Pat, (01)
>The set D can *be* a set of horses. Sets can be
>sets of anything. (02)
True. (03)
>To say that the universe is a set is, literally, to
>say nothing whatever about what kinds of thing are in the universe, (04)
True. But to say that the universe is a set does imply that it is
composed of "things". That is not a vacuous assumption, and it's not
shared by all cultures. (05)
Let me give an example. Suppose I tell you my Aunt Jane's positive
attitude is part of the reason she is a 20year breast cancer
survivor. In order to make sense of that sentence, do I need to
believe there really *is* a set of all possible attitudes, and that
there is a member of this set that really *is* my Aunt Jane's actual
attitude? (06)
That is how I would represent this sentence if I were to construct a
logical theory (using a probabilistic logic, of course) that says a
person's chances of surviving cancer are higher if the person has a
positive attitude. The natural way to do this would be to define a
function that maps a person x to the person's attitude Attitude(x).
The domain of the Attitude function is the set of all persons, and
the range is the set of all attitudes. (07)
I'm willing to accept the following assertions: (1) There is a set of
all people; (2) There is an attribute of persons that we label
attitude; (3) To some degree, we can ascertain whether a person has a
positive attitude; (4) There are interventions we can make to affect
a person's attitude. But I'm not at all sure I agree that the
universe really contains a set of consisting of all the possible
attitudes, and that there really is a function that maps each person
to an element of the set of all attitudes, and that the value of this
function *is* the person's attitude. Maybe there is and maybe there
isn't a real set of all possible attitudes. I don't know. From what
I know of psychology and neurophysiology, I think we're a very long
way from a theory of attitude to which I would give credence as a
faithful representation of how the world really works. I think it's
fairly likely that today's notion of attitude will go the way of
phlogiston, the nonexistent component that was once thought to be
the part of all flammable substances that caused them to burn. (08)
Fortunately, I don't think it is necessary to believe there really
*is* a set of all possible attitudes, and that each person really
*does* have an attitude that is an element of this set, in order to
accept the above theory as a useful representation of reality for
some purposes. We can design questionnaires to measure attitude. We
can correlate questionnaire responses to cancer survival rates. We
can define a conditional probability distribution for cancer survival
given positive attitude and given negative attitude, and estimate the
parameters of this distribution to ascertain the degree to which
attitude affects survival. We can design interventions, measure the
impact of the interventions both on responses to the attitude
questionnaire and on cancer survival rates. We can use the results
of these studies to design clinical interventions aimed at improving
survival rates. Teams of healthcare professionals and statisticians
do this sort of thing all the time. They end up with logical
theories / mathematical models / representations of reality that are
very useful for the practice of medicine. (09)
The theory of attitudes I sketched above is mathematically equivalent
to defining a probability distribution on Tarski interpretations of a
set of logical axioms. In each of these Tarski interpretations,
there is a set of people and a set of possible attitudes and a
function mapping persons to attitudes. Each sentence about attitudes
makes an assertion about elements of or subsets of this set of
possible attitudes. The logical theory is *about* (among other
things) the set of all possible attitudes. (010)
But I can make perfectly good use of this theory, while thinking that
the idea of a set of all possible attitudes is utter nonsense. I may
find this theory very useful for the purpose of representing the
effect of attitude on cancer survival and recommending clinical
interventions that improve survival rates. I can believe this theory
makes assertions that are approximately correct about the real world.
Yet I may think that literally treating these assertions as being
*about* an actual set of attitudes that really exists in the world is
absurd. (011)
It may very well be that our ontologies represent classes of "things"
that do not correspond to real "things" in the world. Maybe some of
our assertions, although they are phrased as sentences about these
"things," are not actually about "real" things. Maybe that's just the
best we can do with language, but our language doesn't map very well
onto what is really there. I don't think one needs to believe that
the Universe really is a set in order to make effective use of
logical theories that represent the Universe as a set. (012)
Kathy (013)
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