David Musick (
Wed, 22 Jan 97 08:27:32 UT

The idea of creating a virtual reality environment with different mathematical
properties than ours was brought up, such as one where 1+1=3. Mathematically
speaking, the operation of addition is simply a function, taking a set of two
input numbers and mapping it to a third. We actually could get more abstract
about this, and talk about any types of sets and their mappings, but it's
unnecessary for this discussion. The way that this function maps these two
numbers to a third number can be fairly arbitrary. So, 1+1 *can* = 3, if we
define '+' appropriately.

It just happens, that in our daily experience, there are certain patterns we
notice about quantity. For most solid things, if we take one of them and then
another one of them and put them together, we have two of them together. If
we take three of them and four of them and put them together, we have four of
them together. So, we notice a certain pattern to the two numbers we bring
together and the resulting sum. This pattern is just a feature of the typical
objects we deal with. And we're so used to dealing with "objects" that have
those properties, that it's difficult for us to imagine a world that didn't
have those properties, since we tend to automatically imagine things in terms
of objects, and objects do indeed have those properties.

Now, if we have something like drops of water, and we take one of them and
another one of them and put them together, we still have one drop of water.
If we take one cloud and another cloud and put them together, we still have
one cloud. Drops of water and clouds aren't really objects, so they don't
have all the properties of objects; thus, there is no reason to expect them to
have the same sort of mathematical properties that objects have.

Whenever we deal with a world that has traditional objects in it, the rules of
arithmetic, as we know them, will apply. If we're talking about virtual
reality and the set of all possible sensory worlds, then nearly all of them
will be very different than ours and the minds which develop in the sensory
systems which experience these sensory worlds will develop different forms of
mathematics than we have. In some, something as basic to us as set theory may
not apply at all, especially if there are no objects in that world and no sets
of "things".

We have such persistent concepts of objects because they are so much a part of
our daily experience. There are certain sensations which persist in our
sensory environment, sensations which we come to recognize and group together
as an object. If I pick an object up and turn it in my hand, it looks similar
from slightly different angles, and that similarity throughout the turning,
where as I turn it, it doesn't change radically over the course of small
angles, draws all those different perspectives together, and I think of that
set of sensations as an object. There is a certain "coherentness" about it.
There is also the fact that different features on it tend to be in the same
relative position to each other and move together (if one feature moves up,
the other ones move up also)

I can imagine (sort of, anyway) worlds where there was just a lot of flowing
images and sounds and there were no objects. Regions of color may come
together or split up and swirl around with other colors and mix in interesting
ways. The user could influence their sensations in certain ways, using a data
suit to input. They would have no bodily representation, like we have here,
but they would have influence over what they experienced, and they would
explore that, like babies explore their influence over their sensory worlds,
and start finding general patterns in it.

Some of the patterns we have found in our sensory worlds are the patterns of
objects, of sensations that consistently go together. These objects relate to
each other in certain, consistent ways, and we can construct an arithmetic,
based on this. Other sensory worlds would have different underlying patterns,
and the minds experiencing them would think in terms of those patterns.

I don't know if a world could be constructed where 1+1=3, where the numbers
represented the number of distinct objects. I think if there are objects,
like we have here, then the arithmetic will have to be the same. But our kind
of objects are certainly not the *only* kind possible.

- David Musick

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