apartness

On Nov 21, 3:44 pm, Marshall <marshall.spi...@xxxxxxxxx> wrote:
On Nov 21, 2:50 pm, galathaea <galath...@xxxxxxxxx> wrote:

hello venkat

i am sorry some of the posters here
are not really very good at math
and they try to hide this by implying others are not very smart

That hypothesis does not fit my observations.

i suspect everyone here is not really very good at math
and they all know it

the deeper you get into it
the harder you realise it is

the ontology is expansive
many different communities
with their own languages and each its own semantics

i suspect everyone here
would benefit to still consider themselves students
especially those who don't currently

there are obviously people on this newsgroup
and people in this thread
who will contribute to the expansion of mathematics
in very interesting ways

many in this thread already have

i just happen to think that much mathematics
is accomplished by people who are not really very good at it

and sometimes
to feel better about whatever of their own insecurities
they may desire hidden
some of the posters here play games of derision
because of failure to meet
their unspecified specification standard

the one they can't even agree on

do not let their flaws affect you
you are asking important questions

I thought Randy's (?) take was much better:
dude's just not used to thinking clearly.

almost every usenetter i have ever read
has shown their thought process broken
at some point

that is it's natural condition at birth
and is how learning is possible

one who is offering the role of teacher
and dismisses earnest discussion with their student
with scoffs of "nonsense"
is a failure as a teacher
is an antiteacher

these questions come up quite regularly around here
so i try not to get too involved or it will eat my time
but a year ago i posted this about the cone of democritus

http://groups.google.com/group/alt.politics/msg/85c90b4e8df1448c

That was an interesting post, but I found the interest
to be purely historical. The lesson I took from it is that
we moderns understand abstraction better than the
ancient Greeks. Which should not come as a surprise.
Much as I respect the Greeks.

abstraction is what rats do
fleas

the 302 neurons of nematodes
abstract their input

there is little evidence that our ability to correlate
a recall symbol to some newly recognised pattern
has increased significantly from the time of the greeks

i think you are talking about formal modelling
building actual formal languages
with syntactic rules for formal manipulation

when you write about being "clear"
i suspect you intend that
venkat is not really doing a good job of axiomatising

formal definitions

building a repeatable computational structure

that could be true
but you are being very _unclear_ about this
by confusing this with abstraction

"dude's just not used to thinking clearly"
applies to all the posters i have seen here

One thing about abstractions: they are not bound
by the natural world. Hence any argument against
any particular abstraction on the basis of it not
matching some observable phenomena is not
a valid one. Neither are abstractions bound by
the limits of anyone's intuition.

abstractions are patterns
in the information that is input our phenomena

that is how our brains function
in the better models we have today

there is no transcendental input channel here

it is all based very firmly in our phenomena

lines are _not_ just collections of points
they cannot be

lines have properties you cannot get from collections of point
properties like limits and orderings
that you never get from _simply_ collecting points

in particular
lines have a topology which are usually hidden in their
specification
and many of the properties that concern you
arise from this extra structure

for the bare bones simplest example i can think of:
if you remove a point from a collection
there is no way to know what to replace in the collection
(the point removed could be any point)
if you remove a point from a line
it is easy to tell what point is missing and replace it

If you describe to me a collection of points and
tell me one is missing and ask me which one,
you have to also tell me what the full collection is.
If you do, then I can tell you which one is missing.
If you don't, I can't, whether the original collection
was a line or not.

And if you hand me a line minus only one point,
and only tell me the original was a line, then
you've still unambiguously described the entire
original collection.

http://plato.stanford.edu/entries/geometry-finitism/

in some geometries over finite fields
i could give a finite list specification
but these are obviously not models of continuous lines

the key here is that _continuity_ of lines
is a topological property
and lines have much more structure
than _only_ collections of points

there is a process of construction
that brings with it a rich set of properties
where we can discuss betweenness
and limits
and apartness
and all the other fun of continua

this is one of the reasons
some consider common models of lines as collections of points
poor symbolism and a type error

in a better formalism
lines are at least a triple (C,T,<)
a collection C of points
constructed by a process which provides
a topology T of open subcollections
and an ordering <
that obeys the axioms of a line

analysing the types of processes that generate such continua
brings up quite legitimate questions
on what it means for points to be separated
along exactly the same lines as the ancient greeks
and if you take meaning seriously
the set theoretic containment itself can get questioned

apartness is a very nonclassical relation
in constructivist geometry
and some of the issues venkat brings up
have a natural interpretation here

(Indeed, I am hard pressed to see how you could
communicate to me any set that was a line minus
a point, in any way *other* than specifying the line
and the point in the usual way, or via some equation.)

I can certainly describe a collection, say, the set
of all pairs (x, y) where y=0, except for (0,0),
together with (0,1), then give you everything
except (0,1), and that is the same exact set as if
I had started with the line y=0 and taken away (0,0).
You can't know which one I started with
unless I tell you.

The fact that a person may instinctively interpolate
doesn't mean the interpolated answer is always
the right one.

this isn't a question of interpolation

given just the set {a, aa}
there is no structure to tell that this was actually
{a, aa, pentagram} without the pentagram

two points determine a line

purely formally
one need only take the boundary

these show a structure beyond collection

one of the problems with extensional specification languages
like set theory
is that they hide much of this additional structure
inside the specification of the collection

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
.

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