You might remember an older xkcd comic ranking various sceinces in terms of their purity. Psychology is just applied biology, biology is just applied, chemistry, chemistry is just applied physics... etc. Math is at the top. (Philosophers would like to think they're the top level, one rank above mathematicians.)
It's all in good fun, and it happens within disciplines as well. There's occasional good-natured sniping between experimental and theoretical physicists, and between the various sub-disciplines at different energy scales like particle physics and solid-state physics.
The story more or less is that knowledge gets generated at one level and passed down to the next for further processing. The mathematicians generate Mathematical Truth, the physicists use it to develop the Laws of Nature, the engineers use those laws to create Useful Devices, and so forth. But though that's the story, it's not quite true. Knowledge flows in all directions. Physics has guided pure math on numerous occasions (just look at Hilbert...or Newton and Gauss for that matter), while engineering and physics are often almost indistinguishable. The large-scale projects in fundamental physics involve collider design that's simply an epic engineering problem, and many times in other fields the most interesting problems in physics and the most interesting problems in engineering are the same thing. There's high-temperature superconductivity, nanotechnology, the intersection of solid-state and materials engineering, pretty much everything involving lasers, and so on.
Which is all to say that the rivalry between disciplines is not so far removed in spirit from the rivalry between the branches of the armed services: at the end of the day we had all better be on the same team.
(Though I have to admit it won't stop me from the occasional engineer joke anyway.)
I'm always annoyed by 'rainbows' that don't have the colors in the order of the spectrum.
Me, I'm always annoyed by five pointed "stars" that are obviously not giant spheres of fusing hydrogen. [/snark]
An engineer, a physicist, and a mathematician are sleeping when at the same time, the wastebaskets next to their beds catch fire.
The engineer wakes and does some quick calculations as to how much water it would take to put out the fire, and pours that plus a lot more water on it.
The physicist wakes, calculates how much water it will take, and pours precisely that amount on the fire.
The mathematician wakes, writes the equations down, and says, "the answer exists" and goes back to sleep.
I know it's probably old and well known, but I thought it was funny.
cf. Phil Anderson's essay "More Is Different" from Science in 1972. The applied disciplines each discover their own emergent phenomena that are either context dependent or just would never plausibly be inferred from the phenomena below them.
Incidentally, I'm a graduate student in economics, which I believe is applied psychology...
Humans seem to have a long history of engineering and using devices that we don't yet completely understand scientifically or mathematically. It's a constant feedback loop between the different levels. Engineers and inventors pass things "up" for processing to scientists just as scientists pass things "down" for processing by engineers.
Electricity, for example, was first a scientific curiosity until the mid 19th century. Then the engineers ran away with it, and made all kinds of interesting devices that pure science couldn't yet fully explain. Just to give some context, the telephone (a rather advanced application of electricity) was invented around 1875. The first time that a scientist proposed the existence of negatively charged particles (what we now call electrons) was around 1879. That's just one example of how far engineering can go with a concept while scientists are still working on explaining it.
And that's a very good thing! If we had to wait for all the mathematical models to be complete and for full scientific understanding before engineering a solution... we'd never have mastered fire. :)
The story more or less is that knowledge gets generated at one level and passed down to the next for further processing.
I see it as a claim about ontology, not about the generation/transmittal of knowledge.
We've discovered that the laws of physics govern chemical processes, which serve as the basis of biology, and so on. However, we obviously knew a lot about biology long before we new about the quantum processes that underlie chemistry. Further, we're clearly never going to solve the SchrÃ¶dinger equation for any biological process.
It's not about knowledge; it's about how things work. The truth of physicalism is perfectly compatible with, e.g., physicists learning from the higher-level special sciences.
An engineer thinks that his equations are an approximation to reality. A physicist thinks reality is an approximation to his equations. A mathematician doesn't care.
And a consumer doesn't care either way as long as the lights come on when the switch is flipped!
A chemist, an engineer, and a physicist go into the loo to take a whiz. How do we tell them apart? The chemist washes his hands first, for obvious reasons. The engineer washes his hands afterward, for obvious reasons. The physicist has no need to wash his hands, for obvious reasons.
I'm reminded a bit of an audio lecture I'd heard recently on the history of mathematics, basically noting how mathematics can be so removed from use that it can easily be forgotten. After ancient Indian mathematicians learned of the trigonometric functions, they quickly developed ways to work out approximations to incredible degrees of accuracy for arbitrarily small values. However, astronomical observations were limited at the time, so no need for that kind of accuracy existed. No one else discovered their methods for a millenia, and the methods in Ptolemy's Almaghest sufficed for most of the world at the time.
That said, I do enjoy far too many mathematician/engineer/physicist/[insert discipline here] jokes.
Given the current attacks on reason, the sciences and the humanities also better realise they are on the same team, and damn quick.