What is the theoretical mathematical justification for differential arithmetic? Throughout undergraduate physics textbooks, you will see informal math with differentials where elements like $dx$ and $dy$ are multiplied around like scalar constants, and differentiation in terms of a variable is treated as analogous to division. What is the theoretical justification for this? I have never seen a formal mathematical argument to say why this can be done, especially not in the textbooks that use it. When I mean formal, I mean an argument from the point of view of rigorous mathematics, not just saying that $\Delta x/\Delta y$ approximates $dx/dy$ so we can treat $dx$ like we would $\Delta x$. Are there any formal proofs available?
An example of the type of differential mathematics I am talking about is used in thermodynamics. https://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation I have never seen the formal justification that undergirds this way of talking about infinitesimal changes and using the differentials like constants.
 A: There is no known logically rigorous justification that works in all instances. But it is immensely useful as a heuristic, and it focuses attention where it needs to be, and it keeps things dimensionally correct (e.g. if $f(x)$ is in meters per second and $dx$ is in seconds, then $f(x)\,dx$ is in meters, and if $s$ is in meters and $t$ in seconds, then $d^2 s/dt^2$ is in meters per second per second, etc.).
A: Although the question is strictly mathematical, there is an important physical/philosophical point to be made which hasn't yet been mentioned, especially since you originally asked this question on the physics website. You say that physical models approximate mathematical models, but it is arguably the other way round! We need to remember that our models are only as valid as our experimental abilities and experimentally continuity doesn't exist: you can measure something very, very, very precisely but it will never be a real number. So one reason physicists are "sloppy" is very straightforward: if you need to model something that you know exists and is finite, you don't need to prove existence theorems, and you also don't need to prove that it converges, which a lot of mathematics is concerned with.
Another reason why physicists use these mathematical idealisations is because they are much more convenient (discrete maths is much harder to manipulate than continuous maths and is also less developed on the whole). Many people like to gloss over this point, but we also need to remember that a lot of rigorous mathematics has been largely inspired by the approximate nature of physics (e.g. distribution theory, calculus, functional analysis, etc) and there are still some concepts that work but are not considered rigorous, such as real-time path integrals, meaning that just because something hasn't yet been proven, it might still be physically useful, relevant and experimentally validated - and I would argue this, rather than mathematical purity, is the main purpose of theoretical physics.
