A Taxonomy of Ambiguities, Paradoxes, Variables
m...@ms.lt
use variables.
I found a book on that very topic by William Byers, "How Mathematicians
Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics".
As I work on this, I'll share my progress at Math Future
http://groups.google.com/group/mathfuture/ and my own group, Living By
Truth http://groups.yahoo.com/gropu/livingbytruth/
This is part of my lifelong quest "to know everything and apply that
knowledge usefully". In 2009, I overviewed my conclusions with my
10-minute video "I Wish to Know"
http://www.youtube.com/watch?v=ArN-YbPlf8M Since then I organized a
"House of Knowledge" of 24 ways of "figuring things out"
http://www.selflearners.net/ways/ Now I'm pulling together all of this
and presenting it as an art show this April at the South Side Community
Art Center, http://www.southsidecommunityartcenter.com, a historic gallery
in Chicago's Bronzeville.
I hypothesize that there are six ways that our minds use variables and
that they correspond to pairs of questions: Whether? What? How? Why?
Together they make for 10 of the ways in the House of Knowledge.
I've been sorting paradoxes listed by Wikipedia:
http://en.wikipedia.org/wiki/List_of_paradoxes
http://www.ms.lt/sodas/Mintys/Paradoksai
Here's a first attempt at sorting the paradoxes:
* The properties of a whole can differ from the properties of the parts
* A contradiction can be contained locally
* Description can restrict freedom
* Attention can affect what is observed
* A statement can negate itself
* A definition is always incomplete and open to multiple interpretations;
in particular, it can be vacuous, having no actual examples
I appreciate very much our efforts to be as complete and inclusive as
possible! I'm trying to get to the germ of contradiction in each paradox
and appreciate it for what it is.
William Byers emphasizes the importance of ambiguity. His perspective is
very helpful. It's interesting that there are remarkable thinkers who
have walked through this territory. For example, Thomas Hobbes's "Table
of Absurdities" includes "category errors" ordered as six pairs from four
categories: bodies, phantasms, accidents and names.
http://en.wikipedia.org/wiki/Absurdity#Hobbes.27_.22Table_of_Absurdity.22
William Byers cites conductor Leonard Bernstein's enormously creative
video lectures from 1973. I'm listening to his 3rd lecture, "Musical
Semantics", where he explains the role of metaphor in music.
http://www.youtube.com/watch?v=8IxJbc_aMTg&feature=related
He interprets, I think convincingly, a passage of Beethoven's as a series
of pleading, cajoling, threatening, etc. and ultimately firmly agreeing.
He shows that it makes a lot of sense to relate such a drama to the music.
He asks, is this what Beethoven felt, and he transfered through his
score? Did Bernstein just make those feelings up? And he says that both
are probably true! and make for a profound ambiguity. We may understand
but can't be sure. The (musical) structure can inspire us to think that
the (semantic) interpretation is in fact correct. And such an ambiguity
is at play perhaps in all of our language, although much less obviously,
when we might not really be understanding how another person is using
certain words. (Which becomes obvious when we're in other cultures, yet
more apparent when we return to our own.) In a sense, my quest is for a
deepest understanding of the universal culture inherent in the limits of
our minds, or any minds, to the utmost detail, such that to violate any of
it would be to no longer know anything at all.
Ambiguity is thus when there are two truths at the same time.
I believe that a "variable" is the delineating of an ambiguity so that we
can agree, given four levels (whether, what, how, why), we can agree as to
which two levels are relevant, are carrying the truths, so that we can
switch back and forth between them. For example, the symbol f(x) means
both a function and an output, and I think that is not accidental. And at
any moment it means one of those OR the other. It's certainly confusing
to those who see it for the first time, however! That's one reason why it
would be helpful to clarify what's going on. My notes:
http://www.selflearners.net/Math/Variable
Paradox is when there are two conflicting truths at the same time.
Paradoxes therefore are great material for figuring out such a taxonomy.
William Byers relates this all to "ideas" including "great ideas" and how
they arise in Mathematics. (I still have to read most of that.)
I realized, while watching Leonard Bernstein's lecture, how "ambiguity"
relates to the distinction I made ("smart" vs. "stupid") in the way I
teach chess lessons. (Maria Droujkova, thank you for your apt terms
"artist" vs. "vendor"! and Kirby Urner, Algot Runeman and Juan, thank you
for your thoughtful replies!)
http://groups.google.com/group/mathfuture/browse_thread/thread/ce12c99dcb71351e
I noted that the challenge of teaching students of very different
abilities at the same time forced me to teach "smart". I taught around a
deep idea (such as "the Pawn is the soul of chess") and students learned
at different levels:
* some learned Whether there are Pawns
* and What a Pawn looks like
* and How to move a Pawn
* but also, Why to move a Pawn.
Thus, multilevel teaching is a teaching of and through ambiguity. Whereas
singlelevel teaching is the removal of any source of ambiguity (leading
ultimately to "multiple choice tests").
This suggests why we don't have answers to the basic questions that I'm
asking and others are as well. There is no single-minded way to teach
multiple-track thinking. And so it isn't taught. We're left with
single-minded teachers raising single-minded students.
We all praised the virtues of this single-mindedness and acknowledged its
central place in our civilization. Driving is a great example.
Certainly, driving in California is much simpler and more intense than
driving in Lithuania. In California, everything is clear - at every
intersection, somebody has to stop (because of a stop light or stop sign
or because they are on the lesser road). But in Lithuania and Europe in
general, you rarely stop. Instead, you yield. Consequently, there are
"right of way" signs that tell you that your road has the right of way and
the one you're crossing doesn't. The essential difference is this: in
California, you can follow the car in front of you and generally, when
they stop, you stop, and when they go, you go. But in Lithuania, you can
never rely on the car in front. They may whiz through an intersection,
but only because it was safe for them; when you get there, there may be a
car with the right-of-way read to slam into your side.
And yet we could be "smart" about teaching simple-minded driving. The
ways things are, drivers have to pass multiple-choice tests to show that
they know, for example, to give a right signal 100 feet before turning.
The standard single-minded teaching teaches them this. But how many
people have any sense of what "100 feet" means? Probably less than 20%.
Instead, consider lectures and discussion of "deep ideas" in driving.
Here are some multi-level topics:
* How to gauge various distances and why they are important for driving
and the rules that govern it.
* A list of the kinds of accidents that people have, from worst to least.
The nature of accidents, their causes and consequences, physical,
financial, psychological, moral and legal.
* Defensive driving.
* How drivers can be alert to each other.
* Signaling our intentions to other drivers.
* The most commonly violated rules. The most frequently given tickets.
* Traffic problems and how drivers can help mitigate them.
Such topics can review all the basic knowledge in the context of memorable
discussion by new drivers, veteran drivers, repeat offenders, and so on.
But we might need some kind of "jury duty" to make them happen. As it is,
we have many taboos that prevent such socializing, certainly with
teenagers. And then we wonder why they are ignorant or incommunicable.
Likewise, where do we start to make such a list of topics in Math ? I
think Maria et al's work with the youngest children is very central. For
example, "symmetry" is a topic for both babies and Nobel prize winners.
Or "fractality" or "proportions" or "exponential growth".
As I work on my art show, I've been collecting questions that we're asking
ourselves, and don't yet know the answer to. My adult math student
Nashawne Ball asks, "Why are systems designed for things to go wrong?"
Now that makes for great math questions! that bring out great math topics!
We're interested that she develop her acting ability and likewise that
question can inspire us. How can she act out the purposes and limitations
of various models, various ways of mathematical thinking?
Thank you for ideas, topics, questions! Also, as I prepare for my show,
I'm creating videos http://www.ms.lt/tv/ to foster my creativity and play
with others, co-create and work towards a shared culture of creativity.
Please do send photos, videos, music etc. that I could include in the
Public Domain.
Andrius
Andrius Kulikauskas
m...@ms.lt
(773) 306-3807
http://www.selflearners.net
Julia Brodsky
This is one of my favorite ones.
Regards,
Julia Brodsky
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Julia Brodsky
www.artofinquiry.net
kirby urner
<< snip >>
> Instead, consider lectures and discussion of "deep ideas" in driving.
> Here are some multi-level topics:
> * How to gauge various distances and why they are important for driving
> and the rules that govern it.
> * A list of the kinds of accidents that people have, from worst to least.
> The nature of accidents, their causes and consequences, physical,
> financial, psychological, moral and legal.
> * Defensive driving.
> * How drivers can be alert to each other.
> * Signaling our intentions to other drivers.
> * The most commonly violated rules. The most frequently given tickets.
> * Traffic problems and how drivers can help mitigate them.
> Such topics can review all the basic knowledge in the context of memorable
> discussion by new drivers, veteran drivers, repeat offenders, and so on.
> But we might need some kind of "jury duty" to make them happen. As it is,
> we have many taboos that prevent such socializing, certainly with
> teenagers. And then we wonder why they are ignorant or incommunicable.
>
Hey, really good list, and eerily exactly what's being shared through
this ODEC organization. Even more synchronistically, my daughter is
right now as I write this in one of those classes, with a driving
lesson later (one hour observing another student drive under
instruction, one hour driving with the other observing).
The instructors dwell on exactly these topics you mention. When
parents come for orientation, they give a lecture on how driving
education has changed since you old people probably learned how, and
then they go over some details.
Lots of talk about the most common accidents, most commonly violated
rules and so on.
> Likewise, where do we start to make such a list of topics in Math ? I
> think Maria et al's work with the youngest children is very central. For
> example, "symmetry" is a topic for both babies and Nobel prize winners.
> Or "fractality" or "proportions" or "exponential growth".
>
I talk a lot about "variables" in this thread. In the little math
language I teach, you have names referring to objects and those
objects come in types. Out of the box, you have builtin or indigenous
types, such as integers and complex numbers, sets with set operations
and so on. You then use those types to define types of your own, of
which you create "instances".
So in this little world (namespace), a "variable" is also referred to
as a "name" and we talk of "=" as "the assignment operator" which
"binds names to objects".
c = Carrot( )
might mean "bind the name 'c' to an instance of the Carrot type".
What makes this math language a little different from some others is
when you then go:
d = c
as conceptually that could mean d names a different entity than c, as
if a copy had been implicitly made such that d and c are now names of
two objects.
However, that's not how it works, such that if I go:
c.color = "orange"
then d.color will likewise be "orange". Why? Because d = c does not
make a copy of anything, but rather d and c are now the name of the
same object.
d is c
will return True, where "is" is another operator for comparing the
identity of two objects. Just being a clone is not enough to pass the
"is" test. d and c have to name the very same object to get True in
this case (in contrast d == c would compare value without requiring
actual identity in memory on "on the heap" as we say).
Anyway, that's just an example of how the word "variable" gets a
particular meaning and spin within a namespace, a language game (like
chess is a language game).
When you first tune in a new language game, one has to set aside
expectations to some degree and just take in the indigenous meanings.
Thinking one knows what "variable" means already can at times help, at
other times get in the way.
This is where training in ethnography comes in useful, as one becomes
more cognizant of one's own prejudices and predilections. All math is
ethno-math.
Kirby
> Thank you for ideas, topics, questions! Also, as I prepare for my show,
> I'm creating videos http://www.ms.lt/tv/ to foster my creativity and play
> with others, co-create and work towards a shared culture of creativity.
> Please do send photos, videos, music etc. that I could include in the
> Public Domain.
>
> Andrius
>
> Andrius Kulikauskas
> m...@ms.lt
> (773) 306-3807
> http://www.selflearners.net
>
mok...@earthtreasury.org
> I'm working towards a taxonomy of ambiguities, paradoxes and the ways we
> use variables.
A huge subject. I have done some work on the nature of delusion as defined
in Buddhism, and specifically on the topics of Cognitive Dissonance (in
which one holds contradictory beliefs, or beliefs that contradict known
facts, and justifies one's defining beliefs more strongly when the facts
refute them) and Learned Helplessness (in which one can be convinced not
to try to deal with the errors of the surrounding society).
My favorite example of a self-contradictory belief comes from Christian
Fundamentalists, who believe that they believe every word of the Bible
(commonly the King James version) literally, but who in fact do not. For
example, I have met nobody who believes that the firmament/sky is solid as
described in Genesis, and there are many other points relating to how one
should behave that are much worse than that. (I took a course in modal
logic in college, dealing with the logic of belief, necessity and
possibility, and the like. It is full of such paradoxes. I also studied
logic and set theory in the absence of Excluded Middle, which results in a
very different approach to paradox from the generally-accepted ZFC version
of logic and set theory.)
> I found a book on that very topic by William Byers, "How Mathematicians
> Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics".
See also
A Budget of Paradoxes, by Augustus de Morgan.
A Budget of Paradoxes, originally published in 1915, is mathematician
Augustus De Morgan's most accessible and entertaining work. Well-known for
his wit, De Morgan takes aim at those people he calls "paradoxers," which
in modern terms would most closely resemble crackpots. Paradoxers,
however, are not crazy, necessarily-rather, they hold views wildly outside
the accepted sphere. If you believed the world was round when everyone
else knew that it was flat, you would be a paradoxer. In this book, De
Morgan reviews a number of books from his own library written by such
"crackpots" who claim to have solved a great many of the puzzles of
mathematics and science, including squaring a circle, creating perpetual
motion, and overcoming gravity. Each is thoroughly put in his place in
ways both entertaining and informative to readers. Skeptics, students of
science, and anyone who likes pondering a puzzle will find this book a
delightful read. British mathematician AUGUSTUS DE MORGAN (1806-1871)
invented the term mathematical induction. Among his many published works
is Trigonometry and Double Algebra (1849).
Many mathematical pioneers were put down as paradoxers in this sense by
many or even most of their contemporaries, or by others later on in other
cultures. To mention just a few of the most prominent:
* The unnamed Pythagorean who discovered irrational numbers
* The oracle of Apollo at Delios who challenged the citizens to double the
size of Apollo's altar
(duplication of the cube, irrationals that cannot be constructed with only
ruler and compass)
* The several inventors of 0 around the world, most notably in India,
China, and South America, and Leonardo of Pisa (Fibonacci), who introduced
the idea into Europe in the form that became established.
* Those who first took negative numbers seriously
* Those who first took complex numbers seriously
* Gauss, Riemann, Bolyai, and Lobachevskii, among the first to take
non-Euclidean geometry seriously, and not just as something to be refuted
* Georg Cantor, for transfinite numbers
* The logicians who showed that the Peano Postulates for counting numbers
have multiple models, and Abraham Robinson for extending their work to
non-standard analysis
* John Horton Conway, for an alternate construction of infinities and
infinitesimals with practical applications in game theory
* Those who hold that 0.99999...is not necessarily equal to 1. (It can be,
but that is a matter of definition of the kinds of number you want to work
with.)
Each of these has been ferociously opposed. For example, legend has it
that the discoverer of the irrationality of square root of 2 was thrown
overboard from a ship for blasphemy. Gauss wrote that he had held back on
publishing his discoveries in non-Euclidean geometry for fear of "the
howls of the Boeotians."
http://christiancadre.blogspot.com/2005/11/back-to-zero-word-from-author-of-1491.html
Back in 967, for example, the monk who became Pope
Sylvester II figured out that his counting would be easier
with a zero sign (it wasn't a zero, as in a circle, but
worked like one). He was accused of trafficking with evil
spirits and forced to abjure it. This kind of thing went on
until at least 1348, when the ecclesiastical authorities of
Padua prohibited the use of zero in price lists, arguing
that prices had to be written in "plain" letters.
Through much of this period European merchants went ahead
and used zero for their accounts, because it was so much
easier. But they hid this from the legal and churchly
authorities. Florentine bankers, prohibited in the 12th
century from using "infidel" symbols, created duplicate
sets of books, one to show the church, one to do your
calculations in. Thirteenth-century archives are replete
with evidence of such bootleg zeroes.
http://en.wikipedia.org/wiki/The_Analyst
And what are [Newton's] Fluxions? The Velocities of evanescent Increments?
And what are these same evanescent Increments? They are neither finite
Quantities nor Quantities infinitely small, nor yet nothing. May we not
call them the ghosts of departed quantities?--George Berkeley, Bishop of
Cloyne
http://en.wikipedia.org/wiki/Georg_cantor
Poincar� referred to Cantor's ideas as a "grave disease" infecting the
discipline of mathematics, and Kronecker's public opposition and personal
attacks included describing Cantor as a "scientific charlatan", a
"renegade" and a "corrupter of youth."
Meanwhile, Cantor himself was fiercely opposed to infinitesimals,
describing them as both an "abomination" and "the cholera bacillus of
mathematics".
Similarly with Copernican heliocentric astronomy, deep geologic time,
evolution, the Michelson-Morley experiments and Special Relativity, the
Ultraviolet Catastrophe and the Quantum theory, plate tectonics, and much
more, but not Cold Fusion or Creation Pseudoscience. These last two fall
under the Wolfgang Pauli rubric, Not Even Wrong.
The paradoxes of religion are also of interest, whether in the form of
Buddhist koans, the Mysteries of the Catholic Church, Advaita Vedanta in
Hinduism, Jewish Kabbalah, Daoism, Sufism, and much more.
A different but somewhat related concept is to take notice of words that
have multiple meanings. According to some psychologists, one of the better
measures of creativity is how many meanings one can think of for words
such as "seal" (17 at Wiktionary), "pitch" (32), "run" (64), and others
like them. However, each language has different sets of such words.
> As I work on this, I'll share my progress at Math Future
> http://groups.google.com/group/mathfuture/ and my own group, Living By
> Truth http://groups.yahoo.com/gropu/livingbytruth/
>
> This is part of my lifelong quest "to know everything and apply that
> knowledge usefully". In 2009, I overviewed my conclusions with my
> 10-minute video "I Wish to Know"
> http://www.youtube.com/watch?v=ArN-YbPlf8M Since then I organized a
> "House of Knowledge" of 24 ways of "figuring things out"
> http://www.selflearners.net/ways/
Would you consider doing an analysis of all of the ways one can be wrong,
and all of the ways of determining that someone (oneself or another) is
wrong? For example, we have all of the logical fallacies; the ways that
data can be incorrect or can be misinterpreted, even if correct; the whole
topic of Cognitive Dissonance, mentioned above; and others.
> Now I'm pulling together all of this
> and presenting it as an art show this April at the South Side Community
> Art Center, http://www.southsidecommunityartcenter.com, a historic gallery
> in Chicago's Bronzeville.
>
> I hypothesize that there are six ways that our minds use variables and
> that they correspond to pairs of questions: Whether? What? How? Why?
> Together they make for 10 of the ways in the House of Knowledge.
I follow Ken Iverson in treating variable names as pronouns, to be used in
as many ways as pronouns are used in human languages, including the
various interrogatives mentioned above. For example:
* Whether? Does this equation have a solution?
* What? Enumerate or otherwise determine the solution set.
* How? What kind of solution are we looking for?
* Why? Can we say that mathematical objects with specified properties exist?
We can also use variables in the definitions of algebraic, logical,
geometric, and other kinds of mathematical object.
> I've been sorting paradoxes listed by Wikipedia:
> http://en.wikipedia.org/wiki/List_of_paradoxes
> http://www.ms.lt/sodas/Mintys/Paradoksai
>
> Here's a first attempt at sorting the paradoxes:
> * The properties of a whole can differ from the properties of the parts
> * A contradiction can be contained locally
> * Description can restrict freedom
> * Attention can affect what is observed
> * A statement can negate itself
> * A definition is always incomplete and open to multiple interpretations;
> in particular, it can be vacuous, having no actual examples
>
> I appreciate very much our efforts to be as complete and inclusive as
> possible! I'm trying to get to the germ of contradiction in each paradox
> and appreciate it for what it is.
>
> William Byers emphasizes the importance of ambiguity. His perspective is
> very helpful. It's interesting that there are remarkable thinkers who
> have walked through this territory. For example, Thomas Hobbes's "Table
> of Absurdities" includes "category errors" ordered as six pairs from four
> categories: bodies, phantasms, accidents and names.
> http://en.wikipedia.org/wiki/Absurdity#Hobbes.27_.22Table_of_Absurdity.22
Also Kant's Antinomies of Pure Reason.
> William Byers cites conductor Leonard Bernstein's enormously creative
> video lectures from 1973. I'm listening to his 3rd lecture, "Musical
> Semantics", where he explains the role of metaphor in music.
> http://www.youtube.com/watch?v=8IxJbc_aMTg&feature=related
> He interprets, I think convincingly, a passage of Beethoven's as a series
> of pleading, cajoling, threatening, etc. and ultimately firmly agreeing.
> He shows that it makes a lot of sense to relate such a drama to the music.
> He asks, is this what Beethoven felt, and he transfered through his
> score? Did Bernstein just make those feelings up?
James Joyce told a reader that all of the ideas he found in the text of
Finnegans Wake were correct, whether or not Joyce had put them there.
Joyce wanted to incorporate all of human literature and religion, among
other things, into the Wake.
> And he says that both
> are probably true! and make for a profound ambiguity. We may understand
> but can't be sure. The (musical) structure can inspire us to think that
> the (semantic) interpretation is in fact correct. And such an ambiguity
> is at play perhaps in all of our language, although much less obviously,
> when we might not really be understanding how another person is using
> certain words. (Which becomes obvious when we're in other cultures, yet
> more apparent when we return to our own.) In a sense, my quest is for a
> deepest understanding of the universal culture inherent in the limits of
> our minds, or any minds, to the utmost detail, such that to violate any of
> it would be to no longer know anything at all.
>
> Ambiguity is thus when there are two truths at the same time.
Tolerance for ambiguity is one of the most important mental qualities,
whether in science, politics, religion, or whatever. In Soto Zen Buddhist
teaching, this can be summed up by always saying, "I could be wrong."
> --
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>
--
Edward Mokurai
(默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر
ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
http://wiki.sugarlabs.org/go/Replacing_Textbooks
kirby urner
<< snip >>
> http://en.wikipedia.org/wiki/Georg_cantor
> Poincaré referred to Cantor's ideas as a "grave disease" infecting the
> discipline of mathematics, and Kronecker's public opposition and personal
> attacks included describing Cantor as a "scientific charlatan", a
> "renegade" and a "corrupter of youth."
>
Yes, it's important to remember that mathematics, philosophy
and all of those are not monolithic subjects in which everyone
pleasantly agrees. As one approaches as an outsider, there
may be this sense of a giant castle or unified facade. This is
more an illusion, a mirage, than anything to much worry about.
I enjoy the ethnographic approach which follows biographies
quite a bit, in order to trace plots. 'Logicomix' uses this to good
effect in telling a Bertrand Russell story, with a lot of important
characters mixed in, including Frege, Keynes, and of course
the incompleteness guy (friend of Einstein's @ Princeton).
Understanding western math requires understanding about how
important it is to be first at something and then to be really
snotty about it with other people, enforcing "intellectual property
rights" etc. This comes from cultures that name things after
individuals, cluttering up the landscape with gazillions of names,
the graffiti of passers through. That's just part of the nomenclature,
like the surface of the moon.
Like in another forum I'm completely against giving the pycon.com
domain name to a non-profit based in the USA and the latter's
craven / cowardly approach to intellectual property, in the sense
of supporting software patents and so many other practices
designed to deprive humans of the fruits of their own ingenuity,
because some corporation has it all locked up in the form of
patents you could never afford. With the deck that stacked against
innovation, I encourage creative businesses to limit their business
dealings with the USA very stringently. If you live there, that
means slowing your shopping to minimal, especially when it
comes to software made in the USA -- pretty unethical to buy
that stuff in my book.
Kirby
kirby urner
> domain name to a non-profit based in the USA and the latter's
> craven / cowardly approach to intellectual property, in the sense
> of supporting software patents and so many other practices
> designed to deprive humans of the fruits of their own ingenuity,
> because some corporation has it all locked up in the form of
> patents you could never afford. With the deck that stacked against
> innovation, I encourage creative businesses to limit their business
> dealings with the USA very stringently. If you live there, that
> means slowing your shopping to minimal, especially when it
> comes to software made in the USA -- pretty unethical to buy
> that stuff in my book.
>
Sorry for the typo, that's pycon.org, not pycon.com
Those of you familiar with the open source / free software movement
know that we consider Richard Stallman one of the philosophical greats
of our time, more prominent that any practicing analytic philosopher I
could name, unless we decide Stallman is analytic (vs. continental).
Our school considers the whole western approach to "priority" to be
ethically slimy and suspect. You really need to wash your hands with
strong stuff after working around these legalistic types sometimes.
Feels so trashy and gross.
Kirby
> Kirby
mok...@earthtreasury.org
> use variables.
Another reference for you: Paradox Lost and Paradox Regained, in
Mathematics and the Imagination, by Kasner and Newman. It also appears in
the third volume of Mathematics and the Imagination, along with The Crisis
in Intuition, by Hans Hahn.