Re: they mention various metaphors and ramble about various concerns (Re: Kuratowski Ordered Pair)

On Dec 21, 9:10 pm, galathaea <galath...@xxxxxxxxx> wrote:
On Dec 21, 2:55 pm, Marshall <marshall.spi...@xxxxxxxxx> wrote:
On Dec 21, 1:22 pm, galathaea <galath...@xxxxxxxxx> wrote:
On Dec 20, 5:56 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

it is regularly pointed out in the foundations of math
that the concept of 2 cannot be defined
without some presumption that the parser already has some bivalence

whether in two neighboring symbols
or two different symbols
or some other embedding of the concept innately

pairs

this is not true of higher naturals

This is just syntax again. This is not "the foundations"
of math; this is the syntax of math. We could as well
"write" expression in trees, assembling them via something
like tinker toys, where we have spokes and edges.
Or any of a variety of other ways. Linear strings
of symbols are convenient; don't mistake convenience
for semantics.

there is no mistake

definitions are syntactical

one tries to make the syntax
obtain the desired semantic properties

Huh who what? Syntax has semantic properties?

"Syntax is syntax and semantics is semantics and never the twain shall
meet."

Hmmm. Doesn't have quite the poetic ring to it one might wish.
Still, the point is, syntax doesn't have semantic properties.
So I can't imagine what you mean.


Whichever one we pick we're probably going to be
able to get to the same places.

(assuming turing completeness of course)

We're just talking data structures, aren't we?
We haven't mentioned manipulation yet, so I don't see
how to fit Turing completeness into the picture.


It's just a question
of how hard it's going to be to get there. If we use
lists, then union is neither commutative nor idempotent.
If we use bags then union is commutative. If we use
sets then it's both commutative and idempotent. Sets
give you more desirable algebraic properties; for these
and other reasons, it is common to use sets as the
foundation. But you can use something else if you want.

this is a circular argument

Huh? It's not any kind of argument, let alone a circular one;
it's just a description of things as they are.


you say commutativity is preferred here
but that is just saying that unordered operation is preferred

I didn't say "preferred"; I said commutativity is "desirable."
And I could drop the "desirable" word and the description would
still be accurate, and as much on-point. Do you disagree that
commutativity tends to make proofs shorter?


it might seem nice algebraically
maybe we can use the same nice properties abelian groups have
right?

but then look at what we are discussing

kuratowski pairs

a, {a, b}

7 symbols to express a fundamental notion
which can be expressed in 3

a, b

You're obsessing about the syntax again. Or are you
trying to establish that the Kuratowski formulation is
more complicated than if we took ordered pairs as
primitive? If so, that's a *given*, something that's
*necessarily* true. Comparing a nonprimitive construction
of something with the taking of that something as
primitive--the primitive will always be simpler, of course.


(we can collect it if it is to appear as a subexpression
in both cases adding two more symbols as is standard)

the kuratowski definition requires repeating a referrent
because it semantically wants collection to obtain an unorder
yet requires the parser to actually use ordered containment

No, that's just incorrect. The semantics don't "want" anything;
that's wrong even if we ignore the anthropomorphism. Never
the twain shall meet.

Again, if the point here is that it's simpler to express ordered
pairs in a system that has ordered pairs as primitive than it
is in a system that doesn't have ordered pairs as primitive,
then yes, of course, Euroduh, totally, it couldn't possibly
be any other way.


what i was trying to state
was that i do not see why it is so difficult to understand
the various positions on ordered pairs mention by a few of the
posters

It's not hard to understand; it's just *wrong* is all.

If they have some aesthetic preference for another formal
system, nobody's holding them back. However when they
start to "ramble about various concerns" etc. about a formal
system, then they are obligated to either substantiate their
concerns formally or else shut up.

there are many reasons to object to a theory
and no one needs shut up

(Boy, I certainly disagree there. There are some people that really,
really need to shut up.)


it is quite legitimate to object for reasons outside a formalism

Sure, sure; I've said as much many times. Even in the post
you're replying to I said so. However nonformal objections do
not count as formal objections. Trying to make one's nonformal
objections appear as formal objections, or not being aware of
or able to express the difference, is a real problem.


take bohmian mechanics

No, thank you; it is not to my taste.

Most of mathematics is not to my taste, as a matter of fact;
my mathematical interests are pretty narrow. (Again, I do not
make the mistake of thinking that this is a statement about
mathematics. It is a statement about what things I find interesting.)

And *none* of physics is to my taste.


much of the motivation for such efforts
are various parsimony principles

it is perfectly reasonable to bring up objections of parsimony
which is all i've really seen in this thread

Well I'm going to go out on a limb here and try to predict
others' behavior. I predict that if someone said "I object
to Kuratowsky's formulation of ordered pairs on parsimony
principles. I want a simpler way to do the same thing" that
people like MoeBlee for example, or even Virgil for that matter,
would say "go for it buddy, and let us know what you find."

The problem isn't people wanting to explore alternatives.
The problem is that they "mention various metaphors and
ramble about various concerns." They talk as if their aesthetic
(or parsimony, or whatever) principles represent a *formal*
flaw in what has already been done. They don't.


Their aesthetic preference for a system in which ordered
pairs are primitive *has no bearing* on the behavior of
any set theory in which ordered pairs are not primitive.
Their sensibilities are not formal objections; they're just
preferences.

when one tries to restrict discussion in this way
it is only to enforce the tide of historicity

Nothing in the quoted paragraph "tries to restrict
discussion." Rather it simply points out the independence
of aesthetic considerations from formal ones such
as consistency. Do you claim otherwise?


this may be safe
conservatism is an engineering principle
but neophobia tends to oppress innovation
and regularly and easily slips to fundamentalism

sometimes people tell others to "shut up"
for instance

Yes, yes, me and my character flaws. We've spoken
of this at length.

As to that "neophobia" thing. I guess you've read a lot of
R.A.W.? Crazy out-of-nowhere idea: did you know Leigh Ann Hussey?


as you alluded to earlier
there exist many systems

i see ordered pairs as being the primitive notion
and unordered pairs as being a construction from ordered pairs
by taking classes of the equivalency quotient (a,b) ~ (b,a)

Fine fine fine. Nothing wrong with that. Just be clear that it
doesn't tell us *anything* about how bags behave, or what
the nature of a list is, or whether Kuratowski pairs are sufficient
for defining maps.

well
it does describe bags parsimoniously with natural bags

Sure.


as even you pointed out in a different post
we can pull out objects from a bag
or point them out
or count them
or whichever process you described

you didn't point out
though
that these actions are orderings

No, these actions are not all orderings.

Some such possible operations will require order
to be there ahead of time. Some will induce order
after the fact. Some will not consider order at all,
and hence will not be affected if there isn't any.
(Empty the contents of the jar into another jar,
for example.)


you tried to make the point that the objects should have an assigned
ordering
constant throughout time

No I didn't. I tried to make the point that neither order,
nor the lack of it, is fundamental to the physical world.
(Not that I consider the physical world particularly
relevant here.)


that is a very platonic approach to mathematics
but has little to do with the _actual_ process of mathematics

Yuh huh.


the actual process of bagging
says that order _doesn't_matter_
that _all_ orders are equivalent

in cognitive development
unordering appears to come after the initial orderings
it requires both object identification and object permanence
and the ability to cognitively associate different orders
and it is an ability to pick up

Okay, you're throwing around Piagetian terms; are you
suggesting that Piaget had any results that show that
infants develop notions consistent with ordered pairs
earlier than they do notions consistent with unordered
pairs?


this is also true in computer science
where addressing is primitive
and unordering is an abstraction layer above

I have never found any argument based on hardware
architecture to have a lick of merit. But even if we
stipulate your idea that unorderedness is somehow
"above" orderedness, that seems like a good argument
to show the *power* of set theory--it deals with the
more difficult issue of unorderedness quite handily.


the word itself
_un_ordering
describes a process of undoing a preexisting order

Ugh. If hardware architecture is a poor surface to which
to adhere an argument, then word dissection is positively
teflon.


mitch mentions this possibility as superposition

i see the ordered pair as intuitive
which appears to be backed by child psychology and neurobiology
action schemas
parsers
and other foundational mechanics for symbology

i don't see the unordered pair as intuitive
and definitely not primitive

it seems a specialised construction to me

You are of course completely free to have your own
aesthetic response to various formal systems. For myself,
I prefer to go a step higher and take relations as the
primitive, a la Codd, and skip "simple" sets entirely.
(Which eliminates tuples and ordered pairs as well.)
I don't confuse my attraction to this idea with some kind
of flaw in other ideas however.

and relations are the fundaments of order
almost directly a pair
(in some collection of ordered pairs)

I said "relations a la Codd." Not binary relations. n-ary
relations, n>=0, with each of n attributes referenced by
name. No order in the rows. No order in the columns.
No order. A very cool abstraction.


indeed
there is a simple translation to the language of topoi

of course your attraction may not be for the same reasons
but is there some drive or principle there?

would it be improper to bring such up?

I am absolutely happy to have a conversation about aesthetics,
assuming we are able to identify it as such. I fucking love
aesthetics. I love movies. I love 19th century literature and
poetry. (And who is your favorite poet, Miss writes-all-her-prose-in-
a-
poetic-form? Ginsberg? Gil Scott-Heron? Mr. e. e. lowercase
himself? Whoever it is, I bet they wore tie-dye at least once in their
life,
he said with gentle amusement.) I love Adult Swim. I love
Impressionism
and Surrealism, but Dada leaves me feeling Nada.

(Somehow I feel that you've been to SFMOMA. Did you see they
have one of Duchamp's "Fountain" there? What is your call:
shocking meta-commentary on the nature of Art itself, or
self-indulgent crap? Or some third possibility? I bet you can
figure out which one is my take.)

Why, this one time I was sitting in my office and in storms a guy
who had recently started at our company, a "name" programmer
quite well known in certain comp.lang circles. He was
very excited because he had just found a very simple class I had
authored that exactly matched a class he had authored at his
previous employer, and the implementation was the same even
to the point of this one teensy tiny detail that was done differently
from how almost everyone would have done. (We had even chosen
the same names for our classes.) He wanted to know, was I
aware of this one obscure corner case in which "our way" of
doing it would produce correct results where the straightforward
way wouldn't? I said no, but in fact the reason I wrote the class
in that particular way was not because of the microminiature
performance advantage I had known of, or the liliputian correctness
advantage, but rather "because it is beautiful." Well if he had
been excited when he came in, you should have him when I
said that. Yes, he said, yes, we are in the business of beauty!

Aesthetics are important. What is the fucking point of living if
everything is just some stupid utilitarian exercise? Might as
well be a mud-dauber wasp at that point.

I think set theory is beautiful, and I think calculus is ugly. That's
why I study the one but not the other. If you think set theory is
ugly, then by all means just shut your eyes. I am sure there are
other choices for foundations, even if they are not currently as
well developed today. As if you would care about that! As if
you run with the herd!


If you want a physical, "evidentiary" metaphor for an
unordered collection, take a big opaque jar and drop
two marbles into it at the same time. Shake them up.
Now reach in, (go ahead and look if you want) and take out
only the first of the two marbles.

You cannot do it, because there is no way to decide
which one is the first. That's because this is an unordered
collection.

oh

i guess i shouldn't answer step by step
after reading up all the back posts

the discussion above applied to this

there is definitely an order
each time i take the marbles out

You might take them out in a particular order.
But that's not the same as saying that one of
them is first and one of them is second, *before*
which is which is determined by your reaching in
the jar. Do the experiment again and you might
not get the same result.

Pour the contents of the jar into another jar. Now
where's that order?


If you, an otherwise intelligent person, really *cannot* comprehend
the idea of an unordered collection (which I don't believe) then
it must be considered a learning disability, a flaw in your intellect,
the same way a colorblind person's inability to distinguish colors
must be regarded as a disability. If you then go on to insist that
*we* can't imagine it either, you would be in the same position
as a colorblind person insisting that color does not exist.

i really have no idea what it means as a primitive

I don't believe it.

I think the problem is that you don't know what it means
to know what something means as a primitive. And so you
expect there to be some connection in your head that
isn't there, but that you expect to be there. But it's your
*expectation* that is deficient, not your understanding.

But if I'm wrong about that, then you're really just colorblind,
and all our babbling about reds and blues and greens is
forever outside your reach. Which wouldn't be a big deal,
really; it's not like it's going to keep you from leading a
rewarding life. But it takes you out of the running as
a critic of set theory.


i know how orders can be viewed as equivalent for a particular purpose
i don't believe order to be innate to objects
(it is innate to the action schemas over structures)

but none of this is a primitive unordered pair

i am quite familiar with quantum mechanics
and indistinguishability
but that is not an ordering principle
and identification is still ordered

and not only that
but these are important points to me

What did you think of my OOP explanation? Did you
just totally not understand it, or rather, was it trivially
obvious?


so many people seems so content to memorise

they seem so happy that they have mastered some formalism
they do not care whether that formalism has meaning
it is an affront to them to even suggest
there ought be a relation to their perceptual realm

they may speak of math as being some kind of symbolic abstraction
but they do not want to address _of_what_ it is an abstraction
they often scorn talk of perception
even though that is the source of conception in neurodynamics

i really don't understand any of it

i'm always left feeling
like the people who claim the most adherence to math
really have the least desire to think about it

memorisation for status has more currency
than exploring for understanding

Darn those humans and their character flaws!


everyone with authority can't be wrong
right?

We're talking about aesthetics. We're talking about picking
primitives for a formal theory. Right and wrong don't enter
in to the question.


Marshall
.