Space and shape

Naively, we think that left and right hands differ in shape. But we just saw that there is actually no intrinsic difference of this kind between the things themselves; there can only be a difference in the way they are entered in a certain kind of space. Hands can differ if their containing space is orientable. If it is not they will all be alike however we enter them in it. We found this out by looking at the spaces defined by a paper strip from outside it. A paper cylinder is orientable, but if we cut it and twist it we can change it into the non-orientable Möbius strip. It alters what things do when they move in the space defined by the strip. This looks as if we can say that the difference in the spaces is a difference in their shapes. The shape of space is to play an explanatory role in our theory of the world. That is what we need to make some sense of. But making sense is just the problem. Can we properly speak of the shape of space?

Now it seems all very well to say that space has a shape when we can regard it as defined by a strip of paper, the surface of a ball or of an arbitrarily far extended table top. We see it from outside in a space of higher dimension and it is visibly twisted, curved or flat from that vantage point.