That does sound interesting. But even if we construct real numbers in
terms of intervals of rational numbers, we would still be taking
rational-valued moments as basic... I suppose it would be possible to
define things starting with intervals, though. But what properties
define an interval of time? With each moment we can associate a
definite physical state. With an interval, we could associate an
average... this average could be taken as basic, constraining
sub-intervals so that their averages (weighted by length) must equal
the total. But that seems quite strange... of course it is not the
only possible way of defining things.
On Sun, Dec 21, 2008 at 12:24 AM, Brent Meeker <meeke...@dslextreme.com> wrote:
> Abram Demski wrote:
>> It sounds like you are saying that probability is useful because it
>> allows us to predict things-- we convert (past) relative frequencies
>> to (future) subjective beliefs. This cannot be denied. But I don't
>> feel like it answers very much... to understand what t means to
>> "predict", I need to understand time already, which is what is being
>> questioned here... What does it mean for a prediction to be more or
>> less reasonable, if all possible futures in fact occur? How does it
>> help me to take the past experimental frequencies, if I know (or at
>> least believe) that all alternatives will take place?
>>>> Mathematically, though, a real-values time variable doesn't eliminate
>>>> moments, it just makes an infinite number of them between any other
>>>> two, with a particular mathematical structure. So the question of what
>>>> makes them "stick together" remains.
>>> They come with a topology which is about the only concept of sticking
>>> together I can imagine.
>> So anything with a topology counts as time?? That doesn't sound right.
>> Or are you saying it is necessary, rather then sufficient?
> No, I'm saying that the time that appears in physics is a variable that
> takes real values and so it has the topology of the real line. That
> topology is continuous so every "moment" has other moments arbitrarily
> close to it which are well ordered. When I think about this it seems to
> capture the idea of "sticking together". If I pick any two times there
> is a dense set of times joining them. Of course time also includes the
> idea of direction. Most fundamental theories of physics are time
> symmetric and the "arrow of time" is tied to expansion of the universe
> by statistics.
> Bertrand Russell wrote a paper in 1935, which is reprinted in "Logic and
> Knowledge" 1956 which considers how instants (i.e. moments) are
> logically constructed from events (which have non-zero durations). He
> shows that "...the existence of instants requires hypotheses which there
> is no reason to suppose true..." It's rather technical, but you might
> find it interesting. I think Russell is right to regard events
> (intervals) as fundmental and instants as idealized constructs.
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