So, I spent some time reading over the thread, and I have some points to make.
First off, let's take the definition of a circle G&W cited, "a closed curve in a plane all of whose points are equally distant from a fixed point in the plane called the center". That is indeed based on observation. Nobody conceived of a circle without first observing something circular in nature. That led to defining it so that we could distinguish something that was circular from something that was elliptical (and other geometric shapes). In effect, even though nobody had conceived of the term "scientific method" yet, that's effectively what people used in order to come up with the definition of a circle. They made an observation, formulated an explanation, compared that explanation to reality, modified it as needed, and then shared it with other people (who checked it themselves and made changes as needed in order to make sure it conformed to reality). Certainly, it wasn't the kind of organized methodology we use today, but it was the same basic process.
The other things he mentioned, deductive reasoning and rational introspection, are themselves dependent upon observations in order to be meaningful. I mean, while it's certainly possible to use them without basing them on an observation (say, off of an opinion), what use would the conclusions be for acquiring knowledge
? All one would be doing is building off of an opinion, which is itself not provable. There are times the opinion might be valid, such as "I like chocolate" - but any conclusions drawn through deductive reasoning or rational introspection based on that opinion are of strictly limited use. And really, even the opinion itself is based on an observation - "I like chocolate because it tastes good".
One of the things we have to be really careful of is to make sure we keep our opinions and the observations that spawned them separate. It's a little too easy to treat them as a single unit, rather than as separate pieces of information. "I like chocolate because it tastes good" is a compound statement, an opinion based on an observation. So the real question is whether we can base opinions on anything besides observations. I don't think we can, at least fundamentally. While it's possible to have an opinion based on an opinion, I think ultimately the first opinion in the chain was based on an observation.
That's where the problems with most religions lie - they're based on observations, like everything else, but observations which were not kept separate from the opinions about the observations. For example, take the Greek myth about why the sun and the moon appear to orbit the Earth. The Sun is pulled by Helios's chariot; the moon is pulled by Selene's chariot. In other words, they don't consider the facts (that the sun and moon appear to orbit the Earth) in isolation - they incorporate the facts and their opinions (that Helios and Selene are responsible) into the explanation. I think if you get right down to it, every religion that ever was is based on that same false paradigm - that you can explain a phenomenon with an opinion based on that phenomenon, without testing that opinion to see if it's right.
On top of that, it's necessary to keep in mind that observations are limited by whatever is used to make the observation. If observing something ten miles away, would you prefer to use your own two eyes, or would you prefer to use binoculars? But it's more than that. Let's take G&W's example of how people observing that the sun revolved around the Earth led most to a false conclusion. While that's certainly true, it's true because their observations were sharply limited. They were not aware of the Earth's rotational movement, the same way that a person riding in a high-speed vehicle with no windows is not aware of its motion until that motion changes. But even then, there's ways to figure out that the sun's apparent motion is based on Earth's rotation - as the ancient Greeks showed long ago.
So you can't just take observations for granted. That's why we have the scientific method. And that's why deductive reasoning and rational introspection, useful as they are, can't compete with it for determining if something is factual or not. You can't disprove the Earth-centric hypothesis (or, indeed, any hypothesis) with deductive reasoning or with rational introspection; you can only disprove it with making observations and testing them.