Should science exclude supernatural explanations?

I've been having a most enjoyable discussion with Garth regarding, specifically, the place of miracles in science, and more generally the demarcation between science and religion. I found myself repeating a dictum that I have often repeated in dialogues and debates with creationists and intelligent design advocates (not that Garth, to my knowledge, falls into either of these categories). I know it's an ugly way to put it, but if science had a "mission statement" it would be this: Seeking natural explanations for natural phenomena. For his part, Garth made the valuable observation that in these kinds of (what he terms) "heartland modernist" debates, many people simply accept such foundational assumptions (e.g. that science excludes supernatural causation by definition) as axiomatic, when perhaps they too should be open to rigorous examination and questioning. (Again--I'm not saying that Garth himself necessarily disagrees with the notion that science excludes supernatural explanations: that's beside the point.)

The marine biologist (and leading light in the "Evolution Wars") Wesley R. Elseberry argues the following:

The modern practice of science is premised upon the radical assumption that the physical universe is comprehensible to humans. That this assumption is radical is supported by the fact that it has not always historically been accepted, that it remains largely unassimilated even today, and that many explicitly reject it since they believe that it denies any reality to theism, mysticism, or even mystery. The modern practice of science also requires that objectivity be approximated, even if it cannot in principle be completely achieved. The practice of science is a pragmatic endeavor whose principle product is the conversion of subjective personal experience into an approximation of objective knowledge concerning physical phenomena. While the subjective appreciation of a role for supernatural causation may be important to personal fulfillment, it does not afford a basis for objective knowledge, nor can it be counted as a means of comprehending the universe in a scientific manner.

Is he right? Or is there a place for supernatural explanations in science?