Wesmorris: Glad to have you throw in your two cents into this debate. You're fun to discuss things with. We cannot properly reference that which both isn't and which cannot be considered to be. That is, a "point of nothingness" can only be held as an erroneous concept. Once one discusses what nothingness is, one cannot hold that there can be points of it. I would disagree. In as much as I hold that points represent real positions. Yet consider what a point of nothingness would entail: It would entail not being there at all. When one says "there is no point" is not one simply saying "there is a point of nothingness"? With zero being a point? Or what?
We can't talk about things with no size? How can we talk about a line, when it has zero width? Or a plane when it has zero thickness? Or a cube when it has zero dimensions in the 4th dimension? Of course a line exists, and of course it has no width, or it would be a plane. Of course a point exists, and of course it has no length, or it would be a line. Your logic is the reason many cultures never invented the number '0'. They thought that the presentation of the absence of a thing was worthless. They were as wrong then as you are now. (I looked over the rest of your post, but its presentation is too weird for me to reply in kind. Please trust that I will remember the things that I posted 10 minutes ago, and that other readers can scroll up to jar their own memories. Playing with each sentence seems like a method for winning a game, or keeping score, not a way to facilitate communication)
Actually, we properly can't. Once it is revealed that these concepts are erroneous. Consider a two dimensional plane suspended in three dimensional space. What would happen if one were to poke one's finger through it? There'd be no resistance, as the object would have no depth. How then can it be connected? I was doing it only to make it so I didn't have to summarize lines and lines. No out of context intended.
Of course I am. Please Register or Log in to view the hidden image! You are speaking of space in particular. There is not "nothing" there, there is an idea there representative of a particular point in space of no size. In reality there is no substinative "there" that we can truly communicate about. If we discuss it, we do it in terms of ideas that are representative of there. In coordinate systems such as those in question, points have no size. They are NOT however, "real" as you seem to mean it, but "representative" of "real". What IS real can only be interpreted via data and processed via a model of some sort, like a coordinate system. That's exactly the thing, and the root of what seems to me to be your misconception. You are holding something "real" to an abstract, when in fact is it only the abstracts that we can discuss or mentally manipulate. Consider a stardard cartesian coordinate system. What are the implications if a point has to have a size? This limits the descriptive power of the system. Of course, every system that is representative of something real and is well thought out, will have a "maximum accuracy" of x decimal places or whatever. But if we speak of the system itself, "infinity" is the maximum theoretical accuracy and as such, a point can't have a size. Perhaps you can arbitrarily choose "hey, anything beyond a google decimal places is useless", but in doing so... well you surely see what I mean. A point has the required size of the application at hand I suppose. If I can't measure beyond .0001 inches, then every point in the system could in a way be said to be +/- .00005 inches or something like that... but even then you're really just admitting to the limitation of your measuring device rather than stating something about the reality of what you're measuring. No. If there is no point, then there is no "x,y". IF there is a point, x=X and y = Y. Note that for instance, the number five is exact to infinite decimal places. Hmm.. it's not that straighforward. Imagine for instance that from a three dimensional being inhabiting 7 U 8 U 9, dim 6 would appear to be a point. Know what I mean?
Wesmorris nailed it. What strikes me as odd is that you complain about the abstract nature of points, lines, and planes... but you use the concept of the infinitesimal in your epistemology. Surely you don't think that quarks are made up of an infinite number of the smallest little things possible, right?
Swivel: Actually, yes. I do not view these things as mathematical constructs, but physical realities. As space shares the same divisibility as any finite number, even though the Planck limit may forever restrain our further investigations beyond the point where photons carry enough energy to induce black holes in the concentration of space.
Wesmorris: Yet think of what a point of no size would entail? If it is of litterally no-size, how can it be referenced? Where would the point be? Abstract notions must be coherent. To speak of anything with no-size yet existing is, to say the least, erroneous. I fail to see how an infinitely small point would have any problem whatsoever in its accuracy? I fail to see how 5 and nothing are linked? I'm afraid I have a hard time grasping upper dimensions. Why would a 7-8-9 dimensional being see 6 as a point?
But... I already did that and explained it to you as well as I could at the time... By coordinates. Wherever the coordinates say, of course the coordinate system is pertinent... Apparently not! HA! Hehe... (teasing you, pardon) Not at all really. It's a reference. What physical size does the word "size" entail? Me too, but you've apparently misunderstood. Read that bit again and see if you can find some sense in it, realizing of course that we cannot speak of "das ding en sich" or whatever (the thing itself, separated by observational distance). We can only model it and test to see if our model fits, to the extent of the accuracy of the tools we have available. As such (since our tools get better and better), the model must have the capacity for infinite accuracy which means points of no size. I've talked with physists types on this board who think plank length is a load of crap idea. If x=5 and y=5 is infinitely accurate, the point it describes is infinitely small. It is analagous to how WE see things... assuming from a 3D + time being such as us sees 2D as planar, 1D as linear and 0D as a point. So to an N dim being, N-3 appears to be a point.
Basically, you're suggesting the quantization of absolute space, which is where my plank length reference came from. But it seems to me that you're mixing it up with the models with which we represent that space. They're really two different discussions, though it's certainly fair to discuss their interaction... which I briefly described in my description of measuring equipment above. If we speak of coordinate systems, we speak of a model that can be applied to anything real or even imaginary. If we apply it to something real, we can only really talk about "reality" to the extent of the accuracy of our measuring equipment. To say that our measuring equipment is accurate to x decimal places in U units, is not to say anything about the limits of the reality we measure. Further, to say a point is zero size is not to say that it is outside space necessarily, as we do not know for certain if space has a minimum size (though it is suspected by many I do believe). To ask: "where does this point of no size exist?", I say "in space, at the coordinates listed". It does not require any size at all to be very useful in terms of the coordinate system being used for the discussion. In fact, a point of no size it generally of far greater analytical power than a point of some arbitrary size, depending of course upon what you're looking to discover.
Wesmorris: Coordinates can only reference a space. On a Cartesian grid, 3,3, -4,6, and 7,-4 are all spaces, not "non-spaces". And certainly the lines do not intersect at "no place"? Good one. Please Register or Log in to view the hidden image! Any size, or rather, the notion of size. Infinite accuracy hardly means no-size. Infinite accuracy would be achieved by reaching the infinitely small far more coherently, as one would still be speaking of position, rather than an absence of position. You agree that any point on any 2d grid can be essentially explained as an intersection of two lines, yes? You also agree that the intersection is a real point of intersection in the grid, whether it be written down or not? And that this would have a space on the grid, no? I agree, in regards to it being an indivisible point of space. I do not agree as I believe there is much in the way of empirical science to show that this is the space where further investigation is currently impossible, owing to the photon litterally turning space into a black hole at that level of concentration of the wave. Agreed - but infinitely small is not zero. Nothing would not be small, as it would cease to have size. So essentially, if we moved one step up, you would imagine our first dimension would be considered a point to a 4th dimensional observer? And this would be held to be fundamental? Well clearly I would disagree, as evidenced by above. We'll be discussing it I am sure!
Dude, "Coordinates can only reference a space. On a Cartesian grid, 3,3, -4,6, and 7,-4 are all spaces, not "non-spaces". " Wholly incorrect. Coordinates reference features in space (such as points, lines, volumes, etc). The space is defined by the coordinate system itself. Coordinate systems are imaginary constructs used by minds to enable them to model things. They do not exist in nature outside of that.
Lines intersect at coordinates, which I've explained clearly, but you seem to be unable to firmly grasp. *bows respectfully* Pardon, I meant "what size is the notion (or meaning) of size?". It's the same as a coordinate. They are meanings, not sizes. They are references to ideas. The limit of x approaching zero is zero. The limit of inifinitley small is zero. The second half of your sentence contradicts itself. You can easily speak of a position of no size by referencing its coordinates. The reference to a coordinate of zero size does not mean its absence. You can define a line by two points on the same coordinate system, which is generally easier but sure. There is no grid unless it's being discussed or at least, imagined, regardless of whether or not it's written. Grids to not exist in "reality". So no, I do not agree with the principle I think you're expressing. No. How do you know? I don't. It would seem the nature of space is quanitization really, given energetic precedent, but regardless... I do not know. Interesting perspective, but obviously flawed I think anyway. To my understanding, a photon in no way "turns space into a black hole at the leven of the concentration of the wave". Light can't escape itself, but I wouldn't define that as a black hole. Oh, and again you're confusing "reality" with "the tools we use to describe reality", and you clearly do not fathom the implications of their interaction. The limit of it is. Have you taken calc? Like ideas do not comprise physical space. References and such. Like coordinates. Yup. I think that's right. No, it's an analagy to describe something that we can't actually test, as we are not able to alter our dimensionality. I'm not arguing from theory here. It's my understanding that this is fact stuff. I just don't think you really understand coordinate systems and their application. Then again, perhaps I'm retarded. *shrug*
Wesmorris: What is a coordinate? A place on a grid. Well properly, it's a universal. That is, a word signifying all sizes as taken abstractly. No finite value can ever reach zero, however. To do so would require a process of division which could somehow reach zero, yet clearly this is impossible. Moreover, as it is a point of conjunction, it is quite obviously not a zero-size area, but infinitely small - they are meeting at a particular space. Coordinates reference a place. How can it have no size if it is a reference to a place? Yes, but what is a coordinate? A position on a grid which is referenced in relation to two axises, as if two lines were intersecting that point, yes? Actually, all coordinate systems are essentially a Cartesian grid, are they not? Whatever the amount of axises we're speaking of. So that though you are correct in saying that there isn't a grid, as in a drawn one, in any real situation, but it can be so expressed as such. Planck space seems more of an observational limit than a real limit. Actually, that's a definition of a black hole. Light being incapable of leaving a space. As it should be clear by now, I don't view the tools and the reality as ultimately that different, assuming that we are discussing tools with objective basis. The limit of the infinitely small is indeed zero - but as with all limits, it never reaches said limit. Nor could it be zero even if it could reach its limit, as I have hopefully proven in this thread (if you want to read some of my posts I've posted before on my rationale). You do realize that coordinate systems are -geometric-, yes? That fundamentally this implies space, yes? Being as geometry is the mathematics devoted to shape. Cartesian coordinates were the great merger of algebra and geometry that was thought impossible until our friend Descartes did his magic. Then I don't grasp its relevance? I'm inclined to the latter belief. Please Register or Log in to view the hidden image!
Wesmorris: Ah, there is our problem. I had an inkling this was such. You misunderstand my entire thesis here in relation to points and space: I am not saying that a mathematical construct has to have space in the sense that we're discussing the grid existing somewhere in physical space. Rather, that it has mathematical space. That is, I am not suggesting that a thought must exist like a rock, or that any arbitrary Cartesian space must represent a position in space we can discuss through our senses. Rather, I am saying that the notion of space in mathematics does not permit of non-spatially extended points, when it can be shown that such is an absurdity brought about by improper conception of dimension.
If you use "place" as synonomous to "exact location" then yes. Otherwise, no. That isn't the answer to the question that was asked. I asked what is the size of that abstract, in reality. Ideas don't have physical size, but can reference such things. Because the place it references is abstract and only representative of something real. Depends on the coordinate system. Essentially? Not sure. I'd guess no. I'd also guess there's lots of coordinate systems of which I'm unaware. Spherical coordinates seem different to me..., but well I guess it depends on what you consider the "essence" of corrdinate system to be. We can generally transform between coordinate systems, but does that make them essentially the same? Does that mean they "exist in reality" outside of abstract constructs? So actually no I don't think they are essentially a cartesian grid. They are essentially "constructs for tracking exact locations". What can be expressed as such? Dunno. If you can show that your equipment is capable of measuring below it, then that might help resolve the issue. No, it isn't. That's a property of it. As it should be clear by now, I don't view the tools and the reality as ultimately that different, assuming that we are discussing tools with objective basis. Ultimately, the idea is to find tools that work exactly as you apparently desire. This is not however, the current case as far as I know. But a point in a coordinate system is not "infinitely small". It's zero size. If 5 is infinitely accurate, that reached infinity and the limit of zero error, meaning zero size. Sure? It fundamentally implies an imaginary, abstract space, yes. What a cat. Trying to help you understand dimensional stuff. As is your perogative. At certain times I agree with you.
Wesmorris: Well as location itself is related to place, yes. By which it can be said they are part of "mathematical space". There is no size to words or concepts, as there of rocks and dogs. Geometric and mathematic concepts do have space, however. Mathematical space, but space nonetheless - although clearly I am not saying they have space as kittens and ponds do. Mathematical space. Well "x" axises, let's say? Well spherical coordinates would essentailly be a three-axial Cartesian grid, potentially in a non-eucliden geometry. Either way, the principles would be the same. But I am not making this into a question of an "essence". By which I mean, this is not a debate on essence like we had on the "essence of mind". But in regards to whether they exist outside of abstract concepts: If something can be measured within a system, does not the system say something about the measured thing? Any physical space can be represented in a Cartesian grid with a suitable number of axes. Well considering a black hole is an object whose escape velocity exceeds light, saying light cannot escape is basically saying a black hole. If something can be completely represented in another way, do not we say that the two are isomorphic? Pardon? It cannot be a point with no size if it discusses a position. WEre this the case, where could we get the coordinate? 1,2, for instance, implies the intersection of two lines at the designated coordinates. This point, then, is the convergence point - which if it were no size whatsoever, could not be converged to. Remembering that I mean mathematical size. Yes. Precisely. The mathematical space of which I have spoken of. Coo'.
Oh.... well then if it's simply mathematical size you're discussing, you'll have to provide a mathematical proof. I cannot, and likely coudn't properly courtique what was presented. You pretty much need a doctor of mathematics, which I am not. I know the stuff far better than the average person, but far worse than a professor of mathematics. I would say from intuition, that the implicaiton of your claim is ridiculous... as no coordinate could be exact. It would have to be for instance listed as (x (+/- something infinetely small or whatever you think it should be), y (+/- something infinetely small or whatever you think it should be)) - in pure mathmatics, which by its definition as far as I understand it, avoids such unnecessary complication by defining integers as exact. Perhaps you could establish a correlary or something establishing the minimum size of your number. In pure mathematics, (2,1) as the intersection you offer, does not actually exist and as such, does not require a physical size. It is a point of reference saying "lines cross here", but as lines have no depth, they do not create a space in their crossing. To say they have a size is to say the numbers describing them are not exact.
