Tuesday, January 8, 2008

P. Dirac about geometric and algebraic way.

Recently, during my search for information about projective space I bumped into an interesting script of P. Dirac with title "Projective Geometry, Origin of Quantum Equations".
The script is made from a talk Dirac gave in 1972. He seemed to talk to a general public, so the talk was rather inform.

In his talk Dirac briefly described projective geometry and argued that projective geometry is more appropriate than Euclidean geometry as a mathematical structure for quantum theory. I was not able to really understand the link between projective geometry and quantum theory, but I believe his view was fundamentally correct in theoretical physics.

What has interested me the most was his philosophical comments about geometry and algebra. In mathematical works you will, according to Dirac, either prefer the algebraic way or the geometrical way. The algebraical way is more deductive where you start with equations or axioms and follow rules to get to interesting results. The geometrical way is more inductive where you start with some concrete pictures and try to find relationship among the pictures.

Dirac put himself into the geometrical camp. But he lamented that his works rather appear more algebraic (e.g. lots of equations) although he used a lot of geometrical methods. The reason for this is that it was pretty awkward to produce and print pictures in published papers. Writing equations was just much easier for him (and probably most other theorists in his time) than drawing pictures. So, it was rather a technique limit that led to heavy algebraic appearance of his works.

That technique limit seems to disappear with the availability of modern computers. I am wandering what impact could it have had if the modern computers were available to those great scientists. On the other hand, in the current academic world the algebraic way still seems to dominate the stage: a paper with a lot of differential equations would be considered more scientific than a paper with a lot pictures.

The mission statement of VisuMap Technologies is unleashing human visualization power for complex high dimensional data. It is indeed our grandiose ambition to revive visualization as first ranking tool to do science, that has somehow lost its glory since Rene Descartes invented his coordinate system. In order to achieve this we not only need new software, new methodologies and theories, but also people who embrace visual way to do scientific research. So, there is still a long way to go!

I feel pleased that Euclidean space has been considered inadequate by Dirac for quantum theory. Since I have felt for long time that the open Euclidean space is not appropriate for generic study of high dimensional data. For instance, some concepts like left/right, centers/peripherals (which make sense in Euclidean space) can not be applied intuitively to high dimensional data. For this reason, I have called RPM (relational perspective map) as MDS on closed manifolds. RPM basically simulates a non-stationary dynamic system on a closed manifold. Interestingly, modern physics offered some useful tools to do that. I have looked in to several low dimensional manifolds as our image space (like sphere, torus and real projective space). Our long term plan is to evolve the image space to more expressive structures to visualize high dimensional data.