I was initially annoyed at Feyerabend but then I do agree with him in some limited areas. The length of this post is perhaps due to my personal interest in science. My thought experiment was to highlight the main area of disagreement and also to discover your views.

My value statement of science - we ought to value accurate predictions of the physical world over inaccurate predictions - is explicitly denied by Wittgenstein, Feyerabend and post-modernism, as I understand them. “My German engineer [Wittgenstein], I think, is a fool. He thinks nothing empirical is knowable - I asked him to admit that there was not a rhinoceros in the room, but he wouldn’t.” (Betrand Russell). I provisionally accept there is not a rhinoceros in the room. Of course, I might discover that rhinoceros can shape shift but so far I have not observed that. In fact, the feeling that there is not a rhinoceros in the room is more real to me than any tricks of language or vague philosophical notions (”here is a hand“, etc). Nietzsche said to deny the physical/actual world, you would also be throwing out any basis for truth or knowledge. ‘[T]he “true world” has been constructed out of contradiction to the actual world: … insofar as it is merely a moral-optical illusion’. (Twilight of the Idols)

I tend to think that, although the physical world seems to exist, we can’t find a basis to attach values to non-physical belief systems. But this is hardly original - it’s just existentialism. I regard your interpretation of LW and F as more post-modern than my point of view. At least we can say “Reality is merely an illusion, albeit a very persistent one.” (Einstein)

“I think the main area of thought is the principle of induction, which is widely supported in empiricism, and which Feyerabend attempts to dismiss.”

I am not sure induction is used in an unquestioning manner in science. If I said “All observed crows are black there fore all crows are black” in philosophy, it would be a potentially incorrect generalization. If in science, I say “All observed crows are black and I will formulate a theory of the blackness of crows”, that is not a generalization because at any moment we might discover a green crow.

We may say “the scientific law of X can never be broken”, which implies we have used inductive reasoning but it is really a lazy simplification of saying “the theory of X has never been disproved and I don’t expect you can too”. Basically, the provisional nature of scientific theories avoids the generalization of inductive reasoning.

“I’d further develop this by saying that a method that makes no predictions of situational outcomes is of less calue than any method (accurate or inaccurate) that does.”

Very deliberately, I made no mention of metaphysics in my thought experiment (or rather I separate it and then made no comment on it - like early Wittgenstein). I am actually surprised you do accept what is essentially the falsification principle at all! What I was trying to say was the value of predictive theories should only be compared to other predictive theories. I am trying to avoid putting a value on non-predictive theories (which you, F and LW would instantly reject).

This is my current view: empiricism has a very restricted (i.e. physical) scope but it is more valuable within that scope than other views. (Remember when I was at a loss over thinking of a disproof of naturalism? lol)

“Feyerabend proposes that to follow one method to the exclusion of others is
a) not very progressive thinking, and
b) contrary to the ‘experience’ of science history …”
I avoided directly talking about methods in my thought experiment but you could say to produce hypotheses, you need one or more methods. I disagree that within science (or more precisely empiricism), has ever changed it’s view on my “ought statement”. It has always valued a more accurate predictive theory over the less accurate. I would agree with Feyerabend that many methods should be employed to produce theories, but the value assessment is constant in my humble opinion.

“Feyerabend is concerned with the smug attitude of some science.”
Feyerabend is correct if he is referring to some popular perceptions of science or particular scientists but the core principle of science does not necessarily imply smugness. Unless all scientists are smug. Are you accusing me of being smug? :)
‘An attitude that could be characterised as saying “we are the only ones that know the answers, and we will share them with you, if you accept our ways.”‘
That is the way science is sometimes perceived but this is of course NOT what science should be about. In fact, science should be the opposite! I can’t find the exact Nietzsche quote, but he talked about the need to reject past theories as insufficient (what he called a “holy nay”) as requirement to write new theories. If science is really dogma, it would be impossible to progress science because we could not question anything.

So, if Feyerabend is criticizing the institutional nature of science we all might be in agreement. Criticising the core value (of accurate predictions are better) of science is laughable - but the anti-realist viewpoint does specifically deny this! (boo hiss to post-modernism when there is an overlap)

“Has a rational method, that functions to guarantee the objectivity of its results.”
Of course, science is more a method for eliminating error rather than guaranteeing correctness.

“And that it, exclusively, produces useful results.”
Exclusively in the physical domain, I believe it does. To disagree with that is anti-realist (which is a valid point of view I guess).

“Feyerabend says that we can see clearly enough that Newtonian mechanics, though ridden with anomalies, took a very long time to get rid of them.” (And allegedly shows science prefers inaccurate theories.)
This is straw man argument in my opinion. All scientific theories to date have what I call “scope”. The scope of a theory is usually stated in the theory itself. For example, Newtonian mechanics does not explain nuclear fusion. It is obvious that no scientific theory has been able to explain everything. (Although this is a goal of some scientists, but that is another story.)

A thought experiment on how scope works:
Scientist A: I have a theory X which explains A, B and C.
Scientist B: Actually, I have some new data which shows that theory X does not explain C in particular circumstances Z.
Scientist A: Um, ok I have to restrict the scope of my theory. It explains A, B and C when C is not in Z.
*a few days later*
Scientist A: I have revised my theory again. It is now called X’ and it explains A, B and C in both “Z” and “not Z”.
Scientist C: Nice. But X’ is more complicated in X. In fact if you assume “not Z”, theory X’ reduces into X. Since I only an interest in “not Z” conditions, I am going to stay using X on the understanding it only applies to “not Z”.
Scientist A: Ok.
Science teacher: Ok with me too. Also since it is easier students to understand X before learning X”, I will teach X first.
An example: X is Newtonian gravity.
A is the fall of objects on Earth
B is nothing in this example
C is the orbit of the planets
Z is orbits close to the Sun (to be exact the perihelion precession of Mercury)
X’ is Einstein’s general relativity.
Conclusion: The persistence of Newtonian mechanics is good enough for most applications and easier to understand than relativity. It is not unscientific to use a theory as long as it is within the scope that it agrees with the physical world.

[the rationalist reconstruction of science argues] “[t]he state should teach science uncritically because it is the one true way.”
I have tried to outline how the direct alternative to science is anti-realism. I imagine most science teachers could not do a lesson on anti-realism without being sarcastic! (But I don’t mean to imply that is what you recommend.) My preference would be to teach what science should be about (keeping an open mind and an empirical basis) and what it should not be about (pseudo-science or dogmatic thinking).

Interesting discussion though. I have some thoughts on tacit acceptance and rejection of science, and the implication of overlap in world views with empiricism, about which I am currently brooding.

Anti Citizen One