Jerry Clark wrote:
>
> Christopher Maloney wrote:
>
> > we can only, each of us, deduce things from the SSA from a first
> > person perspective. We can't take anyone's word for what they've
> > deduced from the SSA. So we shouldn't, for example, take the word
> > of a scientist who's just performed the QS experiment and survived
> > despite thousand-to-one odds, that QS really works. *He* would have
> > a strong reason for believing it, but I would have no additional
> > incentive to believe it.
> >
>
> But if deductions from the SSA can only be made from a first person perspective
> then that makes all deductions from the SSA completely bankrupt because all the
> deductions I've seen made from the SSA are very *general* statements (e.g.
> your one about Life universes not subjectively existing). You can't say that
> deductions are
> subjective unless what they are *about* is also subjective. Or do you believe that
> there is no fact of the matter about whether the probability of finding oneself in
> a Life universe is 0? Is there one probability as far as Chris Maloney is concerned
> and another as far as, say, Al Gore is concerned? Hmmm.
The above makes no sense. Yes, I believe in objective facts. And I also
believe that it is possible for me to make some deductions about those
objective facts by using a first-person reasoning from the SSA. My
deductions would only be probabalistic, of course. That is, they'd only
have a certain probability of being true. But then again, any deductions
I make from empirical evidence is probabalistic.
>
> > So, I think, yes, if my argument is correct, it doesn't matter
> > that a life-SAS would deduce the same thing. I would be correct
> > and (s)he would be wrong. That is, I still do believe that
> > different sorts of SAS's have different measure. Taking Hal's
> > approach and doing away with infinite sets, spoze that there are
> > a total of one billion SAS's in all the universes everywhere, and
> > that ten of them are green, but the rest are blue. Any one of
> > them would be justified in assuming that most SAS's are of the
> > same color, but the green ones who made that assumption would be
> > wrong.
> >
>
> But you are not putting forward an argument, just speculating that there
> might *be* an argument, as far as I can tell. There may be
> an argument for saying that 'one' is less likely to be an SAS in a Life
> universe but I don't think that one can simply say that *I* am not in a
> Life universe therefore it is unlikely. On the same grounds I could
> argue that is extremely unlikely that one is an American citizen.
There were two quite good posts on this topic recently by Jacques Mallah:
http://www.escribe.com/science/theory/m1192.html, and
http://www.escribe.com/science/theory/m1194.html. The difficulty arises
in the confusion between a priori and a posteriori probabilities. For
example, I know that my license plate number is AFT-058. It's not now
valid for me to compute the odds of the car in my driveway having the
license plate AFT-058 as 1/(26^3 10^3).
Likewise, it should not be surprising to me that I'm an American, even
though Americans only make up 1/20 the population of the world.
So this just shows that any conclusions to be derived from the SSA must
have a suitable prior. In my example above, the prior was the assumption
that the any *ratio* of green/blue SAS's was equally likely. Fine. Now
if I were a blue SAS, I would have a sample set of one, from which I could
compute a probability distribution on ratios of colors.
Let R be the ratio of blue to green SAS's, R = b/g. I want to compute
the probability distribution as a function of R on the domain [0,1]:
P(R) P(1 blue | R)
P(R | 1 blue) = ----------------------
P(1 blue)
Now, P(R) is the prior, and I said above that the assumption is that
this is constant over [0,1], and P(1 blue) = 50%, so P(R)/P(1 blue) is
a constant, and is just a normalizing factor.
Clearly, P(1 blue | R) = R. Therefore, with the normalizing condition
that the integral of P from 0 to 1 = 1, we get
P(R | 1 blue) = 2R
Which has an expection value for R of 2/3. That is, given a sample of
one blue SAS, and the prior mentioned above, I would compute that the
probablity of any SAS being blue is 2/3.
>
> >
> > So if the set of life-SAS's is not isomorphic to the set of (3+1
> > dim. pseudo-Riemannian manifold quantum field)-SAS's, then we'd
> > have no a priori reason to assume that the measures of these sets
> > are the same. If the measures are different, then one is larger
> > than the other. My money will be on our set having the larger
> > measure. If the measures
> > are transfinite but of different orders, then I conclude that the
> > probability of finding oneself to be a life-SAS is zero.
I stand by this. Note that I explicitly give a new prior: *if* the
measures of each set are of different orders of transfinite numbers.
Using blue and green again, that would mean that either R=0 or R=1.
We still have the ratio of prior probabilities, P(R)/P(1 blue), is a
constant. But now P(1 blue|R=0) = 0, and P(1 blue|R=1) = 1. The
normalization integral becomes a sum, and we get, simply,
P(R=0 | 1 blue) = 0
P(R=1 | 1 blue) = 1
Which is pretty easy to interpret.
> >
>
> I don't personally believe that our universe *is* a 3+1 dim. pseudo-Riem.
> manifold quantum field, but that's another story.
>
> Where your argument falls down is betting on our set having the larger measure,
> based (I'm presuming) on the evidence that *you* are in the latter sort of world.
> If you wanted to construct a Bayesian proof of your statement you would end up
> showing that you are *minutely* more likely to be in this world.
>
> Once you start talking about transfinite measures I give up. The Bayesian arguments
> that underpin all these anthropic arguments are based on *probability* and
> therefore on *natural* numbers, counting etc.
>
--
Chris Maloney
http://www.chrismaloney.com
"Donuts are so sweet and tasty."
-- Homer Simpson
Received on Thu Dec 09 1999 - 20:14:15 PST