# Re: MATHEMATICS: 1+1=3

Hal Finney (hal@rain.org)
Wed, 22 Jan 1997 07:39:17 -0800

From: "David Musick" <David_Musick@msn.com>

> 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.

This is an intriguing concept. I think the idea is that beings from
such a world might not come up with a mathematics in which 1+1=2. Maybe
they would come up with some other mathematics, equally consistent, but
more useful in their world.

This is a reminder that talking about "a world in which 1+1=3" is
not really the right way to think of it. Mathematics is consistent
within itself irrespective of any given worlds. To ask whether a given
mathematics applies to a world you need to try to set up a mapping
between the mathematical objects and aspects of the physical world.
If the mapping is consistent then the mathematics can be said to apply.

I suspect though that with sufficient effort (and stubborness) you could
set up a mapping between any mathematics and any world. It might not
be very simple, you might have to introduce arbitrary and unnatural
definitions and "objects". In the limit, the mapping would practically
ignore the world and just force its own object definitions which had no
correspondence to any natural physical reality. So in practice we can
judge the degree of applicability of a mathematics by looking at how
complicated the mapping has to be. Unfortunately this assumes a certain
objective reality to this complexity judgement.

In David's world, then, the question is whether we would say that 1+1
does not equal 2 there, or would we say that 1+1=2 there as everywhere,
but beings from that world are unlikely to discover that mathematics?
If the world had a spatial dimension, we could still map numbers to
regions of space (which colors are flowing through) and point out that
these regions can be counted and grouped completely consistently with
the mathematics of the integers. This mapping might be obvious to us,
but practically incomprehensible to the beings who live there. So it
is not clear how to judge whether 1+1 would equal 2 in such a world.

Hal