The Shape of Things – SHAPES 1.0
Proceedings of the First Interdisciplinary Workshop on SHAPES, Karlsruhe, Germany, September 27, 2011. Edited by: Janna Hastings, Oliver Kutz, Mehul Bhatt, Stefano Borgo
If you have ever thought of “shape” as being a simple issue, consider the abstract from “Shape is a Non-Quantifiable Physical Dimension” by Ingvar Johansson:
In the natural-scientific community it is often taken for granted that, sooner or later, all basic physical property dimensions can be quantified and turned into a kind-of-quantity; meaning that all their possible determinate properties can be put in a one-to-one correspondence with the real numbers. By using some transfinite mathematics, the paper shows this tacit assumption to be wrong. Shape is a very basic property dimension; but, since it can be proved that there are more possible kinds of determinate shapes than real numbers, shape cannot be quantified. There will never be a shape scale the way we have length and temperature scales. This is the most important conclusion, but more is implied by the proof. Since every n-dimensional manifold has the same cardinality as the real number line, all shapes cannot even be represented in a three-dimensional manifold the way perceivable colors are represented in so-called color solids.
If shape, which exists in metric space has these issues, that casts a great deal of doubt on mapping semantics, which exists in non-metric space, in a “…one-to-one correspondence with real numbers.”
Don’t you think?
We can make simplifying assumptions about semantics and make such mappings, but we need to be aware that is what is happening.