Number Line - Numberphile
video description
In 0 dimensional space, you only have points. There is no length, area, or space of any sort.
In 1 dimensional space you only one line. You can mark points along that line with points, but they are just markers, abstractions from a lower dimension. The line itself only has length, you can go two directions along the line only. The line has no thickness, takes up no area.
In 2 dimensional space you only have one plane. You can mark points around that plane with points, but they are just markers, abstractions from a lower dimension. The points take up no area. You can demarcate the plane into sections with lines, but they are just markers, abstractions from a lower dimension. The lines take up no area. The plane itself only has width and height, you cannot leave that plane. There is no depth, the plane takes up no space.
In 3 dimensional space. etc.
Points, lines, planes, etc are only markers. They can mark out a subsection of a dimension, but they can start from any arbitrary marker name. You can't take away a marker, you can only change its name. Just as you can't take away 3. 5, you can only decide to call it something else. The concept of the point that previously was named 3. 5 will still be there, still be nameable.
Date: 2022-04-08
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Comments and reviews: 9
iiGetFaded
Numbers are our way of trying to measure the infinite. We want to believe that there is some distance between 0 and 1 however this is paradoxical. In order to have numbers first we must be able to take a point and change its position. 1 represents the smallest change you can make to your position. However thanks to Zeno we understand that you can never change position because before you can move to the position you are going to be you must go half way. In other words when I mark numbers on a number line after I mark 0 what tells me how far to mark 1 away from 0. It doesn't matter how far you mark the 1 because there is no distance on a number line. The number line itself is the concept of a point that is stretched which cannot be done because there is no length in a point to stretch. The conclusion is that the only numbers that exist are 1 and 0. Any other number is our attempt to find a common ground which there is none.
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Numbers are our way of trying to measure the infinite. We want to believe that there is some distance between 0 and 1 however this is paradoxical. In order to have numbers first we must be able to take a point and change its position. 1 represents the smallest change you can make to your position. However thanks to Zeno we understand that you can never change position because before you can move to the position you are going to be you must go half way. In other words when I mark numbers on a number line after I mark 0 what tells me how far to mark 1 away from 0. It doesn't matter how far you mark the 1 because there is no distance on a number line. The number line itself is the concept of a point that is stretched which cannot be done because there is no length in a point to stretch. The conclusion is that the only numbers that exist are 1 and 0. Any other number is our attempt to find a common ground which there is none.
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Topi
Even though there is a rational between two rationals there is not enough rationals to cover the numberline. If you take all algebraic numbers they are not enough to cover the whole number line. The ways to cover it is to imagine some limiting system that finds all the trancendental numbers in beween those algebraic numbers and run with it. That's how reals are constructed and actual measuring possible.
I'd like to know what this guy thinks about Banch-Tarski paradox. My take on it is that it is not paradox as it tries to do something that is not possible. It breaks down a sphere into non-measurable parts and the tries to convince everybody that just doing set addition on these non-measurable blobs you get measure back. Who thinks like that? Like this guy tells measure is not just adding to disjoint sets together. It is a processs that takes much more than that.
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Even though there is a rational between two rationals there is not enough rationals to cover the numberline. If you take all algebraic numbers they are not enough to cover the whole number line. The ways to cover it is to imagine some limiting system that finds all the trancendental numbers in beween those algebraic numbers and run with it. That's how reals are constructed and actual measuring possible.
I'd like to know what this guy thinks about Banch-Tarski paradox. My take on it is that it is not paradox as it tries to do something that is not possible. It breaks down a sphere into non-measurable parts and the tries to convince everybody that just doing set addition on these non-measurable blobs you get measure back. Who thinks like that? Like this guy tells measure is not just adding to disjoint sets together. It is a processs that takes much more than that.
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ED
-So actually you are not adding 1 to 3 to get to 4.
-From 3 you are moving 1 unit to get to 4.
-The units are a wormhole but in the number-line.
-I also imagine the number-line to not be a straight line in this case. I feel the number-line to be more like separated universes where each number is it's own universe.
-Because every number is a point, every universe is 0 dimensional.
-Each wormhole is a formula you are using to get to another universe(number.
-Which means there are more wormholes than universes(EDIT: I'm doubting this after typing the rest below(SECOND EDIT: I'm not having doubts anymore, it must be correct. You can use multiple formulas to get to the same number.
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-So actually you are not adding 1 to 3 to get to 4.
-From 3 you are moving 1 unit to get to 4.
-The units are a wormhole but in the number-line.
-I also imagine the number-line to not be a straight line in this case. I feel the number-line to be more like separated universes where each number is it's own universe.
-Because every number is a point, every universe is 0 dimensional.
-Each wormhole is a formula you are using to get to another universe(number.
-Which means there are more wormholes than universes(EDIT: I'm doubting this after typing the rest below(SECOND EDIT: I'm not having doubts anymore, it must be correct. You can use multiple formulas to get to the same number.
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Chris
At school the teacher had a great big ruler to do stuff on the board with. Took two of us kids to lift. It also had correspondingly thick division lines along the edge, 1cm thick if my memory serves. So that's what the number 3. 5 measures on that ruler: 1cm. Clearly though there's no way that big ruler could have been used to measure that thickness, indeed how can any ruler measure the thickness of its own division lines? That's the problem we're faced with, the paradox of self-measurement. It's not that the number 3. 5 measures nothing, it measures anything depending on the relative fineness of the scales used.
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At school the teacher had a great big ruler to do stuff on the board with. Took two of us kids to lift. It also had correspondingly thick division lines along the edge, 1cm thick if my memory serves. So that's what the number 3. 5 measures on that ruler: 1cm. Clearly though there's no way that big ruler could have been used to measure that thickness, indeed how can any ruler measure the thickness of its own division lines? That's the problem we're faced with, the paradox of self-measurement. It's not that the number 3. 5 measures nothing, it measures anything depending on the relative fineness of the scales used.
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Matthew
Maths has been broken or almost broken so many times. It is like a war torn country that's been occupied by 20 different colonial nations but it can't give up and stop existing.
Like if we broke maths completely, it would still exist. The number line gets proven to be made of nothing but points with nothing between them but more points and points have no length so the number line has no length. we'd still use it. It's like some poor wheelbarrow with a flat wheels, broken axle, full of concrete, but still being used. Poor number line. It has to work at its own funeral and wake up for work the next day.
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Maths has been broken or almost broken so many times. It is like a war torn country that's been occupied by 20 different colonial nations but it can't give up and stop existing.
Like if we broke maths completely, it would still exist. The number line gets proven to be made of nothing but points with nothing between them but more points and points have no length so the number line has no length. we'd still use it. It's like some poor wheelbarrow with a flat wheels, broken axle, full of concrete, but still being used. Poor number line. It has to work at its own funeral and wake up for work the next day.
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weppan
I have a suggestion of a why to think about this problem when a point has no value as described.
If one takes for example three different point value pairs 3. 1-3. 2; 3. 3-3. 5; 3. 6-3. 9 and look at the infinities in-between and say that the infinities move by the same constant. We can from this see that the infinites expand with different rates 3. 6-3. 9(4x2-n) > 3. 3-3. 5(2x2-n) > 3. 1-3. 2(2-n. Decide the ratio numbers and lets call them infi. 1, 2, 3.
This should give you a way to measure points mathematically, when they have no value via the definition of a point.
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I have a suggestion of a why to think about this problem when a point has no value as described.
If one takes for example three different point value pairs 3. 1-3. 2; 3. 3-3. 5; 3. 6-3. 9 and look at the infinities in-between and say that the infinities move by the same constant. We can from this see that the infinites expand with different rates 3. 6-3. 9(4x2-n) > 3. 3-3. 5(2x2-n) > 3. 1-3. 2(2-n. Decide the ratio numbers and lets call them infi. 1, 2, 3.
This should give you a way to measure points mathematically, when they have no value via the definition of a point.
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Peter
Ok so there are two ways of looking at this. The first involves assuming that there isn't a finite number of points between 3 and 4. In other words there are infinite points between 3 and 4. Infinity isn't a quantity and it isn't an amount, it's a potential. So there is an unlimited potential for points between 3 and 4. So it follows that no matter how many points you remove, you don't change the potential. The second way involves looking at 3. 5 as a description of everything leading to 3. 5 and not a description OF 3. 5. So 3. 5 isn't a point but a description. IMO.
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Ok so there are two ways of looking at this. The first involves assuming that there isn't a finite number of points between 3 and 4. In other words there are infinite points between 3 and 4. Infinity isn't a quantity and it isn't an amount, it's a potential. So there is an unlimited potential for points between 3 and 4. So it follows that no matter how many points you remove, you don't change the potential. The second way involves looking at 3. 5 as a description of everything leading to 3. 5 and not a description OF 3. 5. So 3. 5 isn't a point but a description. IMO.
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Maxime
A point is a zero-dimensional concept that has no size, no distance to and from itself, no scale to be applied because there are no OTHER points to create a first-dimensional line. Therefore the point alone is intangible. It can be any number and still mean absolutely nothing without another point sitting elsewhere in space.
Add another conceptual point and we start walking towards the physical reality that we humans know. And even there we're creating a mental compromise because the pure numbers are meaningless on their own.
I love this: D
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A point is a zero-dimensional concept that has no size, no distance to and from itself, no scale to be applied because there are no OTHER points to create a first-dimensional line. Therefore the point alone is intangible. It can be any number and still mean absolutely nothing without another point sitting elsewhere in space.
Add another conceptual point and we start walking towards the physical reality that we humans know. And even there we're creating a mental compromise because the pure numbers are meaningless on their own.
I love this: D
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education
3. 5 - 3. 5 is zero, because zero is our start point. But things either exist, or they don't, and our only way of knowing is by quantifying the information we see all around us. We give form to information, because information is the product of a mind.
Just so happens that counting and measuring is a way of giving form to information that is already there. We quantify things in order to understand them. This has a lot to do with how information theory works in conjunction with physical reality.
reply
3. 5 - 3. 5 is zero, because zero is our start point. But things either exist, or they don't, and our only way of knowing is by quantifying the information we see all around us. We give form to information, because information is the product of a mind.
Just so happens that counting and measuring is a way of giving form to information that is already there. We quantify things in order to understand them. This has a lot to do with how information theory works in conjunction with physical reality.
reply
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