawed
documents. so far so good, then, i suppose.
anyhow, i’m grading linear algebra again
right in here (nothing *but* linear algebra
for something like a *year* now)…
and i’ve only recently become vividly aware
that this “tendency” has penetrated deeply into
the textbooks at this “higher level of the game”.
specifically, i hereby announce that some
satanic force has somehow (even here) replaced
*the sign of set-membership* (
with its mindbendingly-wrong “plain english”
equivalent(s). the perfectly-correct
(and altogether-necessary) symbolism
is now to be replaced, by the edict of
invisible (and mostly unimaginable)
entities, with “x is in S”.
[
this is a good place to skip ahead.
i'm going to geek out slightly here.
you *don't* have to be an adept to follow.
i'm hoping to make a point that can be
at least *partly* understood by math laity.
is the set of natural numbers (more here...
much my most popular post here and probably
my best-read production of all time) "in" the set
of real numbers? loosely, yes. more precisely,
i can easily imagine myself talking to, say, another
teacher about, say, some "property" (like commutativity-
-of-addition; x+y=y+x [for all x & y]) that applies
in the natural numbers. “how do we *know* it applies?”,
i might say. and the answer might come: “because
the naturals are in the reals, and the *reals* enjoy
the property of commutativity”. “good answer!” i would
then reply, and move on to whatever i *really* wanted
to talk about.
again. are all possible probabilities in the reals?
well, yeah, (duh)! in “code”, one has ![[0, 1] \subset {\Bbb R}](http://s0.wp.com/latex.php?latex=%5B0%2C+1%5D+%5Csubset+%7B%5CBbb+R%7D&bg=ffffff&fg=444444&s=0 "[0, 1] \subset {\Bbb R}")
rephrasing, “all the numbers from zero to one (inclusive)
are in the reals” (but also, more precisely,
“the [closed] unit interval is a subset of the set of real numbers”)…
so. now i’m talking to some grad-school dropout (say): “is
the-interval-from-zero-to-one in the reals?” she asks;
“heck, yes” say i, and we get on with whatever we’re really doing.
is “pi” in the real numbers? sure! 
“pi is an element of the reals”.
but wait! being-a-subset is an *entirely different* relation
from being-an-element! pi is simply *not* “in” the reals
in the same way that 
who cares? well, me and a few hundred thousand others or so.
if *you* don’t care? well, that’s why i invited you
to skip this part! read on!
]
the biggest problem from a practical standpoint
(if “how can we make this material better understood”
is a practical question) is simply that students
*hate writing* and at *every opportunity* will
replace “plain english” with (typically very
ill-understood) bits-and-pieces.
*nobody*… no student, no lecturer, no pro
mathematician… is going to write out the phrase
“is a real number” a whole lot more than
twenty or thirty times (in a given sitting-down)
without wanting *some* abbreviation for that
phrase.
and likewise for “is a subset of”… indeed,
*any* sufficiently common phrase *begs for
abbreviation* even in “plain english”
so there it is. mathophiles also… in some sense…
“hate writing”. anyhow, we *love abbreviating*.
“algebra is the science of equations”, i once
heard someone say (explicitly repeating something
he had learned “by rote” from a public-school
teacher during his own schooling… it went on
for another few lines but i didn’t learn that
part from listening to this guy say it three
or four times that one night). and i consider
this to’ve been very well said.
so i’ll hope to return to it.
but first. this history of elementary algebra
at w’edia summarizes the standard dogma of its subject (as i
understand it) well. the evolution from “rhetorical” algebra
(describe *everything* in plain-language words) through “syncopated”
algebra (where “shorthand” symbols [many still common today]
began to replace the most common techical terms… but the
actual *reasoning* was still natural-language based [and so,
by contemporary standards, "informal"]) into
“symbolic algebra” (the “science of equations” as we know it
today: a study of “formal” properties of [carefully-defined!]
symbolic “objects” ["variables" and "equations", for example]).
what w’edia *doesn’t* make much of… but what matters to me
a great deal… is that the emergence of algebra pretty
closely *coincides* in (so-called) *western* history with
the (so-called) renaissance and the (no sneerword necessary)
scientific revolution.
“modern times”, then, *began* when certain humans (*finally*!)
figured out how collections of symbols-on-paper (representing
certain abstractly-defined-objects), produced according to
various “rules”, could be interpreted to reveal previously-obscure
*laws of nature* (so-called). this *was* the “scientific revolution”.
and how does it work? *equations* is how.
so one of the *first* things in understanding
what’s going on the contemporary philosophical
environment is to find out *what an equation is*
(for all literate people): “the equality meaning
of the equals sign”.
when we’re being sloppy, we can confuse “=” with “is”…
but when we actually get to work *using* equations,
we have to *much more precise* to get any value from
the procedure at all. plain-english “is” is *always*
in some sense metaphorical (except in empty utterances
like “it is what it is”)… whereas the equal-sign
rightly-used is as far away from metaphor as we know
how to get.
how does algebra work? (equations and *what* else?)
by *the method of substitution* is how.
“in a context such that A=B is taken as ‘true’,
a properly-written piece of code
including (the symbol) B
*does not change its truth-value*
when (the symbol) A is *substituted* for B.”
this “method” *characterises* algebra.
i first became aware of some its awesome power
in about seventh grade.
and *what* else? “doing the same thing to
both sides of an equation”. and what else?
that’s about it. that’s algebra. the rest
is commentary.
now, for *set theory*, two of the main ideas
are caught up in set-inclusion and set-containment:
and one must be every bit as careful in the use
of these symbols when studying sets as one must be
in the use of the sign-of-equality in studying,
say, polynomial equations (i.e., pretty much,
in algebra).
but about forty percent of the class already don’t
take *equations* seriously. and they’re morally
certain that “sets” are meaningless traps designed
to distract them from “how do i get the *answer*?”.
and the textbook does a great deal to encourage them
to *maintain* this view.
and i don’t like it. please stop.
