Recently I've Posted (what I've Been Calling) Rambles On Foundational

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 andr ew 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 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... $latex \TeX$... 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. can't yet put 'em 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 quit taking on new users. some experimental deal. all for this morning. what's to eat? a couple days ago i discussed a technique for listing all the permutations of a (small) set. essentially this: count the elements. suppose there are n of 'em. there are n! (en-factorial) permutations. our first letter... whatever it is... will be the first letter *of* the *same* number of permutations as will the second letter or *any* letter. so we can say that each letter begins (1/n)th (one-enth) of the list. note that (1/n)*n! = (n-1)! and write out this many copies of each letter in columns. end of step one. for {E,G,B,D,F}... the example i "assigned"... my first inclination is to remark that {E,G,B,D,F} = {B,D,E,F,G} (sets do *not* depend on order)

and so one could begin as follows. B____ B____ B____ B____ . . . B____

D____ D____ D____ D____

E____ E____ E____ E____

F____ F____ F____ F____

G____ G____ G____ G____

D____ E____ F____ G____

(24 rows... there are 5!= 120 permutations to discover and 1/5 *of* 'em start with B: so there are 24 B's. etcetera.) in step two we'll fill in the *second* letter of each permo. consider the B column only (for now). there are four "remaining" letters. each will appear as the *next* letter just as many times as each of the others: this is 1/4 of the total... of 24 B's... and so *six*. "count by sixes" down the first column: 6 D's, then 6 E's, and so on down. BD___ BD___ BD___ BD___ BD___ BD___

D____ D____ D____ D____ D____ D____

E____ E____ E____ E____ E____ E____

F____ F____ F____ F____ F____ F____

G____ G____ G____ G____ G____ G____

BE___ BE___ BE___ BE___ BE___ BE___

D____ D____ D____ D____ D____ D____

E____ E____ E____ E____ E____ E____

F____ F____ F____ F____ F____ F____

G____ G____ G____ G____ G____ G____

BF___ D____ E____ F____ G____ . . . BG___ D____ E____ F____ G____ be sure you see what's going on: all the ideas are in place i think. if it's clear why we're creating 120 entries in 24 rows and why we counted by 6's in this step, everything else should fall into place pretty easily i hope.

count by sixes throughout *each* column, omitting whatever letter has "already been used" of course. BD___ BD___ BD___ BD___ BD___ BD___

DB___ DB___ DB___ DB___ DB___ DB___

EB___ EB___ EB___ EB___ EB___ EB___

FB___ FB___ FB___ FB___ FB___ FB___

GB___ GB___ GB___ GB___ GB___ GB___

BE___ BE___ BE___ BE___ BE___

DE___ DE___ DE___ DE___ DE___

ED___ ED___ ED___ ED___ ED___

FD___ FD___ FD___ FD___ FD___

GD___ GD___ GD___ GD___ GD___

BF___ ... . . . BG___ DG___ EG___ FG___ GF___ each column falls into groups of six unfinished permutations as of now. each such set has two letters present and hence three letters left to "choose from". one-third of six is two so considering "BD" (for example) we'll add *two* E's, *two* F's, and *two* G's. and likewise for *each* of the sets-of-six: count off "missing" letters by twos. BDE__ BDE__ BDF__ BDF__ BDF__ BDF__

DB___ DB___ DB___ DB___ DB___ DB___

EB___ EB___ EB___ EB___ EB___ EB___

FB___ FB___ FB___ FB___ FB___ FB___

GB___ GB___ GB___ GB___ GB___ GB___

BED__ ... . . fill in all the columns by the same procedure. one now has a collection of 60 pairs of three-letter strings. complete the strings by attaching their two "missing" letters: once in one order and once in the other. of the two BDE's, for example,

one becomes BDEFG and the other BDEGF. this completes the whole ordeal. BDEFG BDEGF BDFEG BDFGE BDGEF BDGFE BEDFG . . .

DBEFG ....

the philosphically inclined can assure themselves that the last bit can be thought of as counting "by ones" and "by zeros"; i like to think of this as counting *choices* to be made. more later.

madeline's awake.

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, 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.

/ ^ / /-------@ / \ / \ * / /@ / / /-- ^ \ \ \ \* \ /@ \ / \_______* \ \^ 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 $latex A \leftrightarrow X$ $latex B \leftrightarrow Y$ $latex C \leftrightarrow Z$ "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...) $latex A \rightarrow X$ $latex B \rightarrow Y$ $latex C \rightarrow Z$ $latex A \rightarrow X$ $latex B \rightarrow Z$ $latex C \rightarrow Y$ $latex A \rightarrow Y$ $latex B \rightarrow X$ $latex C \rightarrow Z$ $latex A \leftrightarrow Y$ $latex B \leftrightarrow Z$ $latex C \leftrightarrow X$ $latex A \leftrightarrow Z$ $latex B \leftrightarrow X$ $latex C \leftrightarrow Y$ $latex A \leftrightarrow Z$ $latex B \leftrightarrow Y$ $latex C \leftrightarrow X$ . now.

in the spirit of the
HREF="http://vlorblog.wordpress.com/2009/11/09/notes-for-chapter-zero-modified-byhand-with-considerable-grumbling-from-the-tex-code-but-we-wont-be-doing-thisagain-soon/" rel="nofollow">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}. 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 $latex P\rightarrow E, D \rightarrow I, Q \rightarrow O$ 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.