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Math Forum / Mathematics / Mathematical Logic / October 2008The middle of the string
There's no left or right in a string of beads. And the middle of a string of beads is only dependent on our being able to ascertain left and right. But the string itself doesn't have a left or right: thus, a middle doesn't belong to a string. >There's no left or right in a string of beads. And the middle of a >string of beads is only dependent on our being able to ascertain left >and right. But the string itself doesn't have a left or right: thus, a >middle doesn't belong to a string. You know, it's sad. You seem to have been born several centuries after your time. David C. Ullrich "Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.) > >There's no left or right in a string of beads. And the middle of a > >string of beads is only dependent on our being able to ascertain left [quoted text clipped - 9 lines] > (John Jones, "My talk about Godel to the post-grads." > in sci.logic.) I am sorry David, but you are wrong to insult our forefathers! > There's no left or right in a string of beads. And the middle of a > string of beads is only dependent on our being able to ascertain left > and right. But the string itself doesn't have a left or right: thus, a > middle doesn't belong to a string. In the middle of a string of beads is a neck. > > There's no left or right in a string of beads. And the middle of a > > string of beads is only dependent on our being able to ascertain left > > and right. But the string itself doesn't have a left or right: thus, a > > middle doesn't belong to a string. > > In the middle of a string of beads is a neck. One could also say that the string runs down the middle. > There's no left or right in a string of beads. Of course there is, once you pick it up, once you put one end in your left hand and the other in your right, and stretch it out and look at it, so you can actually obvserve the beads. Strings on the page HAVE ALWAYS ALREADY HAD this done to them. You ARE LOOKING at the evidence of this right now, while reading this message, so if you continue to deny it, that just makes you stupid. > And the middle of a string of beads is only dependent > on our being able to ascertain left > and right. Oh, BULLSHIT. You do NOT need to know left from right in order to tell where the middle is. The middle is wherEVER there is the SAME number of beads on EACH side! You DO have to be able to split the string in two, but you do NOT need to be able to tell WHICH half is left and which is right! Grasp BOTH ends of the string of beads in the SAME hand and let the string hang VERTICALLY! The middle will be the BOTTOMmost bead(or midpoint between beads if there are an even number of beads) and NO concept of left or right will be needed prior, or even imputed afterward! > But the string itself doesn't have a left or right: thus, a > middle doesn't belong to a string. If it has a middle, then it has some beads on one side and some OTHER beads on the OTHER side. What you CALL the sides does NOT matter. The point is that you have a middle IF AND ONLY IF you DO have things on TWO OPPOSITE SIDES OF that middle (and those things match up). >> There's no left or right in a string of beads. > > Of course there is, once you pick it up, once you put one end in your > left hand and the other in your right, and stretch it out and look at > it, Then it's not a string any more. It's now an asymmetric ring. > so you can actually obvserve the beads. No. You observe a hands/beads composite. > Strings on the page HAVE > ALWAYS > ALREADY HAD this done to them. That is why they are not strings. There are only different page styles. > You ARE LOOKING at the evidence of > this > right now, while reading this message, so if you continue to deny it, > that > just makes you stupid. No. The context of a string has been jettisoned. It is no longer a string but is a complex object whose identity is a function of everything in my field of view, and out of view. >> And the middle of a string of beads is only dependent >> on our being able to ascertain left [quoted text clipped - 4 lines] > middle is. > The middle is wherEVER there is the SAME number of beads on EACH side! How do I know I haven't counted the same side twice if I can't tell left from right? > You DO have to be able to split the string in two, but you do NOT need > to be able > to tell WHICH half is left and which is right! How do I know I have split the string in the middle if I can't tell left from right? > Grasp BOTH ends of > the string of beads [quoted text clipped - 4 lines] > and NO concept of left or right will be needed prior, or even imputed > afterward! Thank god we have gravity, hands, etc., that allow us to construct complex objects which have a middle. Because the string won't tell us its middle. That's because A STRING HAS NO MIDDLE! Idiot! >> But the string itself doesn't have a left or right: thus, a >> middle doesn't belong to a string. > > If it has a middle, then it has some beads on one side and some OTHER > beads > on the OTHER side. Unfortunately, a string doesn't have different sides or segments. > What you CALL the sides does NOT matter. A string doesn't have any anyway. > The point is that you have a middle IF AND ONLY IF you DO have things > on > TWO OPPOSITE SIDES OF that middle (and those things match up). Yes, do things like create an entirely different object which, lo and behold, has a middle. > There's no left or right in a string of beads. OK. But there is one end and the other distinct end. > And the middle of a > string of beads is only dependent on our being able to ascertain left > and right. Let's say the middle depends on having two ends then... > But the string itself doesn't have a left or right: thus, a > middle doesn't belong to a string. But it does have one end and another, so it -does- have a middle. Mitch >> There's no left or right in a string of beads. > [quoted text clipped - 5 lines] > > Let's say the middle depends on having two ends then... Yes, ok. We have defined our string as having two ends. That's our definition of a string, we must define it in that way. And two ends MUST have a middle. All I can offer against that objection is that I do not need to specify a middle to have two ends. There is another object that has two ends - a pair of points. But if we define 'ends' as being a property of length, then that would get rid of that counter-example. So it comes down to this: Does a finite line have a middle? Yes it does. So does a string have a middle? Only if it is a line. But I say that a string is not a line. We cannot tell left from right in a string because a string is not a line. And if a string is not a line then it has no middle. So your example refuted my own, but only in the sense that I needed to give a related reason why a string has no middle, viz. a string has no middle because it is not a line, and not because it we cannot determine left from right. In that case, my string of beads example is up the spout. I should have kept to a mathematical string. Wait a minute though. If all the beads are the same on a string of beads then a string of beads has no length. I was right after all. Phew. >> But the string itself doesn't have a left or right: thus, a >> middle doesn't belong to a string. > > But it does have one end and another, so it -does- have a middle. > > Mitch > >> There's no left or right in a string of beads. > [quoted text clipped - 8 lines] > Yes, ok. We have defined our string as having two ends. That's our > definition of a string, we must define it in that way. What? 'must'? Not necessarily. > And two ends MUST have a middle. same response as above. A lot of the progress on your problem would be made much easier if you were a bit more precise, and by that I mean mathematical. Lack of precision allows you to conjecture wildly in multiple directions simultaneously, your choice of direction influenced by metaphorical whim. > All I can offer against that objection is that I do not need to specify > a middle to have two ends. There is another object that has two ends - a > pair of points. But if we define 'ends' as being a property of length, > then that would get rid of that counter-example. but... but then you're talking about something different? > So it comes down to this: Does a finite line have a middle? Yes it does. OK I guess. > So does a string have a middle? Only if it is a line. What? Why would that possibly be relevant? > But I say that a string is not a line. Sure you can say that by fiat...define string, define point, then the rest of us can decide. > We cannot tell left from right in a string because > a string is not a line. And if a string is not a line then it has no middle. [quoted text clipped - 7 lines] > Wait a minute though. If all the beads are the same on a string of beads > then a string of beads has no length. I was right after all. Phew. if all the beads are the same... then a string of beads has no length? really? How does that follow? Mitch >>>> There's no left or right in a string of beads. >>> OK. But there is one end and the other distinct end. [quoted text clipped - 10 lines] > > same response as above. I'm referring to the plain observation that a finite string has length with two ends. It would seem obvious. > A lot of the progress on your problem would be made much easier if you > were a bit more precise, and by that I mean mathematical. Lack of > precision allows you to conjecture wildly in multiple directions > simultaneously, your choice of direction influenced by metaphorical > whim. It isn't accuracy we are striving for. I don't want to know the exact length of the string. We are looking to the everyday principles that inform mathematical reasoning. >> All I can offer against that objection is that I do not need to specify >> a middle to have two ends. There is another object that has two ends - a >> pair of points. But if we define 'ends' as being a property of length, >> then that would get rid of that counter-example. > > but... but then you're talking about something different? Yes. But I need to say why it is different. It is different because a pair of points has no length even though its range (like a finite line) is covered completely by two points. >> So it comes down to this: Does a finite line have a middle? Yes it does. > [quoted text clipped - 3 lines] > > What? Why would that possibly be relevant? A string can have a middle if it is a finite line. >> But I say that a string is not a line. > [quoted text clipped - 15 lines] > if all the beads are the same... then a string of beads has no length? > really? How does that follow? If all the beads are the same I will be unable to differentiate them and make a reliable count. If I can't make a reliable count then I don't have a sequenced line. Woof. > >>>> There's no left or right in a string of beads. > >>> OK. But there is one end and the other distinct end. [quoted text clipped - 13 lines] > I'm referring to the plain observation that a finite string has length > with two ends. It would seem obvious. Define 'string', 'end', and 'middle'. I wasn't thinking about this before but I can immediately apply my own logical pedantry with the single example of the empty string. By my account, I'd guess that to call anything a middle one must have ends respect to which it is a middle, but the empty string or null string (the string of zero length) may have two ends the left and the right (of course they don't go anywhere) but no middle. or maybe the middle of of length zero. I'll leave that up to you. > > A lot of the progress on your problem would be made much easier if you > > were a bit more precise, and by that I mean mathematical. Lack of [quoted text clipped - 5 lines] > length of the string. We are looking to the everyday principles that > inform mathematical reasoning. We may not be looking for accuracy in length, but we sure are trying to find accuracy in definition. Otherwise, how will we know when we've made progress? > >> So it comes down to this: Does a finite line have a middle? Yes it does. > [quoted text clipped - 5 lines] > > A string can have a middle if it is a finite line. 'can' is not 'must' (you were saying 'must' before'; I agree with "...can...if...finite line" though) Are you talking about strings that are like shoe-laces? or like sequences of characters in text? or some curve in space? or what? > >> But I say that a string is not a line. > [quoted text clipped - 19 lines] > make a reliable count. If I can't make a reliable count then I don't > have a sequenced line. Woof. What do you mean by same? indistinguishable? Then logically, you'd have to say that if you have a set of indistinguishable items, then you really only have 1 (or 0) items. You're letting your words come back to you as truth. If you know you have a sequence of beads (0 or more), and you also know that they are indistinguishable (pick bead x, pick bead y, every time you do this x = y), then you have either 1 (or 0) beads. The context that you are missing that most people presume in the culture of mathematics that relegate this misunderstanding to a degenerate boundary case, is that the beads are indistinguishable -up to position-, that is everything is the same about the beads except their position in the string of beads. Likewise with a set of beads, if you have a set of n distinguishable items, one assumes that in a formalization, there is a unique label on each of the n items so that no property of the item (except for the label) is different. As to making a reliable count, I'd say that is built in to the definition of a string, the position of items along the string. Even if they are otherwise identical, the position along the string is enough to allow counting. Mitch > By my account, I'd guess that to call anything a middle one must have > ends respect to which it is a middle, but the empty string or null > string (the string of zero length) may have two ends the left and the > right (of course they don't go anywhere) but no middle. or maybe the > middle of of length zero. I'll leave that up to you. I don't know whether 'a line of zero length' means the absence of a line, or describes a line as being independent of its length, or is a vanishingly small, but not completely gone, line. >> A string can have a middle if it is a finite line. > > 'can' is not 'must' (you were saying 'must' before'; I agree with > "...can...if...finite line" though) It couldn't if the line was a circle. > Are you talking about strings that are like shoe-laces? or like > sequences of characters in text? or some curve in space? or what? I'm taking a string as a sequence of individual, spatially or temporally fixed elements. But I don't think the idea is coherent because in order to fix the elements, I need a higher organising structure (like 'left and right'). >> If all the beads are the same I will be unable to differentiate them and >> make a reliable count. If I can't make a reliable count then I don't [quoted text clipped - 3 lines] > have to say that if you have a set of indistinguishable items, then > you really only have 1 (or 0) items. In fact, I wouldn't be able to tell one from infinity. > As to making a reliable count, I'd say that is built in to the > definition of a string, the position of items along the string. Even > if they are otherwise identical, the position along the string is > enough to allow counting. But once I have counted an element and want to move on to the next element, I don't know whether or not I am counting the next element or the one before it. That's because I can't see the whole string, because to see the whole string would mean that a count has already been tacitly made. Am Sun, 12 Oct 2008 23:54:58 +0100 schrieb John Jones: >> By my account, I'd guess that to call anything a middle one must have >> ends respect to which it is a middle, but the empty string or null [quoted text clipped - 5 lines] > line, or describes a line as being independent of its length, or is a > vanishingly small, but not completely gone, line. It might be considered a (single) point. (Or a set consisting of a single point, especially when we consider lines to be sets of points.) Herb > Am Sun, 12 Oct 2008 23:54:58 +0100 schrieb John Jones: > [quoted text clipped - 12 lines] > > Herb In that case a line is independent of its length. But perhaps what we are doing here is starting out with the concept of a line and then constructing points on it. And THEN, when we have constructed only one point on the line, we say that a line can be a point. Aside from the seeming trickery involved in that, I might note that we don't construct points on a line, we construct positions. And where we have constructed only one position, we still don't construct a line, or even a point. > > By my account, I'd guess that to call anything a middle one must have > > ends respect to which it is a middle, but the empty string or null [quoted text clipped - 12 lines] > > It couldn't if the line was a circle. Possibly. But I 'could' imagine a definition of a line that is a circle that might have a middle. But we're being so fuzzy about everything, trying to extrapolate definitions from single ambiguous examples, I don't think we could tell one way or the other. Take a look at Heath, A History of Greek Mathematics, and the Euclid edition with his comments. Lots of explanation about what 'line' means in particular, but also lost of stuff about defining in general. > > Are you talking about strings that are like shoe-laces? or like > > sequences of characters in text? or some curve in space? or what? [quoted text clipped - 3 lines] > to fix the elements, I need a higher organising structure (like 'left > and right'). How about just 'start element' and 'end element', or just 'natural # position'? That's pretty coherent. > >> If all the beads are the same I will be unable to differentiate them and > >> make a reliable count. If I can't make a reliable count then I don't [quoted text clipped - 5 lines] > > In fact, I wouldn't be able to tell one from infinity. I think you're mixing up counting abilities here. Your counting primitive of using entire distinguishability would just say you have 1 item no matter what. That's all you'd have. If by some other distinction ability you have a greater # of things (even infinity), that's a different counting scheme. Sure you'd be able to tell one from infinity in the first scheme as long as you can distinguish elements in a given way, and also with the other counting scheme. Within a scheme, you'd know the difference between zero, one, and anything else. > > As to making a reliable count, I'd say that is built in to the > > definition of a string, the position of items along the string. Even [quoted text clipped - 4 lines] > element, I don't know whether or not I am counting the next element or > the one before it. Always go in one direction? define string first, then you'll be able to answer these questions. Mitch >> I'm taking a string as a sequence of individual, spatially or temporally >> fixed elements. But I don't think the idea is coherent because in order [quoted text clipped - 3 lines] > How about just 'start element' and 'end element', or just 'natural # > position'? That's pretty coherent. I suppose that we can exchange spatial metaphors (left and right) for temporal or mixed spatial/temporal metaphors (start, end etc) >> In fact, I wouldn't be able to tell one from infinity. > > I think you're mixing up counting abilities here. Your counting > primitive of using entire distinguishability would just say you have 1 > item no matter what. That's all you'd have. In a presentation of indistiguishables I wouldn't be able to make a count at all, not even a count of one. I might be able to count a concept, but I wouldn't be able to count an object. > If by some other > distinction ability you have a greater # of things (even infinity), [quoted text clipped - 3 lines] > Within a scheme, you'd know the difference between zero, one, and > anything else. In fact, I think in a presentation of indistinguishables I wouldn't be able to distinguish zero from one or infinity. A count would be impossible. >> But once I have counted an element and want to move on to the next >> element, I don't know whether or not I am counting the next element or >> the one before it. > > Always go in one direction? I don't have an element in the string that is an 'element of direction'. If each element was also an element of direction then the string would be a line perhaps or would have a structure. But again, I argue that a string cannot support such a structure. It must come from outside, from the operator, who can, for example, distinguish left from right and impose it on the string. > In a presentation of indistiguishables I wouldn't be able to make a > count at all, not even a count of one. I might be able to count a > concept, but I wouldn't be able to count an object. What's the difference between a concept and an object? Mitch >> In a presentation of indistiguishables I wouldn't be able to make a >> count at all, not even a count of one. I might be able to count a [quoted text clipped - 3 lines] > > Mitch A concept wouldn't have any spatial or temporal boundaries or structures. It would not be a sensory object. I would have only a general case. > >> In a presentation of indistiguishables I wouldn't be able to make a > >> count at all, not even a count of one. I might be able to count a [quoted text clipped - 7 lines] > structures. It would not be a sensory object. I would have only a > general case. So is the number 2 a concept or an object? Mitch >>>> In a presentation of indistiguishables I wouldn't be able to make a >>>> count at all, not even a count of one. I might be able to count a [quoted text clipped - 8 lines] > > Mitch Frege would have it that the number 2 is an object. An object that presumably points out two objects. Number per se (but not any named number like 2) looks like a concept. Also according to frege the number 2, like all numbers, have unique identities if they are objects. But the unique identity of the number two is that it is a pair, which is a definable, single object or entity. But then we lose the distinction between a pair and two singletons, so I would say that 2 is not a pair and hence is not an object. > >>>> In a presentation of indistiguishables I wouldn't be able to make a > >>>> count at all, not even a count of one. I might be able to count a [quoted text clipped - 16 lines] > identities if they are objects. But the unique identity of the number > two is that it is a pair, I don't see that. The number 2 describes collections that are pairs (pardon any circularity), but is not necessarily a pair itself. (like the guy over in the sci.math thread who thinks that zero is nothing when really 0 -describes- the null set.) > which is a definable, single object or entity. > But then we lose the distinction between a pair and two singletons, so I > would say that 2 is not a pair the conclusion, yes, 2 is not a pair, but that doesn't follow from the lack of distinction between a pair and 2 singletons (there -is- a distinction?). > and hence is not an object. and that doesn't follow either from all those previous statements, whether right or not. Mitch >>>>>> In a presentation of indistiguishables I wouldn't be able to make a >>>>>> count at all, not even a count of one. I might be able to count a [quoted text clipped - 18 lines] > the guy over in the sci.math thread who thinks that zero is nothing > when really 0 -describes- the null set.) I meant that if 2 has a single identity, then I must know what it is, and I must look for that identity. But I can't say that it's 'twoness'. The only candidate I can think of, for good or bad, is that the single identity of 2 is a pair. >> which is a definable, single object or entity. >> But then we lose the distinction between a pair and two singletons, so I [quoted text clipped - 3 lines] > lack of distinction between a pair and 2 singletons (there -is- a > distinction?). I don't know what the identity of two is, if it's a single identity, except that it might be a pair. The mystics said that numbers had identities, but why Frege said that numbers had identities well I just don't know. >> and hence is not an object. > > and that doesn't follow either from all those previous statements, > whether right or not. I don't know what it could mean for a number to have a single identity. Objects can be regarded as numbers, voided of the differences that make them qualititaively different. What makes a number distinctive in that case is Space, which allows us to discern similar objects. But even that would not mean that numbers have unique identities. > >>>> There's no left or right in a string of beads. > >>> OK. But there is one end and the other distinct end. [quoted text clipped - 13 lines] > I'm referring to the plain observation that a finite string has length > with two ends. It would seem obvious. Furthermore, any string, finite or infinite, with two ends, one end or no ends, has the property that between any two of its members there are at most only finitely many others. This is quite different from a line, ray or line segment. > > A lot of the progress on your problem would be made much easier if you > > were a bit more precise, and by that I mean mathematical. Lack of [quoted text clipped - 22 lines] > > > >> So does a string have a middle? Only if it is a line. My strings have middles whenever they consist of an odd number of items. > > What? Why would that possibly be relevant? > > A string can have a middle if it is a finite line. Strings, unlike lines, have no more than finitely many members between any two of their members. > >> But I say that a string is not a line. Right! > > Sure you can say that by fiat...define string, define point, then the > > rest of us can decide. > > > >> We cannot tell left from right in a string because > >> a string is not a line. And if a string is not a line then it has no > >> middle. The string "abc" has a middle. >>>>>> There's no left or right in a string of beads. >>>>> OK. But there is one end and the other distinct end. [quoted text clipped - 57 lines] > > The string "abc" has a middle. Any 'structure' which has unnamed (unpositioned) elements between any two of its named (positioned) points is fragmented, whether or not there are a finite or infinite number of elements. So while you use the argument that a finite number of elements distinguishes a string line from another (mendellsonian) line, neither of your formulations are, in fact, lines. > I'm referring to the plain observation that a finite string has length > with two ends. It would seem obvious. It IS obvious. It is THEREFORE EQUALLY obvious that it has a middle. If it has an odd number of beads/characters/whatEVER then the middle is just the middle one OF THOSE. And it doesn't even MATTER which way you choose to orient the string, which END you NAME left OR right! The middle IS UNchanged! If it has an even number then the middle is not a bead but it is still a breakpoint BETWEEN two characters/beads/whatEVER, and AGAIN, its location does NOT change EVEN if you reverse left and right! Strings with 2 ends have 2 orientations; every string has a reverse. Strings of length 0 and length 1 are EQUAL to their reverses, though they of course are not the only strings with this property. > Strings with 2 ends have 2 orientations; every string has a reverse. Strings have no orientations or reversals. They are not spatial entities. > Strings of length 0 So now lines can have a zero length. > and length 1 are EQUAL to their reverses, though > they > of course are not the only strings with this property. What about box strings, like numbers on dice. > Strings have no orientations or reversals. They DO SO TOO, DUMBASS. > They are not spatial entities. I NEVER SAID THEY WERE. We are talking about 1st and 2nd here. That IS NOT spatial but it IS STILL one of two opposite orientations (the other being 2nd and 1st of the same two things). > > Strings of length 0 > > So now lines can have a zero length. We ARE NOT TALKING about LINES, DUMBASS! LINES are from the CONTINUUM! THEY have UNcountably many points! STRINGS are DISCRETE! They have AT MOST COUNTABLY many elements! A STRING is A LIST of characters! > What about box strings, like numbers on dice. You GENERALIZE those from THE ORIGINAL LINEAR version by ADDING A SECOND DIMENSION. A box string is a string OF strings. It is a 2-d MATRIX of characters. Equivalently/alternatively, you could take the face of the die as an atomic symbol and DENY that it has bit-strings-in-dimensions as parts. >> Strings have no orientations or reversals. > [quoted text clipped - 3 lines] > > I NEVER SAID THEY WERE. Orientation and reversal are spatial. > We are talking about 1st and 2nd here. There's nothing that makes the left-most element the 'first'. There is nothing that makes any element 'the first'. How can a string, or anything for that matter, tell us which of its elements is 'the first'? The 'first' is not part of the structure of the string, or of the line. The first isn't even part of a count. Same goes for 'start'. It is we who start, not the string or the line. > That IS NOT spatial but it IS STILL one of two opposite > orientations (the other being 2nd and 1st of the same two [quoted text clipped - 4 lines] > > We ARE NOT TALKING about LINES, DUMBASS! Good. So a string isn't a line. Yet you like to think it has a middle. > LINES are from the CONTINUUM! Waffle. > THEY have UNcountably > many points! So lines are made of counted and uncounted points. The uncounted points aren't placed. That means that a line is discontinuous, fragmented. > STRINGS are DISCRETE! They have AT MOST > COUNTABLY many elements! You can count them, yes. But don't think that your count defines the string. Your count defines only itself. > A STRING is A LIST of characters! > [quoted text clipped - 6 lines] > atomic symbol and DENY that it has bit-strings-in-dimensions as > parts. So where's the middle of a box string die with 6 sides? |
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