Darren New wrote:
> Yes. I was phrasing it sloppily there, since we hadn't talked about
> other distributions at that point in the conversation. To clarify, a
> "normal sequence" in the way I'm using it means all possible
> subsequences have the same probability distribution in the asymptote.
> Hence, you can't take a gausian distribution of symbols and expect to
> get a normal sequence from them.

	OK - Agreed.

>>     Agree - other than with the use of the word "will".<G> What I mean is
>> that what you say is how I understand what probability theory says.
> 
> OK. I don't think I know enough math to convince someone else of it. I'm
> just taking it on authority. It would seem to be the sort of thing

	There's really nothing to convince. I'm agreeing with you that
probability theory states the same thing that you state. I have no
_mathematical_ argument against what you're saying (other than the
nitpicking - I agree with your math). I'm just uncomfortable with this
application of probability theory to the "real" world.

	See, in mathematics, everywhere I've seen infinity used with rigor (and
where I understood it), behind all the formalism is basically a
definition of what they mean. Sometimes it is intuitive, but there's no
guarantee that it will conform to reality. Indeed, it is often
impossible to compare it to reality, given the absence of infinities (or
at least our ability to measure them). We can say the cardinality of the
set of natural numbers is that of the rationals, but that's not a
universal truth in the "real" world - it's just a consequence of how we
define cardinality (and numbers).

	It's because of stuff like this that you get the Banach Tarski paradox
(among a bunch of paradoxes). It's due to the Axiom of Choice, which is
quite intuitive to most people. If the BT paradox seems impossible in
the real world, you'll have to give up on the A of C, which seems
counterintuitive.

	Now in probability theory, it's really just mathematics where we're
manipulating numbers and symbols. For cases where we deal with finite
and discrete quantities, probability theory seems to agree with reality.
If it states the probability of something is 0, we interpret that to
mean it will definitely _not_ happen (unless our modeling was off).
Likewise, if it states it is 1, we interpret that to mean it _will_ happen.

	When you now go to continuous distributions, picking a point is, in a
sense, meaningless to the theory. It can only give nonzero probabilities
to intervals, or collections of intervals, etc.** Likewise, probability
theory gives you a 0 for flipping a coin every second forever and never
getting a tails.

	The (usual) interpretation when applied to the real world is that
because it gave me a 0, such a thing can never happen.

	Can it? Can't it? I don't know. I can never know, because I can never
test it. Because of that, I'm wary of results probability theory gives
me for stuff like infinite sequences. I _can_ test it with the real
world for finite events. I'm not saying probability theory is
inconsistent or the math behind it is bad. I'm just saying I shouldn't
blindly accept it as being consistent with the real world just because
it works for finite cases.

	I won't accept that I can't get a forever continuous string of heads
from a coin unless someone can give me a _physical_ reason. There isn't
any - there's only a mathematical one.

<snipped a whole bunch of stuff because I'd just nitpick further>

>> It's not equal to the RHS because the RHS is
>> meaningless. It's undefined.
> 
> It's not meaningless. It's merely not a number. That doesn't mean you

	Well, I guess "meaningless" is vague. It _is_ undefined. About as
useful as dividing by 0.

> But I think I've exhausted my ability to convince you that Shakespeare
> necessarily appears in the output, if the output is infinite and making

	Well, perhaps we're just playing semantic games.

	As I said, I agree that probability theory agrees with you.

	I'm simply wary of using a _purely_ mathematical argument to make
statements about the real world. It invokes no aspect of the real world,
and no laws of physics that I'm aware of. It's just like saying that
yes, you _can_ execute the BT paradox in the real world...

	(As you can guess by now, I'm one of those who feel that the theorems
in mathematics are not necessarily tied to the physical laws of the
universe - other than in the manner that we can think them and
presumably our brains conform to the physical laws...).

** The reason, I suspect, is that theories of integration generally
don't have the integral changing if you happen to remove a point or any
set of measure 0 from the domain of integration.

-- 
"Now we all know map companies hire guys who specialize in making map
folding a physical impossibility" - Adult Kevin Arnold in "Wonder Years"


                    /\  /\               /\  /
                   /  \/  \ u e e n     /  \/  a w a z
                       >>>>>>mue### [at] nawazorg<<<<<<
                                   anl

Post a reply to this message