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.