Dear Matthew, (01)
First of all, I strongly endorse the 4-d view, and I believe that
it is preferable to a 3-d view for many problems. However, I don't
believe that there is any ontology that is ideal for all problems. (02)
JFS>> Whether you have a 3D or a 4D perspective, change still exists, (03)
MW> Not really. 4-Dimensionalism has the effect of making 3D change
> seem static, because it is looked at in a different way, in 4
> dimensions instead of 3. (04)
The basic laws of physics are stated in differential equations, which
are "almost" symmetric with respect to the space and time coordinates. (05)
I use the term 'almost' because entropy creates an "arrow of time",
which breaks that symmetry. In relativistic terminology, the arrow
of time defines a *light cone* that delimits the causal influences.
If you look in the direction of that arrow, you find increasing
entropy. But there is no such distinction in any of the spatial
coordinates. (06)
MW> ... extensionalism in classes is quite natural when you have
> dealt with change in the way that 4-dimensionalism does. (07)
I'll avoid getting into debates about what is 'natural', but I must
emphasize that the distinction is independent of time or change.
In my previous note, I made that point by talking about hypothetical
issues, but the same point can also be made in terms of modal logic. (08)
Second, the distinction can be seen very clearly in examples such as
the types HumanBeing and FeatherlessBiped, both of which have the same
extension, but different intensions. It's irrelevant whether those
two types have the same extension in a 4-d universe or for all time
in a 3-d universe. They are not provably equivalent according to
the usual definitions of the terms. Therefore, they are different
by intension, and only accidentally the same by extension. (09)
JFS>> In any perspective, you must be able to plan for the future,
>> talk about what exists NOW, or what exists in some hypothetical
>> time or place that might never exist anywhere in the 4D universe. (010)
MW> Yes. But in a 4-dimensional world view, all of this can be dealt
> with extensionally, so why wouldn't you? (011)
You wouldn't in either 3-d or 4-d because it's impossible. Many more
things are planned than are ever implemented, and many things that are
implemented have no little or no resemblance to the plans. Therefore,
you must be able to talk about the *type* of airplane because the set
is very likely empty in any or all ontologies. And there is exactly
one empty set: the set of all unicorns is identical to the set of
all airplanes with flapping wings. (012)
MW> Well you can do the usual things with possible worlds to deal
> with That [hypotheticals], so no great problem there. (013)
But we cannot observe, visit, or manipulate possible worlds.
When we reason about possible worlds and entities on our computers,
we are actually using intensional descriptions of the hypothetical
entities to create 'virtual' extensions. (014)
Even with our actual world, it is impossible to deal with extensions
for most of the things we talk about. Census takers are well aware
of the difficulty of enumerating all the people in a single city.
Imagine trying to enumerate or reason with the set of all mice,
flies, or bacteria in a city. We must reason with intensional
descriptions because it's impossible to deal with the extensions. (015)
MW> They [differential equations] are just descriptions of 4D
> objects, just as a quadratic can describe a line in two dimensions. (016)
Yes, indeed. Those equations are *intensional* characterizations
of entities that might or might not exist in any world, independent
of whether the ontology happens to be viewed in 3-d or 4-d terms. (017)
MW> However, a 3D ontology will be predisposed to an intensional
> approach, whereas I find with a 4D ontology an extensional approach
> is more natural. (018)
In all of your examples of hypotheticals and plans, you were talking
about the intensional characterizations. So you were doing what I
was suggesting: talking about intensions. (019)
John (020)
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