The idea that mathematical reality is fundamentally the same as physical
reality is not obviously true, but I don't believe it can be rejected
either as obviously false. The goal in exploring that hypothesis is to
see if we can explain features of physical reality that are otherwise
inexplicable; and also, of course, to see whether the idea is even
coherent or is fundamentally contradictory.
On its face, it is hard to see how numbers, which we think of as rather
simple and static entities, could be anything like the richness and
dynamism of the reality that we see around us. But I can suggest two
reasons to think that this instinct is wrong.
The first is that numbers are really far more complex than they seem.
When we think of numbers, we tend to think of simple ones, like 2, or 7.
But they are not really typical of numbers. Even restricting ourselves to
the integers, the information content of the "average" number is enormous;
by some reasoning, infinite. Most numbers are a lot bigger than 2 or 7!
They are big enough to hold all of the information in our whole universe;
indeed, all of the information in virtually every possible variant of our
universe. A single number can (in some sense) hold this much information.
So our instincts about numbers and their simplicity is not really correct.
Numbers in general are far from simple. They are enormously more
complicated than we could ever comprehend. This should give us reason
to doubt whether our other instincts about numbers are really as obvious
as they seem. For example, nothing could seem more true than that a
number is a static value. But perhaps, in the infinite complexity of
a typical number, there is room for a form of dynamism.
It's useful to consider what is sometimes called the "block universe"
model. This is a way to see that static and dynamic entities are not
really so different. It is a typical way of analyzing things in special
relativity. In this view, time is a fourth dimension. We can map out
the whole universe as a four-dimensional object. A slice through this
object at a given instant of time is the three-dimensional world at
a particular instant.
On the one hand, we know that the universe is dynamic and ever-changing.
On the other, the four-dimensional block universe is a static object.
The apparent dynamism is seen as something of an illusion. There is
no actual passage of time, rather all moments coexist. The future and
the past are merely relative directions like north or south, relative
to some observer.
The block universe (or spacetime) is not the only way to look at things,
but it is one valid way, and it illustrates that even within a static
object (the block universe) there may be the perception of dynamism from
the inside. The point is, then, that conceivably a seemingly "static"
number of sufficient complexity could have similar internal dynamism.
I'll mention one other way of looking at mathematics that can help to
see how physical reality could be thought of as a mathematical object.
There is more to math than numbers. Much more. Another elementary branch
of mathematics is geometry. If it may seem strange to imagine living
in a number, it is perhaps much easier to imagine living in a geometry.
And yet, Descartes showed how to map geometric relations into numerical
ones, via his analytical geometry.
Other types of mathematical objects include the many kinds of logic
that Bruno uses in his analyses. There are also things like theorems
and other highly structured forms of information. In fact, although
we speak of mathematical reality, a better term might be the reality
of logic, or even the reality of information. Mathematics, logic and
similar fields all have to do with information. Each field focuses on
particular kinds of information and the structures that lie within.
Another kind of information is computer programs. We can define an
abstract computer and analyze its effects as it runs abstract computer
programs. This is fundamentally no different a process than analyzing
numbers, logic, geometry or other mathematical objects. But with
computers we tend to make the time dimension explicit, again within
our abstract model. The computer passes through a sequence of states.
And this makes it easier to imagine that our existence might somehow
correspond to the execution of even an abstract computer program.
Again, this is not meant as a proof that physical reality is part of
mathematical or informational reality. Rather, it is meant to dispell the
initial notion that it is a completely absurd proposition. Evidence to
go beyond that, to allow arguing that it really is (or at least might be)
true, has to come from a different direction.
The evidence in favor of the proposition would come from the kind of
reasoning used by physicists to advance multiverse and anthropic models.
Some physicists argue that they can best explain certain observations if
they assume that our universe is just one of many; in some cases, one of
an infinite number. If we end up accepting that an infinite number of
universes "really" exist, having real physical reality, we are much closer
to accepting that all informational objects exist. At least, it's much
harder to argue that the notion that they are one and the same is absurd.
Ideally, we could then start to make predictions based on this
all-inclusive model which could be confirmed by experimental testing.
Obviously we are a long way from being able to do that. I did make some
predictions a few months ago when I was describing my own ideas along
these lines, but they were pretty weak, and I had to make some strong
assumptions, beyond just the basic multiverse model, to get them.
Still, with further work I believe that it will be possible to come up
with improved models and predictions that will allow various multiverse
models to be tested. And it may yet turn out that the most inclusive
multiverse, Tegmark's Level 4 where all mathematical objects exist and
physical existence is just a subset of the mathematical, could be the
model that provides the simplest explanation for our observations.
Hal Finney
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Received on Thu Mar 16 2006 - 20:17:00 PST