KMFE: Can you elaborate on that?
JL: For every thing that happens, there is a true statement that says
that that happens. You take this set of true statements, and it
describes everything. So everything is predetermined.
KMFE: Ah, so, if, say, thing 'x' happens, you say there is a true
statement that says that 'x' happens?
JL: That's
right.
KMFE: And what is that
statement?
JL: 'x' happens, for whatever 'x'
is.
KMFE: That's not a
statement.
JL: Yes it is. 'x' happens, where 'x' is some event. How is that not a
statement?
KMFE: Because you didn't say what 'x'
is.
JL: 'x' could be anything. If 'x' were 'it rains today', the statement
would be 'it rains today happens'.
KMFE: Ah, but you see the key there is you had a way to describe what
'x' is. If you can't do that, you can't form a statement. Now give me a
chance to explain something, because you happen to have wandered into
one of my favorite philosophical
traps.
Now, I don't know much about the real world. So I'm going to start by
talking about abstract things. I want to convince you that it's naive to
assume that "x happens" must be a statement.
Do you know about cardinal numbers? Or countable and uncountable sets?
JL: Yes, I do.
KMFE: All right, good. So let's talk about the real numbers. Because
there are more real numbers than integers. Now we form statements out of
symbols, and we have at most countably many symbols. So the set of
statements is at most countable. That means that there is a real number
about which there is no statement.
It goes deeper than that, in fact. Our brains have at most countably
many thoughts. So there is a real number that it is impossible to even
think about, even given an infinite amount of time.
JL: But you're assuming you have countably many
symbols.
KMFE: Well, it doesn't actually matter how many symbols you have. You
take the power set of the set of symbols, and you've got something too
big. So no matter what there are some "things" you can't form statements
about.
JL: So how does this apply to the real
world?
KMFE: Now, I don't know whether everything is really determined. And I
don't know whether there really are countably many things in the real
world. I suspect you don't either. So at the very least your argument is
inconclusive. But I would like to show that while these true statements
might, in some universes, determine every "thing" that happens, they
wouldn't... really. Not the way we think of things. They wouldn't get
every interpretation of every thing, which is what we really think about
when we think about a thing.
So, say thing 'x' happens, and we have (because we're lucky, mind you) a
statement P that basically says"x happens". And say we have lots of
statements like this, P0, P1... etc., for every thing at every timeand
place.
Now another way to interpret the event at P0 is "not P1", which is
guaranteed to be a statement. Or, if P1 happens to be true then, too,
you could say "P1" instead of "not P1". It's not important, really. The
point is that's another way to interpret what goes on. For example,
you're sitting on the grass. Another way to interpret that is "you're
not eating potatoes". Makes sense, right?
In fact, for every subset of the statements P0, P1... you have a
corresponding interpretation. (Note that if the subset is infinite the
interpretation cannot be phrased as a statement, yet it is clearly still
a distinct way of looking at it). So the number of interpretations is at
least as big as the power set of the number of statements, so, again,
too big.
JL: This all sounds very abstract and contrived.
KMFE: It is, but I don't think it's any more abstract or contrived than
a true statement for every thingthat happens. Mind you statements are a
very human construction. They don't lend themselves to defining the
universe. And if you try to use them that way you have to expect to run
into mathematical problems. And if you're dealing with a mathematical
concept anyway, it's not fair to say that those problems don't matter.
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