And I completely agree with Eric. But we can language it real simply and
intuitively by just looking at what a probability space is. For further
simplicity lets keep it to a finite probability space. (Neither a finite
nor an infinite one says anything about "time".)
A finite probability space has 3 elements: 1) a set of sample points
called "the sample space", 2) a set of events, and 3) a set of
probabilities /for the events/. (An infinite probability space is
strongly similar.)
But what is this "set of events"? That's the question that is being
discussed on this thread. It turns out that the events for a finite
space is nothing more than /the set of all possible combinations of the
sample points/. (Formally the event set is something called a "sigma
algebra", but no matter.) So, an event scan be thought of simply /all
//combination//s of the sample points/.
Notice that it is the events that have probabilities - not the sample
points. Of course it turns out that each of the sample points happens to
be a (trivial) combination of the sample space - therefore it has a
probability too!
So, the events already /have/ probabilities by virtue of just being in a
probability space. They don't have to be "selected", "chosen" or any
such thing. They "just sit there" and have probabilities - all of them.
The notion of time is never mentioned or required.
Admittedly, this formal (mathematical) definition of "event" is not
equivalent to the one that you will find in everyday usage. The everyday
one /does/ involve time. So you could say that everyday usage of "event"
is "an application" of the formal "event" used in probability theory.
This confusion between the everyday "event" and the formal "event" may
be the root of the issue.
Jus' sayin'.
Grant
On 12/14/16 11:36 AM, glen ☣ wrote:
Ha! Yay! Yes, now I feel like we're discussing the radicality (radicalness?)
of Platonic math ... and how weird mathematicians sound (to me) when they say
we're discovering theorems rather than constructing them. 8^)
Perhaps it's helpful to think about the "axiom of choice"? Is a "choosable" element
somehow distinct from a "chosen" element? Does the act of choosing change the element in some way
I'm unaware of? Does choosability require an agent exist and (eventually) _do_ the choosing?
On 12/14/2016 10:24 AM, Eric Charles wrote:
Ack! Well... I guess now we're in the muck of what the heck probability and
statistics are for mathematicians vs. scientists. Of note, my understanding is
that statistics was a field for at least a few decades before it was specified
in a formal enough way to be invited into the hallows of mathematics
departments, and that it is still frequently viewed with suspicion there.
Glen states: /We talk of "selecting" or "choosing" subsets or elements from larger sets. But such
"selection" isn't an action in time. Such "selection" is an already extant property of that
organization of sets./
I find such talk quite baffling. When I talk about selecting or choosing or assigning, I am talking
about an action in time. Often I'm talking about an action that I personally performed. "You
are in condition A. You are in condition B. You are in condition A." etc. Maybe I flip a coin
when you walk into my lab room, maybe I pre-generated some random numbers, maybe I look at the
second hand of my watch as soon as you walk in, maybe I write down a number
"arbitrarily", etc. At any rate, you are not in a condition before I put you in one, and
whatever it is I want to measure about you hasn't happened yet.
I fully admit that we can model the system without reference to time, if we want to. Such efforts
might yield keen insights. If Glen had said that we can usefully model what we are interested in as
an organized set with such-and-such properties, and time no where to be found, that might seem
pretty reasonable. But that would be a formal model produced for specific purposes, not the actual
phenomenon of interest. Everything interesting that we want to describe as "probable" and
all the conclusions we want to come to "statistically" are, for the lab scientist, time
dependent phenomena. (I assert.)
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