Jason Thompson wrote:
>From: "Nick Bostrom" <nick@nickbostrom.com>
>
> > Suppose we know that the true theory of everything is of the form T, and
> > that it has one free parameter, a physical constant k that, on theoretical
> > grounds, can take on any integer value between 1 and 1,000,000. But there
> > is no a priori ground, let's assume, why k should have one of these values
> > rather than another. So by a principle of indifference, we assign each of
> > these possibilities an equal probability. Now consider the specific
> > hypotheses Tn:="T and k=n" for n=1, ..., 1,000,000. Probability calculus
> > then implies that the probability of Tn <= 1/1,000,000. In other words,
> > each of these specific hypotheses must be highly a priori improbable since
> > they are mutually exclusive and their combined probability cannot exceed
>1.
>
>But, again, why should we reason this way at all? Why should we be of the
>belief that since we are able to attach a numerical value to a property of
>the universe, it is therefore a 'variable'? On what sort of theoretical
>grounds can you presume to assign a particular set of integer values to
>physical constants?
In my example, the integers [1;1,000,000] represented the possible values
for some physical constant k. I'm saying that in absence of any reason to
assign one of these possible values of k a higher probability than another,
you ought to assign an equal subjective probability to each of these
possible values. We could have theoretical grounds for thinking that the
value of the constant must be in the interval [1;1,000,000] -- e.g. because
the overall by-far simplest and most elegant physical theory implies that
it has one of those values.
> > Even if it was in some sense physically necessary that
> > k=42925, so that the physical chance of k=42925 is 1, this is compatible
> > with our prior epistemic probability of k=42925 being very small.
>
>Not following you here. Again: why should we reason that since other
>numbers exist, they are (even a priori cognitively) available to said
>constant?
You must distinguish between objective chances and subjective (or
epistemic) probabilities to understand this point. Let me illustrate:
Suppose you toss a coin which you know is biassed, but you have no idea
which way. You rational epistemic probability of Heads in this situation is
50%. This is so despite the fact that you know that the objective chance is
not 50% but something different - maybe 70% chance of Heads, if that's how
the coin is biassed.
Similarly, maybe the physical chance of a particular physical constant
having the value is actually has is 100%; the laws of physics require it to
have that value. We can still consider what your ratinonal epistemic
probability of the constant having that value is, prior to learning that
the physical chance is 100%. (If you know what the physical chance is, then
you should typically set your epistemic probability equal to it - this is
known as the Principal Principle.) And your prior epistemic probability may
well be very different from the physical chance. So why is this prior
epistemic probability relevant? Because it determines which theory you will
end up believing, given that you update your beliefs using Bayesian
conditionalization.
Nick Bostrom
Department of Philosophy
Yale University
Homepage: http://www.nickbostrom.com
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