> "If, without in any way disturbing a system,
> we can predict with certainty the value of
> a physical quantity, there exists an element
> of reality corresponding to this physical
> quantity", wrote once EPR.
> Is there a similar definition, in math?
If, from a set of axioms and rules of inference, we can produce a
valid proof of a theorem, then the theorem is true, within that
I'd suggest that this notion of provability is analogous to the
"reality" of physics. Provable theorems are what we know, within
a mathematical system.
Now, one problem with this approach is that it focuses on the theorems,
which are generally "about" some mathematical concepts or objects,
but not on the objects themselves. For example, we have a theory of
the integers, and we can make proofs about them, such as that there
are an infinite number of primes. These proofs are what we know about
the integers, the "mathematical reality" of this subject.
But what about the integers themselves? They are distinct from the
theorems about them. Maybe we would want to say that it is the integers
which are "mathematically real", rather than proofs about them.
The question in my mind is how to understand Tegmark's theory that the
world is a mathematical structure, something analogous to the integers but
more complex. We actually live within a mathematical object, according
to this view. What does physical reality mean in such a framework?
Would it correspond to mathematical reality, within that one mathematical
structure that we live in?
Or turning to Schmidhuber's model, where the world is a computer program,
what does physical reality correspond to? We have a distinction there
between the program and its output, similar to the distinction in math
between proofs and the objects about which theorems are proven.
If we focus on the output, then that would be the fundamental physical
reality of the universe. We could then "chunk" that output or identify
patterns in it, and those would be real as well. In general, any
computable function which took as input a region of the universe and
produced as output a true/false result would define an element of
Some functions would be much more useful than others, producing "elements
of reality" which are more stable or more predictable. Conserved
quantities, for example, would be elements of reality which were useful
for predictions in a variety of situations. But in principle, I think
all computable predicate functions would have equal philosophical status.