The question: Can we define differentiable functions between (some class of) sets, "without $\Bbb R$"* so that it
- Reduces to the traditional definition when desired?
- Has the same use in at least some of the higher contexts where we would use the present differentiable manifolds?
I was a little bit disappointed when I learned to differentiate on manifolds. Here's how it went.
A younger me was studying metric spaces as the first unit in a topology course when a shiny new generalization of continuity was presented to me. I was thrilled because I could now consider continuity in a whole new sense, on function spaces, finite sets, even the familiar reals with a different metric, etc. Still, I felt like I hadn't escaped the reals (why I wanted to I can't say, but I digress) since I was still measuring distance (and therefore continuity, in my mind) with a real number: $$d: X\to\huge\Bbb R$$
If the reader for some reason shares or at least empathizes with (I honestly can't explain this fixation of mine) the desire to have definitions not appeal to other sets/structures*, then they will understand my excitement in discovering an even more general definition, the standard one on arbitrary topological spaces. I was home free; the new definition was totally untied to the ever-convenient real numbers, and of course I could recover my first (calculus) definition, provided I toplogized $\Bbb R^n$ adequately (and it seemed very reasonable to toplogize $\Bbb R^n$ with Pythagoras' theorem, after all).
Time passed and I kept studying, and through other courses (some simultaneous to topology, some posterior) a new sort of itch began to develop, this time with differentiable functions. On the one hand, I had definitions (types of convergence, compact sets, orientable surfaces, etc.) and theorems (Stone-Weierstrass, Arzelá-Ascoli, Brouwer fixed point, etc.) completely understandable through my new-found topology. On the other hand, the definition of a derivative was still the same as ever, I could not see it nor the subsequent theorems "from high above" as with topological arguments.
But then a new hope (happy may 4th) came with a then distant but closely approaching subject, differential geometry. The prospect of "escaping" once again from the terrestrial concepts seemed very promising, so I decided to look ahead and open up a few books to see if I could finally look down on my old derivative from up top in the conceptual clouds. My expectation was that, just like topology had first to define a generalized "closeness structure" i.e. lay the grounds on which general continuous functions could be defined via open sets, I would now encounter the analogous "differentiable structure" (I had no idea what this should entail but I didn't either for topology so why not imagine it). And so it went: "oh, so you just... and then you take it to $\Bbb R^n$... and you use the same definition of differentiable".
Why is this so? How come we're able to abstract continuity into definitions within the same set, but for differentiability, we have to "pass through" the reals? I realize that this really has to do with why we have to generalize in the first place, so what happens is that the respective generalizations have usefulness in the new contexts, hence the second point in my question statement.
Why I imagine this is plausible, a priori, is because there's a historical standard: start with the low-level definitions $\rightarrow$ uncover some properties $\rightarrow$ realize these are all you wanted anyhow, and redefine as that which possesses the properties. Certainly, derivatives have properties that can be just as well stated for slightly more general sets! (e.g. linearity, but of course this is far from enough). But then, we'll all agree that there's even been a lust for conducting the above process, everywhere possible, so maybe there are very strong obstructions indeed, which inhibit it's being carried out in this case. In this case, I should ask what these obstructions are, or how I should begin identifying them.
Thank you for reading this far if you have, I hope someone can give some insight (or just a reference would be great!).
* If I'm being honest, before asking this I should really answer the question of what on earth I mean, precisely, by "a structure that doesn't appeal to another". First of all, I might come across a new definition that apparently doesn't use $\Bbb R$, but is "isomorphic" to having done so (easy example: calling $\Bbb R$ a different name). Furthermore, I'm always inevitably appealing to (even naïve) set theory, the natural numbers, etc. without any fuss. So, if my qualms are to have a logical meaning, there should be a quantifiable difference in appealing to $\Bbb R$ vs. appealing to set theory and other preexisting constructs. If the respondent can remark on this, super-kudos (and if they can but the answer would be long and on the whole unrelated, say this and I'll post another question).