Intellectual Hierarchy

I'd like to dispel the notion that there is some sort of "applied" linkage between areas of study. It is very easy to believe that "oh, math is just applied philosophy, physics is just applied math, chemistry is just applied physics, and biology is just applied chemistry". This seems to suggest a sort of hierarchy of theoreticalness, with philosophy being at the bottom and most abstract and biology as least. I used to believe this too. But this is quite false. Philosophy is quite different from math in that philosophy can't build on top of ideas. For example, from algebra one can get the idea of infinitesimal calculus, i.e. the hyperreals whereas with philosophy, what can one get from the maximum of say, utilitarianism? As a more specific example, considering the ancient Greeks. The idea of area clearly is the basis for Archimedes' ideas on volume. But Plato's ideas of Platoism were immediately rejected in the next generation by Aristotle, with both ideas being extremely independent. In the entire history of humanity, there seems to be no real progress in philosophy. One could easily believe in Platoism or some other very old philosophy like Diogenes' cynicism today and it'd really be impossible to disprove it or fault you for believing in it. Thus no progress really seems to be made. In math, if you said you only believed in Euclidean geometry (i.e. 2 dimensional) you'd be insane. There is also a different level of assumption. In math, we typically just assume normal logic axioms and the current axioms (ZFC). Thus by doing math we accept the philosophy of dogmatism. On the other hand we have philosophy, in which everything can be questioned. the entire field of epistemology is a testament to this. When one says "proof" in math, we mean "follows from the axioms". When one says "proof" in philosophy you have an entire field dedicated to its study.

Physics isn't just applied math. For one, physics can and does inspire math. Take for instance the Yang-Mill mass problem. This is a 1 million dollar mathematical conjecture about the mathematical underpinnings of a physical model, i.e. this problem was inspired by physics, and its solution will likely involve new mathematical discoveries that are a result of physics. The Navier–Stokes Equation is another example of this. Physics also differs from math in terms of rigor. In physics, we typically just assume calculus to be infinitesimal, despite not being mathematically rigorous. There are also approximations not typically used on mathematics being pervasive in physics, such as the small angle approximation.

Chemistry isn't just applied physics. Consider quantum physics, very clearly influenced by chemistry's results of the subatomic. Now consider chaos theory. Our world is a very chaotic place, a simple fact due to the requirement of measurements (thus always leaving the possibility of some small unmeasured variation causing drastic results). Thus it is impossible for physics to apply at the scale of chemistry since there is just an unpredictable amount of variation. Chemistry is an attempt for us humans to translate the uncomputable and chaotic predictions of physics onto the more reasonable level of molecules. Thus chemistry will always live on a higher abstraction level than physics and they thus are distinct. The argument for biology vs chemistry is very similar, and is even more pronouncing. One can't just say "ahh yes, carbon has 4 bonding sites, bing bang boom life is solved". Even if we have a sort of theoretical explanation for underlying phenomena (such as physics to chemistry or chemistry to biology), that doesn't mean it can be used to understand the extremely complex and different scales of these levels