• keystone
    323


    I'm currently feeling unwell and will reply shortly. Cheers, and thank you for continuing this dialogue.
  • fishfry
    3k
    I'm currently feeling unwell and will reply shortly. Cheers, and thank you for continuing this dialogue.keystone

    No worries, as they say. Get well soon.
  • keystone
    323
    No worries, as they say. Get well soon.fishfry

    Thanks! Mostly better now.

    You can see that if x is in (0,1), then x is in a least one (actually all but finitely many) of the sets (1/n, 1 - 1/n).fishfry

    Allow me to further clarify my position. I can write (0,1) as the union of arbitrarily many disjoint intervals. However, I cannot write (0,1) as the union of infinitely many disjoint intervals.

    I would have to give this some thought. Would it make progress if I stipulate to your metaphysics? I don't know what to say anymore.fishfry

    Do you think you understand my position so far (and perhaps don't agree with it) or do you have no clue what I'm proposing?

    Yes ok, so if you have an alternate way of getting to the same real numbers, what does it matter?fishfry

    I don't have an alternate way of getting to the real numbers. What I lay claim to is the real points, not the real numbers. Consider the ruler depicted below. It features 96 tick marks, to which we can assign 96 numbers. Yet, between each tick mark, there exists a bundle of points to which we can assign an interval. Each bundle has a length 1/16 in. When measuring an object that falls between adjacent tick marks, the best I can do with this ruler is to assign the corresponding interval to the object. My goal is to move beyond the Cartesian coordinate system by separating the concepts of numbers from points. I see no need to assert the existence of numbers.

    UtNXUW7.png

    Not a line, a nested collection of lines. The point zero is (-1, 1), (-1/2, 1/2), (-1/3, 1/3), etc.fishfry

    The fact that the length of each line in your sequence is getting shorter is a red herring. Every single line in your sequence is composed of exactly points. The point count isn't converging to 1. What you've exhibited is not actually a nested collection of lines but an algorithm for generating such a collection (or at least the essence of an algorithm). This distinction is crucial because the algorithm, if executed, doesn't halt. If you chose to execute the algorithm, the best you can do is wait for a long time and interrupt it when the last line produced is sufficiently small. In other words, the output of the algorithm is an arbitrarily small line, not a point.

    Every interval containing a given real number, necessarily contains other real numbers. That's the definition of (not) being isolated.fishfry

    Do you believe individual rational numbers can be isolated? I believe they can. I'm going to use the SB tree to illustrate my view, not because it's essential but because it's familiar. I can cut this tree such that left of the cut is (0,1/2) and right of the cut is (1/2,inf). With this cut, I've isolated 1/2. I cannot do the same for irrational numbers.

    CBMVrXJ.png
  • keystone
    323
    The inability for dimensionless points to be reconciled with the continuum is what motivated Whitehead's point-free geometry, a precursor to the field of Pointless Topology, as for instance formalised using Locales whose distributive law characterizes the meaning of a "spot". (It might be useful to test this law in relation to the SB tree, for both the truncated and infinite version).sime

    Hello Sime, thank you for your comment. It prompted me to explore point-free geometry a bit online. I found this paper that adopts intervals instead of points in its framework, which is quite relevant. I appreciate Whitehead and others' rationale behind their approach, but I must admit, as someone not deeply versed in pure mathematics, I find their concepts a bit challenging to grasp just by skimming. It seems like a thorough reading might be required to truly understand these ideas, something I'm not quite ready to dive into, especially in terms of applying it to something like the SB tree anytime soon.
  • fishfry
    3k
    Thanks! Mostly better now.keystone

    Not bird flu I hope. Jeez the medical propaganda is everywhere these days. Are we all doomed? Like not eventually, but as soon as next week?

    Allow me to further clarify my position. I can write (0,1) as the union of arbitrarily many disjoint intervals. However, I cannot write (0,1) as the union of infinitely many disjoint intervals.keystone

    Correct, but why does that matter? (0,1) is already the disjoint union of open intervals, namely itself.


    Do you think you understand my position so far (and perhaps don't agree with it) or do you have no clue what I'm proposing?keystone

    I literally have no idea what we've been talking about the past several weeks. Which makes me feel foolish sniping at it.


    I don't have an alternate way of getting to the real numbers. What I lay claim to is the real points, not the real numbers.keystone

    But points and numbers are entirely synonymous in this context. The "real line" is just the set of mathematical real numbers.



    Consider the ruler depicted below.keystone

    Your diagrams make my eyes glaze. I appreciate that they're meaningful to you. They are not helpful in terms of communicating with me.


    It features 96 tick marks
    ...
    keystone

    Eyes glazed. I'm sorry. I feel terrible because you put so much work into these diagrams.

    You actually lost me when you distinguished between the real numbers and points; since as far as I know, they are the same thing.



    The fact that the length of each line in your sequence is getting shorter is a red herring. Every single line in your sequence is composed of exactly 2ℵ0
    2

    0
    points. The point count isn't converging to 1. What you've exhibited is not actually a nested collection of lines but an algorithm for generating such a collection (or at least the essence of an algorithm). This distinction is crucial because the algorithm, if executed, doesn't halt. If you chose to execute the algorithm, the best you can do is wait for a long time and interrupt it when the last line produced is sufficiently small. In other words, the output of the algorithm is an arbitrarily small line, not a point.
    keystone

    Actually algorithms are not relevant, because I need not have a finite description for the set of open intervals that descend to a point. It's not a computational process.



    Do you believe individual rational numbers can be isolated?keystone

    Perhaps you are using a different definition of the word. Mathematically, no rational number is isolated, because every interval containing a given rational necessarily contains others.

    But if you have some other definition, you might convince yourself otherwise.

    I believe they can. I'm going to use the SB tree to illustrate my view,keystone

    Eyes glazing like a ham on Easter.

    not because it's essential but because it's familiar.keystone

    This example is useful to you but it is not helpful to me.

    I can cut this tree such that left of the cut is (0,1/2) and right of the cut is (1/2,inf). With this cut, I've isolated 1/2. I cannot do the same for irrational numbers.keystone

    If that's what you mean, then so be it.
  • jgill
    3.6k
    I found this paper that adopts intervals instead of points in its framework, which is quite relevant.keystone

    It also demonstrates the difficulty of trying to do something original and noteworthy in mathematics. It's a very complicated game requiring perseverance and dedication. Are you in it for the long haul?
  • keystone
    323
    It's a very complicated game requiring perseverance and dedication. Are you in it for the long haul?jgill

    Why do you want to know?
  • keystone
    323
    Not bird flu I hope. Jeez the medical propaganda is everywhere these days. Are we all doomed? Like not eventually, but as soon as next week?fishfry

    As a sickly child, when I felt ill, I would imagine myself as heroically fighting severe illnesses, attributing my survival to extraordinary strength. Turns out, I'm just wimp. I was probably just dealing with a common cold last week. Fortunately I wasn't in tune with any of the bird flu news...anxiety doesn't usually help...

    Correct, but why does that matter? (0,1) is already the disjoint union of open intervals, namely itself.fishfry

    The issue revolves around whether the part or the whole is primary.

    If parts precede the whole, then logically, I should be able to union such parts to create the whole, which you acknowledge is not feasible.

    Conversely, if the whole precedes the parts, then I should be capable of bisecting the whole into smaller sections, continuing to do so until I have arbitrarily small parts. This approach is feasible.

    But points and numbers are entirely synonymous in this context. The "real line" is just the set of mathematical real numbers.fishfry

    Yes, individual points are entirely synonmymous with numbers. However, continuous bundles of points are synonymous with intervals. And what I'm saying is that it's these continuous bundles of points (described using intervals) that are fundamental, not the individual points (described using numbers). We start with a continuous bundle of points (described using an interval) and when we cut it (ie. bisect this interval), we not only create smaller continuous bundles of points but also isolate an individual point in between (described using a number). Hence, the individual points and their associated numbers emerge from the bisecting process; they do not exist as independent objects before it. Individual numbers and points are emergent.

    I literally have no idea what we've been talking about the past several weeks. Which makes me feel foolish sniping at it.fishfry

    When I presented that table and you wrote 'I would have to give this some thought' but didn't follow up on it, is it that you don't want to consider an alternate view? The common theme throughout all of my posts (including this one) and summarized in that table is that I believe we must start with the whole and manipulate it to produce the parts. Building (or defining) the whole from the parts is hopeless. Do you understand what I mean by this?
  • fishfry
    3k
    As a sickly child, when I felt ill, I would imagine myself as heroically fighting severe illnesses, attributing my survival to extraordinary strength. Turns out, I'm just wimp. I was probably just dealing with a common cold last week. Fortunately I wasn't in tune with any of the bird flu news...anxiety doesn't usually help...keystone

    That's why the Amish communities weren't hit hard by covid. They don't watch tv :-)

    Correct, but why does that matter? (0,1) is already the disjoint union of open intervals, namely itself.
    — fishfry

    The issue revolves around whether the part or the whole is primary.
    keystone

    How is that issue resolved by the question of whether you can partition the open unit interval into countably infinite many pairwise disjoint open intervals? You can't, but it certainly is the pairwise disjoint union of ONE interval, and so what?

    If parts precede the whole, then logically, I should be able to union such parts to create the whole, which you acknowledge is not feasible.keystone

    I already showed you a non-disjoint countably infinite union of open intervals that equals the open unit interval. And the interval is the disjoint union of its uncountably many points. How many decompositions do you want? Why do you insist on the one decomposition that we can't do?

    Conversely, if the whole precedes the parts, then I should be capable of bisecting the whole into smaller sections, continuing to do so until I have arbitrarily small parts. This approach is feasible.keystone

    You surely can't do that with countably many cuts.

    Yes, individual points are entirely synonmymous with numbers. However, continuous bundles of points are synonymous with intervals. And what I'm saying is that it's these continuous bundles of points (described using intervals) that are fundamental, not the individual points (described using numbers). We start with a continuous bundle of points (described using an interval) and when we cut it (ie. bisect this interval), we not only create smaller continuous bundles of points but also isolate an individual point in between (described using a number). Hence, the individual points and their associated numbers emerge from the bisecting process; they do not exist as independent objects before it. Individual numbers and points are emergent.keystone

    I asked you earlier: Suppose that rather than snipe line by line at this paragraph, I just accept it for sake of discussion. Can we move forward? I just don't see how any of this matters. And your bisection idea doesn't work, you can't get any irrationals that way. But I believe you've already agreed with that.

    When I presented that table and you wrote 'I would have to give this some thought' but didn't follow up on it, is it that you don't want to consider an alternate view?keystone

    I found


    The common theme throughout all of my posts (including this one) and summarized in that table is that I believe we must start with the whole and manipulate it to produce the parts. Building (or defining) the whole from the parts is hopeless. Do you understand what I mean by this?[/quote]

    I found the fitness gym analogy confusing and pointless. But of course your whole approach is pointless (that's a pun) so maybe I'm getting it.

    When it comes to the real numbers, I do think building the parts from the whole is difficult, because you'd need uncountably many cuts. But Dedekind has already built the reals from cuts of rationals, so it can be done. But there are uncountably many cuts.
  • jgill
    3.6k
    It's a very complicated game requiring perseverance and dedication. Are you in it for the long haul? — jgill

    Why do you want to know?
    keystone

    I find their concepts a bit challenging to grasp just by skimming. It seems like a thorough reading might be required to truly understand these ideas, something I'm not quite ready to dive intokeystone

    What you are doing seems to me to be more metaphysics than mathematics. And that's OK. But without studying what is accepted mathematics you have a rough road ahead if you wish to contribute to that discipline. However, there have been amateurs within the last half century who have made significant discoveries. Marjorie Rice. Thankfully, is there to help guide you.
  • keystone
    323
    That's why the Amish communities weren't hit hardfishfry

    LoL.

    Why do you insist on the one decomposition that we can't do?fishfry

    I want to make it clear that a line cannot be constructed from/defined by infinite isolated points (numbers) or micro-lines (intervals). If that's clear then what is a line -->

    ...it certainly is the pairwise disjoint union of ONE interval, and so what?fishfry

    Yes! Forget about declaring that the line is infinite individual things and instead call it ONE thing, ONE bundle, described by ONE interval. This is an important distinction because it frees us from actual infinity allowing for a stronger foundation. We don't need individual numbers to describe a line because we have ONE interval to describe the entire bundle.

    Conversely, if the whole precedes the parts, then I should be capable of bisecting the whole into smaller sections, continuing to do so until I have arbitrarily small parts. This approach is feasible.
    — keystone

    You surely can't do that with countably many cuts.
    fishfry

    I can't do what? How small do you want the bundles to be? I assure you, I can divide them as small as you wish. Of course, I can never cut a line down to indivisible bundles, but I never claimed I could. Why would we even need that?

    And your bisection idea doesn't work, you can't get any irrationals that way. But I believe you've already agreed with that.fishfry

    You're right that I can't execute a cut to isolate an irrational point. However, what I can do is develop an algorithm that defines an endless cutting of the line such that the line segment containing the desired irrational point gets arbitrarily small. As we've agreed, that algorithm is the irrational. There's no need to declare that the algorithm can be run to completion to output an irrational number. The algorithm is sufficient in and of itself. And if I need a number, I can interrupt the algorithm to deliver an arbitrarily narrow interval with rational end-points and I can pick a suitably close rational number within that interval.

    Now, I can't isolate a non-computable this way, but that's not a problem. The non-computable points are not missing from my view. They are included - my line is continuous. The non-computable points just cannot be isolated. But we don't need to isolate them. They fulfill their job being constrained to bundles. Do they not?

    When it comes to the real numbers, I do think building the parts from the whole is difficult, because you'd need uncountably many cuts.fishfry

    You're right that IF I needed to completely cut my line to isolate all points/numbers then it would require uncountably many cuts. My argument is that we don't need completeness. Let's embrace our inability to fully execute a non-halting program. Our inability to isolate everything is a feature of my view, not a flaw. After all, why do you need isolated numbers?

    But Dedekind has already built the reals from cuts of rationals, so it can be done. But there are uncountably many cuts.fishfry

    Let's lay out all countably infinite rationals in an ordered line. How many gaps are there - countably many? What is the difference between a gap and a Dedekind cut? If they are the same, how do we arrive at uncountably many cuts? The answer is that Dedekind doesn't ever execute the cut. Dedekind Cuts only make sense if they correspond to non-halting algorithms which by definition cannot be executed completely.

    I asked you earlier: Suppose that rather than snipe line by line at this paragraph, I just accept it for sake of discussion. Can we move forward?fishfry

    I had asked whether you understood what I was saying and you said you literally have no idea. It's hard to move forward if nothing I'm saying is coming through.

    But of course your whole approach is pointless (that's a pun) so maybe I'm getting it.fishfry

    Ha. My view has points, they're just not fundamental. Points emerge when a cut is made, but the line doesn't come precut and nobody could ever completely cut a line.
  • keystone
    323
    Thankfully, ↪fishfry is there to help guide you.jgill

    Without seeing where I'm going with this, amidst all of my non-technical dialogue, it is admirable that he has stuck around for so long. Suffice it to say that I'm VERY appreciative of fishfry.
  • fishfry
    3k
    I want to make it clear that a line cannot be constructed from/defined by infinite isolated points (numbers) or micro-lines (intervals). If that's clear then what is a line -->keystone

    The real line is the set of real numbers. That's the modern view. Or as Euclid said, "A line is breadthless length."

    Yes! Forget about declaring that the line is infinite individual things and instead call it ONE thing, ONE bundle, described by ONE interval. This is an important distinction because it frees us from actual infinity allowing for a stronger foundation.keystone

    Ok. One thing.


    I can't do what? How small do you want the bundles to be? I assure you, I can divide them as small as you wish. Of course, I can never cut a line down to indivisible bundles, but I never claimed I could. Why would we even need that?keystone

    I don't know. Why is any of this important? You claim you can get to every real number via cuts, but only finitely or countably many of them. And that's not true. And the reason we need it is so we can have the real numbers.

    You're right that I can't execute a cut to isolate an irrational point. However, what I can do is develop an algorithm that defines an endless cutting of the line such that the line segment containing the desired irrational point gets arbitrarily small.keystone

    You could not do that, for the reason that there are only countably many algorithms, and uncountably many irrationals. You haven't got enough algorithms.

    As we've agreed, that algorithm is the irrational. There's no need to declare that the algorithm can be run to completion to output an irrational number.keystone

    There aren't enough algorithms to define all the reals. Most reals are noncomputable.

    The algorithm is sufficient in and of itself. And if I need a number, I can interrupt the algorithm to deliver an arbitrarily narrow interval with rational end-points and I can pick a suitably close rational number within that interval.keystone

    Actually that is untrue. There aren't enough algorithms.

    Now, I can't isolate a non-computable this way, but that's not a problem. The non-computable points are not missing from my view. They are included - my line is continuous. The non-computable points just cannot be isolated. But we don't need to isolate them. They fulfill their job being constrained to bundles. Do they not?keystone

    Well, if you want to see it that way, I suppose so. But without the noncomputables, the real line is not continuous. So they are there but they're not there?

    I ask again: Suppose instead of endlessly sniping at your ideas, I just agree. What then? What is the point of all this?


    My argument is that we don't need completeness.keystone

    Well then the intermediate value theorem is false. Calculus would collapse.

    Let's embrace our inability to fully execute a non-halting program. Our inability to isolate everything is a feature of my view, not a flaw. After all, why do you need 2ℵ0
    2

    0
    isolated numbers?
    keystone

    They'e not isolated, I've explained that several times. But we need them for completeness! The completeness of the real numbers is the defining property of the real numbers. Without completeness they're not the real numbers.


    Let's lay out all countably infinite rationals in an ordered line. How many gaps are there - countably many?keystone

    Uncountably many. Each gap represents an irrational.

    What is the difference between a gap and a Dedekind cut?keystone

    A Dedekind cut is a pair of sets of rationals. A gap is a non-rational point. They amount to the same thing, expressed differently.

    If they are the same, how do we arrive at uncountably many cuts?keystone

    Because there are uncountably many reals and only countably many rationals.

    The answer is that Dedekind doesn't ever execute the cut. Dedekind Cuts only make sense if they correspond to non-halting algorithms which by definition cannot be executed completely.keystone

    That's just nonsense. It's wrong. But again, what is the end game of all this? Suppose I stop objecting and ask you, what is the point? Actually I have asked you several times recently.

    I had asked whether you understood what I was saying and you said you literally have no idea. It's hard to move forward if nothing I'm saying is coming through.keystone

    Well you've made progress. You did say something I understand, namely that Dedekind cuts must correspond to non-halting algorithms. The problem is that it's utterly false. It's just a massive misunderstanding on your part.

    Ha. My view has points, they're just not fundamental. Points emerge when a cut is made, but the line doesn't come precut and nobody could ever completely cut a line.keystone

    Ok. Fine. Now what?
  • fishfry
    3k
    Thankfully, ↪fishfry is there to help guide you.jgill

    No good deed goes unpunished :-)
  • keystone
    323
    I'm really tempted to respond to all your latest comments, but you're getting impatient, so I'll hold back and move forward.

    Well then the intermediate value theorem is false. Calculus would collapse.fishfry

    In later posts, I aim to demonstrate that calculus not only remains intact with my perspective, but is actually built on firmer foundations. However, before we advance I'm going summarize the essentials so far. If you understand what I'm saying (even if you don't agree) we'll be ready to proceed.

    1) Initial Composition: My line consists of the same points and numbers as the real line. However, initially, the continuous points bundle together to form a line, and the continuous numbers bundle together to form an interval. Thus, we begin with a single object (a line) described by a single interval.

    2) Isolation Through Cuts: A point/number can only be isolated from the line through a cut. Until the cut is executed, it is meaningless to refer to the point/number as an independent entity.

    3) Rational Cuts: A rational cut corresponds to isolating a rational by bisecting the line.

    4) Irrational Computable Cuts: An irrational computable cut corresponds to a non-halting algorithm that isolates an irrational computable within an arbitrarily small interval. This cut cannot be executed completely.

    5) Irrational Non-Computable Cuts: These cuts don't exist. Irrational non-computables cannot be isolated.

    6) Completeness: All the points are there from the start (bundled in the line) so in a sense the line is complete. However, it is impossible to fully cut the line such that all points/numbers are isolated so in a sense the isolated points/numbers are incomplete.

    PLEASE try to understand the following example (including the figures!). This is essential for me to make any progress explaining why calculus doesn't collapse with my view. Notice that in these 1D examples the figures contain the same information as the unions. It contains no additional information, but when we move to 2D, the figures become much more significant.

    1) I start with a line (-inf,+inf)

    ArLyaOw.png

    2) I execute a rational cut at 0 such that it's now:
    (-inf,0) U 0 U (0,+inf)

    qHWulX9.png

    3) I then plan an irrational computable cut corresponding to π such that it's now:
    (-inf,0) U 0 U (0,π-ε1) U π-ε1 U (π-ε1,π+ε2) U π+ε2 U (π+ε2,+inf)
    where:
    π is the familiar irrational number and ε1 and ε2 are arbitrarily small positive numbers. Their independent values are not important as they are never used in isolation. What's important is that π-ε1 and π+ε2 are rational numbers and π lies within the arbitrarily narrow interval (π-ε1,π+ε2).

    aDkMZLk.png

    Do you follow what I'm saying?
  • fishfry
    3k
    I'm really tempted to respond to all your latest comments, but you're getting impatient, so I'll hold back and move forward.keystone

    This isn't going anywhere. And as I've mentioned, the real numbers are categorical[/math]. That means that up to isomorphism, there is only one model of the real numbers. If you have a construction of the real numbers, it produces the same real numbers as anyone else's construction.

    So I will stipulate that you have a construction of the real numbers (though I don't think you do). But then, so what? I keep asking you that. You have an alternate construction of the real numbers. But whatever you have constructed is exactly the same thing as the regular old real numbers.

    Just like you can use Dedekind cuts or Cauchy sequences. You can have a preference for one or the other, but it makes no difference.


    In later posts, I aim to demonstrate that calculus not only remains intact with my perspective, but is actually built on firmer foundations. However, before we advance I'm going summarize the essentials so far. If you understand what I'm saying (even if you don't agree) we'll be ready to proceed.keystone

    Well if you have the intermediate value theorem and the least upper bound property -- ie, completeness -- then what you have, whatever it is, is isomorphic to the standard real numbers. Perhaps you have better philosophy, or a story you find more satisfying. But logicians have shown that the real numbers are categorical. There aren't any alternate models.

    1) Initial Composition: My line consists of the same points and numbers as the real line. However, initially, the continuous points bundle together to form a line, and the continuous numbers bundle together to form an interval. Thus, we begin with a single object (a line) described by a single interval.

    2) Isolation Through Cuts: A point/number can only be isolated from the line through a cut. Until the cut is executed, it is meaningless to refer to the point/number as an independent entity.

    3) Rational Cuts: A rational cut corresponds to isolating a rational by bisecting the line.

    4) Irrational Computable Cuts: An irrational computable cut corresponds to a non-halting algorithm that isolates an irrational computable within an arbitrarily small interval. This cut cannot be executed completely.

    5) Irrational Non-Computable Cuts: These cuts don't exist. Irrational non-computables cannot be isolated.

    6) Completeness: All the points are there from the start (bundled in the line) so in a sense the line is complete. However, it is impossible to fully cut the line such that all points/numbers are isolated so in a sense the isolated points/numbers are incomplete.[/qouote]

    Without disputing point by point (for example, you're wrong about noncomputables, because if you don't have the noncomputables, you can't have completeness), it doesn't matter. If you have the real numbers, they're the same real numbers.

    And your claim of completeness but no noncomputables is inconsistent. One of those has to go.
    keystone
    PLEASE try to understand the following example (including the figures!). This is essential for me to make any progress explaining why calculus doesn't collapse with my view.keystone

    No, I don't need to. If calculus works, then you have the standard real numbers.

    Perhaps you are building the constructive or intuitionistic real line. Did I already mention those earlier? If so, they have a theory of computable completeness that lets them finesse the issue. Perhaps you mean that.


    Notice that in these 1D examples the figures contain the same information as the unions. It contains no additional information, but when we move to 2D, the figures become much more significant.keystone

    Skipping the next bits ...

    π is the familiar irrational number and ε1 and ε2 are arbitrarily small positive numbers.keystone

    No such thing as an arbitrarily small positive real number. Not in the standard reals, not in the constructive reals, and not in the hypereals. That's because if epsilon is a positive real number, then

    0 < epsilon/2 < epsilon.

    It's essential that you understand that. The real numbers are a field. You can ALWAYS divide by 2.

    Their independent values are not important as they are never used in isolation. What's important is that π-ε1 and π+ε2 are rational numbers and π lies within the arbitrarily narrow interval (π-ε1,π+ε2).keystone

    There is no such thing as an "arbitrarily narrow interval." I believe I've identified the exact flaw in your thinking.


    Do you follow what I'm saying?keystone

    I think I've refuted it. Two points to sum up:

    1) The real numbers are categorical. Any two models are isomorphic; and

    2) There is no such thing as an arbitrarily small positive real. That's because the real numbers are a field, in which you can always divide by 2.
  • keystone
    323
    So I will stipulate that you have a construction of the real numbers.fishfry

    I don't have an alternate construction of the complete set of isolated real numbers.

    Well if you have the intermediate value theorem and the least upper bound property -- ie, completeness -- then what you have, whatever it is, is isomorphic to the standard real numbers.fishfry

    I don't have the intermediate value theorem or the least upper bound property.

    No, I don't need to. If calculus works, then you have the standard real numbers.fishfry

    I acknowledge that for the bottom-up view, calculus requires the complete set of isolated real numbers, the intermediate value theorem, and the least upper bound property to "work"...I use quotes because it also requires some mental gymnastics. However, that's just not the case for the top-down view. It works perfectly in absence of all of the above...including the mental gymnastics.

    No such thing as an arbitrarily small positive real number.fishfry

    Consider the following Python function:

    def small_number_generator():
        n = 1
        while True:
            print(n)
            n /= 2
    

    If executed, this function will print 1, 1/2, 1/4, 1/8, 1/16, and so on to no end. By saying that there is no smallest positive number you are essentially acknowledging that this program does not halt. I agree with that. What I'm saying is that for any positive number you provide, x, I can run the code in a finite amount of time to print out a number smaller than x. In other words, it has the potential to print out a number as small as you want but it cannot actually print out the smallest positive number, any more than it can halt. Do you understand this distinction?

    Moreover, without ever executing the program I can describe the function's potential. Assuming it will run for at least a little while, at any time the last number it will have actually printed, ε, necessarily cuts the line (0,1) as depicted below:

    SpMizCV.png

    Again, until we execute the function ε doesn't hold an actual value. In this illustration, ε is simply a placeholder. The fact that I drew it approximately 1/3 between 0 and 1 is inconsequential. All that can be said is that if executed, ε will correspond to a point somewhere between 0 and 1. That's how you should interpret the drawing.

    In this light, I ask that you revisit the example from my last post and see if you understand step (3) where I plan an irrational computable cut at π. I specifically wrote plan there instead of execute because I wanted to focus on the potential of the cut, as I have done for the program illustrated above.

    But then, so what? I keep asking you that.fishfry

    I keep trying to advance forward but your responses continue to either directly or indirectly show that you're not following. If you don't understand what I'm illustrating when I plan an irrational computable cut at π then you won't understand my 2D illustrations that demonstrate that the IVT and the LUB property are not required for the top-down view.
  • fishfry
    3k
    I don't have an alternate construction of the complete set of isolated real numbers.keystone

    Then what on earth are you doing?


    I don't have the intermediate value theorem or the least upper bound property.keystone

    You claimed completeness. Do you now retract that? Or have a private definition?

    I acknowledge that for the bottom-up view, calculus requires the complete set of isolated real numbers, the intermediate value theorem, and the least upper bound property to "work"...I use quotes because it also requires some mental gymnastics. However, that's just not the case for the top-down view. It works perfectly in absence of all of the above...including the mental gymnastics.keystone

    How does it work? What does it do?

    No such thing as an arbitrarily small positive real number.
    — fishfry

    Consider the following Python function:

    def small_number_generator():
    n = 1
    while True:
    print(n)
    n /= 2

    If executed, this function will print 1, 1/2, 1/4, 1/8, 1/16, and so on to no end. By saying that there is no smallest positive number you are essentially acknowledging that this program does not halt.
    keystone

    If executed, such a program will eventually output the same number over and over, until its computing resources run out. You are factually wrong and I hope you can see why. Even so, there is no smallest positive real number, and you have not provided an argument.

    I agree with that. What I'm saying is that for any positive number you provide, x, I can run the code in a finite amount of time to print out a number smaller than x. In other words, it has the potential to print out a number as small as you want but it cannot actually print out the smallest positive number, any more than it can halt. Do you understand this distinction?keystone

    What? There is no smallest positive real number.

    Moreover, without ever executing the program I can describe the function's potential. Assuming it will run for at least a little while, at any time the last number it will have actually printed, ε, necessarily cuts the line (0,1) as depicted below:keystone

    Not even wrong. You are talking nonsense.


    Again, until we execute the function ε doesn't hold an actual value. In this illustration, ε is simply a placeholder. The fact that I drew it approximately 1/3 between 0 and 1 is inconsequential. All that can be said is that if executed, ε will correspond to a point somewhere between 0 and 1. That's how you should interpret the drawing.keystone

    There is no smallest positive real number. I see that you are confused about this.

    In this light, I ask that you revisit the example from my last post and see if you understand step (3) where I plan an irrational computable cut at π. I specifically wrote plan there instead of execute because I wanted to focus on the potential of the cut, as I have done for the program illustrated above.keystone

    There is no smallest positive real number. You have convinced me that you are simply confused about this point. The real numbers are a field. You can always divide by 2.


    I keep trying to advance forward but your responses continue to either directly or indirectly show that you're not following. If you don't understand what I'm illustrating when I plan an irrational computable cut at π then you won't understand my 2D illustrations that demonstrate that the IVT and the LUB property are not required for the top-down view.
    keystone

    Your most recent exposition postulated a smallest positive real number. There is no such thing. There is nothing for me to follow.

    But even so. I have repeatedly asked you to give me the big picture. Give me something.
  • jgill
    3.6k
    I acknowledge that for the bottom-up view, calculus requires the complete set of isolated real numbers, the intermediate value theorem, and the least upper bound property to "work"...I use quotes because it also requires some mental gymnastics. However, that's just not the case for the top-down view. It works perfectly in absence of all of the above...including the mental gymnasticskeystone

    What is an isolated real number?

    Show us elementary calculus from the top down. I am curious.
  • keystone
    323
    You claimed completeness. Do you now retract that? Or have a private definition?fishfry

    As I said earlier, I've got the points bundled into a continuous line, but not all of the points can be isolated. So if by 'completeness' you mean a line without gaps then my line is complete. However, if by 'completeness' you mean a line that can be described as the disjoint union of infinite points/numbers then my line is incomplete.

    If executed, such a program will eventually output the same number over and over, until its computing resources run out. You are factually wrong and I hope you can see why.fishfry

    What you are essentially saying is that a turing machine cannot operate on an infinite memory tape since such a tape cannot exist in a finite world. Ok, you're right.

    Even so, there is no smallest positive real number, and you have not provided an argument.fishfry

    I largely agree but I would phrase it as 'there is no smallest possible positive number'. This distinction is important if numbers are emergent but it's not worth discussing at this time.

    Let me rephrase my argument to address these points you've made.

    • 1) Assume that any computer running the program will eventually overflow and stop printing numbers.
    • 2) If we limit our scope to computers that print at least two numbers, for any given computer there will always be a last number printed (let's assign this property of a given computer the label e).
    • 3) For a computer with few resources, e might equal .
    • 4) For a computer with more resources, e might equal .
    • 5) In our observable universe, there is a smallest possible value for e.
    • 6) However, it is possible that in an existence outside of our observable universe e can be even smaller.
    • 7) Instead of placing a finite upper bound on the size of a computer, it is convenient to assume that a computer can be arbitrarily large but must necessarily be finite. To be clear, I'm not claiming that there is an infinite computer. I'm simply remaining agnostic on the bounds of finite computers.
    • 8) Thus, e can be arbitrarily small but must necessarily be a positive number. To be clear, I'm not claiming that there is a smallest positive number. I'm simply remaining agnostic on the bounds of finite computers.
    • 9) e is not a property of the program, but instead a property of the computer executing the program.
    • 10) To describe the program, we must talk in general terms. Let ε be a possible value for e.
    • 11) ε necessarily lies on the open interval (0,1).
    • 12) I illustrate this description of the program as follows:
    SpMizCV.png


    Is that more clear now?

    But even so. I have repeatedly asked you to give me the big picture. Give me something.fishfry

    As always, I'm grateful for this discussion and I'm certainly not complaining, but I hope you see that I have to walk a very thin line with you. I can't talk too high level as you will ask for the beef, I can't show figures as they will make your eyes glaze over, I can't use analogies because my analogies don't stick, and when I try to talk technical you often skip over or misunderstand my ideas. Of course, it doesn't help that I'm not a trained mathematician. Again, I'm extremely grateful for this discussion, just trying to put things in perspective.

    Here's the next 4 steps of my plan:

    • 1) Get you to agree to my use of ε in the computer example (including understand the illustration).
    • 2) Get you to agree to my use of ε in the pi example (including understand the illustration).
    • 3) Progress to 2D, where the Cartesian Coordinate system is replaced with a top-down alternative, and the zeros of y=x^2-2 have a very different meaning. Illustrations become important here which is why we need to get past 1) and 2) first.
    • 4) Top-down interpretation of calculus.

    Hopefully this plan will at least give you confidence that I'm heading somewhere with all of this...
  • keystone
    323
    What is an isolated real number?jgill

    With the top-down view we start with a continuous bundle of real numbers forming a line. In other words, we have one object (an interval), not objects (isolated real numbers). We can isolate numbers as an afterthought by cutting the line (something like a Dedekind Cut). For example, I can isolate 1 by cutting (0,2) to produce (0,1) U 1 U (1,2). Rational numbers can be isolated in this fashion. One can develop an algorithm to isolate irrational computable numbers (akin to a Cauchy sequence of intervals) but the cut cannot be executed. Non-computable numbers cannot be isolated in any sense. In other words, they will forever be stuck interior to an interval.

    Show us elementary calculus from the top down. I am curious.jgill

    I hope to show you...just need to get to step 4 of the plan I outlined to fryfish.
  • fishfry
    3k
    As I said earlier, I've got the points bundled into a continuous line, but not all of the points can be isolated. So if by 'completeness' you mean a line without gaps then my line is complete. However, if by 'completeness' you mean a line that can be described as the disjoint union of infinite points/numbers then my line is incomplete.keystone

    If there are no gaps, it's complete. That's the informal definition. The official definition is that every Cauchy sequence converges. You haven't defined Cauchy sequences so I don't know.

    What you are essentially saying is that a turing machine cannot operate on an infinite memory tape since such a tape cannot exist in a finite world. Ok, you're right.keystone

    Right, all physical computers have bounded resources.

    I largely agree but I would phrase it as 'there is no smallest possible positive number'. This distinction is important if numbers are emergent but it's not worth discussing at this time.keystone

    Hmmm, murky.

    Let me rephrase my argument to address these points you've made.
    ...
    Is that more clear now?
    keystone

    I believe you are doing computer math. Well known. I believe you are describing IEEE-754 floating point numbers. There is a smallest and largest possible value. It's actually not even hardware dependent, it's in the standard.

    https://en.wikipedia.org/wiki/Fixed-point_arithmetic

    Or you might be describing fixed point numbers.

    https://en.wikipedia.org/wiki/Fixed-point_arithmetic

    Either way you're doing computer arithmetic, where there is a minimum smallest interval and even a largest possible number.

    Way way short of being the real numbers. So you can do approximate calculus or discrete calculus that way if you like. I think this is what you are getting at.

    Am I close?

    As always, I'm grateful for this discussion and I'm certainly not complaining, but I hope you see that I have to walk a very thin line with you.keystone

    It's ok, I have my second wind I think. Especially now that I know you're just doing computer arithmetic, fixed or floating point.


    I can't talk too high level as you will ask for the beef, I can't show figures as they will make your eyes glaze over, I can't use analogies because my analogies don't stick, and when I try to talk technical you often skip over or misunderstand my ideas.keystone

    Yes! Now we're communicating :-)

    Of course, it doesn't help that I'm not a trained mathematician. Again, I'm extremely grateful for this discussion, just trying to put things in perspective.keystone

    If you're doing computer arithmetic or some variant, we're on the same page.

    1) Get you to agree to my use of ε in the computer example (including understand the illustration).
    2) Get you to agree to my use of ε in the pi example (including understand the illustration).
    3) Progress to 2D, where the Cartesian Coordinate system is replaced with a top-down alternative, and the zeros of y=x^2-2 have a very different meaning. Illustrations become important here which is why we need to get past 1) and 2) first.
    4) Top-down interpretation of calculus.
    keystone

    I'll stipulate that you can do discrete calculus on a computer.

    https://en.wikipedia.org/wiki/Discrete_calculus

    Hopefully this plan will at least give you confidence that I'm heading somewhere with all of this...keystone

    Well if we're doing computer arithmetic and some variant of discrete calculus, that's interesting to know. What do you think?
  • jgill
    3.6k
    It's ok, I have my second wind I think. Especially now that I know you're just doing computer arithmetic, fixed or floating point.fishfry

    A point of clarity. Thanks. Calculus started with discrete, then moved to infinitesimal, then with technology back to discrete in some sense.
  • keystone
    323
    Well if we're doing computer arithmetic and some variant of discrete calculus, that's interesting to know. What do you think?fishfry

    Discrete calculus is certainly important to my view but it's not what I'm talking about.

    There are two steps to a cut:
    Step 1: Planning the cut with an algorithm
    Step 2: Executing the plan by completing the algorithm

    EXAMPLE: Let's cut (0,4) by π.

    Step 1: My plan is illustrated as follows.
    TQOT11q.png

    • π is the familiar irrational number
    • ε1 and ε2 are placeholders for positive numbers which can be as small as your computer allows
    • (π-ε1,π+ε2) describes a line with rational upper and lower bounds. This line can be as small as your computer allows. Ultimately, I want to call this line pi. To distinguish it from the point/number, π, I'll call this line π.

    Step 2: It is impossible to execute the plan to infinite precision. This is where discrete calculus comes into play. On one computer, the execution of the plan might look as follows.

    TEBHeej.png

    On a different computer step 2 might look a little different. However, regardless of the computer, the structure of the execution always corresponds to the structure of the plan as captured by the following graph (after all, what good is a plan if the execution looks nothing like it):

    ynUemWu.png

    where 0|2|4|6 are points and 1|3|5 are lines.

    This graph doesn't seem to add much value in 1D systems but when we get to 2D systems it becomes much more useful.

    • With the top-down view, the plan and it's execution are distinct steps such that π remains a line, no matter how powerful your computer is.
    • With the bottom-up view, the plan and it's execution are equivalent such that π collapses to a point. I believe this is an unacceptable and an unnecessary leap of thought akin to claiming that there is a last term in a Cauchy Sequence.
    • Although step 2 is incredibly useful for applied mathematics, that's not what I'm concerned with. I'm solely concerned with step 1 and I believe step 1 is what is of interest to pure mathematicians.
    • Counter to standard belief, I believe calculus is about plans (not their execution) and I believe it's unknowingly been this way all along.
    • For example, when a mathematician describes π they always describe the algorithm, they rarely talk about the algorithm's execution...unless referring to a Pi Recitation Contest...
  • jgill
    3.6k
    So, you replace pi with a tiny line segment whose length depends upon a computer. So, changing computer affects this small interval.

    I believe step 1 is what is of interest to pure mathematicians.keystone

    I am one of those and I doubt your claim, but there may be others who find it of interest. I don't see anything of substance here so far, but I may be missing the point.
  • keystone
    323
    So, you replace pi with a tiny line segment whose length depends upon a computer. So, changing computer affects this small interval.jgill

    Yes. This is the whole idea behind discrete calculus, right? The analyst discretizes their model such that their machine can manage the computation.

    I am one of those and I doubt your claim, but there may be others who find it of interest. I don't see anything of substance here so far, but I may be missing the point.jgill

    To be fair, the 1D case isn't particularly exciting. Things get much more interesting in 2D. I've concentrated on the 1D case because it provides a simpler framework to establish (though I say "simpler" with some irony, as it's taken longer than expected to reach this point).
  • jgill
    3.6k
    Things get much more interesting in 2Dkeystone

    I recommend you move on to 2D. Just a thought.
  • fishfry
    3k
    A point of clarity. Thanks. Calculus started with discrete, then moved to infinitesimal, then with technology back to discrete in some sense.jgill

    Is that right? I'm not sure if I have studied that, the pre-Newton and pre-Leibniz developments. You could be right.
  • fishfry
    3k
    Discrete calculus is certainly important to my view but it's not what I'm talking about.keystone

    What about computer arithmetic, fixed and floating point representations, smallest and largest possible values?

    There are two steps to a cut:
    Step 1: Planning the cut with an algorithm
    Step 2: Executing the plan by completing the algorithm
    keystone

    All that followed went way over my head.




    ε1 and ε2 are placeholders for positive numbers which can be as small as your computer allowskeystone

    So you are doing normal math except within the limits of a finite computational space. If not fixed/floating point, something else. But computer arithmetic regardless.


    (π-ε1,π+ε2) describes a line with rational upper and lower bounds. This line can be as small as your computer allows. Ultimately, I want to call this line pi. To distinguish it from the point/number, π, I'll call this line π.keystone

    You cannot telescope down to pi on computer-limited representations of numbers. If you mean that your number pi is actually a little interval around pi with approximation bounds given by the limitations of your computer representation, I'm fine with that.

    Don't see the point though.

    With the top-down view, the plan and it's execution are distinct steps such that π remains a line, no matter how powerful your computer is.
    With the bottom-up view, the plan and it's execution are equivalent such that π collapses to a point. I believe this is an unacceptable and an unnecessary leap of thought akin to claiming that there is a last term in a Cauchy Sequence.
    Although step 2 is incredibly useful for applied mathematics, that's not what I'm concerned with. I'm solely concerned with step 1 and I believe step 1 is what is of interest to pure mathematicians.
    Counter to standard belief, I believe calculus is about plans (not their execution) and I believe it's unknowingly been this way all along.
    For example, when a mathematician describes π they always describe the algorithm, they rarely talk about the algorithm's execution...unless referring to a Pi Recitation Contest...
    keystone

    I didn't follow much of this. Ok check that. I re-read it twice and I do not understand any of it. I'm genuinely sorry I can't be of more help. BUT ... I have an idea:

    If you reject the noncomputable reals, what you have is the constructive real numbers, and the calculus based on them is called constructive real analysis.

    All the explanations on the web immediately get into technicalities involving intuitionist logic, and seem like heavy reading.

    My understanding is that they are just doing calculus using the computable numbers. There is one trick I know in the subject, which is that when you want to see if a sequence converges, you have to have a modulus of convergence which is a function that lets you know how well the sequence is converging.

    I'm not explaining that very well, because I don't understand it very well. In fact I believe I understood it for a small while a few years ago, but I seem to have lost it. There were some advocates of constructive analysis and intuitionist logic on this site a few years ago, but they seem to have drifted away. Cauchy sequences that never converged, as it were.

    But I wonder if this is what you are getting at. The numbers you define consist of a sequence and a function (or plan, or algorithm) saying how the function converges. Or something.

    You know, I do think you have some good intuitions about things ... but you keep rejecting all my examples. Computer math, constructive analysis, etc.

    Still ... tell me what it all means. I have had conversations with proponents of constructive analysis, and at one point I put a little effort into trying to learn it, but in the end, I didn't really get the hang of it.

    It's popular these days because it's good for doing computer math and automated proof checking. There's a lot of research in related areas.
  • keystone
    323
    What about computer arithmetic, fixed and floating point representations, smallest and largest possible values?...So you are doing normal math except within the limits of a finite computational space. If not fixed/floating point, something else. But computer arithmetic regardless.fishfry

    I'm sure you can appreciate the problem of substituting rational number approximations of irrational numbers too early in a computation. The best approach is to do all the manipulation first and only perform the substitution at the very end when the computation is required. I would rephrase this as follows:

    • Step 1: Manipulation of real numbers
    • Step 2: Computation based on rational numbers (approximations)

    This is analogous to the 2-step cutting process I outlined in my previous post. In both cases, step 2 is crude and done using computer arithmetic. It's the realm of applied mathematicians and not of interest here. I'm solely concerned with step 1.

    You cannot telescope down to pi on computer-limited representations of numbers. If you mean that your number pi is actually a little interval around pi with approximation bounds given by the limitations of your computer representation, I'm fine with that.fishfry

    That's pretty much what I'm saying! But instead of talking about any particular computer (which only becomes relevant to step 2), I want to remain in step 1 and talk in general terms. As such, would you allow me to say that π is (π-ε1,π+ε2), and that the value of ε1 and ε2 only need to be determined in step 2? And the following figure is simply saying that π is somewhere between 0 and 4.

    TQOT11q.png

    Don't see the point though.fishfry

    If you say that the above figure makes sense to you, then I can show you a 2D figure, and the benefits and consequences of my perspective will hopefully become clear.

    If you reject the noncomputable reals, what you have is the constructive real numbers, and the calculus based on them is called constructive real analysis.fishfry

    If the noncomputables reals can describe continua it is because below the surface they rest upon a more fundamental scaffolding which can describe continua in and of itself. So no, I'm not interested in constructive real analysis. I'm interested in good ol' real analysis. I just want to place it on a stronger philosophical foundation. I think my perspective will become clearer to you when explained in 2D.

    I'm genuinely sorry I can't be of more help.fishfry

    From the outside, it may seem like this conversation isn't progressing, but your reluctance to accept my informal ideas has highlighted areas where I need to strengthen my arguments. So, you are indeed helping me a lot.
bold
italic
underline
strike
code
quote
ulist
image
url
mention
reveal
youtube
tweet
Add a Comment