Re: Mathematical existentiality and triviality.


Doug Goncz wrote:
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



That's right - I'm playing with uniqueness. And my usage of the word
trivial is rather narrow, or specialized, as there are things such as
"trivial solution of a differential equation" which dont satisfy my
usage of the word trivial, I think it would actually be nontrivial
according to my usage in this context.

What I'm trying to grapple with is how mathematical existence could be
(seemingly) arbitrary, and I just cant wrap my head around how that
would jive with logic and set theory etc.

Another example is that 0/1 = 0/2 = 0/3 = ......= 0/n = ......m * 0/n,
and you can construct all kinds of equations and relationships which
are all = 0, and the whole thing is perfectly trivial. You can actually
set 0/5 as being DNE, or "does not exist", and arithmetic remains
intact because you still have 0/4 (for example), and if you ever create
a 0/5 you can simply substitute with 0/4 to get 0 exactly as you would
have if 0/5 still existed.

If we create a rule that 0/5 itself does not exist, and we allow
substitution, nothing really changes.

On the other hand, substitution only works because 0/5 = 0/n for all n,
and so the designation of DNE does not really seem valid in the first
place, unless you use limits maybe ?

.

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