• keystone
    296
    You’re suggesting that my line, which already extends in space, requires additional points, which themselves individually have no length, to actually have length. I wish you could appreciate the irony in your position.

    The length of that union is zero, if the intervals are restricted to rationals. Do you agree with that point?fishfry
    If an interval corresponds to a set of points (and nothing else) then I agree that an interval containing only rationals has no length.

    No points. So they're all the empty set? I'm not supposed to push back on this?fishfry
    Our problem is that you are only allowing points in your sets. Suppose I introduce a new concept called 'k-interval' to define the set of ANY objects located between an upper and lower boundary. Would you then consider allowing objects other than points into the set?

    Why on earth do you troll me into arguing with your points, then admitting that you agree with me in the first place?fishfry
    I wanted to show you that even if I cut my unit line to contain all rational points between 0 and 1 that there would still be stuff in between the points -- continua. Perhaps I used the wrong tactic by talking about an idea which I don't support. I did say at the start of the paragraph that it was impossible but maybe I could have been clearer.

    I very much believe in the continuum, which is pretty well modeled by the standard real numbers.fishfry
    Yes, you believe in continua, but not as 'objects in and of themselves'. You believe that continua can't exist in the absence of points. Please confirm.

    I am at an utter loss as to what you have been getting at all this time. Can you get to the bottom line on all this? So far I get that your "continua" are either empty or have length 0. Or that they somehow have length 1, despite being composed of only rationals.fishfry
    My preference is that you accept non-points into sets, however, if you're unwilling to do that then here's an alternate approach. To move this conversation forward, let's say that when I say 'a line', you can think to yourself that I'm referring to 2^aleph_0 points (which somehow assemble to form a line), and I'll think to myself that I'm simply referring to a line (which cannot be constructed from points). In other words, you can go on thinking that points are fundamental and I'll go on thinking that lines are fundamental. How does that sound to you? All I need from you really is to allow me to restrict my intervals to those whose bounds are rational (or +/- infinity). Could you let that fly? ...Just to see how far my position can go in the absence of the explicit use real numbers (I'm fine if in your eyes their use is implied but I just won't ever mention them)...

    So for example, can you allow me to say that there are 5 objects depicted below? You can go on thinking that 2 of the objects are composite objects and I'll go on thinking that all 5 objects are fundamental (they're either 0D or 1D continua).

    y3CBveX.png
  • fishfry
    2.9k
    You’re suggesting that my line, which already extends in space, requires additional points, which themselves individually have no length, to actually have length. I wish you could appreciate the irony in your position.keystone

    Likewise.


    If an interval corresponds to a set of points (and nothing else) then I agree that an interval containing only rationals has no length.keystone

    But you are the one saying that you only have rationals.

    Our problem is that you are only allowing points in your sets.keystone

    In standard set theory, the only thing that sets can contain is other sets. We can call them points but that's only a word used to convey geometric intuition. Actually sets don't contain points, they contain other sets.

    Suppose I introduce a new concept called 'k-interval' to define the set of ANY objects located between an upper and lower boundary. Would you then consider allowing objects other than points into the set?keystone

    I don't know anything about set theory with urlements.

    Why on earth do you troll me into arguing with your points, then admitting that you agree with me in the first place?
    — fishfry
    I wanted to show you that even if I cut my unit line to contain all rational points between 0 and 1 that there would still be stuff in between the points -- continua. Perhaps I used the wrong tactic by talking about an idea which I don't support. I did say at the start of the paragraph that it was impossible but maybe I could have been clearer.
    keystone

    Only adds to my annoyance level. But that's a low bar so no worries.

    Yes, you believe in continua, but not as 'objects in and of themselves'. You believe that continua can't exist in the absence of points. Please confirm.keystone

    Too deep for me. I don't even know what that means.

    My preference is that you accept non-points into sets,keystone

    So set theory with urelements? I don't know much about that subject past the definition.

    however, if you're unwilling to do that then here's an alternate approach. To move this conversation forward, let's say that when I say 'a line', you can think to yourself that I'm referring to 2^aleph_0 pointskeystone

    You only believe in rationals. Where are you getting these things?


    (which somehow assemble to form a line), and I'll think to myself that I'm simply referring to a line (which cannot be constructed from points).keystone

    If you have a line and you have the rationals, you will get the real numbers by Cauchy-completing the line.

    In other words, you can go on thinking that points are fundamental and I'll go on thinking that lines are fundamental. How does that sound to you?keystone

    Your idea is not coherent. If you start with a line (a thing you have declined to define) and say it's populated by the standard rational numbers by cuts, then you can construct the standard real numbers.

    All I need from you really is to allow me to restrict my intervals to those whose bounds are rational (or +/- infinity). Could you let that fly?keystone

    By bounds you mean endpoints? So now you are backing off entirely from your last half dozen points, and saying that your ontology consists of intervals with rational endpoints. But the real numbers are indeed present inside the intervals after all? Is that what you are saying?

    What are these lines of yours, anyway?

    ...Just to see how far my position can go in the absence of the explicit use real numbers (I'm fine if in your eyes their use is implied but I just won't ever mention them)...keystone

    You can't get anywhere as far as I can see.

    So for example, can you allow me to say that there are 5 objects depicted below? You can go on thinking that 2 of the objects are composite objects and I'll go on thinking that all 5 objects are fundamental (they're either 0D or 1D continua).keystone

    You haven't given a coherent definition of these objects. All along you've been saying they are intervals of rationals. That's at least coherent, even if such intervals lack all properties of being continua.

    But now only the endpoints are rational, leaving me baffled as to what those objects are.

    ps -- A forum member once pointed me to the ideas of Charles Sanders Peirce (correct spelling) who said that the mathematical idea of a continuum could not be right, since a true continuum could not be broken up into individual points as the real numbers can.

    Perhaps you are getting at some idea like that. Here's one link, you can Google around to find others if this interests you.

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

    His ideas on continuity:

    https://plato.stanford.edu/entries/peirce/#syn
  • keystone
    296
    But you are the one saying that you only have rationals.fishfry
    No, I'm saying that within an open interval, such as (0,0.5), lies a single objects: a line. Absolutely no points exist with that interval. If you say that 0.25 lies in the middle of that interval, I will say no, 0.25 lies between (0,0.25) and (0.25, 0.5). And what this amounts to is cutting (0,0.5) such that it no longer exists anymore. In its place I have (0,0.25) U [0.25] U (0.25,0.5).

    In standard set theory, the only thing that sets can contain is other sets. We can call them points but that's only a word used to convey geometric intuition. Actually sets don't contain points, they contain other sets.fishfry
    Let's move away from directly using sets to describe the path. Instead, we'll describe the path using a graph, and then define the graph with a set.

    I don't know anything about set theory with urlements.fishfry
    Urelements are indivisible 'atoms'. The lines that I'm working with are divisible.

    You only believe in rationals. Where are you getting these things?fishfry
    That is not what I believe. I can define a line using no rationals: (-inf,+inf). I see this line as a single object (a line). It is not populated by rational points. It is not populated by any points for that matter. I've drawn it for you below in between points at -inf and +inf. To walk this path from -inf to +inf you don't need limits, you just walk the corresponding graph from vertex 0 to vertex 1 to vertex 2.

    9Z23toS.png

    You would call this green line the 'real number line'. You see this as 2^aleph_0 points. You believe that to walk any length on this green line you need limits. I understand what you're saying. We're just starting from different starting points. You're starting from the bottom and I'm starting from the top.

    By bounds you mean endpoints? So now you are backing off entirely from your last half dozen points, and saying that your ontology consists of intervals with rational endpoints. But the real numbers are indeed present inside the intervals after all? Is that what you are saying?fishfry
    Yes, I mean endpoints. I used the term 'bounds' because it is a more general term that applies to higher dimensional analogues. I'm searching for a way to keep this conversation going so it doesn't end prematurely out of frustration. Currently, I don't believe I can persuade you that a continuum can exist without points. However, I've come to realize that convincing you of this isn't necessary. Here’s my new approach:

    1) Start at the bottom
    2) Build up to the top
    3) 'Start' at the top
    4) Approach the 'bottom' from the top

    I see this equivalent to starting at the bottom of the S-B tree -> working our way to the top of the tree -> then proceeding back down to approach the bottom. I know you won't see it that way, which is fine. But to be clear, I still believe things are very ugly at the bottom filled with pumpkins. Nevertheless I do understand how mathematicians think things work at the bottom and if starting at the bottom is the only way you'll allow me to get to the top then I'll go with it. I understand your criticisms of starting at the top, I just don't accept them. Once you allow me to get to (3) I endeavor to show you that (3) and (4) alone fully satisfy our needs and if I'm careful (e.g. by only allowing for rational endpoints) that (1) and (2) are not only superfluous but undesirable. Is that a fair approach?

    But now only the endpoints are rational, leaving me baffled as to what those objects are.fishfry
    Yes, the endpoints are rational, and the object between any pair of endpoints is simply a line. It doesn't go deeper than that. I understand you see that line as a composite object consisting of 2^aleph_0 points. However, I view the line as a primitive object. Clearly, our starting points differ. To move the discussion forward, could we agree to a compromise where we refer to a line as a "composite" object? This way, by including composite it's evident that I recognize your perspective, but the quotes indicate that my viewpoint doesn't necessitate this classification.

    A forum member once pointed me to the ideas of Charles Sanders Peirce (correct spelling) who said that the mathematical idea of a continuum could not be right, since a true continuum could not be broken up into individual points as the real numbers can.fishfry
    I agree with this point. The issue has been the lack of viable alternatives. I see that Peirce was suggesting the use of infinitesimals, and you're aware of my stance on those—the one from the comment where you thought I was just trolling.
  • fishfry
    2.9k
    But you are the one saying that you only have rationals.
    — fishfry
    No, I'm saying that within an open interval, such as (0,0.5), lies a single objects: a line. Absolutely no points exist with that interval. If you say that 0.25 lies in the middle of that interval, I will say no, 0.25 lies between (0,0.25) and (0.25, 0.5). And what this amounts to is cutting (0,0.5) such that it no longer exists anymore. In its place I have (0,0.25) U [0.25] U (0.25,0.5).
    keystone

    I'm afraid I don't know what a line is, absent the real numbers, unless you mean the original line of Euclid, "A line is breadthless length." I'm not a scholar of Euclid so I really can't say.

    I mentioned Peirce to you because it seems to me that you are interested in the "top down" definition of a continuum. I'm deeply unqualified to discuss the matter. I can only give you the standard mathematical interpretation, which is unsatisfying to both of us. I don't know enough about the philosophy of the continuum to comment.

    Let's move away from directly using sets to describe the path. Instead, we'll describe the path using a graph, and then define the graph with a set.keystone

    Sigh. Your pictures don't help. What is a line? What does the notation [0, 0.5] mean?

    I don't know anything about set theory with urlements.
    — fishfry
    Urelements are indivisible 'atoms'. The lines that I'm working with are divisible.
    keystone

    You said that your sets contains things other than sets. You just keep making up your own terminology. I don't think we are making any progress, and I no longer know what we are discussing.

    You only believe in rationals. Where are you getting these things?
    — fishfry
    That is not what I believe. I can define a line using no rationals: (-inf,+inf).
    keystone

    That directly contradicts what you said earlier. And I don't know what your notation means.

    I see this line as a single object (a line).keystone

    What is a line?

    It is not populated by rational points. It is not populated by any points for that matter.keystone

    It's empty? We're going in circles.

    I've drawn it for you below in between points at -inf and +inf. To walk this path from -inf to +inf you don't need limits, you just walk the corresponding graph from vertex 0 to vertex 1 to vertex 2.keystone

    Ok.


    You would call this green line the 'real number line'. You see this as 2^aleph_0 points. You believe that to walk any length on this green line you need limits. I understand what you're saying. We're just starting from different starting points. You're starting from the bottom and I'm starting from the top.keystone

    Ok. We're going in circles. I have no idea what you're talking about.

    Yes, I mean endpoints. I used the term 'bounds' because it is a more general term that applies to higher dimensional analogues. I'm searching for a way to keep this conversation going so it doesn't end prematurely out of frustration.keystone

    I'm not qualified to discuss the philosophy of the continuum with you, except as it relates to the standard mathematical real numbers.


    Currently, I don't believe I can persuade you that a continuum can exist without points.keystone

    I'm perfectly willing to believe it, I just don't know anything about it.

    However, I've come to realize that convincing you of this isn't necessary. Here’s my new approach:

    1) Start at the bottom
    2) Build up to the top
    3) 'Start' at the top
    4) Approach the 'bottom' from the top
    keystone

    Was this supposed to be helpful?

    I see this equivalent to starting at the bottom of the S-B tree -> working our way to the top of the tree -> then proceeding back down to approach the bottom. I know you won't see it that way, which is fine. But to be clear, I still believe things are very ugly at the bottom filled with pumpkins. Nevertheless I do understand how mathematicians think things work at the bottom and if starting at the bottom is the only way you'll allow me to get to the top then I'll go with it. I understand your criticisms of starting at the top, I just don't accept them. Once you allow me to get to (3) I endeavor to show you that (3) and (4) alone fully satisfy our needs and if I'm careful (e.g. by only allowing for rational endpoints) that (1) and (2) are not only superfluous but undesirable. Is that a fair approach?keystone

    Jeez man ...

    But now only the endpoints are rational, leaving me baffled as to what those objects are.
    — fishfry
    Yes, the endpoints are rational,
    keystone

    Two seconds ago you denied this.

    and the object between any pair of endpoints is simply a line.keystone

    What is a line?

    It doesn't go deeper than that. I understand you see that line as a composite object consisting of 2^aleph_0 points.keystone

    I did not say that, and there are other characteristics that a line must have. I am perfectly willing to adopt your ontology, if only you will state it clearly.

    What is a line?

    However, I view the line as a primitive object.keystone

    Ok. Euclid again?

    Clearly, our starting points differ. To move the discussion forward, could we agree to a compromise where we refer to a line as a "composite" object? This way, by including composite it's evident that I recognize your perspective, but the quotes indicate that my viewpoint doesn't necessitate this classification.keystone

    You could move this forward by telling me what a line is. But I don't think I'm helping anything by sniping at your ideas in frustration.

    A forum member once pointed me to the ideas of Charles Sanders Peirce (correct spelling) who said that the mathematical idea of a continuum could not be right, since a true continuum could not be broken up into individual points as the real numbers can.
    — fishfry
    I agree with this point. The issue has been the lack of viable alternatives. I see that Peirce was suggesting the use of infinitesimals, and you're aware of my stance on those—the one from the comment where you thought I was just trolling.
    keystone

    Just giving a reference to what you seem to be getting at. A continuum that can't be divided into points.
  • keystone
    296
    unless you mean the original line of Euclid, "A line is breadthless length."fishfry
    Yes!!! I agree with Euclid's definition of lines and points. I appreciate that he provides foundational definitions of both as separate, fundamental entities. Thanks for pointing this out.

    What is a line? What does the notation [0, 0.5] mean?fishfry
    Euclid also said that "The ends of a line are points." When I describe a path as 0 U (0,1) U 1:
    (0,1) corresponds to the object of breadthless length and
    0 and 1 correspond to the points at the end.

    It seems that some people intepret Euclid as saying that a line without endpoints extends to infinity. I do not think this is necessarily the case. While (-inf,+inf) is a valid line, I believe (0,1) is also a valid line in and of itself.

    ---------------------

    Please give the following figure a chance as it captures a lot of what I'm trying to say:

    3s7gEJF.png

    1) In the first row, we have a line with two endpoints, totaling three objects.
    2) I can represent this path as a graph composed of three connected vertices. Notice that the lines and points are all represented as vertices in the graph.
    3) I want to put the information from all vertices into a set. That's 3 objects in one set. Not just the end points.
    4) When the runner travels from 0 to 1, they don't run a path composed of infinite points. They walk the graph, which in this case is the journey from vertex 1 to vertex 2 to vertex 3.
    5) If you cut the line, you'll end up with five objects: the two endpoints, a middle point, and the two line segments in between. This is what we have in the second row.
    6) The length of the points is zero. In fact, no matter how many times we cut the line, the total length of the points will always be zero.
    7) The total length of the line segments is one. In fact, no matter how many times we cut the line, the total length of the segments will always be one.
    8) Notice that in the second row, the interval (0,1) is not present because it has been cut.
    9) We can continue cutting the line indefinitely, and one particular sequence of cuts is depicted across the subsequent rows.
    10) Notice the pattern in the columns labeled 'length of lines' and 'path length'. As we progress downward, we approach the familiar geometric series.
    11) Unlike the bottom-up approach, which requires a limit to make the summation total one, the top-down approach results in a summation that totals one at every stage.

    I'm not qualified to discuss the philosophy of the continuum with you, except as it relates to the standard mathematical real numbers.fishfry
    I believe that someone even as intelligent and knowledgeable as yourself is not qualified to discuss the bottom-up philosophy of a continuum because it is flawed. I'm 100% certain you have the capacity to understand, discuss, and criticize the top-down philosophy.

    That directly contradicts what you said earlier.fishfry
    You're right, I did say that the endpoints were necessarily rational numbers. (-inf, +inf) has no endpoints. While there are scenarios where it is useful to include points at infinity, for this discussion, let's agree that the points at -inf and +inf are not real points. I'm only using infinity as a shorthand. I should have been clearer.
  • fishfry
    2.9k
    unless you mean the original line of Euclid, "A line is breadthless length."
    — fishfry
    Yes!!! I agree with Euclid's definition of lines and points. I appreciate that he provides foundational definitions of both as separate, fundamental entities. Thanks for pointing this out.
    keystone

    Didn't I ask you about this several posts ago? Ok, Euclid's line.

    What is a line? What does the notation [0, 0.5] mean?
    — fishfry
    Euclid also said that "The ends of a line are points." When I describe a path as 0 U (0,1) U 1:
    (0,1) corresponds to the object of breadthless length and
    0 and 1 correspond to the points at the end.
    keystone

    Ok so you are doing classical Euclidean geometry (not modern Euclidean geometry, please note).

    It seems that some people intepret Euclid as saying that a line without endpoints extends to infinity. I do not think this is necessarily the case. While (-inf,+inf) is a valid line, I believe (0,1) is also a valid line in and of itself.keystone

    Euclid would not recognize that notation; and at this point in our conversation, neither do I. You have variously stated that (0,1) contains only rationals, or that it may even be empty.

    In view of my new understanding that by line, you mean Euclid's line, what does the notation (0,1) mean? Euclid did not have numbers as we know them.

    Please give the following figure a chance as it captures a lot of what I'm trying to say:keystone

    Utterly baffled. Utterly. Baffled. No idea what it means. 0, 0 + 0, 0 + 0 + 0, no idea what I am supposed to glean from that. And by the way, what is this "+" symbol? Have you defined it? Is this the standard + of the rational numbers?

    I feel terrible ignoring these diagrams that you put so much work into, and that hold so much meaning for you. I wish I could be more helpful. I don't mean to just continue to snipe at you. It pains me. I just don't know what you are saying and have no idea how to respond.

    I believe that someone even as intelligent and knowledgeable as yourself is not qualified to discuss the bottom-up philosophy of a continuum because it is flawed.keystone

    I never claimed to be able to discuss the philosophy of the continuum. On the contrary, I've admitted that I can't. Except, that I know a bit about the real numbers, and they are the standard mathematical model of the continuum. And that counts for something.

    I'm 100% certain you have the capacity to understand, discuss, and criticize the top-down philosophy.keystone

    Possibly, but not the inclination. If I could dispatch a clone, I'd have him read Peirce. I'm not a philosopher of the continuum. I'm not a philosopher at all. I only come to this forum to clarify people's mathematical misunderstandings. And it's a full time job :-)

    You're right, I did say that the endpoints were necessarily rational numbers. (-inf, +inf) has no endpoints. While there are scenarios where it is useful to include points at infinity, for this discussion, let's agree that the points at -inf and +inf are not real points. I'm only using infinity as a shorthand. I should have been clearer.keystone

    Ok. So far, your line is Euclid's original line. Leaving undefined, your notation (0,1), which you have repeatedly pointed out is NOT the open unit interval of real numbers.

    ps -- Ok I took another look at your picture. You correctly note that the sum of the lengths of the points is 0. But then you say that the sum of the lengths is 1, and I'm not sure how that follows.

    Since your intervals are entirely made up of rationals, the total length must be 0.

    Where is the extra length coming from?

    I'm willing to let you say that the length of the interval (0,1) is 1 even though it's only made of rationals. I'll stipulate that for sake of discussion, even though it's hard to understand how it works.

    But what does it all mean? I'm lost and dispirited. It's not my role in life to feel bad about myself for endlessly sniping at your heartfelt ideas.
  • keystone
    296
    Didn't I ask you about this several posts ago? Ok, Euclid's line.fishfry

    Sorry, I didn't appreciate the point when you first mentioned it. Yes, I'm starting from classical Euclidean geometry.

    And by the way, what is this "+" symbol? Have you defined it? Is this the standard + of the rational numbers?fishfry

    Yes. Formally the arithmetic is performed as described here (https://www.sciencedirect.com/science/article/pii/S1570866706000311) but informally it's performed using the standard method we teach kids. The formal and informal results are equivalent.

    what does the notation (0,1) mean?fishfry

    It describes the line's potential. I'm going to provide a shorthand answer involving real numbers that I don't want you to take literally. If this explanation lands, great, otherwise forget it.

    • No points exist on lines, including the unit line (0,1). To put it another way, there are no 'actual points' present on that segment. (Actual vs. potential is discussed below).
    • Cutting line (0,1) in two will introduce an 'actual point' between the two resulting line segments. That point will have a rational coordinate between 0 and 1.
    • In my last post, I noted that -inf and +inf are not 'actual points' but rather are used as helpful shorthand. I should have called them 'potential points'.
    • With a similar shorthand, we can say that on line (0,1) exist 'potential points', which have real number coordinates between 0 and 1.
    • The rational 'potential points' can become 'actual points' through cuts.
    • The irrational 'potential points' are permanently confined to their 'potential point' status.
    • I want to reiterate that 'potential points' don't actually exist. They're just a fiction that may help us comprehend the potential in continua. If you don't think potential points are a useful concept we can just drop.
    • The interval "(0,1)" describes the potential of the corresponding unit line.

    Since your intervals are entirely made up of rationals, the total length must be 0. Where is the extra length coming from?fishfry

    The length of a line comes from its potential.

    I'm lost and dispirited. It's not my role in life to feel bad about myself for endlessly sniping at your heartfelt ideas.fishfry

    Sometimes it’s a bit frustrating when my explanations don’t connect, but this conversation is exactly what I need right now, so please don’t feel bad. I'm very appreciative that you've stuck around.
  • keystone
    296
    You correctly note that the sum of the lengths of the points is 0. But then you say that the sum of the lengths is 1, and I'm not sure how that follows.fishfry

    Path Length = Length of Lines + Length of Points
    Path Length = Length of Lines + 0
    Path Length = Length of Lines

    So referring to row 3 of that figure...
    Path Length = Length of Lines
    1 = 1/2 + 1/4 + 1/4
  • keystone
    296
    But what does it all mean? I'm lost and dispirited.fishfry

    Looking ahead from a top-down perspective, I aim to delve deeper into the concept of irrationals. While "potential points" might serve as a useful fiction to understand the potential of a line, irrationals should be grounded in real meaning, not tied to fictional constructs. Therefore, let's explore this further. I believe that discussing irrationals is most effective within a two-dimensional context. In this post let me set the stage for 2D continua and later we'll look at :

    1) Start with a 2D continuum with a domain of (0,3) and a range of (-1,2) as depicted below.
    tBEge0i.png
    2) Cut this continuum with y=x^2-2 and y=0 as depicted below.
    • In 2D, cuts are curves and points lie at the intersection of cuts.
    • Please note that in 2D, intervals are no longer used. (x,y) corresponds to 2-dimensional co-ordinates.
    • Red objects are 0D continua (points)
    • Green objects are 1D continua (curves)
    • Yellow objects are 2D continua (surfaces)
    • The goal here is to determine what the ? should be labelled.
    EWRz8Ev.png

    4) From the bottom-up view, there are points in this figure. Comparatively, from the top-down view, there are only 25 objects: 9 points, 12 curves, and 4 surfaces. These objects are indexed below. If any one of these objects is removed, the system will no longer be continuous.

    lkcAkvS.png

    5) The connectivity of these objects can be represented with a graph as depicted below. When discussing the distance between objects, we must discuss it in the context of graphs (i.e. the shortest path between two vertices).

    1pLGIIp.png

    6) Walking from one object to the another does not require limits. Simply walk the graph, one vertex at a time. You can always travel from one vertex to another in finite steps.

    7) If we make another cut to our system, we'll need an entirely new graph.

    Does this all make sense so far?
  • fishfry
    2.9k
    Didn't I ask you about this several posts ago? Ok, Euclid's line.
    — fishfry

    Sorry, I didn't appreciate the point when you first mentioned it. Yes, I'm starting from classical Euclidean geometry.
    keystone

    Ok. I wanted to say that I'll stipulate to your non-rigorous conception of a continuum of being made of tiny little continua "all the way down," with no need for actual points, if that's your idea. I think this is what Peirce is getting at.

    In any event to save us some time, I'll stipulate to your vision, even if it's a bit contradictory and not totally clear.

    So then what?

    And by the way, what is this "+" symbol? Have you defined it? Is this the standard + of the rational numbers?
    — fishfry

    Yes. Formally the arithmetic is performed as described here (https://www.sciencedirect.com/science/article/pii/S1570866706000311)
    keystone

    LOL You are committed to that idea, I'll give you that.

    but informally it's performed using the standard method we teach kids. The formal and informal results are equivalent.keystone

    I'll stipulate to arithmetic on the rationals, I think we can agree on that.

    what does the notation (0,1) mean?
    — fishfry
    uu
    It describes the line's potential.
    keystone

    Ok. With my earlier stipulation, I could say:

    (a) What? That makes no sense; or

    (b) I sort of get it. The line contains a frothing sea of tiny little micro-continua that are not points. Is that about right?

    I'm going to provide a shorthand answer involving real numbers that I don't want you to take literally. If this explanation lands, great, otherwise forget it.

    No points exist on lines, including the unit line (0,1). To put it another way, there are no 'actual points' present on that segment. (Actual vs. potential is discussed below).
    Cutting line (0,1) in two will introduce an 'actual point' between the two resulting line segments. That point will have a rational coordinate between 0 and 1.
    keystone

    Well here you are in trouble. If you allow "cuts" then à la Dedekind we have the real numbers. But you don't want to go there so ok. There are cuts but not so many as to allow the reals.

    In my last post, I noted that -inf and +inf are not 'actual points' but rather are used as helpful shorthand. I should have called them 'potential points'.
    With a similar shorthand, we can say that on line (0,1) exist 2ℵ0
    2

    0
    'potential points', which have real number coordinates between 0 and 1.
    keystone

    Oh my. You are now allowing the reals? Ok. Maybe that's good.


    The rational 'potential points' can become 'actual points' through cuts.
    The irrational 'potential points' are permanently confined to their 'potential point' status.
    I want to reiterate that 'potential points' don't actually exist. They're just a fiction that may help us comprehend the potential in continua. If you don't think potential points are a useful concept we can just drop.
    The interval "(0,1)" describes the potential of the corresponding unit line.
    keystone

    I sort of get your thinking. Not sure where you're going but I'll stipulate to all this, even with the vagueness.

    Of course all mathematical entities are fictional, so I can't see what the difference is between and actual and a fictional point. Once you stipulate to fictional points, they become actual by virtue of being used and accepted. Just as negative numbers and imaginary numbers once did.

    The life cycle of a mathematical idea is is:

    Regarded as impossible ---> Fictional but useful ---> Normal everyday stuff.

    Same path taken by non-Euclidean geometry. Impossible, then Riemann's curiosity, then Einstein's platform for general relativity

    So once you admit a "fictional" entity you might as well grant it actual status, since you will eventually.

    Since your intervals are entirely made up of rationals, the total length must be 0. Where is the extra length coming from?
    — fishfry

    The length of a line comes from its potential.
    keystone

    But here's the thing. I said earlier that in the standard unit interval, the length is stored in the irrationals.

    You are saying the exact same thing, but changing the name of irrationals to "fictionals." I don't see how that changes anything. You just changed their name but they're the same irrationals.


    Sometimes it’s a bit frustrating when my explanations don’t connect, but this conversation is exactly what I need right now, so please don’t feel bad. I'm very appreciative that you've stuck around.
    keystone

    Ok. I'm adopting a less rigorous and more intuitive sense of what you are saying.

    But you have also met me more than halfway. You have agreed finally that there ARE irrationals on the line, and that they carry, or store, the length. You just call them fictionals instead of reals. But they are the same thing.

    [==== next post ====]

    Path Length = Length of Lines + Length of Points
    Path Length = Length of Lines + 0
    Path Length = Length of Lines

    So referring to row 3 of that figure...
    Path Length = Length of Lines
    1 = 1/2 + 1/4 + 1/4
    keystone

    I'll stipulate that the length is stored in the fictionals, which I'll continue to think of as the reals till you claim otherwise.
  • keystone
    296
    I'll stipulate to your non-rigorous conception of a continuum of being made of tiny little continua "all the way down," with no need for actual points, if that's your idea. I think this is what Peirce is getting at.fishfry

    I also think that's what Peirce was getting but that's definitely not what I'm getting at. Remember when I "trolled" you by introducing a scenario involving infinitesmals? I believe that approach aligns with Peirce's thinking and I believe it's wrong.

    The line contains a frothing sea of tiny little micro-continua that are not points. Is that about right?fishfry

    You keep trying to concieve of my line as something built from smaller more fundamental elements (before points, now infinitesimals). It is not built from anything. (0,1) is one object - a line. The smaller elements emerge from the line, not the other way around.

    Well here you are in trouble. If you allow "cuts" then à la Dedekind we have the real numbers. But you don't want to go there so ok. There are cuts but not so many as to allow the reals.fishfry

    I'm not allowing a single real number. We can partition the S-B tree at a rational node (e.g. 1/2), but we cannot partition it at a real node (because real nodes don't exist).

    hBJD2ya.png

    Of course all mathematical entities are fictional, so I can't see what the difference is between and actual and a fictional point.fishfry

    Just as you don't grant infinity actual status as a natural number, I don't grant irrational points actual status as points. After all, infinity and irrational points are inseparably linked in the S-B tree, since irrational points become actual points at row infinity. If there is no actual row infinity, there are no actual irrational points.

    You are saying the exact same thing, but changing the name of irrationals to "fictionals." I don't see how that changes anything. You just changed their name but they're the same irrationals.fishfry

    The difference is that you believe individual irrationals can be isolated, whereas I think we can only access irrationals as continuous bundles of fictional points. A mathematical 'quanta' if you will. In a 1D context, I refer to this continuous bundle as a line. And if we cut a line, we have two lines (i.e. two bundles of fictional points). No matter how many times we cut it, we will never reduce a bundle down into individual points. Since we can only ever interact with these bundles, it is meaningless to discuss individual irrationals - they are fictions. The bundles are not. Do you see the distinction?
  • jgill
    3.6k
    It is not built from anything. (0,1) is one object - a linekeystone

    How do you propose to pass from a finite line to a circle, say? If you are considering topological transformations, how can you express them? Sorry for butting in, but I remain curious.
  • keystone
    296
    How do you propose to pass from a finite line to a circle, say? If you are considering topological transformations, how can you express them? Sorry for butting in, but I remain curious.jgill

    Welcome back. I've transitioned from topology to graph theory, which (in this context) maintains similar concepts but is much simpler. To convert a path graph into a cycle graph, I would use vertex identification. Not sure what you're getting at. And really this is beyond the scope of what I'm covering. Right now, I'm just focused on reinterpreting the Cartesian coordinate system.
  • fishfry
    2.9k
    I also think that's what Peirce was getting but that's definitely not what I'm getting at. Remember when I "trolled" you by introducing a scenario involving infinitesmals? I believe that approach aligns with Peirce's thinking and I believe it's wrong.keystone

    It's the only way I can make sense of what you're saying.

    You keep trying to concieve of my line as something built from smaller more fundamental elements (before points, now infinitesimals). It is not built from anything. (0,1) is one object - a line. The smaller elements emerge from the line, not the other way around.keystone

    [0,1] is standard mathematical notation for a particular set of real numbers. You can't fault me for bringing my preconceptions about that notation. You should use less suggestive notation.

    How about "L". If you say, "I have a line, I call it L," then I can't come back and challenge you about that notation.

    You want it both ways. You think you can traverse the line from 0 to 0.5 to 1, freely borrowing our high school intuitions about the real number line. And when I bite on that bait, you say, "Oh no it's not the real line!"

    You call your line [0,1], you treat it as if it's the usual unit interval, and then you object when I believe you!


    I'm not allowing a single real number. We can partition the S-B tree at a rational node (e.g. 1/2), but we cannot partition it at a real node (because real nodes don't exist).keystone

    Ok back to no real numbers.


    Just as you don't grant infinity actual status as a natural number,keystone

    It's not a natural number. It's not 0 and it's not the successor of any number. I can PROVE "infinity", whatever you mean by that, is not a natural number.

    I don't grant irrational points actual status as points. After all, infinity and irrational points are inseparably linked in the S-B tree, since irrational points become actual points at row infinity. If there is no actual row infinity, there are no actual irrational points.keystone

    Last post you started believing in the real numbers, but you called them fictitious. Now you're backing off even that.

    The difference is that you believe individual irrationals can be isolated, whereas I think we can only access irrationals as continuous bundles of 2ℵ0keystone

    Whatever that means.

    fictional points. A mathematical 'quanta' if you will. In a 1D context, I refer to this continuous bundle as a line. And if we cut a line, we have two lines (i.e. two bundles of 2ℵ0keystone

    Ok. Can't agree, can't disagree.

    fictional points). No matter how many times we cut it, we will never reduce a bundle down into individual points.keystone

    That's true of the real numbers as well. You know, I think you are just coming to understand the nature of the standard mathematical real numbers.

    If you start with the real line and cut it any number of times, you can never isolate a point that way. Do you agree?

    Since we can only ever interact with these bundles, it is meaningless to discuss individual irrationals - they are fictions. The bundles are not. Do you see the distinction?keystone

    No, because the real numbers have the same property. I cut [0,1] in half, I get [0, 1/2) and [1/2, 1]. I'm arbitrarily placing 1/2 in the second segment.

    If I cut [0, 1/2) in half, I get [0, 1/4) and [1/4, 1/2), and so forth.

    No number of cuts will ever isolate a point.

    So I think what is happening here is that you are coming to a better intuition of the standard real numbers. Because you can keep cutting the unit interval in half and you will never isolate a point.
  • keystone
    296


    Throughout our conversation, my perspective and how I express it have greatly developed, leading me to believe it's best to reformulate and clarify my position. I'll be on a short holiday for the next few days, and I'd also like to take the necessary time to gather my thoughts before responding. For now, let me make two points:

    • The essence of my perspective (top-down) remains the same, although it requires some minor adjustments.
    • Having to reformulate my view underscores the significant value I've derived from our conversation—thanks once more!

    I'll reach out again in a few days. I look forward to continuing this discussion. Enjoy your weekend!
bold
italic
underline
strike
code
quote
ulist
image
url
mention
reveal
youtube
tweet
Add a Comment

Welcome to The Philosophy Forum!

Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.