Canute, leeaus and possibly MacM: You guys really need to study up on the meaning of the word "dimension". Dimension refers to the number of numbers you need to specify to locate a point in space. A line has one dimension because to locate any point on a line you need only one number - e.g. the distance from one end of the line to the point. A plane or the surface of a sphere is two dimensional. To locate a point on the surface of a sphere, for example, you need to specify 2 numbers, such as lattitude and longitude. Space has three dimensions. Spacetime has four dimensions. Statements like "the sphere is the first dimension" make no sense.
Sorry but I don't understand that so can't get what follows. Yeah, post the link if there's stuff there that's relevant.
I agree with all that. It is sloppy to say that the sphere is in a dimension since it is outside the normal co-ordinate system. However the suggestion is that there is one degree of freedom more than we normally assume within our normal spacetime co-ordinate system. It is all too tempting to call this additional realm a dimension. In a way it's just saying that the expansion of the universe, which can be described (badly imo but never mind) as if it were the surface of an inflating sphere, is explained by the existence of an additional co-ordinate system at right angles to the normal three/four, and which actually has some of the properties of a sphere when conceived in normal 3D terms, or in its effects on our 3D spacetime. That it has this property explains why we use a sphere in our usual metaphors and analogies for spacetime expansion. God knows if that will make any sense to you.
leeaus: I don't see how you get that dividing an infinite line leaves you with finite length. If I were to divide such a line into three parts, I could get one finite section, but the other two would still be infinte. Dividing it into two parts, I'd be left with two infinite sections.
Thanks JR for agreeing in a round about way. The number of numbers brought down to what it is happens to be one thing. What you probably need to understand is that there is no such thing as a smallest point. What you are calling a point in space is always a three dimensional point. Thus if you can’t get down smaller than the sphere the sphere is the first dimension. Like you are referring to Cartesian coordinates but each coordinate is always of a three dimensional nature its self isn’t it. Malikri if you can make sense of two infinite distances you are going well. Infinite distance encompasses all distance. This may or may not interest you Canute. http://home.iprimus.com.au/siewk/INFINITE CONVERGENCE/ It is there mainly for why water rises against the direction of the moon gravity. But if you have reasoned logic about why 1 + 1 = 2 would be interested. Did mention it was anti mathematics. leeaus
Dimensions James R., I hadn't entered this discussion but it appears you anticipate that my view would be lacking. Can't imagine why you would think that but in any case I'll give it for your comment. I only see one (1) spatial dimension. The (3) dimensions that we refer to as x,y, and z are for our convience mathematically but they are the same (1) dimensional property or quality. A one dimensional finite line rotated through every possible steradian enscribes a spherical volume. Knowing to believe only half of what you hear is a sign of intelligence. Knowing which half to believe will make you a genius.
JR Appreciate your insertion of the word convenience (spelt as it is) but what we are onto here is a higher level of thought than convenience. The finite line that you are using to describe a sphere it self must be three dimensional in reality. In your mind it can have zero width and breadth but in mechanical reality if you allow a length zero width and breadth it doesn’t exist. So its just a mind game your doing with your self. Not something that can be demonstrated in reality unless of course you produce a non imaginary length with zero width and breadth. You will always find that a period of higher level disciplined thought about the topic leads to the conclusion that the sphere is the first dimension. If you wish to stick solely with imagination without checking in with reality, enjoy your self. Leeaus
leeaus: <i>JR Appreciate your insertion of the word convenience (spelt as it is) but what we are onto here is a higher level of thought than convenience. </i> Sorry, you've lost me. I didn't use the word "convenience" anywhere. <i>The finite line that you are using to describe a sphere it self must be three dimensional in reality.</i> A line is a mathematical idealisation. It has length, but zero width and height. It is one dimensional. <i>In your mind it can have zero width and breadth but in mechanical reality if you allow a length zero width and breadth it doesn’t exist.</i> If you're saying there are no real-world objects which are one dimensional, then I agree with you. <i>So its just a mind game your doing with your self.</i> You too, it seems, since we agree. <i>You will always find that a period of higher level disciplined thought about the topic leads to the conclusion that the sphere is the first dimension.</i> As I said above, that statement is meaningless, unless you mean to redefine the word "dimension". If you're doing that, please post your preferred definition. I've posted mine.
Sorry JR My mistake. It was not your post. Mac was quoting your post. Appreciate your view of a dimension is to specify position using numerical language. Geometrically speaking though what you call a dimension, a point on a line, is its self three dimensional. Thus before you can call length a first dimension you have to accept that length its self is made of the three dimensional points. Thus the first dimension is the point or the sphere. Sorry again for not noting whose post it was. leeaus
Hi leeaus, As James R pointed out, this is a matter of how you define a dimension. You are right in saying that one or two dimensional objects cannot exist in our real world, but as pointed out, this is a mathematical abstraction (and it turns out that some thin surface effects can be neatly explained using two-dimensional mathematical models). However, I think we are better off sticking to the mathematical definition of a dimension, being the number of independent coordinates required to uniquely determine a position on the geometry in question. Otherwise, whatever you write will be confusing for aproximately the rest of the world, as they all interpret dimension in the mathematical sense. "Thus before you can call length a first dimension you have to accept that length its self is made of the three dimensional points." From a mathematical point of view, the concept of length can be defined without the need of three dimensions (one is enough). Bye! Crisp
Dimensions are strange things. They are talked about in physics and in woodwork, in science fiction and in tales of the supernatural, in religion and in quantum cosmology. Yet it is quite hard to know what we mean by ‘dimension’. Can we travel to other dimensions? Does my washing machine have more than three dimension? Does God exist in another dimension? Physics suggests that existence has precisely 11 dimensions, some of which are tangled into little loops. Daily experience suggests that there are only three, with time thrown in as an odd sort of one. Are we all talking about the same thing? What is a ‘dimension’ and how many of them are there really? My modern science dictionary says that a ‘dimension’ is “the number of coordinates needed to specify the position of a point in some space of possibilities”. What this definition says (bear with me) is that if you want to specify the position of a point within a two dimensional graph you need two co-ordinates. There are two dimensions in which the point can vary its position, in other words two aspects to its space of possibilities, the x axis and y axis. This is how we know that it is a two dimensional graph. In order to specify the precise position of a point within the graph we require two measurements, one along each axis, and therefore two values to specify the point’s complete co-ordinates. This is true because we defined our graph as being two dimensional. Likewise, if we now assume that our point is a star adrift in three dimensional space, we require three measurements to fix or specify its position, since its ‘space of possibilties’ has three dimensions, that is how we define it. (Let’s ignore time for now). Now all this is obvious and you may wonder why I am plodding through it. This is surely stuff we all know. However there is something odd about this definition. It is not phrased in everday language, not what you're average Joe would call everyday language anyway, but that is not what makes it odd. Its oddness lies in the fact that even a little thought shows that as a definition it is completely devoid of all meaning. It is a pefectly circular bit of self-reference that tells us nothing whatsoever about dimensions. This is obvious if you consider how you would conceive of a ‘dimension’ with the help of this definition. You find that it does not help at all. In fact ‘the number of coordinates needed to specify the position of a point in some space of possibilities’ is not something you can look at or conjure up in your mind at all. It is just the number of coordinates needed to specify the number of coordinates needed to specify how many dimensions there are in some possibility space in which someone has specified how many coordinates are needed in order to specify the position of some point within it. It is not a ‘thing’ at all. It is an explanation of how to relate our words to each other when we are talking about dimensions. We must therefore discuss dimensions without the help of my dictionary. I take a dimension to be a degree of freedom in possibility space in which values can vary. Spatial dimensions are a particular case, but hot and cold, big and small, are equally dimensions, degrees of freedom. Leeaus's first, and my fourth, spatial dimension, is spherical. It is a co-ordinate system and therefore a dimension. However it is a very odd co-ordinate system. By one view it is a point and by another a sphere. However these are equivalent topographical metaphors, not a description. It is a degree of freedom with the mirrored mathematical properties of both a point and a sphere, not a shape in spacetime. The idea is that if spacetime is the 3D (+1) surface of a sphere then all points on the sphere are geometrically and ontologically underlain by the sphere, and have more than three spatial co-ordinates. (Leeaus - did I get your view right?)
Canute perhaps the ingredient you search valiantly for is the right angle. The coordinate approach that Crisp suggests be used so as we all are on the same line of reasoning is mathematically sound but technically weak. It dismisses the fact that all coordinates require coordinates to be specified. If the right angle did not exist, the nothing of religious theories could exist. If the right angle exists in what would otherwise be nothing you have the first dimension. A sphere. Hope this helps your efforts to get the bottom of what a dimension is. Leeaus
Dimensions are just directions in which you can move, while showing no change in the other directions... ex: I can move in X without changing Y and Z. It doesn't matter how you define them, as long as this still holds true (and you acount for all possible movement).
"It dismisses the fact that all coordinates require coordinates to be specified." Ok, "the number of coordinates you need to specify" is not the proper mathematical way of defining a dimension, usually this is done through a basis for vector spaces etc etc... But I would like to add that you do not need coordinates to specify what you mean by a coordinate. What you need is a coordinate system and a reference point in that coordinate system (and ofcourse a metric that defines how you measure distances). These three are unrelated and can freely be changed by another version (you can freely pick the reference point, coordinates and -- to some extend -- the metric). The mathematics stays the same. Bye! Crisp
Almost Canute but not quite. Persol’s definition is probably the most succinct with respect of mathematical logic. However when it comes to the geometry of a straight line, it doesn’t get there. Persol’s straight line direction has to be relative to the other two dimensions to be a straight line. Like a line of zero width and breadth has nothing to coordinate its self against. Which in terms of geometry brings the right angle into any concept of direction. If each dimension is relative to the other two direction wise, then it follows that a single direction is not an original dimension. That the whole shebang of the three is the original dimension. That is what are regarded as the three dimensions are lines at right angles to each other within a total three dimensional point. However it probably isn't worth worrying about unless you give geometry precedence over arithmetic. If you use geometry as you home base, first and foremost what exists is the right angle. If all you want to do is measure distance you haven't really got to worry about that and Persol's definition is succinct. regards to your quest leeaus
Canute: You were doing well until you got to this part: I don't understand what this is supposed to mean at all.
Leeaus, "Persol’s straight line direction has to be relative to the other two dimensions to be a straight line. Like a line of zero width and breadth has nothing to coordinate its self against. Which in terms of geometry brings the right angle into any concept of direction." Aha, I think I now fully understand your point of view. You cannot say that a line has zero height and zero width and a certain length if you have not defined height and width. Ok, but this is incorrect Please Register or Log in to view the hidden image!. You can define one dimension mathematically without the need for other dimensions - all you need to do is (for example) specify how to measure distances. If you say that the distance between two points x and x+dx is given by dx, then you have defined the idea of "length" in one dimension without resorting to other dimensions. But in this case it is not allowed to speak about width and height (the other two dimensions) as they are not defined in this context. From the point of view of the "real world", you are right: our perception of a line is something that has a certain length, but no width/height --- a statement about three dimensions. This is not surprising I think, as our brain is used to working in three dimensions (?). Bye! Crisp
