• boethius
    337
    Things are getting real here on the philosophy of mathematics forum. It's time to throw my own ball into the game (... or is two balls).

    A few threads deal directly or indirectly with the construction of the reals.

    When I was younger, and studying math, I spent a lot of time on foundational topics and had a keen interest in things like the construction of the reals.

    It was only when I had to work on real problems, invariably using a computer that only deals with a small set integers, that I started to have my doubts about the real numbers.

    I largely agree with fishry's assessment here:

    We can formalize the process of filling in the holes with various technical constructions of the reals. There are several, the two best known being Dedekind cuts and Cauchy sequences. The details aren't of interest. The point is that it can be done within set theory and it allows us to found calculus in a logically rigorous way, something that escaped Newton and Leibniz. We can also axiomatically define the reals as "the unique Cauchy-complete totally ordered infinite field." When you unpack the technical terms, you end up with an axiomatic system that's satisfied within set theory by the Dedekind cuts or Cauchy sequences. It's all very neat. One need not believe in it or care. It must be frustrating to you to both not believe in it, yet care so much!fishfry

    My beef is that the real numbers are introduced too early in education. Infinite processes and essentially 100% of numbers being infinitely complex are, though perhaps can be dealt with abstract rules, too difficult to conceptually grasp for most high school students. Moreover, most students studying high-school level calculus will not go on to study pure maths. Any applied math does not need real numbers to solve any real world problem. A finite sequence that gets one as close as required within the experimental error is sufficient for any real world engineering and, of course, is represented by a rational number.

    Introducing the real numbers in high school, at best, imposes an unnecessary conceptual burden on students making learning harder (often contributing to "I don't get it" and choosing not to study math further), but worse, it breaks the chain of reasoning. High school mathematics simply posits the real numbers as the "kind of number calculus operates in" and does not go into how to construct them. The natural numbers and rational numbers are clear how to make them, and the chain of reasoning is smooth introducing natural numbers and then rational numbers and proofs built on these kinds of numbers.

    Simply positing the real numbers creates a discontinuity in reasoning that is the anti-thesis of what mathematics is about. Filling in this reasoning gap is simply a waste of time for high-school level calculus, and, more importantly, the average teacher introducing calculus would not have the knowledge and skills to do so.

    Why the conceptual burden is high, is that even when just dealing with "normal" real number behaviour such as mapping all the points between 0 and 1 to all the points between 0 and 10 000, there is simply no way to imagine what is happening (unlike most other high-school math that one can, after some effort, "see it" and "get it"). There is of course no real world problem where this feature of the real number system is needed.

    However, it's also necessary to deal with questions like "why can't we just have rational numbers with infinite numerator and denominator; seems unfair to allow infinite decimal expansion but not infinite numerators?". Of course, we can have rational numbers with infinite numerator and denominator but it's then needed to explain how these aren't the "real" rational numbers we're talking about when we say the square root of two is irrational. Even if this explanation is successful (which I doubt the average high school teacher would be able to answer adequately) there's the followup problem of avoiding the claim that the rational numbers aren't finite, as there's clearly an infinite amount of them as they are countable by the infinite sequence of natural numbers. But ... if they aren't finite how do we avoid infinite numerators and denominators? If we're claiming the natural numbers aren't complete enough in some sense to simply go onto simply include every decimal expansion injected into every possible fraction (just take the decimals expansion of pi and place them over something else, like the decimal expansion of e to create a new rational number that comulates perfectly), then what do we make of the real number with a decimal expansion of the sequence of natural numbers, i.e. 0.12345678910111213...; is this not a correspondence between the a real number and natural numbers which we can then go onto make rational numbers with this natural number as numerator or denominator; if it is a real number with a decimal expansion "as big" as the natural numbers, why can't we make all the other real numbers through a process of permuting numbers in the natural number sequence along the way, and every time we permute we count 1? If we're "not allowed" to simply claim all the natural numbers is a new infinite natural number, why are we "allowed" to make infinite decimal expansion; they both seem very much the same process of assembling individual numbers on a line.

    I'm aware these questions have answers. My point that I'm making is that these questions are perfectly natural for a student encountering real numbers for the first time to ask. In the rest of mathematics questions about proofs have answers; there's an answer to why theorems work (otherwise they're not theorems) and answering every critique possible of a theorem is what (in my view) mathematics is.

    However, I do not believe it's possible for the average high school teacher, much less the average high school student, to be able to answer the above questions about the real numbers. The main problem, in my view, is that students simply can't imagine how big infinity is and do not have the prerequisite knowledge to represent what (maybe) is infinity with symbols.

    So, my challenge is if someone can construct the real numbers in a concise and clear way that the average student starting calculus in high school would easily understand for then transcendental constants like pi to make perfect and clear sense and all the tricky questions above perfectly clear answers (just as clear as in geometry or proofs about discrete numbers).

    If not, then my recommendation is to drop the real numbers and do calculus in the numerical regime of "as small as you want ... but not infinitesimal". I realize that pi is no longer pi in this regime, but some ornament could be added to all our precious transcendental constants to indicate that it is representing an approximate value (of "as approximate as needed", to be defined in the algorithm that will provide a numerical result); then, for students who go into pure maths, the process of how to remove those ornaments can be learned and students can then transcend to things like the purest e. Every other student that continues with math working essentially in the computer, those ornaments then serve as a constant reminder pi really isn't in the computer in any sense and thinking so is a mistake that will mess numerical recipes from time to time, and an additional step will be required of talking with the pure mathematics department if ever "real pi" is needed to solve a real world problem (they may lose count waiting, but that's the point).

    This approach not only relieves the burden of the real numbers, but also teaches essential habits of numerical computation. It's also a critical educational error to solve equations with real numbers and then just "type into a calculator" to get a numerical answer; this isn't how numerical computation is done properly and makes learning both the pure math parts that are being attempted (however inadequately) as well as the numerical procedure to "get an answer" both simply wrong understanding of the maths involved.

    TD;LR: we should teach ZF in high school and then add C later for pure maths students.
  • fishfry
    1.1k
    TD;LR: we should teach ZF in high school and then add C later for pure maths students.boethius

    What??

    First, the technical construction(s) of the reals are taught only to math majors in a class called Real Analysis. Nobody who's not either a math major or someone taking that class as an elective ever sees the technicalities. Perhaps they teach high school kids advanced real analysis in high school in Russia or China but truly I doubt that very much.

    Second, if your complaint is with pedagogy it's not about math. And certainly there are many problems with the way math is taught. But that is not a beef with the real numbers. In fact I didn't see you present any beef with the real numbers. I only saw you beef with the teaching of the real numbers; a subject on which I'm in complete agreement with you in the large, if not necessarily every detail. When I'm emperor of the world the first thing I'm going to do is send all the math curriculum boards off to Gitmo. I've long held that idea.

    And third, what does the axiom of choice have to do with anything? It's certainly not needed to define or construct the reals. Teach ZF to high school students? What, teach ordinals and cardinals to high school students? 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.

    as small as you want ... but not infinitesimalboethius

    That IS the essence of the real numbers. There are no infinitesimals in the standard real numbers. I think perhaps you had an unhappy calculus class, as most students do. They don't teach the technical definition of the real numbers in calculus. Perhaps what you're unhappy about is that you didn't have a more rigorous class in calculus. But that's not done because calculus is a service class for a lot of physics, engineering, economics, and other majors. The math majors have to make the best of it till real analysis class.

    In fact this is the very reason there's so much confusion about the real numbers. Nobody is ever taught what the real numbers are simply because it's not relevant to anyone's profession unless they're going to be a math major. So even technical professionals like physicists and engineers don't know what a real number is. And then when you get to online discussion forums, you get a lot of confusion on the subject.

    I appreciate that you agreed with what I wrote, but what you wrote didn't have much if anything to do with it. I think what you are calling for is better teaching of the reals to high school students, which I'd also like to see. But to go into the full technical details is way beyond high school students.

    [Disclaimer: Poster quotes me in AGREEMENT and I give him a hard time. I'm a terrible person. Forgive me].
  • jgill
    233
    Good explanation. I seem to recall from long ago a study by Piaget on the ability of young people to understand calculus. I may be mistaken but it seems that in general the age of fifteen was a benchmark, with those below that age experiencing a lot more difficulty with the subject. Of course there are spectacular exceptions. As for the intricacies of the real number system, I wonder. :chin:
  • fishfry
    1.1k
    As for the intricacies of the real number system, I wonderjgill

    It's worth noting that the pedagogy retraces the history.

    Newton developed calculus to study the motions of the heavens. He was not able to drill down a logical explanation of his methods even to the standards of rigor of the day. He had no idea what his derivative (what he called a fluxion) was. If we contemplate the expression (in Leibniz's notation, which was better than Newton's) , what exactly is it? If the numerator and denominator are not zero, that's NOT the derivative. If they are, then the expression is not mathematically defined.

    The philosopher George Berkeley sarcastically called Newton's fluxions, "The ghost of departed quantities." What a great line. If these guys came back today they'd all be on the Internet flaming away at each other.

    Newton struggled with the logical nature of infinitesimals and limits. In fact his publications clearly show that he didn't ignore the issue at all, but was rather keenly aware of the problem and tried hard over his lifetime, without success, to come up with a good explanation.

    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.

    This was a very great achievement of humanity, one not very well appreciated. We don't teach the history of math. In addition to "Pull down the exponent and subtract 1" that we beat students over the head with, it would be great if we could impart the sense of mathematicians working on this problem of defining limits for 200 years before they worked it out.

    It's no surprise that these are extremely subtle concepts to understand. So in high school and college we just tell people that real numbers are mysterious "infinite decimals," whatever that means, and no harm is done. And in calculus class we show people what limits are but we can't really be precise, so mostly we focus on techniques, which the physicists and engineers and economists will need for their work.

    The development of the real numbers and the limit concept in the 19th century is one of our greatest intellectual achievements. I wish there were more appreciation of it.
  • boethius
    337
    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.
  • ssu
    1.9k
    My beef is that the real numbers are introduced too early in education. Infinite processes and essentially 100% of numbers being infinitely complex are, though perhaps can be dealt with abstract rules, too difficult to conceptually grasp for most high school students.boethius
    The problem is that no math course has enough time to really take the time. Usually it's just "here's the proof, there, I showed it to you, now use this algorithm".

    About pedagogy, when I started first grade in my country in the late 70's they had the wonderful idea to start math education with set theory. I found it a bit confusing then (I had a problem to learn the various Venn diagrams and their relations to addition and subtraction etc.) and I remember that the teachers weren't happy about the reform either. Well, that was the 70's and now they teach things to my children in the first grade basically the same way they taught things to my parents (and even grandparents).

    Real numbers are one of those things that at first glance seem to be easy, but aren't at all. Just as, well, set theory.
  • A Seagull
    172
    In teaching maths, I think it is important to make a clear distinction between pure (abstract) maths and applied (to the real world) maths. It is the conflation of the two that causes problems.

    Of course children first learn maths with the conflated maths; counting sheep etc. But perhaps around the time they enter secondary school ( around age 13) the distinction needs to be emphasised.

    In pure maths one is dealing entirely with the manipulation of symbols following particular rules. Here the symbols used whether for integers, reals or imaginary numbers have no more connection to the 'real world' than any other; they are all abstract.

    Then for the application to the 'real world' (applied maths) one takes a particular part of mathematics and applies a mapping between the abstract symbols and concepts that apply to the 'real world'.

    With this clear distinction the complications of maths fade away.
  • boethius
    337
    With this clear distinction the complications of maths fade away.A Seagull

    Yes, this is basically what I am advocating, though with much heavier emphasis on the applied part in secondary school (and applied maths will still have plenty of symbol manipulation in it's own right and plenty of theorems that apply to both pure and applied maths).

    Though more specifically I am singling out real numbers as the particular problem; though maybe there are others. Likewise, perhaps there is a pedagogical approach that accomplishes both, as you seem to be suggesting, but my feeling is that you can't really do pure maths without set theory, which as points out was a failed experiment to teach children.

    However, if there's some simple explanation of the real numbers and all the questions that naturally arise from infinite decimal expansion, then I'd be proven wrong on this particular point.



    Did this set theory experiment simply not work at all, or did it produce some small cadre of math geniuses?
  • StreetlightX
    4.6k
    The OP seems to me to be a particularly bad idea. Part of the excitement of learning is to learn what you do not know - to understand one's own ignorance so as to all the better situate what you do know. To keep away even the introduction of the reals in order to coddle the apparently effete minds of the young would leave them with an impoverished understanding of the natural and rational numbers themselves. One of the more exciting moments in my math education in high school was learning about imaginary numbers, even if the work we did with them was incredbily basic. It spoke to a wider world of number, and ramified back upon the little I did know of the rest of math, and made me appreciate it in a new light.

    It strikes me as both condesending and and an injustice to the excitment that math can elicit by treating kids as idiots just because they can't engage in construction. It seems a way to suck any exploratory spirit out of math, and kill any joy that might be gleaned from it.
  • A Seagull
    172
    my feeling is that you can't really do pure maths without set theory,boethius



    Why set theory? Set theory is pretty uninteresting really, apart from Venn diagrams which are fun and useful. I presume that you are referring to the idea that set theory provides the 'foundation' to mathematics. But pure mathematics is abstract and doesn't need any foundations apart from its axioms which introduce the symbols and define the rules. (And admittedly these axioms are more implicit than explicit).

    And as for the real numbers, they become necessary when one looks to divide (for example) 10 by 4. (10/4). although the task is in the domain of integers the answer is outside. It could be written as a fraction 2 1/2 and that is what ancient mathematicians did. They considered that all numbers could be written as integers or fractions or a composite of the two. It was quite a shock to them when they came to realise that the square root of two could not be expressed as a fraction! There was no alternative except to introduce real numbers.
  • boethius
    337
    I presume that you are referring to the idea that set theory provides the 'foundation' to mathematics.A Seagull

    More or less. There are alternative foundations, but set theory is the main one.

    But pure mathematics is abstract and doesn't need any foundations apart from its axioms which introduce the symbols and define the rules. (And admittedly these axioms are more implicit than explicit).A Seagull

    This is also true for applied mathematics.

    Applied mathematics also has definitions and axioms, just focusing on those that generally have real world scientific application (applied maths is a subset of pure maths).

    There are not infinite sets of anything in the real world (real world of scientific investigation at least); so how to deal with infinite sets can be excluded from applied mathematics. For me this is the main difference; there is no need to learn calculus in the real number system and then calculus in a finitist numerical system appropriate for a computer. It's, in my view, only historical accident that learning calculus with real numbers first seems to make sense. One can learn first a finitist numerical system for doing calculus.

    And as for the real numbers, they become necessary when one looks to divide (for example) 10 by 4. (10/4). although the task is in the domain of integers the answer is outside.A Seagull

    This is the basic thing my pedagogical program would get rid of.

    You never need an "exact" (i.e. infinite decimal expanded) value of 10/4 in an applied problem (first because it's 2.5, but I assume you intended an example like 1/3 or then an obtuse reference to infinite trailing zeros).

    The numerical regime basically refers to replacing all calculations that can go on forever in a finitist setup tailored for the computing machine doing the calculations (values can be arbitrarily large or small, not infinite, and not more than can fit in the computer ... with a whole bunch of caveats) with algorithms that can be carried out to the required precision (the series sum or whatever the algorithm is, and a halting condition); in other words, those significant digits from physics class, can form the axiomatic basis of a completely adequate calculus for applied problems.

    Learning the axiomatic setup of numerical computation rigorously is (until someone shows me how easy real numbers can be) a far better use of students time leading to, I believe, a better understanding of maths for both future applied maths and pure maths students. Understanding finitist maths well, I would argue is the correct basis to then going beyond finitism for students interested to do so; likewise, I would argue a more rigorous use of finitist maths wherever it is adequate in physics and other sciences is far better than a lazy use of more powerful models.

    In other words, real numbers are not necessary when dealing with 1/3, or any calculation that can in principle go on for an arbitrary length; approximate solutions are fine for any real world problem.

    The reason I'm posting here in logic and math and not politics, is not simply because of the theme but because my contention has a counter example of a very clear and simple presentation of the real numbers that high school teachers and students would find of appropriate effort to fully grasp. If there is such explanations that are graspable by the average student, then I'd capitulate.
  • Pfhorrest
    965
    As I recall my math education, the concept of the "real numbers" wasn't even introduced until we got to "imaginary numbers" to juxtapose them with.

    We had just "numbers" (natural numbers), constructed by counting (the successor function). We could do addition and multiplication on them just fine and didn't need to worry about any other kinds of numbers.

    Those then became "positive numbers" when contrasted with "negative numbers", which were introduced to fill out the set of numbers that could be constructed through subtraction, which were again just "all the numbers".

    Those then became "whole numbers" (integers) when contrasted with "fractions", which were introduced to fill out the set of all numbers that could be constructed through division, which were again just "all the numbers".

    Those then became "rational numbers" when contrasted with "irrational numbers", which could be made in a bunch of different ways; there wasn't just one kind of operation that resulted in irrational numbers sometimes. Between rational and irrational numbers, that was again just "all the numbers", and we never had to worry about having one method of constructing any given one of them, just that there was the stuff that could be constructed through division and the stuff that couldn't.

    Those only became "real numbers" when contrasted with "imaginary numbers", which were introduced to fill out the set of numbers that could be constructed through taking roots, which were, by this point, finally not treated as "all the numbers" but as a set of their own, the "complex numbers", suggesting that the reals are still considered the normal set of all numbers, and the complexes are considered some kind of weird superset made of pairs of numbers, not just numbers simpliciter.
  • fishfry
    1.1k
    So if you want to get back on track, answer my questions concerning the real numbers iboethius

    Perhaps you could state them succinctly. I prefer not to wade into this. You have an ax to grind and I've only succeeded in upsetting you. If you'll list some clear questions I'll do my best to respond. My sense is that you didn't actually read your own post. If you did, you'd realize that you have no beef with the real numbers, only with their teaching in high school. I share many of your concerns in that regard.

    You did state that ZF should be taught in high school and ZFC in college. That does not make sense to me at all. ZF is a fairly sophisticated system. I wouldn't teach a full course in ZF to high school students except to the most mathematically motivated and talented students. ZFC actually adds very little in terms of complexity or teachability. If anything, the axiom of choice regularizes infinite sets and eliminates a lot of problems. For example absent choice, there's an infinite set that's not Dedekind-infinite. Surely you don't mean to claim we should be teaching this to high school students.
  • boethius
    337
    Perhaps you could state them succinctly.fishfry

    The whole point of my post is that high school students would have no way of stating their questions succinctly as you demand, but they are in my view meaningful questions to ask.

    You could argue that they aren't meaningful questions and can just be dismissed not warranting an answer, or you could "not wade into it" as you suggest to yourself post-wading.

    You have an ax to grind and I've only succeeded in upsetting you.fishfry

    I have no axe to grind. But you very much seem to have an axe to grind with projected axe grinders.

    However, the continuous debate around this topic here on philosophy forum inspired me to post my own personal beef, which is that simply positing the real numbers without constructing them nor dealing with all the non-intuitive questions that can arise with completed infinities such as infinite decimal expansion, is poor pedagogy.

    But, I'll play your game; perhaps it will satisfy the OP as all my questions will have simple and clear answers that an average high school student will have no problems understanding with a little effort.

    Let's start with infinite numerators, denominators and exponents.

    Instead of accepting the conclusion that root 2 is irrational (not a rational number), I'm going to solve root 2 using infinite denominators and numerators.

    Where do I get these infinite natural numbers to make my rational solution to root 2? I simply take suitable real numbers and take out the decimal symbol and insert those infinite digit expansions into polynomials to represent values that solve my problem exactly, which I admit, I was unable to accomplish with any solution using finite natural numbers I could name.

    Fairly simple procedure.

    Please demonstrate how this infinite numerator and denominator either does not get counted in Cantor's diagonal proof, does not represent an irrational value, or there is something preventing finding and placing all the digits of suitable real numbers into a numerator and denominator to solve for root 2.

    An infinite normal digit expansion (which I'll choose to use as suitable) is neither odd nor even, as is well known, and so there's no contradiction of division by 2 as is usually concluded in the run-of-the-mill finitist approach to proving the irrationality of root 2; I can just keep that 2 coefficient around no problem and divide by 2 to get rid of it. Therefore, root 2 is rational.

    If I am given these infinite expansion of digits, seems I should be able manipulate and place them where I want if I have some procedure to do so (unless given suitable axiomatic conditions preventing me to doing what I want).

    What axioms does a high school student possess to avoid the above issue of concluding root 2 is rational? If none really (if only because the reals aren't even constructed to begin with, just posited as a given) then I think we agree about the OP.

    Followup question (as I believe this is what interests you to demonstrate) what axioms does a university student possess to avoid the above issue and how?
  • fishfry
    1.1k
    The whole point of my post is that high school students would have no way of stating their questions succinctly as you demand, but they are in my view meaningful questions to ask.boethius

    We're in deep and complete agreement on this. The mathematical definition of the real numbers is far beyond high school students; in analogy with the difficulties Newton and Leibniz had, which needed to wait 200 years for resolution.
    Instead of accepting the conclusion that root 2 is irrational (not a rational number)boethius

    Instead of accepting the conclusion that root 2 is irrationalboethius

    This fact has a proof, 2300 years old dating to Euclid's written account; but most likely at least several hundred years older than that.

    The subject of the square root of 2 is part of my response to @Metaphysician Undercover in the bijection thread. I'm drafting a response that will expound at length on the mathematics and the mathematical philosophy of the square root of 2. I hope to corral my thoughts into a postable screed within a week or so. Meanwhile please forgive my lack of comment today on the mathematical existence of a positive real number whose square is 2. I believe in such a real number and I believe it's not the ratio of integers and I believe that it's computable, hence encodes only a finite amount of information despite its endless and patternless decimal representation. A real number is not its decimal representation. I hope you believe these things too.

    ps -- Note well The irrationality of the square root of 2 does NOT introduce infinity into mathematics. All the irrationals familiar to us are computable, and have finite representations. The noncomputable reals do introduce infinity into math; but plenty of people who don't believe in noncomputable reals nevertheless DO believe in the square root of 2. Namely, the constructive mathematicians.

    Please demonstrate how this infinite numerator and denominator either does not get counted in Cantor's diagonal proof, does not represent an irrational value, or there is something preventing finding and placing all the digits of suitable real numbers into a numerator and denominator to solve for root 2.boethius

    Euclid's proof of the irrationality of has nothing at all to do with Cantor's discovery of the uncountability of the reals. The rest of this paragraph, I confess, is not intelligible to me.

    What axioms does a high school student possess to avoid the above issue of concluding root 2 is rational?boethius

    None whatever. In high school we mumble something about "infinite decimals" while frantically waving our hands; and the brighter students manage not to be permanently scarred for life.

    The teaching of mathematics in the US public schools is execrable. How many times do I have to agree with you about this? I'd gladly join you in a protest down at the local school board, but it would do no good. The educrats have bought off and bamboozled the politicians. The teaching of math in the US is stupid by order of the government. I wish I could do something about it.

    Followup question (as I believe this is what interests you to demonstrate) what axioms does a university student possess to avoid the above issue and how?boethius

    A university student in anything other than math: None.

    A well-schooled undergrad math major? Someone who took courses in real and complex analysis, number theory, abstract algebra, set theory, and topology? They could construct the real numbers starting from the axioms of ZF. They could then define continuity and limits and I could rigorously found calculus. It's not taught in any one course, it's just something you pick up after awhile. The axiom of infinity gives you the natural numbers as a model of the Peano axioms. From those you can build up the integers; then the rationals; and then the reals. Every math major sees this process once in their life but not twice. Nobody actually uses the formal definitions. It's just good to know that we could write them down if we had to.

    So, as I've noted already, people who study math in university get all their conceptual questions about the real numbers and the nature of calculus answered. The physicists, engineers, economists, pre-meds, and everyone else, do not. The formal constructions aren't important as long as you know the rules for manipulating real numbers. Even the mathematicians just use the real numbers according to the rules of addition and subtraction and so forth. The formal constructions are to demonstrate that the real numbers have logically valid mathematical existence. [As always please don't jump in just to point out that this is not necessarily actual existence. I quite agree].

    Buildings and bridges are made of quarks. But architects and bridge builders don't need to know particle physics. Likewise nobody needs to care about the formal definition of the real numbers; except that if they ask, they can honestly be told that there is one. And of course it's all on Wikipedia.

    https://en.wikipedia.org/wiki/Dedekind_cut
  • tim wood
    3.6k
    Old high-school student bumping along in the wake of this thread. A question or two: mention is made of of constructing real numbers. I suppose that means identifying particular examples or indicating how any can be found. I can identify more reals than I'll ever use, being integers and fractions. Irrationals a different story. I know about the square root of (2), & etc. Beyond that, in simplest possible terms and not too long, what does it mean to construct a number? If it means to associate a name with a concept (like "square root of 2" with the square root of two), that implies stumbling across a needed number, recognizing it as such and what it is, and naming it - resulting in, in theory, a partial listing of the names of some irrational numbers.

    But this isn't what the words "constructing a number" suggest to me. Any light for the darkness, here? Not too bright, please!
  • fishfry
    1.1k
    But this isn't what the words "constructing a number" suggest to me. Any light for the darkness, here?tim wood

    Construct in this context means build a thingie within set theory that behaves exactly like we want our thingie to do.

    For this purpose, the construction of Dedekind cuts (linked earlier) is a construction of the real numbers. But how do I know that? It's because we first write down properties we want the real numbers to have; then everyone can use them, even though we don't know whether they exist mathematically. The construction shows that they do.

    Let me expand this in full gory detail.

    Here are the axioms for a totally ordered, Cauchy-complete, infinite field.

    https://sites.math.washington.edu/~hart/m524/realprop.pdf

    This PDF lists the properties of the real numbers. It's much better than the Wiki link on the same topic.

    Briefly you can do arithmetic: a + b = b + a, and multiplication distributes over addition and ab = ba, and if b is nonzero then the quotient a/b exists. Then you have the order relations: for any two distinct reals either a < b or b < a.

    Now the rationals satisfy these properties; so we need one more special property that fully characterizes the real numbers: The Least Upper Bound property, which says that every nonempty set of reals bounded above, has a least upper bound. This is also known as the completeness axiom. It says there are no "holes" in the real numbers.

    Example. The rationals don't satisfy the LUB property. The nonempty set does not have a least upper bound! This came up earlier. This is why the rationals aren't a mathematical continuum.

    The real numbers -- whatever they are -- SHOULD have this property.

    Now as far as we know, there is no such abstract object that satisfies these properties. Maybe the real numbers don't exist. But it doesn't matter. We can just use their properties. We can do physics, engineering, calculus, etc ... even if the real numbers don't exist. Just by using their properties!

    But now a someone comes along and calls the entire enterprise null and void because for all we know, we're talking about something that doesn't exist. But if you believe in the rationals, you must believe in Dedekind cuts; and the set of Dedekind cuts satisfies all the real number properties. So we are justified in calling the set of Dedekind cuts the real numbers.

    Having seen this construction once in our lives, we are confident that the real numbers are mathematically legitimate, because we can build an object using set theory that behaves exactly like the real numbers. We now forget all about it; till someone asks, "Oh yeah? How do we know there is any such thing as the real numbers?" Then we show them.

    tl;dr: The real numbers are anything that satisfies the list of real number properties. But is there even any such thing at all? Yes. Within ZF set theory we can build up a set that has exactly these properties. We can in fact do this in several different ways. They're all isomorphic. We call any of these models the real numbers.

    Was this helpful?
  • aletheist
    1.2k
    The teaching of math in the US is stupid by order of the government.fishfry
    I laughed out loud at this. It reminds me of my favorite line from the movie, The Incredibles 2, when Bob is trying to help Dash with his homework: "Why would they change math? Math is math!"
  • tim wood
    3.6k
    Was this helpful?fishfry
    Very, and very clear. Especially this.
    Construct in this context means build a thingie within set theory that behaves exactly like we want our thingie to do.fishfry
    Because this is the answer to a question I'll ask pro forma below.

    I'm familiar with the axioms (I'm a child of new math, if you know what that 1960s fad was), but the lub - well, that's not so clear. Maybe because it's too obvious - that happens.

    From online
    "Let S be a non-empty set of real numbers.
    1) A real number x is called an upper bound for S if x ≥ s for all s ∈ S.
    2) A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S."

    1) is pretty clear. With respect to integers only, given the set (1,2,3) 3 is an upper bound. Now here's maybe the question the answer to which will help me out. 3 is an upper bound, but is it correct to say that all x, x>3 are also upper bounds, and that 3 is the least upper bound?

    And the idea that the rationals do not provide a lub for the square root of two simply means that although there is no end of upper bounds, for any upper bound that seems a candidate for the lub, a better candidate can always be found, in the rationals. If this is it, then I understand the lub.

    Here the pro forma question, though it's evolved since the first paragraph. And even this you've already answered. It seems to me that to question the existence of a measure, or distance, or number corresponding to the square root of two is the greatest nonsense, because it is so easily modeled, and a fortiori any other irrational. Almost as easily is π modeled, so with transcendentals.

    Nor is a numeral for any of these lacking, if by "numeral" is meant a name. Of course an exact decimal representation is a problem, But then so is my idea of the perfect woman (pace wife). But it appears that the proof of these things ignores the obvious: irrational numbers are easily proved to exist. For me, I guess, the question is, what is (was) the problem? What the need for the thingie? (If for some arcane application, that's enough of an answer: likely I could not follow anything more than that.) And ty, btw.
  • jgill
    233
    Some years ago the New Math was in vogue. As a mere instructor at the time I was given a text on College Algebra having a lengthy first chapter devoted to an axiomatic approach to the subject. It was not a good experience for instructor or student. :worry:
  • alcontali
    999
    Then for the application to the 'real world' (applied maths) one takes a particular part of mathematics and applies a mapping between the abstract symbols and concepts that apply to the 'real world'.A Seagull

    Yes, I think so too.

    Furthermore, the mapping back to the real world must go through the regulatory framework of an empirical knowledge discipline, such as science. Direct application of mathematics to the real world should be discouraged, because mathematics does not seek to create such regulatory framework for empiricism, while such framework is clearly needed.

    Therefore, real-world considerations are the domain of downstream users of mathematics, such as science, engineering, and so on. Mathematics itself should stay clear of those, in order to preserve its purity.
  • alcontali
    999
    Why set theory? Set theory is pretty uninteresting really, apart from Venn diagrams which are fun and useful.A Seagull

    In my opinion, the most successful offshoot of set theory is relational algebra, for which the canonical language is SQL:

    The main application of relational algebra is providing a theoretical foundation for relational databases, particularly query languages for such databases, chief among which is SQL. The relational algebra uses set union, set difference, and Cartesian product from set theory, but adds additional constraints to these operators.Wikipedia on relational algebra

    Relational algebra is massively big. Very little modern software can do without.

    Furthermore, separate from relational algebra (which is a niche application), there is a strong trend to moving to executable (general) set-theoretical expressions in modern programming. The flagship library in this regard is certainly underscore.js.

    Set theory is an incredibly invasive species which, over the last two decades, has increasingly invaded the practices of contemporary software engineering. Set theory is slowly but surely turning into the primary foundations in programming. In fact, it is so intuitive that few people actually realize that all of that is almost pure ZFC set theory.
  • fishfry
    1.1k
    I'm familiar with the axioms (I'm a child of new math, if you know what that 1960s fad was),tim wood

    Yes, that was an attempt to teach set theory in grade school. Needless to say the teachers were confused and the students were confused. Big fail. Now we have Common Core. The teachers are confused and the students are confused. In the US, public school students learn math in spite of the curriculum, not because of it.

    but the lub - well, that's not so clear. Maybe because it's too obvious - that happens.

    From online
    "Let S be a non-empty set of real numbers.
    1) A real number x is called an upper bound for S if x ≥ s for all s ∈ S.
    2) A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S."

    1) is pretty clear. With respect to integers only, given the set (1,2,3) 3 is an upper bound. Now here's maybe the question the answer to which will help me out. 3 is an upper bound, but is it correct to say that all x, x>3 are also upper bounds, and that 3 is the least upper bound?
    tim wood

    Yes. And for example 1 is the least upper bound of the set .9, .99, .999, .9999, etc.

    And the idea that the rationals do not provide a lub for the square root of two simply means that although there is no end of upper bounds, for any upper bound that seems a candidate for the lub, a better candidate can always be found, in the rationals. If this is it, then I understand the lub.tim wood

    Yes. Again consider where is the rationals. No matter what upper bound you pick, there is no least upper bound in the rationals. So the rationals are not complete. (Or Cauchy-complete, or topologically complete, to distinguish this from other uses of the word complete).

    But that set does have a least upper bound in the reals. In fact every nonempty set of reals bounded above has a least upper bound. That fact uniquely characterizes the real numbers among all totally ordered fields.


    Here the pro forma question, though it's evolved since the first paragraph. And even this you've already answered. It seems to me that to question the existence of a measure, or distance, or number corresponding to the square root of two is the greatest nonsense, because it is so easily modeled, and a fortiori any other irrational. Almost as easily is π modeled, so with transcendentals.tim wood

    Yes definitely. There are so many ways to develop the square root of 2. You don't even need least upper bounds. You can do it with pure algebra. If is a purely formal symbol that means nothing, but has the property that , then consider the set of all formal expressions



    Define addition componentwise, and multiplication using the usual distributive law. Then it's easy to see that is closed under addition and multiplication, and that multiplication distributes over addition, etc.

    What about inverses? It's not immediately obvious, but in fact if then



    https://math.stackexchange.com/questions/821260/inverting-ab-sqrt2-in-the-field-bbb-q-sqrt2


    In other words our set is a field (add, subtract, multiply, divide) that contains a square root of 2.

    But, just as with the real number earlier, all we've done is define some formal symbols that have the right properties. Can we construct such a field using set theory? Yes. If you have seen some abstract algebra, here is the construction.

    You start with , defined as the set of all polynomials in one variable having integer coefficients. We can add, subtract, and multiply any two such polynomials as we learned in high school. So is a commutative ring.

    Now the set of all multiples of the polynomial is an ideal in this ring. Ideals in rings are analogous to normal subgroups in group theory. You can "mod out" by an ideal to get another ring. In this case the ideal generated by , denoted , is a maximal ideal; and therefore (it's a theorem) that when you mod out by you get a field. And what field do you get? You get a field isomorphic to our field of formal made-up symbols.

    This is a bit of abstract algebra that most people haven't seen unless they took that course. But the point is that we can whip up a field containing a square root of 2 in the usual two ways: We can invent it as a make-believe set of formal symbols that behave according to some rules; AND we can construct such a beast within set theory.

    This gives all the mathematical existence it needs.

    Apologies if this exposition was a bit too much abstract algebra. But perhaps someone reading this took a course in groups, rings, and fields but forgot this beautiful construction, which we can sum up in one equation:



    On the left we have a made-up collection of formal symbols that mean nothing; on the right, we have a concrete realization of that thing cooked up within set theory.

    Again, note that this construction parallels how we define the real numbers. We make up a formal system using some rules; and then we show that such a thing can be built out of spare parts within set theory. That is mathematical existence.

    Nor is a numeral for any of these lacking, if by "numeral" is meant a name. Of course an exact decimal representation is a problem, But then so is my idea of the perfect woman (pace wife). But it appears that the proof of these things ignores the obvious: irrational numbers are easily proved to exist. For me, I guess, the question is, what is (was) the problem? What the need for the thingie? (If for some arcane application, that's enough of an answer: likely I could not follow anything more than that.) And ty, btw.tim wood

    Seems that way to me too. Our friend @Metaphysician Undercover, who must be a neo-Pythagorean, is mightily vexed by the fact that the square root of 2 is (a) a commonplace geometric object, being the diagonal of a unit square; and (b) doesn't happen to be the ratio of any two integers.

    What of it? Humans got over this about 2500 years ago.
  • fishfry
    1.1k
    Some years ago the New Math was in vogue. As a mere instructor at the time I was given a text on College Algebra having a lengthy first chapter devoted to an axiomatic approach to the subject. It was not a good experience for instructor or student.jgill

    Yup. New math, new new math, Common Core. One educrat failure after another. I have no idea what the answer is.
  • boethius
    337
    We're in deep and complete agreement on this. The mathematical definition of the real numbers is far beyond high school students; in analogy with the difficulties Newton and Leibniz had, which needed to wait 200 years for resolution.fishfry

    Yes, we're in agreement.

    And, as I mentioned in the OP, I also agree with your position that there's no "problem" in the real number system, axiom of choice, well ordering, cardinals and the like; at least not some trivial contradiction I'm aware of.

    My questions do have answers, and I'm only trying to demonstrate here that the answers are incredibly tricky and go far beyond high school maths.

    I think continuing the debate is a good way to bring up how tricky these ideas are, and why simply positing the real numbers without the axiomatic framework to avoid these problems is bad pedagogy.

    ps -- Note well The irrationality of the square root of 2 does NOT introduce infinity into mathematics. All the irrationals familiar to us are computable, and have finite representations. The noncomputable reals do introduce infinity into math; but plenty of people who don't believe in noncomputable reals nevertheless DO believe in the square root of 2. Namely, the constructive mathematicians.fishfry

    Yes, it is quite clear to conclude root 2 is irrational in a finitist constructive approach.

    My procedure only kicks in once I'm given the real numbers and have access to completed infinite decimal expansion. Given this, I now double back and ask "can I use these new values to prove root 2 is rational in my new system of rules that includes sets of completed infinite decimals".

    I am now no longer satisfied by the proof by contradiction that originally brought me to believe root 2 was irrational, as I can solve the equation with values that are neither odd nor even. I can also now do the same thing to solve exactly for the roots of any polynomial that I was previously unable to do.

    Euclid's proof of the irrationality of 2‾√2 has nothing at all to do with Cantor's discovery of the uncountability of the reals. The rest of this paragraph, I confess, is not intelligible to me.fishfry

    Where this relates to Cantor, is that if I simply "have the real numbers" and can use my procedure to prove root 2 is rational (because I just have them and have no axiomatic system to prevent me from doing it), then I should be able to count it in Cantors diagonal proof. If I complete the count of the rationals I will "eventually get" to this rational number with infinite numerator and denominator; it's got to be there somewhere.

    None whatever. In high school we mumble something about "infinite decimals" while frantically waving our hands; and the brighter students manage not to be permanently scarred for life.fishfry

    Yes, we totally agree. The purpose of my questions is that none of these (what I view) as quite intelligible questions you can start to ask once you "have" the real numbers can possibly be answered in the context of high school maths in a reasonable amount of time. Therefore, it is a mistake to simply posit the real numbers in high school and only serves to mystify mathematics rather than build clear understanding of how the next idea relates to the previous ideas.

    The teaching of mathematics in the US public schools is execrable. How many times do I have to agree with you about this?fishfry

    This is what the OP is about, so from my point of view every time there's agreement on this point I am very satisfied.

    The reason I'm not attacking as contradictory real numbers, Cantor's proofs, AC, in the other threads is because I know I won't succeed. I can only make a muddle of it here in the context of the lack of suitable axioms and understanding at the high school level, which as you've pointed out tool the smartest people hundreds of years to figure out how to prevent wild proliferation of contradictions.

    A university student in anything other than math: None.fishfry

    Yes, if anyone was having doubts about my recommendation that the real numbers in high school is bad pedagogy, take a long look at this statement.

    A well-schooled undergrad math major? Someone who took courses in real and complex analysis, number theory, abstract algebra, set theory, and topology? They could construct the real numbers starting from the axioms of ZF. They could then define continuity and limits and I could rigorously found calculus. It's not taught in any one course, it's just something you pick up after awhile. The axiom of infinity gives you the natural numbers as a model of the Peano axioms. From those you can build up the integers; then the rationals; and then the reals. Every math major sees this process once in their life but not twice. Nobody actually uses the formal definitions. It's just good to know that we could write them down if we had to.fishfry

    Yes, I agree, but my question is not how the reals are constructed. My exercise here starts with having the reals already.

    My question is how exactly does one prevent the reals from breaking previous proofs by contradiction.

    For, if we use proof by contradiction to establish the irrationals, then create the reals as existing between rationals, but then with the reals we have completed infinite decimal expansion and can go back and invalidate the proof by contradiction by just injecting suitable decimal expansions to solve the roots of the root 2 polynomial, then all the reals are now rational and there's no reals between rationals, and we now no longer have the reals, because they're all rational.

    We then re-check Cantors proof that the rationals cannot count the reals and simply conclude that if "counted high enough" we would eventually go through all the rationals with infinite decimal expansions taken from the reals as numerator and denominator.

    For me, constructing the reals isn't the tricky part, it's preventing the above things happening. Why it's way beyond high school math is that it's not at all intuitive what proofs by contradiction mean and mathematical induction means, when going from a finite to an infinite regime.

    Solvable ... but extremely tricky.

    Then once it's solved by preventing infinite decimal expansions from corresponding to natural numbers (that for every real number represented by infinite decimal expansion, there is not a natural number that simply lacks the decimal point, and that the reals are not "onto" the natural), then the next step is even more careful preventing the assertion that all the natural numbers placed after a decimal point do not correspond to a real number; we cannot simply take the completed set of natural numbers as a natural number, with even a single natural number corresponding to the decimals of a real number we can still carry out the scheme.

    Asking high school students to understand that decimal expansion does not represent a natural numbers lacking a decimal point, is far beyond a reasonable task. For, both natural numbers and real numbers seem very much at first viewing just a list of numbers that you can continue as long as you like; there is no way to intuit some difference beyond "as far as you can practically go in any universe somewhat similar to ours given any amount of time".

    Which we already agree on; I'm continuing the "high school devils advocate" simply to demonstrate, with your help, how far away from "simple, intuitive steps" resolving any of these problems are.

    Now, if I had a simple clear answer to these kinds of contradictions that took hundreds of years to build up frameworks to prevent, I'd say so. My purpose here is to check that no one else on PF has a simple answers either.
  • ssu
    1.9k
    Did this set theory experiment simply not work at all, or did it produce some small cadre of math geniuses?boethius
    Lol. Well, they took it back so I guess that the cadre was very small. And as Fishfry commented earlier, this experiment wasn't just limited to my country (Finland), but the US too. I'd suspect that we copied the 'new trends' during those progressive times from the US. From the viewpoint of teaching small children math, starting with counting sheep is the way to go. It is the natural way, I'd argue.

    In teaching maths, I think it is important to make a clear distinction between pure (abstract) maths and applied (to the real world) maths. It is the conflation of the two that causes problems.

    Of course children first learn maths with the conflated maths; counting sheep etc. But perhaps around the time they enter secondary school ( around age 13) the distinction needs to be emphasised.
    A Seagull
    I think this is a bigger philosophical problem for mathematics. Basically mathematics has evolved from the necessity of counting, calculating and computation. Hence 'applied math' came first. Abstract mathematical thought has emerged only later. This makes us start mathematics from the natural numbers.
  • Pfhorrest
    965
    Starting mathematics from the natural numbers is pretty natural. If you begin with nothing but the empty set and the sole sufficient operator of joint denial, the simplest new operator you can build is disjunction, and the simplest thing you can disjoin with the empty set is the set containing itself, and hey look that’s the first iteration of the successor function and if you keep doing that you end up with the natural numbers.
  • Mephist
    303
    However, it's also necessary to deal with questions like "why can't we just have rational numbers with infinite numerator and denominator; seems unfair to allow infinite decimal expansion but not infinite numerators?". Of course, we can have rational numbers with infinite numerator and denominator but it's then needed to explain how these aren't the "real" rational numbers we're talking about when we say the square root of two is irrational.boethius

    I think the main thing to understand here is that decimal numbers with infinite decimals can be considered as an extension of "regular" decimal numbers (finite list of digits), but infinite natural numbers (infinite list of digits) cannot be considered as an extension of "regular" natural numbers, since you cannot define on them arithmetic operations compatible with the ones defined on the "regular" natural numbers. Then, you can't build fractions with infinite integers because you cannot build infinite integers in the first place. In my opinion this is quite easy to understand. Did I miss something?

    I think the infinities and infinitesimals of mathematics are the things that make it become more "magic" and interesting. The problem with teaching in my opinion is more related to the fact that the "magic" of the fact that infinities and infinitesimals really work is not explained, or worse, explained by pretending to have a simple logical explanation that, however, is not part of the school program.
  • A Seagull
    172
    Starting mathematics from the natural numbers is pretty natural. If you begin with nothing but the empty set and the sole sufficient operator of joint denial, the simplest new operator you can build is disjunction, and the simplest thing you can disjoin with the empty set is the set containing itself, and hey look that’s the first iteration of the successor function and if you keep doing that you end up with the natural numbers.Pfhorrest

    It is a lot simpler just to start with the natural numbers as axioms. Introducing set theory just complicates things and achieves nothing.
  • boethius
    337
    I think the main thing to understand here is that decimal numbers with infinite decimals can be considered as an extension of "regular" decimal numbers (finite list of digits), but infinite natural numbers (infinite list of digits) cannot be considered as an extension of "regular" natural numbers, since you cannot define on them arithmetic operations compatible with the ones defined on the "regular" natural numbers. Then, you can't build fractions with infinite integers because you cannot build infinite integers in the first place. In my opinion this is quite easy to understand. Did I miss something?Mephist

    I'm not building with infinite integers, I'm building with the infinite decimal expansion representation of real numbers and simply pruning off the decimal symbol. Sure, if we simply define integer as "not this" then it's not building an infinite integer, but it is building something that I can then do things with if I'm not prevented from doing so.

    Now, clearly if the proof by contradiction of irrational numbers is constrained to using "regular" natural numbers or integers, I have no qualms. It checks out.

    However, if we switch regimes to one where we now have access to the infinite digit expansion of real numbers, we can revisit every proof in the previous regime with our new objects; and now, revisiting the root 2 proof is irrational I am able to solve it with these new objects and not arrive at a contradiction as oddness / eveness is no longer defined upon which the classic proof by contradiction depends. This is what I am doing.

    Am I prevented from doing this full stop? Am I unable to find a "suitable decimal expansion" to solve my problem? What exactly is preventing me from doing this, that is what I would consider a suitable answer in the context of learning maths. Given these infinite decimal expansion, I want to use them as what ways I see fit, unless I'm prevented by some axiom. Lot's of things may have been, and still are, true in the previous setup before I had these objects, but in the new setup where I can make use of these objects in equations, I want to take full advantage, and revisit every proof by contradiction as well as every mathematical induction proof.

    Broad features and themes involved in rigorous proofs elsewhere I do not consider a good answer for learning math. For me, "learning math is" understanding the proof oneself, not understanding that others elsewhere have understood something.

    Again, I am discussing high school students level of understanding and what's reasonable in terms of capabilities, time and relevance.

    So your answer doesn't explain why I cannot do my method.

    Moreover, your approach, would seem to me, to imply that the decimal expansion representation of a real number cannot be counted; is this your implication? or would you say the digits in a real number are countable?

    Also, how do you maintain infinite sequences can be completed, there are no infinite integers, the sequence of integers is infinite, simultaneously within the system suitable for high school level maths. Do we simply elect not to use our "complete the infinity tool" on the integers, and add this axiomatically? What axioms do they have to work with? Do they know enough set theory do make a model that avoids all these problems, or do they have another suitable basis?
  • boethius
    337
    I think the infinities and infinitesimals of mathematics are the things that make it become more "magic" and interesting. The problem with teaching in my opinion is more related to the fact that the "magic" of the fact that infinities and infinitesimals really work is not explained, or worse, explained by pretending to have a simple logical explanation that, however, is not part of the school program.Mephist

    Yes, I agree with you here.

    I'm not against touching on the infinity subject; there could be a whole class on it for students wanting to go into pure maths.

    I think we agree that it's bad pedagogy to simply posit the reals with no explanation and no time or ability to answer very expected and natural questions. Instead of curiosity leading to better understanding, it leads to confusion and a sense maths is "because we say so", which is the exact opposite sense students should be getting. Students would be better served by a less ambitious (not actually having irrationals and transcendentals as objects) but more rigorous calculus in the numerical regime, which would make a much more solid foundation for students going on to use applied maths, who can simply stay in this regime (as they will likely be solving every problem with the computer), and better serving pure maths students as well (that mathematics is rigorous, and extensions are made to do new things in a rigorous way).
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