-- The link to reality, as I see it, then is essentially in the axioms. For some this link is irrelevant, because the games to be played (intellectually) in such universe are gratifying in themselves, like for real mathematicians, say. -- For others, this link motivates moving in such intellectual universe at all, as it were: not for the fun of it, but for the use of it in another universe, say that of practical life applications.

But here confusion galore, again: I once went to a math-conference on " Semigroups and their applications " (Prague '96). This title attracted me since I had spent several years on the applications of finite semigroups as "sequential closure" of FSM's (Finite State Machines) in the hope to find ways of decomposing sequential behaviour (as specified by an FSM) into a network of smaller FSM's, by factoring their closure (=semigroup). This seemed a promising approach, after deriving the 5 Basic State Machines (those with semigrp of order 2) and recognizing them as indeed essential & sufficient for all "practical purposes" (of digital network synthesis). For me this was an eye opener.
About the conference on "applications" (I had no occasion to interpret the program properly beforehand): most papers were on Category Theory, and not-to-be-followed by a simple engineer with some discrete math background. -- My point: it all depends on the context what is the meaning of words like application or real or ideal or infinite ( countable or not? ).

From the discussions on decimal representations (re: .9999* =?= 1.000* or: Cantor's diagonal) I recently aquired a healthy distaste for the infinities beyond countable : my private right. I'll advisedly keep my tendency to stay with objects that can be generated constructively, and represented uniquely, and compared for equivalence or equality, if need be.

This means: stay with a generative view of the math-spaces relevant for my other interests, like synthesis ( design ) of digital ccts & systems. See for instance my recent suggestion for Cantor's "diagonal", which I assume implies a square & countable w x w table for the suggested comparison of countable length strings. Cantor's reals & diagonal: not both complete (.htm) This allows to know precisely the "missing codes" in an assumed complete table -- which turns out to be not complete after all. With the quite interesting bootstrap -like generation function of form 2^(2^(2^(2^(.....(2^2))))...) -- countable of course! So in fact negligeable with respect the "real infinity".
Roughly : . . Each finite structure is as a prefix of infinity, (quite sufficient for me).

All this, due to my insistence on generation, yields a countable subset of the "reals", no surprise (of course). Which suits me fine: a "real" that cannot be displayed, not even in countably much time/ space, does not really interest me, for the reasons explained above (qua motivation). -- "Mathematics" is a multi-dimensional world, just like "engineering" (EE, Chem-eng, Mechanics, Civil-eng, Aircraft-eng, Roads/Waterways-eng, etc...), so nothing new under the sun. . It just requires to be aware of the psychological motivations of people who work in the various kinds of disciplines, and learn to know who/how to ask questions, and who/how to discuss, in this NG.

====================================================================
 Re: Why the attack on Cantor's Theory?          sci.math 1mar99:
[**-----------------------------------------------------------------
-- Josh Scholar:
Attacks on the use of the infinite in mathematics are neither new nor
necessarily naive and have enjoyed much more respect from the
mathematical community over the years than this over the top ridicule
would suggest.

If you've never heard of Intuitionism or of Luitzen Brouwer you might
start check any book on the history of Mathematics.  You might check
out the next link - http://werple.net.au/~gaffcam/phil/brouwer.htm

Also, formalizing numerical analysis by analysing limited precision
number systems is a useful topic.  I've read articles about people
working on the subject precisely from the point of view of
understanding what can be computed by a machine. Once again the idea
deserves and is getting more respect from mathematicians than the
sniggering in this news group would suggest.
------------------------------------------------------------------**]

NB: Amen.
    The Finite (say Gauss' residue arithmetic;-) and the Infinite
    (say Cantor's diagonal argument) are complementary.
Taking only one of these as the "Real" Math is rather childish and
unbalanced. It does require a double focus though, because they are
essentially different -- as are the Discrete and the Continuous ;-)

However, for solid logic and essentially trouble-free math results,
I bet any time on the Finite ...;-)
------ The Finite as a [prefix-] projection of the Infinite --------
------ (object/semantics)          (compositionrule/syntax) --------

 Still a lot needs to be done there: ever seen a good general method
of attacking Diophantine eqn's? (re: Hilbert's 10-th problem, 1900).
Or a general structure theory (all details) of any finite semigroup,
as sequential closure of a Finite State Machine (=deterministic
computer model, Mealy 1956)?
So get cracking, finally. Don't be shy, the century is almost gone ;-)

Ciao, Nico Benschop -- http://www.iae.nl/users/benschop
 "Structure of Constant Rank State Machines,
          and the five Basic State Machine Types" :
  http://www.iae.nl/users/benschop/c-ranksm.pdf  <----------<<<<<<<
http://www.iae.nl/users/benschop/cantor.htm (Finite Sequential view)
http://www.iae.nl/users/benschop/fewago.htm (Fermat, Waring, Goldbach)

=======================================================
(Re: Cantor & Uncountable & Infinity)  sci.math 1mar99 :
-------------------------------------------------------

The whole idea of some Ultimate Infinity, not to be transcended, is
quite immature since it considers only quantities, and not qualities.
  Mapping Everything onto a "real" line is quite absurd:
  don't there exist Graphs, (Semi-)Groups, Languages, etc..etc?

Even in Cantor's case: comparing 2^N with N is comparing two distinct
TYPES of objects -- Flattening all onto a single line, with 'density'
or some other measure supposedly capturing the lost essence is short
of ridiculous, or to put it mildly: ill advised. The powerset 2^N of
Peano's sequentially generated naturals N = {+1}* has a totally
different structure, namely that of a Boolean Lattice (intersection
and union) - given some axioma that these finite concepts do make
sense for infinite objects. So what?

Why count & compare dissimilar things? Are you sensibly comparing the
Alhambra with the sandgrains it consists of eventually...?  Whole
generations of mathematicians are raised on Cantor's diagonal, and
still new generations revive it in the P?=NP arena -- All this about
some form of infinity, *without* considering differences in 'quality'
cq 'type-of-structure'..

The limit IMHO is this: For a realistic person, there are a^n distinct
strings of length n over an alphabet A of size |A|=a. And if it makes
you happy: let this also hold for infinite strings over some infinite
alphabet ;-)
  There is an algebraic structure S (semigroup of all transformations
of a set T), defined over a set T of n symbols (states or letters)
that contains maximally n^n objects (functions of T into T), which
even for finite n has a general structure/ decompostion that is not
understood fully. Groups [n!] and Lattices [2^n], Languages [a^n] and
Arithmetic and Logic, they all fall under this denominator (the
associative algebra of strings under concatenation, including Graphs
and Matrices and StateMachines, etc... ;-).

Can't we decide to waste a bit less intellectual energy on pressing
these all onto a single "real-line" - And rather spend some effort in
distinguishing the various types of objects, with corresponding
composition operations, and build some integrated understanding of all
these things and their structure - Finite first... and Infinite later
- after we at least understand them in-the-finite?  Otherwise, doing
the infinite before the finite, might suggest some urge to escape,
would'nt it ?-) --------- (voer voor psychologen;-) --------

Just a suggestion...


Re: A formal definition of a problem, and how I can compute it.

. . . Computer 'proof' of God.
. . . On foundations: Aristoteles' logic, Spinoza, Clarke, Boole, Shannon.


From: Nico Benschop
Subj: 
Re: Absolute Proof of God: Part I
Date: 13 Oct 1999
To  : Fred Galvin 
News: sci.math

Fred Galvin wrote:
>
> On Mon, 4 Oct 1999, Clive Tooth wrote:
>
> > Off-topic crap.
>
> It may be crap (I didn't read it), but I don't see how it can be off
> topic, as the existence of God has consequences for the foundations
> of mathematics.  ...[*]
>      For instance, if God exists--I assume we're not talking about
> Mercury or Thor here, but the omnipotent omniscient God of Christian
> theology--then the axiom of choice must be true, as the God would be
> capable of choosing an element from every nonempty set, well-ordering
> the universe, etc.; and questions such as the true size of the continuum
> would have a definite answer known to the Almighty. Historically, great
> mathematicians from Euler to Goedel have offered proofs of the existence
> of God (all right, maybe Euler was just kidding), and theological
> considerations are said to have played a role in Cantor's thinking
> about infinite sets.

Indeed, in fact George Boole in his "Laws of Thought" (1854),
  [ where he introduces his (set-) algebra as an idempotent branch of
  [ arithmetic (for properties x, y of an object: x^2=x, so x(x-1)=0 -->
  [ with roots x=1, x=0; and commutative xy = xy : no order dependence)
mentions that a main motivation for him to develop this binary logic
was to formalize & simplify the age-old logic of Aristoteles, as employed
by Spinoza (17-th century Dutch philosopher) and more recently Clarke
(spelling?) for their God_existence 'proofs' -- And to check these proofs.
       (IIRC: Boole himself was son of a clergyman/minister in Ireland)

His conclusion: those arguments were 'circular' (the result was in the
premises;-) No surprise: axioms are assumptions, cannot be 'proven'...
Nowadays we use  Boolean algebra for more practical purposes, like the
optimal design of combinational logic circuits, after Claude Shannon in
1938, as a student in MIT, noticed the isomorphism between Boolean(+,.)
and switch_connection(parallel,series). That took some 85 years !
    (BTW:  how long after 1928 Schushkewitch' paper on the detailed
     structure of finite simple_semigroups will it take to develop
     associative function composition algebra (=finite semigroups)
     for optimal sequential logic design in computer engineering
     and digital network theory, for synthesis purposes ?-)  ..[2]
--
Ciao, Nico Benschop  - http://www.iae.nl/users/benschop ..ref[2]
"The Structure of Constant-Rank State Machines" (.pdf) 

--- Principle of Dynamic Balance: -One is Always Halfway Anyway (AHA) ---

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx1.1xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Subject: Re: philosophical question
   Date: Sun, 02 Apr 2000
   From: Nico Benschop
  Org'n: Amspade Research
Newsgrp: sci.math
---
Rajarshi Ray wrote:
>
> I want to ask you guys a question that is not exactly mathematical
> but leaning toward philosophy :-
>
> Do you believe, like Pythgoras and Kronecker, that the only things
> that are truly significant are the whole numbers i.e. the sequence
> 1,2,3... and that everything else is somehow derivative of that
> concept? If so do you consider this to be an actual feature of the
> universe itself, whatever that means, or do you think it is a feature
> of our own human minds?  Just some thoughts I'd like to share!
> --
> Reality is that which refuses to go away when I stop believing in it.
>                                                      Phillip K. Dick

I think that the (positive;-) integers were a first step towards
abstraction, which allows expressing insight about more than special
cases, regarding objects around us (that come first: I strongly believe
that we learn 'bottom up' ... from the special to the more general).

Furthermore (IIRC) Descartes mentioned as essential for reasoning:
there is a need to distinguish things & concepts sharply from each
other, before we can ever hope to reason reliably about them. This
assumes, I think, a binary outcome of our research on a topic, say a
proposition being either true or false (in those days: God exists or
not;-) Hence not something like: in this context that particular truth
holds, while outside that context the whole concept is irrelevant, or
cannot even be defined properly. -- (so don't discuss 'intuition' or
'feeling' at all ... if you want to be taken seriously, that is ;-(

This assumption of discrete reasoning & its outcome, perfectly fits the
integers. Moreover they have a linear (1-dimensional) model a la Peano,
one-generator N = (+1)*, which nicely fits the single-focus linear way
of step-by-step proof that most mathematicians prefer (so no parallel
or multi-dimensional development of concepts & proof please!)

The alternative: continuous variables, have always posed severe problems
in trying to be 'exact' a la Descartes (re Cantor, the Reals essentially
being a 2-dimensional concept: his 0/1 strings {0,1}*, and diagonal ;-)

In my view, the sharpest dichotomy  in math & physics is that between
adherents of the Discrete (particles, integers, finite) versus the
Continuous (waves, probabilities, reals). My guess is that only if
_both_ are taken as essential, a bridge connecting them can be found
(re my homepage http://www.iae.nl/users/benschop/cantor.htm ); Then a
constructive & realistic model of the world around us can be built.
At least there should be a balance between objects & abstraction, or:
between semantics (=representation) & syntax, the finite & the infinite.

Ciao, Nico Benschop - http://www.iae.nl/users/benschop

----( a Natural Law: product of quantity and quality = constant )----
----( a Problem:  quantity and quality are orthogonal concepts  )----

Subject: Re: Ludicrous speculation
   Date: Sun, 16 Apr 2000
   From: n_f_benschop
  Org'n: Amspade Research
   News: sci.math
------
In article <8d9qi3$3gt$1@nnrp1.deja.com>,
  ca314159  wrote:
> In article <38F6E337.35B58DEB@chello.nl>,
>   Nico Benschop  wrote:
> > > NB:
> > > > A paradox is a subjective contradiction in two ways of
> > > >'understanding'.... usually a mark of something missing in our
> > > > models. IMHO this has nothing to do with 'objective' nature
> > > > (in sofar as this is possible, that is: if we can understand
> > > > it without contradictions. I guess there _is_ no such thing
> > > > as objective nature without a model;-)
> > > > -- Ciao, Nico Benschop
> > > > -- http://piazza.iae.nl/users/benschop/math-use.htm
> >
> > goodwillhunting wrote:
> > >
> > > I am not sure that things should always make logical sense in
> > > physics. We see human behaviour which is highly illogical, and
> > > even though we are talking about humans, we are still looking at
> > > the behaviour of energy - in the form of a human being.
> >
> > True. On the other hand, what seems illogical from one prespective,
> > may be perfectly logical using another 'model', creating a paradox.
> > Like the wellknown particle/wave controversy about a model of light.
>
>     Simpson's paradox is an interesting example in statistics
>     of apparent contradiction.
>
> > Similarly, human behaviour is so varied that what is 'apparently'
> > very illogical, may turn out to have a perfectly logical
> > explanation from another point of view (re Sherlock Holmes;-)
>
>      It's interesting that Conan Doyle was quite unlike his
>      Sherlock Holmes in later life and Lewis Carroll spoofs
>      on logic were contrasted with Charles Dodgson as logician.
>
> > My point (partially) is that one should try to be flexible enough
> > to allow several viewpoints, that is: not be prejudiced to just a
> > single focus. This is quite against the established causal model
> > in Western thinking/science/math (with a linear cause-effect
> > sequence, and a striving for a "theory-of-everything" and a
> > minimal axiom base).
>
>     Logic is largely causual and so temporal. It might be
>     said that analogy is acausual and spatial as it refers
>     to a synchronicity of states in a state space.

In my thinking frame (state machine synthesis;-) this would be:
   causal :: sequential :: logic   :: arithmetic :: Peano induction
  acausal :: parallel   :: analogy :: morphism :: 'resonance'

> The west being more time optimized is "technological" while the
> east seems more optimally analogical and dominated by
> non-phonetic spatially optimized "ideogrammatic" languages.
>   It takes a long time to write each chinese character but they
> say alot more and are less specific in their meaning or "state".
> The space-time trade offs of formal languages come to mind.
>
> > A "multi-focus" model is not necessarily 'illogical' - although
> > Kepler's dual-focus ellipse model of planetary motion did stir
> > quite some commotion (question: with the Sun in one focus,
> >              _what_ is in the other focus? ;-)
>
>    The "quantum mirage":
>       http://www.almaden.ibm.com/almaden/media/mirage.html
>       http://www.almaden.ibm.com/almaden/media/image_mirage.html
>    just like the floating penny Miracle Mirage toy ?

Thanks Robert, nice & interesting reference!

>  Analogic seems to have received alot less attention
>  ^^^^^^^^          as a science than logic.

Indeed. It is the creative part :
                  ^^^^^^^^
     _where_ do your idea's come from?   From analogy....

>      "Superpositions" of concepts in idioms are alot like
>      superpositions physical concepts like "wave" and "particle"
>      and many theories in linguistics have physical counterparts
>      like Diachronic/Synchronic analysis, "Linguistic Relativity"
>      theory, and "Translation Indetermincy".
>
>      Curious phenomena like synchonicity and numerology probably
>      in some cases have a 'rational' basis in that twilight zone
>      between logic and analogic:
>
>       http://groups.google.com/groups?oi=djq&selm=an_611181761

on 666 (the number of the Beast)=  # 'Computer' = # 'HolyBible' !-)

>  [...]
>  My question is, can a curriculum be developed to educate
>  students in the rational use of logic and analogy without
>  them getting lost in philosophical speculation.

Nothing wrong with 'philosofical speculation' and 'analogy'
as one side of that coin;-)    As long as it is balanced by
the other side that we know all too well, some 4 centuries
since Kepler, Galileo, Descartes, Newton, Huygens, etc...

---- AHA : One is Always Halfway Anyway ---- (between past & future)

>  It seems fortunate that some people are inherently logical like
>  scientists, while others are inherently analogical like artists.
>  If everyone were "quantum", human energy would be spread thin
>  (like the Zen of buttering bread, in the movie "Diva")
>
> > --
> > Ciao, Nico Benschop -- http://www.iae.nl/users/benschop/search.htm
> >

Subject:  Re: What works in undergraduate mathematics?
   Date:  Tue, 2 May 2000
   From:  Nico Benschop
  Org'n:  Amspade Research
Newsgrp:  sci.math
--------
Allan Adler wrote:
>
> This seems to fall into the class of questions of the form, "Doc, it
> hurts when I go like this," and the answer is "Don't go like this."
> [...]
> You can't fix it. It was never meant to work. It is not a solution
> to the problems of education or scholarship. It is merely the only
> wheel in town.
> What I think is that it should stop being the only wheel in town.
> This is difficult as long as everyone is persuaded only by the one
> rule of inference (money) that pays their bills. [...]
>
> I say, "Out of the universities and into the streets!".
> Work where you must, but also leave the workplace behind and teach
> and learn in society itself.
> There is a need for adult role models to participate in intellectual
> activity for its own sake and to take it seriously. A math professor
> who never thinks about anything but math or mathematicians or the
> politics of the math department is not setting a good example.
>
> How to do this is a matter for continued experimentation. [...]
> I don't pretend to have much success in this area.
> My primary student is myself and I'm trying to succeed
>             with that student where the system failed.
> One of my experiments is my sporadic journal Labyrinths :
>    http://www.swiss.ai.mit.edu/~adler/LABYRINTHS/labyrintro.html
> Another is a set of activities I am trying to get off the ground
> in Bowling Green, KY. For some current information on that, see:
>       http://www.accessky.net/wwwwwww (there are 7 w's)
>
> For me, the question is not how to get the great bureaucracies
> to work; there are already enough people working on that.
> I see the question as:
> ** How to nourish the neglected area of non-bureaucratic education.**

This is the core of the matter, and in other countries called (among
others): "education permanente" or "continued education". It has to
do with motivation: _why_ learn something of which you really have
not much (or any) clue about its applications & possibilities?

Only later in life an intelligent person gets the urge to learn what
others have learned & presented in books, theaters, musea, films, &c.
_Then_  you pick what you need, and who knows: something more, just
for the heck of it - just because it is interesting and fun.
Quote (Albert Einstein):
  "It is a miracle that curiosity survives formal education."

Surely, there is a 'bootstrap' involved here: the chicken-and-egg
problem of what comes first? This is at the center of many problems
of learning & solving, especially in the Western world:
    Creativity is NOT a linear thing, but cyclic (roughly;-)

That is: different/opposit aspects present themselves simultaneously,
  and it is good to develop a multi-focus sensitivity to disparate but
essential aspects of a (any) whole... To bring analysied & linearized
material to a youngsters doorstep and expect him/her to study this for
the first 20 years or so is asking for trouble, really.

To study math for many years as a 'pure' discipline, _without_ the
benefit of doing/living a self-supporting life is a rather abstract
model of life, which is breaking-up any motivation of all but the
abnormally concentrated...

> This is a concept in search of a definition, and for the right to be
> called education even if it isn't funded. One can get a glimpse of
> what it might mean by considering the example of libraries and
> museums, which pretty much leave people free to follow their own
> curiosity. The institutions themselves might have their own
> bureaucracies, but patrons are largely shielded from them, unlike in
> schools where the bureaucracies are as much a part of the problem as
> a part of the solution. -- Allan Adler,  ara@zurich.ai.mit.edu

The little I learned of discrete math (finite Semigroups, Arithmetic,
Boolean logic) is self-taught, as I needed it in my job (digital VLSI
design research), starting with sequential logic with the (Finite)
Internal State Model (FSM: Mealy/ Moore, Hartmanis/ Stearns, Birkhoff/
Bartee, Clifford/ Preston) and then _only_ those chapters that seemed
relevant.     And not to forget my favourite "The development of
Mathematics" by E.T.Bell, a marvellous motivating & critical book [*],
with enough detail (upto mid last century;-) and A.Reny: "Dialogues in
Mathematics".    [*] http://www.iae.nl/users/benschop/quotes.htm

My guess is that at least some of these writer pairs are a combination
of Math/Engineering (correct me if I'm wrong here;-), precisely what
I'm talking of: a multi-disciplinary outlook, a balance of form &
content, of semantics & syntax, of representation & abstraction.
Just some thoughts...  --- NFB.

 Subj: Re: Does probablity depend on knowledge of what happened before ?
 Date: Wed, 21 Jun 2000 ( sci.math )
 From: Nico Benschop
Org'n: Amspade Research
------------------
Doug Magnoli wrote:
>
> Oh my, oh my, oh my....
>
> This is the same argument I had with my eighth grade teacher who
> contended, while we were studying genetics, that if the family
> next door had 4 girls, the next child would likely be a boy.
> (We're assuming 50% prob for each gender here, and her rationale
> is the same as yours). I asked her, 'if you flip a coin and it
> comes up heads 5 times, what's the next toss likely to produce?'
> She answered, 'tails.'
> 'And if I flip the coin and it comes up heads 5 times, and then
> I walk away, leaving the coin on the table, and you come along
> and flip it, what's likely to occur?'
> She said, 'it's 50% probability of heads or tails.'
> 'Why the difference?,' I asked.
> 'Because we're starting over.'
>
> And that's the point, because with every toss we're starting over.
> The coin has no memory, it has no history.      ...[*]
> If we know that it's a fair coin, the probability of heads is
> always  1/2, no matter what's gone before. -- Doug Magnoli
>
> Pagadala wrote:
>
> > In a tossing experiment with a fair coin, if i were asked to
> > guess the outcome of next toss, and given the information that
> > all (or most of) preceding results were 'heads', i would go for
> > 'tail' as my guess for next outcome. Is the probability of next
> > outcome dependent on knowledge of preceding results?  ...[&]
> > -------- Pagadala

Re [*] Precisely: crucial is the concept of 'independence', which
   has to do with memory. It's difficult for persons with memory(;-)
   to imagine a process without it (as we _assume_ inanimate objects
   and processes to be)... a typical western 'objective' assumption.

Just thinking of Kafka (IIRC;-) who describes someone (himself?-),
after gruelling experiences, trying to 'continue as if nothing
happened'. ... The coin presumably does, we certainly don't...
              (if not mangled or molten;-(
--
Ciao, Nico Benschop -- http://piazza.iae.nl/users/benschop/ic8basic.htm

Subject: Re: Teaching math without numbers?
   Date: Mon, 10 Jul 2000
   From: Nico Benschop
  Org'n: Amspade Digital Research
Newsgrp: sci.math
---------
Niklas Dougherty wrote:
>
>  (Sun, 09 Jul 2000 04:36:18 GMT) wrote:
>
>  This seems difficult, teaching math without numbers. And I hope
>  you're not suggesting that children be introduced to these
>  variables and whatnot before they can even count to ten. I'll tell
>  you what's odd: why have I never seen a proof of the fact that the
>  order in which a finite set of objects is counted does not affect
>  the count? I have seen proofs of other "obvious" mathematical truths,
>  but not this one. In other words, I've never seen it proven that
>  there are as many letters in "abcde" as in "ebadc" or "dbace" or
>  in any other arrangement of those letters. This bugged the heck
>  out of me when I was 8 years old "how do I KNOW there are still
>  twelve marbles here, no matter what order I count them in?"

Strange, it never ocurred to me that the order of counting them would
make a difference. Just as the way you throw apples in a bag would'nt
make a difference in their combined weight, or...

Is'nt math _based_ on such intuitions, like:
               addition is associative and commutative;
is'nt that a more formal expression of the same?

> It can be proved in various way. For instance, one can prove that a
> permutation (switching two elements in a finite set) is a bijection,
> and that the set P_n of all permutations is a symmetric group on
> n elements of the order n!, which gives that you can count your
> 12 marbles in 12! ways and still come out with a set of order 12,
> containing the same elements (since permutations are bijective).
>
> Normally, one considers the fact: 1+1+1+1+1+1+1+1+1+1+1+1 = 12
> as axiomatically associative, so that the way that you count a
> set's  items is irrelevant.

Indeed, 'associative' is necessary for 'iteration' to be unique:
  aa.a = a.aa = a^3 (re 'counting').

Where actually does 'commutative' come in?
I guess that's also there:  a^2.a = a.a^2   <--->  2+1 = 1+2.
(can't get rid of introducing more complex concepts to 'explain'
the trivial & obvious ;-)

> Using formal set theory, you can prove this on
> a still more elementary (?) level. --  N.
          ^^^^^^^^^^^^^^^^^^^^^^^^
Interesting, the "marbles counting" intuition is for me the more
elementary (I guess your question mark tends to agree;-)
Reminds me of that quote:
      Math is: how to make simple things difficult.

Ciao, Nico Benschop --- http://www.iae.nl/users/benschop

Subject: Re: Memorization comes before understanding
Date: 10 Jul 2002
From: Nico Benschop
Org.: Digital Research

glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) wrote:
>
> William Hale  wrote:
> > Patrick Reany  wrote:
> > [cut]
> >> The objective proof of                                     [1]
> >>      having attained "understanding" of a concept          [2]
> >> is the ability to solve problems that require the concept. [3]
> >
> > This is not correct.
>
> T'is.

Indeed, but why? Can you elaborate? Your 'yes' and my 'yes' to
  agreeing with [3] as charactristic of [2] may be different...

What I mean is: your 'understanding' may still be different
from my 'understanding', although on the surface we seem to agree.

In other words, what we observe is always filtered by something
like the sum-total of our previous experiences, which filter-bank
is likely to be quite different for you and me. Of course, we
both 'know' enough of the English language, we both are raised
in a Western society, at schools that roughly aim for similar
values to be programmed into our little brains, etc. etc.

These unmentioned assumptions form the basis for our communication,
and quite some 'memory' is involved in such 'filter-bank'. I'm
 just reading "The Wholeness of Nature" (1996, UK) by Henri Bortoft
(philosofer) http://www.dialogonleadership.org/Bortoft-1999cp.html
   "In 1972 I came to know Goethe's work just by accident.
    Goethe's point was to develop a different kind of seeing,
    a seeing that strives from the whole to the parts.
    That was very close to Bohm's hologram."

In essence: there is no objective observation of 'facts',
we always interprete (often unconsciously) what we see / read,
hence filter it, while putting it in our 'memory' with some
label as to its context and value. The concept of TRUTH is
also very much filtered personally, and dependent on context.

E.g.: Goethe did not understand, or rather disagreed, with
  Newton's theory of colours (being purely numerical, quantitative)
while he (Goethe), as artist, was missing something very essential
namely the 'quality' of colours and their value in our visual
image of the world we live in, and the physiology of our eyes,
yielding a colour-in-context of other colours and shapes
(re: a 'random' collection of spots (a la Rohrschach) that, after
 some observation time, many people agree to represent a giraffe)

So, as to 'memory' and 'understanding': the terms into which
you provide a further decomposition / explanation (or rather
  'embedding' into a larger context of personal experiences)
will probably depend on _your_ context & expectations.

For me, 'memory' in this context would correlate with
  'association' and 'resonance' of certain 'features'
  (re feature extraction in OCR: optical character recognition)
while 'understanding' has to do with 'able-to-apply' in
a 'relevant' context (recognized as such by my filterbank;-)
                     (some bells went ringing...)

Towards more math-like terms:

 understand => similar => resonance => analogy => (iso)morphism

In fact: the two concepts of memory and understanding are
             not so clearly separated in my mind, really...

  (in my early schooldays I hated learning 'by rote' the
   words & rules of the several foreign langages that we
   in the Netherlands must learn: French, German & English.
   So it was a relief to me that e.g. all those trig formula's
   could be derived easily from very few basic ones; hence my
   preference for a 'generative' view of things & FSM's &c :-)
[*]  countable = A* and uncountable = B*, with |A|=1, |B|>1.

-- NB - http://www.iae.nl/users/benschop/math-use.htm
        http://www.iae.nl/users/benschop/cantor.htm    [*]