Re: Mathematics: how to start again

On Fri, 12 Sep 2008 17:28:12 -0700 (PDT), Mistress Helios wrote:
On Sep 12, 9:30 am, Dave Seaman <dsea...@xxxxxxxxxxxx> wrote:
On Fri, 12 Sep 2008 04:43:25 -0700 (PDT), Mistress Helios wrote:
On Sep 11, 10:12 pm, Dave Seaman <dsea...@xxxxxxxxxxxx> wrote:
On Thu, 11 Sep 2008 18:54:20 -0700 (PDT), Mistress Helios wrote:
On Sep 11, 5:25 am, Frederick Williams <frederick.willia...@xxxxxxxxx>
wrote:
v4vijayakumar wrote:

Give an example of a topological space that is both connected and totally
disconnected.

I don't know what mathematically that means, but is it possible
something to be both connected and disconnected..?? If it is connected
then it is not disconnected, or, if it is disconnected then there is
no connection. :-)
Consider the segment [0,1], and to each point we assign a probability
of 1/2 that the point exists. Consider the points, as stated, and not
the expected length.
Then, there is a nonzero probability that the whole thing is totally
connected, and a nonzero probability that the whole thing is totally
disconnected.
The only problem is that this solution requires existential
indeterminacy - therefore, the solution itself is wholly questionable
as to whether it is even math or not. This is indeterminate, and so
why am I posting in sci.math ? Because it _might_ be math !
It also might be nonsense. Again, that's indeterminate.
I believe that was stated properly - but you should also probably not
listen to what I say. Im sure there is a perfectly logical answer to
the question, as opposed to mine which is indeterminate as to whether
it is logical or not.

Good grief.

You are working much too hard.  Forget probabilities and just look at the
definitions.  It should be obvious.
Well, if one is doing mathematics proper, then the empty set is a good
candidate.

The empty space works, but there is also a nonempty solution.  It's
unique, up to homeomorphism.

I've been trying to train myself to think differently and so the most
obvious answer to me really is spacetime. But I cannot claim that I am
doing math, only that it might be math. Im not designing airplanes or
anything so I have the freedom to think about silly things like that.

If you are looking at anything other than the simple mathematical
definitions of the terms, then you are looking in the wrong place.
When all else fails, start by writing down the definitions.


If one is doing mathematics, yes. I agree.

This is sci.math, after all.

Im pursuing something else, something that might be math, but might
not be math.

If you have zero bananas and zero oranges, then bananas are oranges ?
Is zero bananas even a banana ? Relying on definitions cannot resolve
this.

Wrong. There is indeed a nonempty solution, and it follows directly from
the definitions.

[ nonsense snipped ]

One strange thing is that I posted two different solutions to the OP's
question. One based on existential indeterminacy, and one based on
unadulterated mathematics. Both solutions make sense, and as one would
expect - the empty set had a hand in this. To me, this merely
reinforces the view that indeterminacy is indeterminate, hence the
third chapter of the book.

You posted one solution; the empty topological space. There is, however,
a nonempty solution.

Definition. A _component_ of a topological space (X,T) is a nonempty
subset of X that is both open and closed.

Definition. A space is _connected_ if its only component is the space
itself.

Definition. A space is _totally disconnected_ if each of its components
consists of a single point.

Now let's suppose that (X,T) is a nonempty space that is both connected
and totally disconnected.

Question 1. How many components does the space have?
Answer 1. One, because it is connected.

Question 2. How many points are in each component?
Answer 2. One, because the space is totally disconnected.

Question 3. How many points are in the space altogether?
Answer 3. One, by answers 1 and 2.

Indeed, every singleton space is clearly a solution. No hocus-pocus or
handwaving is required. The definitions alone are sufficient.


--
Dave Seaman
Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
<http://www.indybay.org/newsitems/2008/03/29/18489281.php>
.

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