Do you have any book recommendations for a nonstandard analysis treatment of probability? I'm really only familiar with the measure theory approach, myself. (In fact, I've been known to say that probability is the study of measurable functions with finite, nonzero integral over the real line.)

The standard is Nelson's Radically elementary probability.

Vovks and Shafer's game-theoretic approach is interesting in that related approaches like bandit models and online learning have recently picked up in popularity.

Nonstandard Analysis is an interesting side note. But pulling out the axiom of choice to differentiate x^2 is a bit much IMO. (Yes, I'm aware that there are different ways to construct the nonstandard model. But the subtleties needed to really understanding NSA are substantial. I far prefer the little-o approach that Knuth recommends.)

Perhaps you are more familiar with Robinson's derivation but Keisler has an axiomatic approach that makes calculus easier for students than regular calculus (besides by the time you are doing continuous differentiation you are already calling on some far out concepts like real numbers and inifinity).

And real understanding takes time no matter what you do, the best one can do is start off in a manner such that the tools required to understand well enough are not more complicated than the subject matter itself.

Yes, pedagogically you can just use infinitesmals after an appropriate hand wave. Most students will ignore the complicated details and just absorb the infinitesmal picture. If you take the attitude that students just need to learn the formulas, then it doesn't matter what approach you take.

However you've created difficulties for the day that they dive back in and try to really learn the subject. Because as they go to learn what a real number really is, and what its properties are, and about pathological functions, they also have to learn about the hyper-reals and a complex model-theory construction that (in both variations that I am aware of) requires choice.

There are a variety of other pedagogical choices that do not present such barriers to comprehension. (Note, by no means am I a fan of the limit approach. I know full well that it goes over the heads of the students, and I see no point in having the person at the front blather on about stuff that the class is not able to expected to understand.)

In any case my biggest complaint about how Calculus is taught is this. I think people come out of a first Calculus course without understanding the tangent line properly. If you don't understand the tangent line, Calculus is a mass of formulas. Despite how easy it is symbolically to jump directly from the tangent line to the derivative, I think that a solid week should be spent on the tangent line (calculating it for more complicated functions, finding applications, etc) until it is every student understands it well enough that they are ready to take the leap of looking at the slope and building a function out of it.

I do not agree. This is because the text treats infinitesimals and epsilon-delta at the same time (or after). By that time the material is familiar enough that epsilon delta is not so difficult. Kind of like learning python before assembly. The student would also get some predicate logic much earlier than normal which is a great boon. As for the reals, how many people end up studying that in depth? All I remember is being proud of doing something related to the awesomely named Dedekind Cuts.

Anyone that can follow the proofs required to construct the reals can easily do so for the hyperreals. I am not an expert in the area and it's been years since I studied at depth but I know there are methods which avoid the need for model theory. I do not recall them invoking the axiom of choice but I could be wrong on that.

But the point is not that the learner gets a Hyperreal only approach but a varied exposure. I think that is the key - much more important than understanding tangents (what happens to visualization at dimensions of 4?) I think that learning all of the disparate parts at once and letting the student have the time to become comfortable (linear maps, derivatives, surfaces, groups) is what would be best. Allowing them to drift and backtrack and then whenever they felt ready would take whatever appropriate exams to show mastery of each area. The exams would also allow for more interesting problems.

It would take longer but the end product would be a far more cohesive understanding than the mishmashed nature of the current historical siloed approach. You've done machine learning right? I think one can take something from there about learning. The brain is just a much more advanced version of those basic objects: more and varied examples is better than curated and small examples. It won't overwhelm anyone unless they're overly impatient and then, math is probably not for that personality type.

I have been through constructions of the reals, and the proofs for the hyperreals. It has been years, but I am confident that without much difficulty I could probably give you every proof that is necessary from scratch.

The proofs for the hyperreals involve a lot more machinery than the proofs for the standard reals. That is my educated opinion based on knowing both sets of proofs and constructions. (Of course this need not make infinitesmals a pedagogical disaster - very few students actually care much about learning the proofs.)

As for the model theory approach, I am intimately familiar with the ultrafilter construction. It uses choice. I know there is a second construction which I am not familiar with, but from what I've read it also requires choice. Both involve model theory. That's a mighty big sledgehammer for a pretty small fly.

Incidentally Dedekind cuts can be understood as follows. The set of reals can be equated with the set of points where you can cut the rationals into two. More precisely if X is a nonempty subset of the rationals with an upper bound, we get a cut of the rationals into the set of upper bounds of X, and things that are not an upper bound of X. Any two subsets can be considered equivalent if their set of upper bounds is identical. An equivalence class of subsets is a real number.

For any cut you can generate a unique set A of rationals that are not upper bounds, and a set B of rationals that are upper bounds. When you do this, all rationals in A are less than all rationals in B, and A does not contain an upper bound.

Conversely if we have a partition of the rationals into 2 non-empty sets A and B such that all members of A are less than all members of B, and A does not contain an upper bound. Then we have a cut of the rationals.

So there is a 1-1 correspondence of reals to places we can cut the rationals to partitions of rationals with that property.

Those partitions of the rationals are called Dedekind cuts.

(This is one of two constructions of the real numbers. The other, Cauchy sequences, turns out to generalize more usefully in the field of topology.)

Hey, you are right. the method I was thinking of does require model theory to justify its axioms (IST). It had been waved away as you put it, so that the core could be focused on. But you don't really need to understand why the axioms are justified any more than most people understand the axioms of ZFC (excepting those like you of course). And if the outcome is a better first intuition of calculus, I don't think it is accurate to label it a fly.

I did find out that there is a constructive approach though, so the axiom of choice is not actually necessary. www.math.ucla.edu/~asl/bsl/0403/0403-001.ps