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Can one "do calculus" in weak formal systems like primitive recursive arithmetic or elementary function arithmetic? Can one at least explain the physics of classical mechanics in an elementary formal mathematical system like the ones described above ?

I admit this is a vague question so I leave to you the task of giving this question a reasonable rigorous interpretation (if any). I feel that with lot's of work , one can reproduce most of the results of calculus in the form of multivariable inequalities between some integers.

Amr
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    There are really two questions here: (1) how can we use the *language* of just natural numbers and "basic" functions on natural numbers to talk about the objects of calculus?, and (2) fixing such a translation mechanism, what sorts of axioms do we need to *prove* the (appropriate translations of the) basic results of calculus etc.? Which is your main focus? (Incidentally, the topic of **reverse mathematics** is relevant for each of these - note that the first-order part of $\mathsf{RCA_0}$ is $\mathsf{I\Sigma_1}$ and hence its provably total functions are exactly the primitive recursive ones.) – Noah Schweber Oct 14 '21 at 23:42
  • @NoahSchweber I admit the vagueness of my question. I just wanted to know if this question was considered elsewhere in the litereature – Amr Oct 14 '21 at 23:44
  • @NoahSchweber I will give you an example of what I would consider as doing calculus in finitist mathematics. Take for example Taylor's series of sine function. I imagine that the result can be stated in finitist mathematics as an inequality between integers (the inequality will come from using the remainder term, taking x in sin(x) to be rational then multiplying by the product of all denominators). I hope that's clear ? – Amr Oct 14 '21 at 23:48
  • @NoahSchweber My focus is not to add anymore axioms/rules to these fintisitc weak formal systems, but rather to see how much of calculus can be done in these formal systems – Amr Oct 14 '21 at 23:49
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    So for specific "nice" functions like $\sin$, we can indeed do everything in $\mathsf{PRA}$ or indeed much less. This is because $(1)$ a continuous function's entire behavior is bound up in a couple of relations/functions on $\mathbb{Q}$ (basically *approximation* and *modulus* functions), and $(2)$ for "nice" functions these maps are primitive recursive or indeed even better. The topic of **computable analysis** is relevant here. However, this runs into serious trouble if we want to talk about proving results about functions in general, e.g. the fundamental theorem of calculus. – Noah Schweber Oct 14 '21 at 23:50
  • Actually my example of sin(x) is a bad example, because how can one make sense of sin(x) in finitist formal language in the first place – Amr Oct 14 '21 at 23:53
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    It's at this point that a richer language becomes pretty necessary. This is where **reverse mathematics** enters the picture. The base theory $\mathsf{RCA_0}$, per my previous comment, is pretty finitistically acceptable since its provably-total functions are merely the primitive recursive ones. However, the richness of its language lets us quantify over continuous (or otherwise "simply codeable") functions on $\mathbb{R}$ (once we apply the appropriate translations) which is not something $\mathsf{PRA}$ can do (for purely linguistic, not strength-based, reasons - $\mathsf{PA}$ can't either!). – Noah Schweber Oct 14 '21 at 23:53
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    Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/130532/discussion-between-noah-schweber-and-amr). – Noah Schweber Oct 14 '21 at 23:53
  • Nilsen's "radically elementary probability theory" seems to be an effort along the line. But he had to admit non-standard analysis everywhere. Whether this is more finite than conventional math is for debate. – Just a user Oct 15 '21 at 02:45

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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.

Noah Schweber
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  • Thanks for the answer, it was nice talking to you in chat room. Question: Let S be any statement about equality of primuitive recursive functions provable in ZFC , must S also be provable in PRA – Amr Oct 15 '21 at 00:45
  • @Amr Certainly not. Consider the function $f(x)=$ the number of proofs of $0=1$ in $\mathsf{PRA}$ of length $ – Noah Schweber Oct 15 '21 at 00:47
  • Hmmm. Aha!. Logic is amazing – Amr Oct 15 '21 at 00:49
  • So the purpose of Harvey's grad conjecture is really to tell us that if infinite mathematics happens to be inconsistent, then we are still safe because the interesting results will be reproducible in finitist mathematics, kind of like a weaker version of hillberts dream – Amr Oct 15 '21 at 00:54
  • So like you said there are finitist sentences provable in ZFC but not provable in finitist mathematics itself. What makes you have faith that one can import the idea of a Taylor series of some function into finitist mathematics but with lots of tedious work? Is it some kind of faith similar to that required for the church turing thesis ? – Amr Oct 15 '21 at 05:23
  • @Amr "So like you said there are finitist sentences provable in ZFC but not provable in finitist mathematics itself. What makes you have faith that one can import the idea of a Taylor series of some function into finitist mathematics but with lots of tedious work?" That's a total non-sequitur - while there are sentences about natural numbers (I'm not sure what a "finitist sentence" is) is provable in ZFC but not in PRA (I'm not sure what "finitist mathematics" exactly is), that has no bearing on whether specifically one can provide finitist translations of Taylor series of specific functions. – Noah Schweber Oct 15 '21 at 05:25
  • Are you asking why I'm certain that $\mathsf{CODE}(\sin)$ is primitive recursive? – Noah Schweber Oct 15 '21 at 05:28
  • Yes I am. My other question is "Is Harvey's grand conjecture" something like the Church Turing Thesis ? – Amr Oct 15 '21 at 06:44
  • A finitist sentence is statements about the natural numbers expressible in PRA – Amr Oct 15 '21 at 06:57