Re: Mathematical existentiality and triviality.
- From: "Doug Goncz" <DGoncz@xxxxxxx>
- Date: 2 Jul 2006 02:04:00 -0700
RadicalLibertarian@xxxxxxxxxxx wrote:
There exists an delta s.t. epsilon/2 = delta.
What does that mean exactly - "there exists" ? How does that work
exactly ?
If existence is to have any meaning (in a mathematical context), then
there must be a nonexistence. Otherwise the concept itself is
meaningless.
Of course, this could be illustrated symbolically by writing an upside
down "A" with a slash "/" through it.
Then you can say things like " There exists no x s.t. x = x+1". All
well and good.
What I really need to know is how triviality fits into the picture.
For example, let a be an arbitrary point in R. Now construct a
collection of trivial points from the point a as follows. Let b1 b2
b3....b(n) be points in R s.t. |a-b(n)| = 0 for all n. So, each of the
b(n) = a, hence they are trivial because any of the b(n) can simply be
disragarded and replaced by the point a because a = b(n) for all n.
The question is whether the b(n) neccesarily even exist. Certainly,
uniqueness is intentionally violated.
Can you say - "There exists a trivial point b(n), bla bla bla", or does
it not exist at all ? Or, is it actually the case that existence in
this case is arbitrary ?
Can that be generalized ? Can you say that the existence of any trivial
object is arbitrary ?
Hello, whoever. "Radical Libertarian" thou art.
I read, and John Baez wrote in "This Week's Finds in Mathematrical
Physics" (week 234):
Indeed, in the Middle Ages, music was part of a "quadrivium" of
mathematical arts: arithmetic, geometry, music, and astronomy.
These were studied after the "trivium" of grammar, rhetoric and
logic. This is why mathematicians scorn a result as "trivial"
when it's easy to see using straightforward logic. When a
result seems more profound, they should call it "quadrivial"!
Try saying it sometime: "Cool! That's quadrivial!" It might
catch on.
Does that help? Were you talking about a point in the Cartesian plane?
One might write of a point between A and B that "If A#B, there
trivially exists a point C between A and B..." but as far as it being
arbitrary,,,,I dunno.
No, on rereading the above I see you are indeed talking about points in
R, not R^2. So you are writing about uniqueness and what we might call
arbitrariness or (with a giggle) arbitrariality. On that I can say that
while positing some collection of points b(n) might present a conundrum
of identical, differently labeled points, the *set* of all such points,
plus the point a, does contain a single element, and it matters not a
whit how we identify that element. My opinion, and just had sets in
Discrete Math.
Doug Goncz
Replikon Research
Seven Corners, VA 22044-0394
Doug
.
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