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What do you know about the boundaries between different bodies of knowledge, e.g. bodies of science. I think it's a common question, e.g. "where's the boundary between mathematics and physics".
But what kind of generally agreed formal theories exist for grouping different knowledge areas?
The first European classification of sciences is due to Aristotle, notably in his work Metaphysics. On the first level he distinguishes theoretical, practical (e.g., ethics, politics) and producing sciences. On the second level he distinguishes, e.g., three kinds of theoretical sciences. They are classified according to their subject (Met. VI, 1026 a13f.):
Of course, mathematics is still considered a separate science, but its definition differs from the characteristics above. Secondly, today the original domain of physics comprises all natural sciences.
Probably today's main border is between the humanities and natural science. But interdisciplinary work becomes more and more important. E.g. in the domain of cognitive science, some researchers vote for cooperation even across this border.
There is a strong sense within the recognised set of sciences, that what really determines the boundaries are the points where emergent phenomena become more relevant than their sources.
There is a level where the actual locations of hydrogen ions are strictly relevant, and one where the notion of acidity takes over. There is a level where Brownian motion is a relevant phenomenon, and a boundary beyond which where temperature is a more useful concept that means the same thing in a more abstract sense. There is a level where cells are important and a level where tissues and organs are relevant, and then another where all that matters are whole bodies.
Of course, these boundaries do not align well. We know that in basic chemistry we are going to attend to acidity and temperature, rather than ion positions and Brownian motion. But addressing different specialities within the discipline of chemistry, say crystallization or the details of polar vs nonpolar solvents, we might be using one of the two more emergent concepts and one of the two more basic ones.
But, in general, we can separate out towers of increasingly emergent disciplines, for example from nuclear physics, to chemistry, to biology, to psychology, to sociology, in a way that makes sense. Of course then even in that one string, there are lots of interior details like microbiology and a few other additional layers between chemistry and biology, and things like animal behaviorism lying between biology and psychology.
And obviously this is not a set of strictly linear successions. It branches. Psychology in particular lacks coherence and forks into general, personal and 'rational' psychology (among others) and will have sociology as its natural successor in one direction, while in two others it can be seen as basic to anthropology and ethnology, or to humanistic disciplines like history and literature.
But as a general way of deciding which science you are doing, you can rely upon how many and which among the concepts basic to your thinking are actually emergent phenomena too abstract to make strict sense in other sciences.
I would push this sense to its logical extreme.
I think the same pattern applies to a great degree outside of science. Philosophy's great divisions, for instance tend to cohere largely around what you are willing to treat as emergent concepts, and what you are going to attend to at a level of greater detail. Ethics is not free of logic or epistemology, but it takes basic understanding and motivation as an emergent phenomenon that relies upon them. Politics depends upon feelings and intuitions that arise out of personal ethics as emergent phenomena. And a similar network of chosen levels of attention can map out the rest of the subdisciplines.
Some of these boundaries are much more contentious, but can really be seen in the same light. One way of looking at those things is that they are places where some things are deeply emergent, and others are very low-level at the same time.
The very contentious boundary you bring up first, where mathematics meets advanced physics like relativity and quantum mechanics can be seen as a place where we depend deeply upon mathematics, as emergent from psychology and logic, to tell us what can be reasonably understood and communicated, while stripping away a lot of layers of emergent concepts and focussing on absolute simplicity in every other sense.
The first thinkers on nature were the Ionian cosmologists; and the first thinkers on mathematics were the Pythagoreans - their thinking was also explicitly religous; with for example, Pythagoras even being seen as a religous leader. They too, given Platos thinking on the One can be seen as cosmologists.
Quite commonly it's understood that the rational and enquiring spirit logos crystallised out of the inconsequential religous mythos; but one could argue along with Webers The spirit of Capitalism and the Protestant Ethic as something that was a fundamental constituent of thinking then.
Mathematics first became explicitly involved (and religion slowly disengaged) in Physics in the modern sense with Galileo; and the calculus of Liebniz and Newton - but now with multiple affinities; still, the subject divisions are visible - for example, calculus is heavily used in Physics; it's justification by analysis, though of interest to mathematicians is not usually seen to be of interest to physicists.
The major divisions of knowledge is posited by Badious division of the Understanding along the positive (truth-producing) axes of love, politics, arts and sciences; and the negative (critical) spirit of philosophy; these should be seen as not literally pure, though disciplines have an arete (virtue) that have deeper affinities.
For example the structuralism of Bourbaki (in mathematics) and of Levi-Strauss (in anthropology) had such an affinity, that Levi-Strauss had to explicitly deny this causal possibility - securing his discipline on the semiotics of Saussure.
Another concept that is useful to understanding this division, this individuation amongst the unitary body of knowledge, is the concept theorised by the anthropologist Mary Douglas, degout which regulates the potential and possibility of pollution; for example this is implicit in the interdisciplinary philosopher (in poetics, politics, theology) Kearneys criticism of the Academy on avoiding interdisciplinarity as it risked the purity of disciplines.
Another philosopher who ignores disciplinary boundaries is Deleuze; his very prose using the transgressive tactics of the artistic avant-garde: symbol and image, montage, cut up, disorientation and aporias suggestive in Mallarmes poetics, for example.
Still, it's important to recognise I think that Aristotle, usually considered as the first scientist also wrote on politics, ethics and physics; and in a substantial way; as did Lucretious who inherited the atomic theory of Democritus mediated by Epicurus.
I claim that a theory, model, or piece of work belongs to some field "X" if it answers a question related to the field "X", and if it satisfies some expected criteria required by field "X" (like, rigor, possibility to make predictions, etc).
Ex: if you're trying to answer "what is the nature of such mathematical objects and its properties?" then you're doing mathematics, even if what you're working with is based on physics. Now, there it can appear that answering such mathematical question leads to some physical insights, but this requires some reinterpretation that is never free to obtain.
There is no answer without question. Keep that in mind.
