Re: What is this thing known as "=" ?


T.H. Ray wrote:

T.H. Ray wrote:

T.H. Ray wrote:

T.H. Ray wrote:

T.H. Ray wrote:
Huang wrote:

It may be possible to prove that
existence
implies
uniqueness, and when
somebody succeeds at this, then you
will
have
much
more convincing
existence proofs.


Of course, existence does imply
uniqueness,
if
you
take
uniqueness to be observability, and
you're
talking
about the eigenvalues of a quantum
state.

Because the statement is meaningless
outside
any
context
in which existence is defined, however,
it
doesn't
convey
the generalized mathematical meaning
that
you
assign to
it. Provide an example of an existence
proof
that
you
think would be more convincing were
your
assertion
proved, and I will show you exactly
what I
mean.

Tom


Well, (I think it was) Hegel who said
some
things
about uniqueness
which sounded right, I'll try to dig it
up. I
cant
remember - this
stuff is really pretty old - regardless,

Consider the unit cube. Call it Cube_1.
Then
consider
any other unit
cube which is identical to Cube_1, call
such
cube(s)
Cube_2,
Cube_3,...Cube_n.

Clearly, all such cubes, being identical
to
Cube_n,
all such cubes may
be collapsed into a single cube Cube_1.
The
uniqueness of the object is
impled by it's existence (unproven) and
so
all of
these other Cube_n's
are trivial.

Same thing would hold for the reals,
including
zero.
Even though the
actual physical quantity which is
represented
by
zero
is "physically
trivial", the number zero would still be
unique
because it is the
number which exists and is apart from
what it
represents.


You have just told me that all numbers are
identical
to zero. Now, try to do mathematics.

Tom


That is just plain silly.

Yes, it is. It also what you said. You don't
know
that
it is what you said -- but that's your problem.

Tom


I did'nt say that all numbers were identical to
zero.
I did not say
that.

First, you need to ask - what is a number. A
number
is an abstract
representation of a quantity. A quantity of what
?
Bananas, trees,
goats, whatever.

The number itself is a mere representation of a
quantity. Usually, it
is attatched to aome kind of collection of
(objects)
which it
represents. In pure math you just cancel
(objects)
from both sides of
the equation and you play with numbers alone -
abstracted from any
physical signifigance.

However, numbers were invented by people who were
counting physical
things like chickens and goats.

So, you can put (objects) back into each side of
an
equation if you
wish, something like

5 * (goats) = 2 * (goats) + 3 * (goats)

Ok ? Ok.

Next, zero goats is the same as zero chickens so
that

0 * (goats) = 0 * (chickens)

Zero is still a real number, it's unique
(unproven in
my opinion), but
what it represents is a non-object, or better
still,
a "trivial
object".

Why trivial ? Because zero goats is
indistinguishable
from zero
not-goats. And, Harris (1.1) states that the
existence of a trivial is
indeterminate.

If you have zero goats, then you still have
goats.
You just have zero
of them.

However, having zero goats means that you dont
have
goats.

Both are TRUE.

So, hopefully you can see that what I said is
more
along the lines of
"chickens are goats" than "all reals are equal to
zero".

And I would insist that chickens really ARE
goats,
whenever you have
zero of each !


GOOD DAY.


And that, my friend, is precisely the logical
equivalent
of saying that all numbers are identical to zero.
I
am reminded of some dialogue in the movie National
Lampoon's Christmas Vacation, between the Chevy
Chase character and the little girl, something
like:

"How do you feel about Christmas coming?"
"Shittin' bricks!"
"You shouldn't use language like that."
"I'm sorry. Shittin' rocks."

The counting numbers (natural numbers) would remain
the
same whether one called them 0_1,0_2,0_3... or
1,2,3...

That is why, in the Dedekind-Peano axioms of
arithmetic,
the terms "number," "zero" and "successor" are left
undefined.

Tom


Dedekind factored out the (physical object) from both
sides, he's
working with numbers alone, absent of what they might
represent.

There are three different kinds of physical objects.
Cuurently, math
only acknowledges two different types, but there are
three.

Modern math would say that you either have a banana,
or you do not.
Banana exists, or it does'nt. It's a dichotomy.

I'm saying that there is a third type, which is the
trivial banana.

(1) A banana in your lunchbox exists. It is existent.
(2) A banana which is also a lemon does not exist. It
is nonexistent.
(3) The abscence of any object is a trivial banana.
Existence is
indeterminate.

I think that this approach actually explains
uniqueness in such a way
that there can be no doubt that the reals are
distinct. Any non-unique
number must be trivial. Were talking about
quasi-existence, where the
existence of the trivial is indeterminate.

I think it reinforces uniqueness of the reals.


If you insist. I don't find any mathematical sense
in it. In mathematics, existence is always given a
specific context; "quasi existence" or "existence of
the trivial" doesn't have any mathematical context. In
fact, I can't even parse these concepts within any other
context with which I am familiar. Unless, perhaps, the
concepts are themselves pseudo quasi trivial. You
suppose?

Tom


My usage of the word trivial is very specific. Basically, consider any
unique object, the unit cube for example. Any other unit cube which is
identical to this cube must be trivial.

Or, if you had a dollar bill, you could make the claim that it is
really 1,000 dollars, but that 999 of them are trivial.

So, a trivial object is an identical clone of a unique object. You
might say that it simply "does not exist", however ! - there is no way
to distinguish or determine if a given object is the original or a
clone. So, you have quasi-existence. It is a third existential type.

So, my usage of the word is pretty narrowly defined, but quite simple.

Any unique object can have as many trivial clones as you wish, they are
all trivial, and the original object retains it's uniqueness. None of
these clones exist, but it is impossible to determine if the existent
specimen is the original or a clone, so the existence of any given
clone cannot be negated.

So, trivials are quasi-existent. I dont think that they are teaching
this anywhere, but it does make sense.

Hopefully that made sense. Otherwise, I'm probably really going nuts
because it sure seems to make sense to me. : )

What you probably wont believe is the quasi-existence of uncertainty. I
have a hard time with that myself. I have a hard time believing that a
probability or an uncertainty could have a quasi-existent manifestation
as a real tangible object - but this is suggested by the double slit,
and it follows from the triviality of randomness.

.

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