Unsurprisingly, the answer is "**yes, depending what you mean**." Relevant topics include **arithmetization**, **computable analysis**, and **reverse mathematics**.

The first point of vagueness is exactly what "finitistic mathematics" means. There are various candidates for this one might propose - in the setting of first-order arithmetic we have $\mathsf{I\Sigma_1}$ and the substantially-weaker $\mathsf{EFA}$, and in the setting of second-order arithmetic we have $\mathsf{RCA_0}$ and its various weakenings.

- Note that there's a horrible terminology issue here: both "first-order arithmetic" and "second-order arithmetic" here refer to languages in
**first-order logic**. Terms like "one-sorted arithmetic" and "two-sorted arithmetic," or "arithmetic" and "analysis," respectively would be much better; unfortunately the above usage is entrenched. **Boo I say.**

Even beyond the specific choice of theory, though, there's a clear "super-vagueness:" what *language* are we even posing this theory in? Strength of theory and strength of language are a priori unrelated: there are very strong theories in very limited languages and very weak theories in very rich languages. For example, $\mathsf{PA}$ has much greater logical strength than $\mathsf{RCA_0}$ (as measured e.g. by the two systems respective provably total functions), but the language of $\mathsf{RCA_0}$ is richer than that of $\mathsf{PA}$ in a very important way. The linguistic aspect of finitization is more subtle in my experience, so let me focus on that first. **For simplicity, I'm going to imagine that we only ever care about continuous functions defined on all of $\mathbb{R}$ going forward.**

"Calculus results," broadly speaking, fall into two categories: results about specific continuous functions of interest, and general results about the class of all continuous functions as a whole. Now a priori a single real number lives at the same "type level" as a whole set of natural numbers, and so a function on real numbers like $\sin(x)$ lives at the same "type level" as a whole *set of sets of natural numbers*. Fortunately, however, continuity comes to the rescue:

Fix an appropriate bijection $b:\mathbb{Q}^4\rightarrow\mathbb{N}$. Given a continuous $f:\mathbb{R}\rightarrow\mathbb{R}$, let $$\mathsf{CODE}(f)=\{b(a,b,c,d)\in\mathbb{Q}^4:\forall x\in\mathbb{R}(\vert x-a\vert<b\implies \vert f(x)-c\vert<d)\}.$$ Then $\mathsf{CODE}(f)$ determines $f$ uniquely amongst all continuous functions $\mathbb{R}\rightarrow\mathbb{R}$, and basic operations on continuous functions correspond to arithmetically definable operations on their $\mathsf{CODE}$s.

This gives us two useful tricks. First, "naturally occurring" continuous functions - including $\sin(x)$ - have very simple (e.g. primitive recursive or better) $\mathsf{CODE}$s. Consequently results about specific such functions can be faithfully stated in the language of $\mathsf{EFA}$, and usually proved there too. Second, even when we look beyond specific functions of interest, we're still able to make things surprisingly concrete: via $\mathsf{CODE}$ing a continuous function on $\mathbb{R}$ is "morally equivalent" to a set of natural numbers, and so we can faithfully ask what general calculus theorems are provable in (say) $\mathsf{RCA_0}$. But we can't get all the way down to pure arithmetic, and this isn't a matter of strength: $\mathsf{PA}$ isn't an appropriate vehicle for this sort of thing either, despite being stronger than $\mathsf{RCA_0}$ in many senses.