A philosophically minded person asks a good question properly, and Dawkins succumbs to delivering more of a comeback than an answer.
A more frank answer running less risk of sounding evasive might be,
“Of all that Science holds true, it places its greatest faith in “natural née supernatural”, the fundamental axiom that reality is not fundamentally absurd; that it works in certain ways, and that those ways can be understood. With this in mind, Scientists are not superhuman – naturally, they take many scientific (and unscientific) descriptions at face value like all of us, because they’d never find the time to personally test each one in detail. A few basic axioms are at the heart of each science – and some are shared. From these, Science relies heavily on logic and math to build theories that should predict and explain all known phenomena within that science. They must be testable, and if evidence contradicts them, the axioms may even be discarded! This monumental skepticism is the mark of Science, far moreso than the axioms. (Some of which are so simple that you would accept them the instant you understood them, and some surprisingly unintuitive and mind-boggling.) It’s clear that we have found a good way to learn how reality works. Other, even better ways may appear, and they will succeed Science or become integrated with Science.”
The way the person phrases the question makes him sound educated enough to already hold many of Science’s axioms true and already suspect that distrusting Science as a whole is fruitless and misguided. Naturally, it’s perfectly legitimate to question axioms; point out that logic contains legitimate paradoxes; use Gödel’s proof of incompleteness and inability of consistency in formal logic, the foundation for mathematics. But focusing on these is to look at the weak end of Science’s explanatory power! The testability, the objectivity, and the successful deduction and generalization from evidence is what makes Science uniquely reliable and trustworthy. In other words: if a theory is perfect, and a single solid piece of evidence or objective experimental phenomenon surfaces that goes against the theory, then the perfect theory is questioned. Mere empiricism is what makes it stronger, not weaker. And Science is aposterioricity City!
This much is true of the hard sciences. But some sciences are quite far removed from current reality (as in, having accessible, clear evidence) by time, space, or level of abstraction. The reason Geology, Cosmology, or the Social Sciences can predict and explain as much as they do is due in part to the axioms and world views they share with the hard sciences. They can be questioned practically all the way down to these shared axioms and world views, and are therefore more vulnerable targets of distrust and attack. But because they share axioms not yet refuted by evidence and world views formed from evidence, it’s just this interfacing and interlacing of evidence that justifies at least the core of the weaker hard sciences and soft sciences.
The remaining most effective way to question the validity of a science, assuming evidence is not shaky, must then be to question its axioms. The fulcrum on which the scales of faith hang seems to be the fundamental axiom. It always seemed to me that if you declare some parts of reality supernatural, you automatically say reality cannot, needs not, ought not be understood.
I always thought this an avertible disaster; an avoidable danger – the danger of leaving the field wide open to recidivist lazy mystic censors of wit.
It’s hard for me to see how such a claim can be made without simultaneously and justly looking weaker than Science, and that if you are interested in the supernatural, you can do no better than to discover it through Science. For that is exactly what has happened many times in the few centuries Science has let the evidence lead you to truth and discovery.
Now, I realize Dawkins is human too, and I’ve had almost 50 minutes to think and write this answer, so it’s unfair. I just thought it was a perfectly good question that deserved a better answer than it got. :)