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

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

the von neumann definition of 2 in set theory is
{{}, {{}}}
for example

If we want to think of anything other than singleton
items, we need some kind of collection. To build
collections, we need some minimal collection construct.
We could use ordered pairs as our collection primitive.
We could call them ordered pairs or cons cells or
whatever. We can spell "right" as "cdr" if we want.
We can use bags, or lists. We can use relations.
We can use simple binary trees. We can use
ridiculously over-complicated under-featured trees
inflicted on us by the W3C, if we are masochists
and/or don't know any better.

Or we can use sets.

yes
various mereologies provide different semantic properties

we could look more generally at various topoi
and use the subobject classifier to build containments

this is all standard
which i've written about quite often

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

(assuming turing completeness of course)

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

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

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

(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

if one were to
as i
desire a mathematical foundation as a science
then parsimony is certainly a useful tool
though i absolutely agree it is not a criterion for correctness

that is how the principle of parsimony is treated in science

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

it is quite legitimate to object for reasons outside a formalism

take bohmian mechanics
this is a theory of quantum mechanics
that takes realism seriously

wigner's friend is not a formal paradox in the copenhagen
interpretation
but it shows that the copenhagen interpretation
lacks a certain form of property meaningfulness
at least at a systems level

there are different approaches to resolve this

a variety of quantum logical approaches
consistent histories
many worlds
and realist theories like bohmian mechanics

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

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

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

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

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

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

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

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

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

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)

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?

Meanwhile, I see no need for set theory to emulate your personal
opinions in metaphysics. I don't take set theory to even be about such
metaphysical concerns.

then you are a firmly in the camp that foundations are meaningless

i have never understood that position
i've always felt foundations were there
to describe the physical process called mathematics
and like other scientific descriptions
requires evidentiary metrics over models

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

or each time i count them

or each time i describe the contents of the jar

and it still assumes the ordering for parsing
but claims an unordered semantics?

I don't know what you mean by 'unordered semantics'.

me neither!

well
i understand my point on the semantics
i just don't understand the meaning in the semantics itself
but that might just confuse you more

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 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

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

it's obvious that it's my fault

i have all the signs of crankdom

i've been arguing with teachers
ever since grade school

everyone with authority can't be wrong
right?

i mean
if your not convinced of this
my theory on star people should be plenty evidence..

http://groups.google.com/group/alt.philosophy/msg/77f2d0b0a22c2b44

seriously
no one should take my *** seriously

i am only a corruptor

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galathaea: prankster, fablist, magician, liar
.