Wednesday, April 10, 2013

Geometrical Psychology

Imagination receives the stream of Consciousness, and holds apart and compares the different experiences

I was reading about Benjamin Betts’ unusual diagrams of consciousness, collected in the 1887 tome Geometrical Psychology, or, The Science of Representation, a predecessor to Julian Hibbard’s geometric diagrams of love. This reminded me of something I’d read about years ago – the question of whether mathematics in a different universe may vary from the one we grew up to either love or hate.

If there ever is a universe out there with a non-Euclidean space, or if that universe has more than the three spatial dimensions we know so well – it entirely possible that the geometry of that universe is unlike ours.

Euclid's five postulates, the axioms of geometry below would therefore be different from ours.

1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

So we are able to see how geometry may be different in another universe, but can we push this notion to think that arithmetic may also differ from that of ours. After all, isn’t arithmetic so pure, so fundamental that it surely can’t be different? For example, let's say we have a box in our universe and we place a cat in this box. We then put another cat into the box. We now have two cats in this box, and we would use this fact to generate one of our axioms of arithmetic in this universe: 1 + 1 = 2

But now let's say we’re in a different universe with a different geometry. Again, we have a box and we again place a cat into the box. We then add another cat into our box. But in this universe physical objects behave differently so that when two objects (cats) touch they merge to become a single object. So we now have only the one cat in our box. In which case the mathematics developed in this universe has the axiom: 1 + 1 = 1

It’s possible the whole system of mathematics developed in a different universe may differ from ours because it’s based on axioms different from what we’ve formulated. Those axioms would be based on physical axioms of a universe different in geometry from ours. These, to the inhabitants of that universe will be completely natural and obviously correct.