November 29, 2018 at 1:56 pm #6710Paul HowardParticipant
If I remember correctly, I’ve read that time is circular, that time has a tree structure, and that trees don’t have loops. These seem to be conflicting facts. How should this be understood?December 2, 2018 at 8:10 am #6711
Both space and time are loops and trees. Loop means it is a closed region of space and time. An example of a closed region of space is a house, and an example of a closed region of time is an hour. The hierarchy is that a house is inside and a city and an hour is inside a day.
A more correct way to think about it is a house is a “unit” of space and an hour is a “unit” of time. Therefore, even though space and time can be subdivided we can understand in terms of larger units. Without these units, “hour” and “house” would not be individually meaningful; we would reduce these units always to smaller and smaller units in a reductionist theory.
One consequence of this hierarchy of space and time is the uncertainty relations in atomic theory. The atomic particle’s time is said to be “uncertain” because it is spread in time. Just like you can say that the year is 2018 but the year is not yet completely finished. Similarly, I can say that I’m in a city but I’m outside my house. There is hence a distinction between a position state and a classical position. The position state is like being in the house, and the classical position is like being in a particular room of the house. Atomic theory speaks about the position state which is certain, but the classical position is uncertain. However, experiments have shown that we can fix the position state to an ever-increasing degree of certainty, which is like saying that after I fix my position as being in the house, I can further fix the position of being in a room.
The hierarchical position and time are needed because of meanings. Just like you can say that the time is half past 12, but you haven’t said which day we are talking about. Similarly, you can say that I’m in the bedroom, but you haven’t said which specific house I’m in. So ‘bedroom’ is not universally meaningful; it is meaningful in relation to a particular house. Likewise, half past 12 is not meaningful universally; it is meaningful in relation to the particular day. The day is meaningful in relation to the month, the month in relation to the year, etc.
So, what we call the ‘cycle’ of time is actually a unit of time like a day that begins and ends, and the hierarchy of time is that the day is inside a month, which is inside a year, etc.December 4, 2018 at 2:38 pm #6714Paul HowardParticipant
That makes sense. Thank you.
I was thinking before that loop means a circle, but then I remembered when I was a kid having a toy car race track with a loop. The loop had an entrance and exit that were adjacent, rather than being an actual circle. This makes sense to me because, although a house is a closed region of space, it must have one or more doors. Hierarchical loops give the impression of an extended spiral, like the threads of a wood screw, but then it’s not clear where the doors would be.
The idea of looping space reminds me of a Pac Man game where exiting the screen on one side is followed by entrance on the opposite side, except perhaps on a different board or level.December 5, 2018 at 12:14 pm #6716
There are three descriptions of space–(1) a tree, (2) lotus petals, and (3) spherical. These appear to be confusing normally but you can think in terms of the circle limit diagrams. http://mathworld.wolfram.com/PoincareHyperbolicDisk.html
The center of the circle is the root and the triangles are the branches. You can also think in terms of a lotus flower. Also, because the triangles toward the circumference become smaller, you can never reach the circumference, and this is an analog of Zeno’s paradox.
In terms of numbers, if the center is visualized as 1, and the successive emanating leaves as fractions of 1, then the fractions get smaller and smaller, and they are all parts of the center, and yet different from the center, and there is potentially no limit to dividing, and yet no matter how many times you divide, you never actually finish dividing. So, there is a “circle limit”, which means that space is infinite, and yet there is a boundary to this infinity that cannot be crossed.
Then again you can view this picture as the flattened projection of a sphere, in which the center is the north pole and the circumference is the south pole. The exception is that if you begin from the north pole, you can never reach the south pole, but you can get closer and closer to it.
If you draw this kind of geometry with colors, then the north pole will be white, and the south pole will be black. That type of drawing is called a “color sphere”. White is the full color and black is lacks color, but you can never get pure black because it could never be seen.
So this kind of picture is useful in visualizing the tree with closed spaces which then spawn more closed spaces. And even though you can imagine infinite divisibility you can never reach that infinite division. This applies both to space and time. This kind of space is hyperbolic but if you insist that equal distance is traveled in equal time, then the same thing becomes a sphere. In other words, if you presume that the world is uniform the tree becomes a sphere.
This is the approximate sketch, but it needs formal mathematics (for semantics) in which we are able to represent the branches as parts of the whole and yet separate from the whole. I still don’t understand it well enough but if I do someday I may be able to write that geometry.December 26, 2018 at 5:29 pm #6730Danakeli DasiParticipant
You wrote, “This applies both to space and time. This kind of space is hyperbolic but if you insist that equal distance is traveled in equal time, then the same thing becomes a sphere. In other words if you presume that the world is uinform the tree becomes a sphere.”
So does this mean that one’s presumptions held during perception can literally alter the experienced or perceived picture of reality (say of our Earth) from being a tree to being a sphere (& vice versa)?January 1, 2019 at 5:08 am #6745
Maybe it did not come out the way I wanted it to. What I meant is that the senses perceive only one type of sensation — e.g. color, form, and size by the eyes. Each of these forms a three-dimensional space. Typically when we speak of three-dimensional space in science we just mean in terms of ‘size’. We don’t think about color and form as a separate space. But there are separate spaces and in each space, we measure one property — e.g. color or size. But the mind combines all these spaces into a single space, so we can say that there is a single space but it is mental, not sensual. This mental space is like a tree with its root at the center and leaves at the periphery. The hyperbolic plane represents this idea. This plane is actually infinite in the sense that you can never reach the boundary; the ‘units’ of space and time become shorter as you move outward, which means you travel the same distance in the same time, so if you try to escape this space, you can get closer and closer to the boundary but never reach the boundary.
Now to our observation, it would seem that the universe is infinite or this space is infinite IF we think in terms of a linear geometry — i.e. that space and time uniform. But factually space is bound — i.e. it has a definite boundary. To understand this boundary we have to change the geometry to hyperbolic and say that the universe is finite or bounded, but you can never go outside the universe no matter how far you go in any direction. In this space, as you move out in some direction, you travel less and less but your clock moves faster and faster. So you age quickly and travel less and you think that you have gone a lot of distance but you haven’t moved much. You can say that as you move outward your vehicle is like a boat tied to the shore.
But if you want to think that your boat is not actually tied and you are moving as fast as before then space will be spherical. Light will actually take a longer time to come from the periphery so you will get the sense of a ‘perspective’ that some part is farther away from your eyes, which leads to the perception of roundness. It is not just a belief but also a sense experience. The only problem is that there will be a part of the earth that we can never reach no matter how hard we try. This part corresponds to — in the spherical model — the periphery or the ‘south pole’, assuming the center of the space is ‘north pole’. We don’t actually know what the actual north and south poles are from the hyperbolic geometry mapped to our present earth spherical model. If we know that properly then we can do geography and identify that point on earth where we can never reach, although we can get closer and closer to that point from all directions.January 2, 2019 at 12:44 am #6747Danakeli DasiParticipant
Oh, yes, this is more clear to me now. Very interesting! Thank you.
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