The dilemma “discrete or continuous universe” is as old as philosophy. Now it is central to modern physics, a field whose practitioners don’t care much about philosophy.
As a mathematician, hence belonging to the “learners” pythagorean school — cite from wikipedia on pythagoreanism:
According to tradition, Pythagoreanism developed at some point into two separate schools of thought, the mathēmatikoi Μαθηματικοι (“learners”) and the akousmatikoi Ακουσματικοι, (“listeners”).
— I shall strike back and accuse modern physicists of lack of imagination in tackling the discrete-continuous dilemma.
In the same time, and that is the more interesting part, I advance the following thesis:
Reality emerges from a more primitive, non-geometrical, substratum by the same mechanism the brain uses to construct the image of reality, starting from intensive properties (like a bunch of spiking signals sent by receptors in the retina), without any use of extensive (i.e. spatial or geometric) properties.
Therefore understanding vision may give us new ideas for physics.
1. for the lack of imagination part, I argue that making an experiment (which in particular may probe the discreteness or continuity of a piece of reality) is like making a map of a territory. However, there are mathematical results which put a priori bounds on the accuracy of any map (aka Gromov-Hausdorff distance), thus making irrelevant the distinction between a discrete or a continuous territory. See this for an introduction, also see this for the particular case of the Heisenberg group.
2. for the thesis part, I shall explain why it is a reasonable speculation based on the same mathematical results.
This is based on the paper arXiv:1011.4485.