In response to another question during the show, Dr. Pamela Gay offers an understandable explanation for the impossible size of Planck lengths, which are 10-20 times the diameter of a proton. While that size can be approached using the language of mathematics, it's impossible to appropriate as a physical unit - things that small cease to be discreet and can only described in terms of probabilities, "quantum foam" in her words. I chuckled at her co-host's suggestion that the idea had little in common with philosophy. The notion that one could forever divide one's progress in two and never arrive at one's destination is paradox made famous by Zeno.
Dr. Gay also maintains the astronomy blog Star Stryder and teaches at Southern Illinois University at Edwardsville.