Second, if your complaint is with pedagogy it's not about math. — fishfry
Yes, did you even read my post, this is my complaint.
I have no issue with real numbers "existing" in whatever sense mathematicians using the real number system want to believe. I am not convinced that "the true infinity" or "the true continuum" is captured by these symbolic systems, but I agree with you when you say mathematicians need not care and usually don't care; you can use a different system if it suits your style or problem.
I even cite your own words on this subject and express my agreement.
Your not giving me a hard time, you just have poor reading skills of prose; but I don't mind that, you don't make any claims to be able to understand non-formal arguments and perhaps have formal reasons to believe this task is impossible.
The reason I presented my arguments in prose is because that's the sort of thinking a high school student will be equipped with starting to use the real numbers.
My challenge is that: is there any answers to these prose questions that doesn't involve an entire university course, which maybe not even enough. As someone who's taken these university courses and who works with math in my day job building numerical models, you seem to claim I don't understand these issues. Even if it was true, which I doubt, isn't this more evidence to my point?
It would be fun to teach ZF to SOME high school students, the especially mathematically talented ones. The mainstream, no. I wonder what you are talking about here. Again, the axiom of choice is not needed to defined or construct the reals. — fishfry
Again, terrible reading comprehension; mathematicians not learning any humanities really is a problem.
I do not claim the axiom of choice is needed to construct the reals.
My argument is above the tdlr which doesn't mention the axiom of choice. My tdlr is an over simplification of my argument in a recommendation that I believe most people who understand this subject and have good reading comprehension would get.
Which you seem to agree with, that ZF can be taught at a high school level, which is my recommendation. I think you would agree that most high school students would not be prepared to deal with C (which for me, is what then makes the real number system mathematically interesting; unless there's been some breakthrough since I last looked at this topic that C is no longer required).
It's worth noting that the pedagogy retraces the history. — fishfry
This is basically our difference.
I disagree that the pedagogy retraces the history. If it actually did, maybe I'd have less of an issue.
Newton did not have the real numbers to do calculus as you note, yet high school calculus students simply start with the real numbers.
Dating from 1687, the publication of Newton's Principia, to the 1880's, after Cantor's set theory and the 19th century work of Cauchy and Weirstrass and the other great pioneers of real analysis; it took two centuries for the smartest people in the world to finally come up with the logically rigorous concept of the limit. For the first time we could write down some axioms and definitions and have a perfectly valid logical theory of calculus. — fishfry
You realize you're just adding more weight to my contention in the OP here?
If you need to read Principia mathematica and two centuries of the smartest people to understand the real number system ... maybe this is too much of an ask to high school students?
Do you agree?
If not, my challenge is that you explain the answers to my questions in a way that a high school teacher and then students would understand. If you can't, just agree with my OP rather than try to prove your smarter than me, which I so far not seeing any evidence for: going off on random tangents, not addressing the point of the OP, cowardly hedging your own complaints etc.
For instance, I did not define "infinitesimal", it's just a word that I find perfectly suitable to use to refer to series converging to a point (i.e. the distance becomes infinitely small). My use of infinitesimal was to contrast using prose (using words most people here would understand) the definitions one would find in numerical calculus compared to what we usually just call calculus; not to conjure up 17th century philosophical debates.
To lift from wikipedia because I do basic "google the subject matter" research when engaging in internet debates.
From the wikipedia page on infinitismals:
Logical properties
The method of constructing infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used. We consider here systems where infinitesimals can be shown to exist.
In 1936 Maltsev proved the compactness theorem. This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalise them [...]
There are in fact many ways to construct such a one-dimensional linearly ordered set of numbers, but fundamentally, there are two different approaches:
1) Extend the number system so that it contains more numbers than the real numbers.
2) Extend the axioms (or extend the language) so that the distinction between the infinitesimals and non-infinitesimals can be made in the real numbers themselves.
[...]
In 1977 Edward Nelson provided an answer following the second approach. The extended axioms are IST, which stands either for Internal set theory or for the initials of the three extra axioms: Idealization, Standardization, Transfer. In this system we consider that the language is extended in such a way that we can express facts about infinitesimals. The real numbers are either standard or nonstandard. An infinitesimal is a nonstandard real number that is less, in absolute value, than any positive standard real number.
Followed immediately by a section called "Infinitesimals in teaching":
Calculus textbooks based on infinitesimals include the classic Calculus Made Easy by Silvanus P. Thompson (bearing the motto "What one fool can do another can"[12]) [...]
Another elementary calculus text that uses the theory of infinitesimals as developed by Robinson is Infinitesimal Calculus by Henle and Kleinberg, originally published in 1979.[16] The authors introduce the language of first order logic, and demonstrate the construction of a first order model of the hyperreal numbers. The text provides an introduction to the basics of integral and differential calculus in one dimension, including sequences and series of functions. In an Appendix, they also treat the extension of their model to the hyperhyperreals, and demonstrate some applications for the extended model.
So, not only is infinitesimal perfectly fine mathematical jargon to talk about things "infinitely small" in both a technical and a general sense (as wikipedia starts the article by saying: "In mathematics, infinitesimals are things so small that there is no way to measure them"), but it is a common notion (according to wikipedia) used to introduce students to calculus, as it's intuitive.
This, my contention is, is a pedagogical mistake unless there are answers to all the very normal questions students can have about the real number system (that are as easy to grasp as other associated concepts being introduced). I have yet to see them.
Why is there so much debate around these infinite related questions such as cardinals and continuums here on philosohy forum? And not about questions like solving the quadratic equation or any number of other theorems? Because, in my view, it takes very specialized knowledge to understand modern mathematics modelling of these questions, which as you say, need not bother anyone that specialists are building such systems, but it is bad mathematical pedagogy to introduce to students concepts that they are unable to fully grasp and have zero need for any of the tasks at hand; it serves only to mystify mathematics rather than build understanding.
An analogy would be introducing Euclidean geometry in the context of Reiman manifolds or rotation in quaternians because that's what the cool kids in university do, with neither having any basis to have any clue what a Reiman manifold or quaternion really is nor ever needing the extra things Reiman manifolds or quaternions provide to address the Euclidean problems being asked to solve; now, I understand why concepts got inverted historically (since we were computationally extremely limited until recently), in the development and subsequent teaching of calculus as opposed geometry (pending an answer to my questions), my point is it's now a completely fixable conceptual problem in our teaching methods: that finite computation is a much more basic concept than the real numbers, real analysis, metric spaces and so on (i.e. real numbers are not required for any high school level problem and there's no need to introduce them until they are actually needed).
Now, I'm not saying these issues should be kept secret or something, there could be extra material for students who want to get into it; but I see no high school level problem that is not perfectly addressed in the numerical regime which is far easier to understand; you can really "see" and "get" how a computer functions in principle and why algorithmic approximations that truncate at a suitable number of steps yields answers to real world problems that students can visualize even at a high school level; there is nothing remotely as difficult conceptually as an infinite decimal expansion. It's also critical to understand not just the algorithm that converges on the desired constant but under what conditions are correct to end such an algorithm for any given applied mathematical problem, which is what the vast majority of high school math students will be going into: engineering, computer science, programming, chemistry and even accounting requires intuition of the strengths and limitations of machine computation (i.e. what kinds of problems require special attention to the the finite nature of the computer, in terms of memory, floating point representation, iteration steps, economizing computer resources and so on; and what kinds of problems one can just paste code from stack overflow and let it ride).
So if you want to get back on track, answer my questions concerning the real numbers in a way that a high school teacher and student understands. I've claimed to understand the answers to these questions, but you seem to be arguing it's all too complicated for me and that you will explain to me why I don't in fact understand the issue and you're going to demonstrate that. Well, if this is true, I'll be the first to benefit from your addressing the point in the OP. I eagerly await.