I have a following puzzle.
Let us have a simplice of volume-tetrahedron. Its volume is some number. Let us have some process where we require this volume in limit to reach 0 ( the volume) - e.g differential dP/dV.
Question: what happens with the form? what will be the form of the 0 volume?
Tetrahedron, sphere, point (what is that?) etc?
The same question applies to triangle as area simplice as its area is reduced to 0, and straight line as 1D simplice as its lenght is contracted to 0.
Will it still be a line with 0 length, or a point with 0 dimensionality?
Thank You in advance.
AFAIK, when you shrink something to zero size it cannot have any geometric properties that distinguish it from other 'types' of points.
For example, we can define a process where you take a triangle (or a tetrahedron, it doesn't really matter) and consider the vertexes getting closer together until they meet. When this happens the sides will have zero length and the vertexes will be on top of one another.
Now think about a square or cube in the same way. In the limit that the length of the sides goes to zero they are exactly the same object.
In this way you can argue a point with zero dimensionality and a line in the limit that the length goes to zero are identical.
Now this isn't actually very physical, because of the uncertainty principle so objects like strings can never shrink to zero size. I can't claim to know much about string theory though. Ask me again in a few years.
For example, we can define a process where you take a triangle (or a tetrahedron, it doesn't really matter) and consider the vertexes getting closer together until they meet. When this happens the sides will have zero length and the vertexes will be on top of one another.
Now think about a square or cube in the same way. In the limit that the length of the sides goes to zero they are exactly the same object.
In this way you can argue a point with zero dimensionality and a line in the limit that the length goes to zero are identical.
Now this isn't actually very physical, because of the uncertainty principle so objects like strings can never shrink to zero size. I can't claim to know much about string theory though. Ask me again in a few years.
You have two situations
1. The shape has non-zero length sizes. It has form and volume regardless of how samll you make it.
2. The object has 0 length sides, so has no form. It is a point.
The limit of the volume is 0, but that is meaningless to the physical property of an ever shrinking object. The limit cannot be reached for any real value. There is no "transition" as such.
Just my opinion.
An interesting example of infintesimal form is the mandelbrot set. Fractint can be used to zoom in on the boundary of the set. When one has zoomed to 10^26, the original set has expanded to the side of the Universe. there are animations of zooms to 10^89. There is no end. Of course, there is no concept of limit that can be applied to the set and the region around it as a whole. Just the trajectories of points within the set (which are either cyclic, converging, or chaotic).
http://www.fractal-animation.net/ufvp.html
Search for E+89 (the original wmv is the best, with sound)
1. The shape has non-zero length sizes. It has form and volume regardless of how samll you make it.
2. The object has 0 length sides, so has no form. It is a point.
The limit of the volume is 0, but that is meaningless to the physical property of an ever shrinking object. The limit cannot be reached for any real value. There is no "transition" as such.
Just my opinion.
An interesting example of infintesimal form is the mandelbrot set. Fractint can be used to zoom in on the boundary of the set. When one has zoomed to 10^26, the original set has expanded to the side of the Universe. there are animations of zooms to 10^89. There is no end. Of course, there is no concept of limit that can be applied to the set and the region around it as a whole. Just the trajectories of points within the set (which are either cyclic, converging, or chaotic).
http://www.fractal-animation.net/ufvp.html
Search for E+89 (the original wmv is the best, with sound)
So You are all saying the form will vanish together with dimensions.
hmm. How about law of e.g. conservation of information and derivatives/integrals?
The size of the volume does not contain information about its form, neither do the length of its edges.
It is the functional dependency between the variables involved in defining form that is important.
Now we know that the limit of a function depends on its form - e.g if function is x^2, than d(x+dx)^2)/dx when dx reaches 0 is 2x.
That means, even when size vanish, the ratio related to the properties of the curve x^2 remains.
Now, the same must be true in case of reducing 1 dimension, 2 dimension 3 dimensional from some volume to 0.
It is like writing : df(x,y,z) / dxdydz = ???? This something that is left will be related to form, so I can not accept that form vanishes without trace.
if we apply integration to it, we may even reconstruct the previous form with accuracy limited by constant, so in fact, we can reconstruct the FORM exactly, while we can not reconstruct the VOLUME - we do not know from which volume we started decreasing the tetrahedron; but we know it was tetrahedron from the derivative which is left after volume goes to 0.
Once we also know initial conditions, we are able both to regain the Volume or other parameters together with FORM by the means of remembering the integration interval and start point- that is definite integral.
So I answered my question. Form remains, volume does not. Thank You for input
hmm. How about law of e.g. conservation of information and derivatives/integrals?
The size of the volume does not contain information about its form, neither do the length of its edges.
It is the functional dependency between the variables involved in defining form that is important.
Now we know that the limit of a function depends on its form - e.g if function is x^2, than d(x+dx)^2)/dx when dx reaches 0 is 2x.
That means, even when size vanish, the ratio related to the properties of the curve x^2 remains.
Now, the same must be true in case of reducing 1 dimension, 2 dimension 3 dimensional from some volume to 0.
It is like writing : df(x,y,z) / dxdydz = ???? This something that is left will be related to form, so I can not accept that form vanishes without trace.
if we apply integration to it, we may even reconstruct the previous form with accuracy limited by constant, so in fact, we can reconstruct the FORM exactly, while we can not reconstruct the VOLUME - we do not know from which volume we started decreasing the tetrahedron; but we know it was tetrahedron from the derivative which is left after volume goes to 0.
Once we also know initial conditions, we are able both to regain the Volume or other parameters together with FORM by the means of remembering the integration interval and start point- that is definite integral.
So I answered my question. Form remains, volume does not. Thank You for input
@Ivars
I guess for me to understand that I will need a differentiatible form for a tetrahedron.
y = x^2 is a continuous function.
A sphere is a continuous function x^2 + y^2 + z^2 = 1
A cube is not continuous.
So, while your analogy is intriguing, I wonder how to implement it. Maybe you transform it into another space first?
Anyway, if you think about the derivitive as the operation as dxdydz APPROACHES zero, you still can have form and the volume also APPROACHES zero. In the limit, it is zero, but it seems like taking a limit is not a conserving operation.
For example, what is the limit of x^2 as x goes to zero. All form is lost. What is the limit of the volume of a cube as xyz all go to zero. It's a different sort of operation.
So, I think a cube can shrink forever, and will never go to zero. It cannot shrink to zero.
The 0.9r = 1 argument is similar. The limit of the sequence 0.9 0.99 0.999 ... is 1. But, 1 is not a member of the sequence. The limit of a shrinking cube is not a cube.
My math sucks, I'm just guessing.
I guess for me to understand that I will need a differentiatible form for a tetrahedron.
y = x^2 is a continuous function.
A sphere is a continuous function x^2 + y^2 + z^2 = 1
A cube is not continuous.
So, while your analogy is intriguing, I wonder how to implement it. Maybe you transform it into another space first?
Anyway, if you think about the derivitive as the operation as dxdydz APPROACHES zero, you still can have form and the volume also APPROACHES zero. In the limit, it is zero, but it seems like taking a limit is not a conserving operation.
For example, what is the limit of x^2 as x goes to zero. All form is lost. What is the limit of the volume of a cube as xyz all go to zero. It's a different sort of operation.
So, I think a cube can shrink forever, and will never go to zero. It cannot shrink to zero.
The 0.9r = 1 argument is similar. The limit of the sequence 0.9 0.99 0.999 ... is 1. But, 1 is not a member of the sequence. The limit of a shrinking cube is not a cube.
My math sucks, I'm just guessing.
QUOTE (meBigGuy+Nov 9 2007, 09:55 AM)
@Ivars
I guess for me to understand that I will need a differentiatible form for a tetrahedron.
y = x^2 is a continuous function.
A sphere is a continuous function x^2 + y^2 + z^2 = 1
A cube is not continuous.
So, while your analogy is intriguing, I wonder how to implement it. Maybe you transform it into another space first?
hej MeBigGuy
I guess You make it up from from continuous Fourier transform, and than take derivative from that as tetrahedron or cube volume goes to 0.
In 2 D, you can take triangular function , its Fourier transform and than derivative from that as dA ( area) go to 0.
I do not know what is Fourier transform of dA, but something will be definitely left, as Fourier transform of e.g triangular function involves sin^2 (pi w)/(piw)^2 which definitely does not vanish after differentiation.
So the form of shape with sharp corners is preserved in it Fourier transform derivatives as its volume or area goes to 0.
Which means its FORM actually is phase dependent superposition of waves.
Now Fourier transform is not the only way to represent some shape, there could be other orthogonal basis - which leads to a conclusion that form of cube or tetrahedron is actually hidden in phase information between its constituents, not the actual shape of orthogonal functions used.
I guess for me to understand that I will need a differentiatible form for a tetrahedron.
y = x^2 is a continuous function.
A sphere is a continuous function x^2 + y^2 + z^2 = 1
A cube is not continuous.
So, while your analogy is intriguing, I wonder how to implement it. Maybe you transform it into another space first?
hej MeBigGuy
I guess You make it up from from continuous Fourier transform, and than take derivative from that as tetrahedron or cube volume goes to 0.
In 2 D, you can take triangular function , its Fourier transform and than derivative from that as dA ( area) go to 0.
I do not know what is Fourier transform of dA, but something will be definitely left, as Fourier transform of e.g triangular function involves sin^2 (pi w)/(piw)^2 which definitely does not vanish after differentiation.
So the form of shape with sharp corners is preserved in it Fourier transform derivatives as its volume or area goes to 0.
Which means its FORM actually is phase dependent superposition of waves.
Now Fourier transform is not the only way to represent some shape, there could be other orthogonal basis - which leads to a conclusion that form of cube or tetrahedron is actually hidden in phase information between its constituents, not the actual shape of orthogonal functions used.
Ivars if you think of it as being of 'scales' all of it disappears :)
form defines function.
A shell is as much the emptiness 'inside' it as the 'form' defining it.
so while i find your thoughts really cool i would like it to go one step further.
Welcome to the house of illusions
form defines function.
A shell is as much the emptiness 'inside' it as the 'form' defining it.
so while i find your thoughts really cool i would like it to go one step further.
Welcome to the house of illusions
QUOTE (yor_on+Nov 9 2007, 09:29 PM)
Ivars if you think of it as being of 'scales' all of it disappears
so while i find your thoughts really cool i would like it to go one step further.
Welcome to the house of illusions
hej yor_on
Welcome:)
I think all this exactly definable in mathematical terms still but may require some twisting of maths.
e.g. taking of futher derivatives of x^2 eliminates form after 2 steps, while x^3/2 has infinite depth of forms. This seems strange, not to say wrong. On other hand, by taking fractional derivatives if x^2 which are not multiples of 2 we again can continue till infinity.
So the depth of FORM becomes dependant on space we deal with it in and the way we reduce the VOLUME when taking limit.
The question is then is there an absolute mathematical space where each unique form will have its ultimate description, or , from other point of view, are there some fundamental non-reducible forms every other form consists of.
Math seems to be illusion anyway, but FORM and VOLUME seems to be reality. So perhaps math is not an illusion afterall, or it is real illusion, or reality is illusion, whatever pleases us best as long as it works to explain logic behind workings of Nature.
so while i find your thoughts really cool i would like it to go one step further.
Welcome to the house of illusions
hej yor_on
Welcome:)
I think all this exactly definable in mathematical terms still but may require some twisting of maths.
e.g. taking of futher derivatives of x^2 eliminates form after 2 steps, while x^3/2 has infinite depth of forms. This seems strange, not to say wrong. On other hand, by taking fractional derivatives if x^2 which are not multiples of 2 we again can continue till infinity.
So the depth of FORM becomes dependant on space we deal with it in and the way we reduce the VOLUME when taking limit.
The question is then is there an absolute mathematical space where each unique form will have its ultimate description, or , from other point of view, are there some fundamental non-reducible forms every other form consists of.
Math seems to be illusion anyway, but FORM and VOLUME seems to be reality. So perhaps math is not an illusion afterall, or it is real illusion, or reality is illusion, whatever pleases us best as long as it works to explain logic behind workings of Nature.
Sorry, but I don't buy any of your fourier transform stuff. If you believe it, show me with the math.
I repeat
If you think about the derivitive as the operation as dxdydz APPROACHES zero, you still can have form and the volume also APPROACHES zero. In the limit, it is zero, but it seems like taking a limit is not a conserving operation.
For example, what is the limit of x^2 as x goes to zero. All form is lost. What is the limit of the volume of a cube as xyz all go to zero. Taking the limit is a different sort of operation from taking a derivitive..
I am saying that a cube can shrink forever, and will never go to zero. It cannot shrink to zero. It cannot lose its form.
If you perform an operation that makes it go to zero, it loses its form.
The only way to get something to go to zero is to multiply by zero or subtract. Taking a limit has no physical analogy. It is not a process to make something go to zero (as the original problem stated)
I repeat
If you think about the derivitive as the operation as dxdydz APPROACHES zero, you still can have form and the volume also APPROACHES zero. In the limit, it is zero, but it seems like taking a limit is not a conserving operation.
For example, what is the limit of x^2 as x goes to zero. All form is lost. What is the limit of the volume of a cube as xyz all go to zero. Taking the limit is a different sort of operation from taking a derivitive..
I am saying that a cube can shrink forever, and will never go to zero. It cannot shrink to zero. It cannot lose its form.
If you perform an operation that makes it go to zero, it loses its form.
The only way to get something to go to zero is to multiply by zero or subtract. Taking a limit has no physical analogy. It is not a process to make something go to zero (as the original problem stated)
QUOTE (Ivars+Nov 8 2007, 02:05 PM)
Now we know that the limit of a function depends on its form - e.g if function is x^2, than d(x+dx)^2)/dx when dx reaches 0 is 2x.
You mean [(x+dx)²-x²]/dx.
You mean [(x+dx)²-x²]/dx.
QUOTE (Ivars+Nov 8 2007, 02:05 PM)
y = x^2 is a continuous function.
A sphere is a continuous function x^2 + y^2 + z^2 = 1
A cube is not continuous.
A sphere is a continuous function x^2 + y^2 + z^2 = 1
A cube is not continuous.
You mean 'differentiable'. For instance, the sphere always has a smooth shape, while a sphere has sharp points and edges.
y = |x| is continous but it isn't differentiable at x=0.
Ivars, don't even get me started on how much BS you're saying with that crap about Fourier transforms!! You complain that I push you too much when you talk about maths, that you're trying to learn, yet you persist in talking about things you have absolutely no clue about. Why do you pretend you know anything about Fourier transforms? Or simplex constructions of 'volumes' in various dimensions?
You've bearly said anything correct or viable in this thread, you're just spouted nonsense.
y = |x| is continous but it isn't differentiable at x=0.
Ivars, don't even get me started on how much BS you're saying with that crap about Fourier transforms!! You complain that I push you too much when you talk about maths, that you're trying to learn, yet you persist in talking about things you have absolutely no clue about. Why do you pretend you know anything about Fourier transforms? Or simplex constructions of 'volumes' in various dimensions?
You've bearly said anything correct or viable in this thread, you're just spouted nonsense.
Ok,Ok
But the original question remains:
What happens with arbitrary FORM when volume (area, length) goes to 0? In continuous mathematical space?
Do not say it disappears as it obviously does not in case when form is defined by a differentiable function.
E.g. in 1 D FORM would be defined by a curve. A derivative of this curve is obtained by reducing both variable ( unit) and function depending on variable defining curve to 0. So length of curve and measure used goes to 0 simultaneously. But the ratio/functional dependence between these 2 remains, and allows to reconstruct curve (FORM) by integration but not the exact positioning of it.
The other question I posed is why do some functions disappear ( nth derivative =0 ) to a non-re-construable form after finite number of steps, while others does not?
Some, like e^x even does not notice the consecutive differentiations. I know mathematically why, but where does the info about FORM is stored that it is impossible to loose it no matter what?
Some, like sinx , cos x, e^ix behave cyclically, shifting phase by 90 degrees after each differentiation.
Why should some FORMs behave like that, while others disappear leaving no trace? (e.g d^3/dx^3 (x^2) = 0, and no one knows that it was x^2 this time).
But the original question remains:
What happens with arbitrary FORM when volume (area, length) goes to 0? In continuous mathematical space?
Do not say it disappears as it obviously does not in case when form is defined by a differentiable function.
E.g. in 1 D FORM would be defined by a curve. A derivative of this curve is obtained by reducing both variable ( unit) and function depending on variable defining curve to 0. So length of curve and measure used goes to 0 simultaneously. But the ratio/functional dependence between these 2 remains, and allows to reconstruct curve (FORM) by integration but not the exact positioning of it.
The other question I posed is why do some functions disappear ( nth derivative =0 ) to a non-re-construable form after finite number of steps, while others does not?
Some, like e^x even does not notice the consecutive differentiations. I know mathematically why, but where does the info about FORM is stored that it is impossible to loose it no matter what?
Some, like sinx , cos x, e^ix behave cyclically, shifting phase by 90 degrees after each differentiation.
Why should some FORMs behave like that, while others disappear leaving no trace? (e.g d^3/dx^3 (x^2) = 0, and no one knows that it was x^2 this time).
sin, cos, tan, exp are all defined by infinite series, they can never be differentiated away. Polynomials, by definition, have only finitely many terms in their Taylor expansion (they are their Taylor expansion) and so will terminate.
There's no 'conversation of information', when you differentiate something, you are asking how it smoothly changes at a particular point. Consecutive derivatives then compute how that change changes. Some functions eventually have constant changes of changes of changes of ..... of changes.
And dV is not "volume is zero", but the change in volume is very small. dP/dV means "What's the ratio of the small amount pressure changes by when I change the volume by a small amount?", not "What's the pressure when volume is zero".
Shrinking a ball to a point make it a point. There doesn't need to be any way of reconstructing it from a sphere, though at least a solid ball and a point are topologically the same (both solid, compact, simply connected spaces). A sphere and a point are not, so you, technically, cannot morph one into the other in a well defined way.
There's no 'conversation of information', when you differentiate something, you are asking how it smoothly changes at a particular point. Consecutive derivatives then compute how that change changes. Some functions eventually have constant changes of changes of changes of ..... of changes.
And dV is not "volume is zero", but the change in volume is very small. dP/dV means "What's the ratio of the small amount pressure changes by when I change the volume by a small amount?", not "What's the pressure when volume is zero".
Shrinking a ball to a point make it a point. There doesn't need to be any way of reconstructing it from a sphere, though at least a solid ball and a point are topologically the same (both solid, compact, simply connected spaces). A sphere and a point are not, so you, technically, cannot morph one into the other in a well defined way.
Kind of fun to speculate around isn't it :)
" The only way to get something to go to zero is to multiply by zero or subtract. Taking a limit has no physical analogy. It is not a process to make something go to zero (as the original problem stated) "
I guess that is a mathematical statement MBG?
Because in QM everything seems to lose its form when magnified, does it not?
Or do you know any 'real' particles, keeping its 'form' under magnification?
What we see as 'form' seems more to be a matter :) of scale to me.
Which doesn't mean that there can't be 'forms' that will be intact under all circumstances.
But then you will have to introduce some 'other' factor that sort of create that 'stability'. Fractals are funny that way. Could there be a universe solely built on fractals? I don't think so, because then we would have to accept ' from nothing comes all' which frankly makes me somewhat uncomfortable ::))
" The only way to get something to go to zero is to multiply by zero or subtract. Taking a limit has no physical analogy. It is not a process to make something go to zero (as the original problem stated) "
I guess that is a mathematical statement MBG?
Because in QM everything seems to lose its form when magnified, does it not?
Or do you know any 'real' particles, keeping its 'form' under magnification?
What we see as 'form' seems more to be a matter :) of scale to me.
Which doesn't mean that there can't be 'forms' that will be intact under all circumstances.
But then you will have to introduce some 'other' factor that sort of create that 'stability'. Fractals are funny that way. Could there be a universe solely built on fractals? I don't think so, because then we would have to accept ' from nothing comes all' which frankly makes me somewhat uncomfortable ::))
QUOTE (AlphaNumeric+Nov 10 2007, 11:08 AM)
And dV is not "volume is zero", but the change in volume is very small. dP/dV means "What's the ratio of the small amount pressure changes by when I change the volume by a small amount?", not "What's the pressure when volume is zero".
Shrinking a ball to a point make it a point. There doesn't need to be any way of reconstructing it from a sphere, though at least a solid ball and a point are topologically the same (both solid, compact, simply connected spaces). A sphere and a point are not, so you, technically, cannot morph one into the other in a well defined way.
Physically perhaps dV can be very small, but when Euler defines derivatives in his Foundations of Differential Calculus he is very specific that for correct application of differentials one must understand that (p.vii):
And
And
the ratio is only correct when dx=0, so it is of absolute importance that these differentials are ABSOLUTELY nothing, and vanish simultaneously, so we can conclude nothing from them except that their mutual ratios reduce to finite quantities.
If these infinitely small quantities , which are neglected in calculus, are not quite nothing, then necessarily an error will result. Those quantities that shall be neglected must surely be held to be absolutely nothing.
Geometric rigor shrinks from even so small an error. That comparison, which is the concern of differential calculus, would not be valid unless the increments vanish completely.
So, in differentiation, the only valid way to apply it correctly is by looking at vanishing completely, not just small or relatively small increments.
I was not asking about discrete approximations to FORM, but what happens to geometrical FORM of curve, shape, volume when its length, area, volume is reduced to 0.
A point has no structure. A derivitive is not a point. The derivitive is related to the relationship of a point to adjacent points. You cannot take a derivitive of a point.
Again, within reality, things cannot shrink to zero. You have to subtract to actually get zero. When you subtract all of something, form is lost. There is not even a point.
Shrinking a ball to a point make it a point. There doesn't need to be any way of reconstructing it from a sphere, though at least a solid ball and a point are topologically the same (both solid, compact, simply connected spaces). A sphere and a point are not, so you, technically, cannot morph one into the other in a well defined way.
Physically perhaps dV can be very small, but when Euler defines derivatives in his Foundations of Differential Calculus he is very specific that for correct application of differentials one must understand that (p.vii):
QUOTE
Differential calculus is a method for determining the ratio of vanishing increments that any function takes on when variable, of which they are functions, is given a vanishing increment.
Integral calculus is a method of finding these functions from the knowledge of the ratio of their vanishing increments
Integral calculus is a method of finding these functions from the knowledge of the ratio of their vanishing increments
And
QUOTE (->
| QUOTE |
| Differential calculus is a method for determining the ratio of vanishing increments that any function takes on when variable, of which they are functions, is given a vanishing increment. Integral calculus is a method of finding these functions from the knowledge of the ratio of their vanishing increments |
And
the ratio is only correct when dx=0, so it is of absolute importance that these differentials are ABSOLUTELY nothing, and vanish simultaneously, so we can conclude nothing from them except that their mutual ratios reduce to finite quantities.
If these infinitely small quantities , which are neglected in calculus, are not quite nothing, then necessarily an error will result. Those quantities that shall be neglected must surely be held to be absolutely nothing.
Geometric rigor shrinks from even so small an error. That comparison, which is the concern of differential calculus, would not be valid unless the increments vanish completely.
So, in differentiation, the only valid way to apply it correctly is by looking at vanishing completely, not just small or relatively small increments.
I was not asking about discrete approximations to FORM, but what happens to geometrical FORM of curve, shape, volume when its length, area, volume is reduced to 0.
It says 'vanishing INCREMENT'
By your logic, y=x^2 only has a derivative when x=0. Wrong, it has a derivative when the dx in (x+dx)^2/dx goes to zero. See the difference?
There's a difference between f'(x) and f'(0). f'(x) is the derivative, f'(0) is the derivative evaluated at x=0.
If you can't understand your own quotes, why are you even bothering to read such books?
By your logic, y=x^2 only has a derivative when x=0. Wrong, it has a derivative when the dx in (x+dx)^2/dx goes to zero. See the difference?
There's a difference between f'(x) and f'(0). f'(x) is the derivative, f'(0) is the derivative evaluated at x=0.
If you can't understand your own quotes, why are you even bothering to read such books?
Not only at x=0, but it is possible of course to have derivative at x=0. So it is possible to have FORM preserved as f'(x) when argument is also 0 if derivative is defined there.
I used continuous when I should have used differentiatable, and Ivars quoted me. I knew it when I did it, but chose to say it that way anyway. A cube is continuous.
I question whether arbitrary form can go to zero in a continuous way. I can take the limit of when it goes to zero, and talk about that. But, I'm thinking it (a cube, for example) cannot progress all the way to zero. We can conceptualize about it at 0.
I still think that taking the derivtive of a function is a different thing completely.
A thing can shrink infinitely, and never become zero. The limit, which it can never become, can be zero. There is no form in the limit of a volume.
Maybe I'm missing the point, as I think I have said this already.
QUOTE
What happens with arbitrary FORM when volume (area, length) goes to 0? In continuous mathematical space?
I question whether arbitrary form can go to zero in a continuous way. I can take the limit of when it goes to zero, and talk about that. But, I'm thinking it (a cube, for example) cannot progress all the way to zero. We can conceptualize about it at 0.
I still think that taking the derivtive of a function is a different thing completely.
A thing can shrink infinitely, and never become zero. The limit, which it can never become, can be zero. There is no form in the limit of a volume.
Maybe I'm missing the point, as I think I have said this already.
QUOTE (meBigGuy+Nov 11 2007, 06:32 AM)
A thing can shrink infinitely, and never become zero. The limit, which it can never become, can be zero. There is no form in the limit of a volume.
Maybe I'm missing the point, as I think I have said this already.
hej MeBigGuy
We are trying to define a point here, so no one really can miss:)
I was trying to imagine what happens with FORM if we reduce the volume to 0 ( if we speak about 3D). I could not, because:
Some on said in limit 0 anything will become point, but what is the FORM of point? Will everything end up being the same point regardless where it started?
I brought in derivatives because we know they help to solve 0/0 if we can differentiate the functions involved at the required points.
That means even 0 has some sub structure which can be revelaed by taking a derivative of a function whose value is 0.
So i thought, derivative of a FORM, if FORM is a function (e.g surface of a cube) of coordinates, could also remain valid when volume goes to 0.
But I could not state it porperly, and still something is missing.
My idea is, of course, that there are :
a) something left, which we can call pure form, after volume vanishes depending on where it started, in any coordinate system and space
there must be some forms that never vanish in this case in any coordinate system and in any space
By space I mean the way of differentiating. Usually, linear space is used, where increment in varibale is dx and in function df= f(x+dx)-f(x).
However, by using e.g fractional space where we use dx^(m/n), negative space (-dx), irrational space nth root(dx) , exponential space e^dx, logarithmic ln dx , sinusoidal sin dx etc we can arrive at very different outcomes of differentiation.
The question is, are there any forms which would always remain after whatever number of whatever differentiations are performed.
Or perhaps they are non-differentiable in no space, non-reducable. Cube might be one of them. I will try to find some references.
Maybe I'm missing the point, as I think I have said this already.
hej MeBigGuy
We are trying to define a point here, so no one really can miss:)
I was trying to imagine what happens with FORM if we reduce the volume to 0 ( if we speak about 3D). I could not, because:
Some on said in limit 0 anything will become point, but what is the FORM of point? Will everything end up being the same point regardless where it started?
I brought in derivatives because we know they help to solve 0/0 if we can differentiate the functions involved at the required points.
That means even 0 has some sub structure which can be revelaed by taking a derivative of a function whose value is 0.
So i thought, derivative of a FORM, if FORM is a function (e.g surface of a cube) of coordinates, could also remain valid when volume goes to 0.
But I could not state it porperly, and still something is missing.
My idea is, of course, that there are :
a) something left, which we can call pure form, after volume vanishes depending on where it started, in any coordinate system and space
there must be some forms that never vanish in this case in any coordinate system and in any spaceBy space I mean the way of differentiating. Usually, linear space is used, where increment in varibale is dx and in function df= f(x+dx)-f(x).
However, by using e.g fractional space where we use dx^(m/n), negative space (-dx), irrational space nth root(dx) , exponential space e^dx, logarithmic ln dx , sinusoidal sin dx etc we can arrive at very different outcomes of differentiation.
The question is, are there any forms which would always remain after whatever number of whatever differentiations are performed.
Or perhaps they are non-differentiable in no space, non-reducable. Cube might be one of them. I will try to find some references.
Maybe the question should be if 'zero' exist :)
If it does that would mean that there should exist 'areas' consisting of 'nothing'.
Kind of hard to imagine.
Can you say that two meters to the left and three meters to the right we have an un existing area of.. what?
Zero is a ingenious invention but no different from the forms containment of 'space' i think :) ahh like a vessel containing ??
So while being a complement to what we can 'see' its nonetheless impossible to define in 'real' spacetime ::))
If it does that would mean that there should exist 'areas' consisting of 'nothing'.
Kind of hard to imagine.
Can you say that two meters to the left and three meters to the right we have an un existing area of.. what?
Zero is a ingenious invention but no different from the forms containment of 'space' i think :) ahh like a vessel containing ??
So while being a complement to what we can 'see' its nonetheless impossible to define in 'real' spacetime ::))
QUOTE
That means even 0 has some sub structure which can be revelaed by taking a derivative of a function whose value is 0.
A point has no structure. A derivitive is not a point. The derivitive is related to the relationship of a point to adjacent points. You cannot take a derivitive of a point.
Again, within reality, things cannot shrink to zero. You have to subtract to actually get zero. When you subtract all of something, form is lost. There is not even a point.
QUOTE (Ivars+Nov 11 2007, 05:16 PM)
I brought in derivatives because we know they help to solve 0/0 if we can differentiate the functions involved at the required points.
No, they are limits. They don't 'solve' 0/0, there's nothing to solve.
No, they are limits. They don't 'solve' 0/0, there's nothing to solve.
QUOTE (Ivars+Nov 11 2007, 05:16 PM)
That means even 0 has some sub structure which can be revelaed by taking a derivative of a function whose value is 0.
No, it means that some functions can go to zero, it doesn't mean anything about 0.
Ivars, why, why, why do you persist in talking such BS?! Come on, really, WHY?. You know you're just making it up as you go along, that you have no clue about the Zeta or Gamma functions. You bearly understand derivatives! Yet you continue to talk about maths you have no grasp of and try to pretend you have applied it to physics?
Go on, give me the answers to question 1. It'll take you 5 minutes to work out the solutions to those if you know how, they are childs play compared to the Zeta function or vortex flows.
Put your clearly quite ignorant brain where your mouth is. Or shut up.
Ivars, why, why, why do you persist in talking such BS?! Come on, really, WHY?. You know you're just making it up as you go along, that you have no clue about the Zeta or Gamma functions. You bearly understand derivatives! Yet you continue to talk about maths you have no grasp of and try to pretend you have applied it to physics?
Go on, give me the answers to question 1. It'll take you 5 minutes to work out the solutions to those if you know how, they are childs play compared to the Zeta function or vortex flows.
Put your clearly quite ignorant brain where your mouth is. Or shut up.
If you are constructing calculus from infinitesimals (not limits) than there may be some utility in the notion of vectors of zero magnitude --- but still having direction. But I don't see how you could apply this to geometric figures as once they've been "scaled to zero" then the process would not be "reversable" to scale the figure back up to the original. I.E., information is lost.
QUOTE (AlphaNumeric+Nov 12 2007, 05:45 PM)
Go on, give me the answers to question 1. It'll take you 5 minutes to work out the solutions to those if you know how, they are childs play compared to the Zeta function or vortex flows.
Put your clearly quite ignorant brain where your mouth is. Or shut up.
Hej Alphanumeric
skipping the grammar corrections
I looked at equations, and , I agree, those should be easily solvable, but I have to look up some differential equations book. I intentionally avoided infinitesimals as long as possible, but perhaps now it is time to involve them.
May be it is time to learn more details, may be You are right. I feel a lot of animosity around when trying to speculate about such sacred things as Zeta function/vortexes/tetration based on intuition only.
So You definitely succeeded to shut me up - for a few days. And gave something to think about.
Thanks.
Put your clearly quite ignorant brain where your mouth is. Or shut up.
Hej Alphanumeric
skipping the grammar corrections
I looked at equations, and , I agree, those should be easily solvable, but I have to look up some differential equations book. I intentionally avoided infinitesimals as long as possible, but perhaps now it is time to involve them.
May be it is time to learn more details, may be You are right. I feel a lot of animosity around when trying to speculate about such sacred things as Zeta function/vortexes/tetration based on intuition only.
So You definitely succeeded to shut me up - for a few days.
And gave something to think about. Thanks.
QUOTE (Ivars+Nov 13 2007, 08:15 AM)
I looked at equations, and , I agree, those should be easily solvable, but I have to look up some differential equations book.
What?!
You have to look in a book for those?! If you had bothered to read the books you claim to have, to understand the maths you talk about, you'd know those questions are high school level! That sheet is from the first term of the first year of a university course. They are practically revision for 18 year olds!
If you cannot instantly say "I know how to solve that" then you have absolutely zero clue about differential equations. Hell, you should be able to give solutions to some of them off the top of your head.
What?!
You have to look in a book for those?! If you had bothered to read the books you claim to have, to understand the maths you talk about, you'd know those questions are high school level! That sheet is from the first term of the first year of a university course. They are practically revision for 18 year olds!
If you cannot instantly say "I know how to solve that" then you have absolutely zero clue about differential equations. Hell, you should be able to give solutions to some of them off the top of your head.
QUOTE (Ivars+Nov 13 2007, 08:15 AM)
I intentionally avoided infinitesimals as long as possible, but perhaps now it is time to involve them.
There's nothing to do with infinitesimals here, derivatives are defined by limits of elements in the Reals or the Complex numbers. It's the questionable interpretation of Newton's notation which leads to people mistakenly thinking that taking derivatives requires knowledge of infinitesimals. Almost noone, including university maths students, ever does anything involving infinitesimals, it's a misconception.
QUOTE (Ivars+Nov 13 2007, 08:15 AM)
May be it is time to learn more details, may be
Wrong. That sentence implues you already know some details. I don't see any evidence of that.
QUOTE (Ivars+Nov 13 2007, 08:15 AM)
I feel a lot of animosity around when trying to speculate about such sacred things as Zeta function/vortexes/tetration based on intuition only.
Nothing to do with them being 'sacred', you wax BS about something you obviously have no clue about. You've seen a pretty picture of the Zeta function's modulus in the complex plane and you just make up a load of crap about it.
If you stopped talking so much about things you know nothing about, maybe you'd get less animosity?
How precisely do you go about talking 'intuitively' about something like this when you don't understand any of the equations, concepts or notation? You can't, you are left to just make stuff up.
I remember reading pdfs like that before I took the course and trying to interpret them in a way which I could link to things I'd already done. More often than not I was wrong, because too many new concepts were required to grasp such things. That pdf comes from a 3rd year course on complex analysis. You just admitted to having to look in a book to do homework questions most 1st years from that university find incredibly easy.
Do you see how stupid you make yourself appear in the eyes of people who know the actual workings of the concepts you keep name dropping? Or is that what you hoped wouldn't happen, that if you used complicated enough words you'd read on Wikipedia noone would be able to tell you're full of nonsense?
If you stopped talking so much about things you know nothing about, maybe you'd get less animosity?
How precisely do you go about talking 'intuitively' about something like this when you don't understand any of the equations, concepts or notation? You can't, you are left to just make stuff up.
I remember reading pdfs like that before I took the course and trying to interpret them in a way which I could link to things I'd already done. More often than not I was wrong, because too many new concepts were required to grasp such things. That pdf comes from a 3rd year course on complex analysis. You just admitted to having to look in a book to do homework questions most 1st years from that university find incredibly easy.
Do you see how stupid you make yourself appear in the eyes of people who know the actual workings of the concepts you keep name dropping? Or is that what you hoped wouldn't happen, that if you used complicated enough words you'd read on Wikipedia noone would be able to tell you're full of nonsense?
hej Alphanumeric
Whatever logical basis You may seem to feel You have behind the kind of reasoning You demonstrate, it does not tell me I can not speculate with whatever ideas I want to, including in this forum. I have also got a lot of responses that WORKED FOR ME, and still find few new people.
And I will continue to exchange ideas with those people. It gives me pleasure, however You will try to prove rationally there can not be any pleasure.
Your style of communications does not work for me, except for minor disruptions of focus when I may stop and think for 5 minutes perhaps this was not a bad idea, he seems so clever, lets be nice to him. It never works.
So I would advise You to shut up as much as I will shut up because You do not add value to our discussions. You do not belong. You seem unable to understand that people might have different aims in communication, learning and different ways of learning, and You show no promise of being able to find out each persons motives, and support him in those.
Which basically means no big change will happen
Whatever logical basis You may seem to feel You have behind the kind of reasoning You demonstrate, it does not tell me I can not speculate with whatever ideas I want to, including in this forum. I have also got a lot of responses that WORKED FOR ME, and still find few new people.
And I will continue to exchange ideas with those people. It gives me pleasure, however You will try to prove rationally there can not be any pleasure.
Your style of communications does not work for me, except for minor disruptions of focus when I may stop and think for 5 minutes perhaps this was not a bad idea, he seems so clever, lets be nice to him. It never works.
So I would advise You to shut up as much as I will shut up because You do not add value to our discussions. You do not belong. You seem unable to understand that people might have different aims in communication, learning and different ways of learning, and You show no promise of being able to find out each persons motives, and support him in those.
Which basically means no big change will happen
So you think meaningful communication can come from several people discussing a topic they are utterly ignorant of?
Why do you think that is a more worthwhile endeavour than spending 10% of that time actually reading the work of other people?
You want to discuss ideas, so why do you avoid reading the discussion of people who have successfully proven they understand such material? I'm not after someone to be 'nice' to me, as you seem to imply in your post, I'm after you to bother to show you have done things you claim, to have read books and learnt material you want to think you have. It's no skin off my nose if you don't bother to learn the maths and physics you like to think you know, I actually continue to expand my understanding and knowledge, I don't care if you wallow in ignorance for the rest of your life. My criticism is for your benefit. But it's not like I've learnt that material myself, engage in research and help teach maths to university students or anything.
No, wait. That's exactly what it's like! You want to discuss maths with everyone but mathematicians.
Wow, what excellent reasoning and logic there Chuckles. Had 5 minutes to look in a book to 'remind' yourself how to do those questions yet?
Why do you think that is a more worthwhile endeavour than spending 10% of that time actually reading the work of other people?
You want to discuss ideas, so why do you avoid reading the discussion of people who have successfully proven they understand such material? I'm not after someone to be 'nice' to me, as you seem to imply in your post, I'm after you to bother to show you have done things you claim, to have read books and learnt material you want to think you have. It's no skin off my nose if you don't bother to learn the maths and physics you like to think you know, I actually continue to expand my understanding and knowledge, I don't care if you wallow in ignorance for the rest of your life. My criticism is for your benefit. But it's not like I've learnt that material myself, engage in research and help teach maths to university students or anything.
No, wait. That's exactly what it's like! You want to discuss maths with everyone but mathematicians.
Wow, what excellent reasoning and logic there Chuckles. Had 5 minutes to look in a book to 'remind' yourself how to do those questions yet?
hej Alphanumeric
That is part of learning, very important one, to find someone with whom You can discuss Your notions on the level of Your understanding. No one can be totally ignorant in things that are not solved yet and therefore called philosophical or metaphysical. You seem to think physics and engineering is the same, so specialization is all it takes.
But it is wrong. As it has been proved, this all it takes: Early math skills matter
I qualify, don't know about You
That is part of learning, very important one, to find someone with whom You can discuss Your notions on the level of Your understanding. No one can be totally ignorant in things that are not solved yet and therefore called philosophical or metaphysical. You seem to think physics and engineering is the same, so specialization is all it takes.
But it is wrong. As it has been proved, this all it takes: Early math skills matter
I qualify, don't know about You
You want to discuss ideas, so why do you avoid reading the discussion of people who have successfully proven they understand such material? I'm not after someone to be 'nice' to me, as you seem to imply in your post, I'm after you to bother to show you have done things you claim, to have read books and learnt material you want to think you have.
I want to discuss with people who either accept me or my ideas, and are at my level. Otherwise, it is impossible. I do not want to prove to anyone I have read anything, I am just using some quotes and names I find interesting at that moment.
Well, perhaps You sincerely believe so. No one could have guessed it from your tone. I would rather call it DICTATE and DEROGATE in a way that You will always find and nail a mistake;Yeah! - which is never helpful, as it paralyzes initiative. But You will learn about it later in your real life.
Well, perhaps You sincerely believe so. No one could have guessed it from your tone. I would rather call it DICTATE and DEROGATE in a way that You will always find and nail a mistake;Yeah! - which is never helpful, as it paralyzes initiative. But You will learn about it later in your real life.
No, wait. That's exactly what it's like! You want to discuss maths with everyone but mathematicians.
I would love to, but can not- so what to do now? I discuss with other people. With pleasure.
QUOTE
So you think meaningful communication can come from several people discussing a topic they are utterly ignorant of?
That is part of learning, very important one, to find someone with whom You can discuss Your notions on the level of Your understanding. No one can be totally ignorant in things that are not solved yet and therefore called philosophical or metaphysical. You seem to think physics and engineering is the same, so specialization is all it takes.
But it is wrong. As it has been proved, this all it takes: Early math skills matter
I qualify, don't know about You
QUOTE (->
| QUOTE |
| So you think meaningful communication can come from several people discussing a topic they are utterly ignorant of? |
That is part of learning, very important one, to find someone with whom You can discuss Your notions on the level of Your understanding. No one can be totally ignorant in things that are not solved yet and therefore called philosophical or metaphysical. You seem to think physics and engineering is the same, so specialization is all it takes.
But it is wrong. As it has been proved, this all it takes: Early math skills matter
I qualify, don't know about You
You want to discuss ideas, so why do you avoid reading the discussion of people who have successfully proven they understand such material? I'm not after someone to be 'nice' to me, as you seem to imply in your post, I'm after you to bother to show you have done things you claim, to have read books and learnt material you want to think you have.
I want to discuss with people who either accept me or my ideas, and are at my level. Otherwise, it is impossible. I do not want to prove to anyone I have read anything, I am just using some quotes and names I find interesting at that moment.
QUOTE
My criticism is for your benefit.
Well, perhaps You sincerely believe so. No one could have guessed it from your tone. I would rather call it DICTATE and DEROGATE in a way that You will always find and nail a mistake;Yeah! - which is never helpful, as it paralyzes initiative. But You will learn about it later in your real life.
QUOTE (->
| QUOTE |
| My criticism is for your benefit. |
Well, perhaps You sincerely believe so. No one could have guessed it from your tone. I would rather call it DICTATE and DEROGATE in a way that You will always find and nail a mistake;Yeah! - which is never helpful, as it paralyzes initiative. But You will learn about it later in your real life.
No, wait. That's exactly what it's like! You want to discuss maths with everyone but mathematicians.
I would love to, but can not- so what to do now? I discuss with other people. With pleasure.
@Ivars
You fight with AN, but totally ignore my points.
You fight with AN, but totally ignore my points.
QUOTE (meBigGuy+Nov 14 2007, 09:33 AM)
@Ivars
You fight with AN, but totally ignore my points.
hej MeBigGuy,
Excuse me. I got sidetracked, and lost a little bit of my initial idea in details. Lost orientation slightly, so to say.
Again, within reality, things cannot shrink to zero. You have to subtract to actually get zero. When you subtract all of something, form is lost. There is not even a point.
I 100% agree that in physical reality there is neither infinitesimals, nor infinity, but somehow lately I have got the feeling that there is still a need to incorporate, tie the mathematical nothings and infinities into the physical world around us, perhaps via sub quantum substances and their phase transformations. That is what I am trying to express, how the ideal world of mathematics is connected to the real world of quantized matter and radiation. And, later, life and consciousness.
The idea about the form came from the fact that I was trying to imagine void- space without anything, without change- which seems indiscernible from absolutely ideal continuous solid. Then I thought but can this ideal solid still be particulated somehow, like have a lattice- e.g. tetrahedral -which is the simplest form of quantized volume, as triangle is of area.
Or, is there a possibility to imagine such solid and imaginary lattice it is there, but its not expressed. Then I thought is there in mathematics something which would behave so that if you reduce the size of the cell forming a lattice , something will be left even after You reduce the cell size to 0?
Surely You can not take a derivative of a point, but derivative exists at any point as a value y(xo) , and as functional dependence it exists everywhere it is defined, in each point as y(x).
To me it seemed, may be wrongly, that derivatives give answers about the FORM of curves and surfaces (surrounding volumes) even if there is no curve or surface - the information about the form is contained in derivative. The more complicated the form is, the more derivatives there exist. All derivatives together should tell all about the form in general= so 2 surfaces z1 (x,y) and z2(x,y) with all derivatives equal should have the same form, generally?
Now if we know only derivatives, we can create a space where only derivatives are present, not the form itself. Such a space would define all forms that can be created from these derivatives up to a constant, meaning this space would not tell where exactly relative to coordinate system chosen the surface is located. Which is kind of a principle of relativity- where ever You chose to place the surface , its form is independent from the choice of the beginning of coordinates.
But is seems i have not been able to make my point clearly.
You fight with AN, but totally ignore my points.
hej MeBigGuy,
Excuse me. I got sidetracked, and lost a little bit of my initial idea in details. Lost orientation slightly, so to say.
QUOTE
A point has no structure. A derivative is not a point. The derivative is related to the relationship of a point to adjacent points. You cannot take a derivative of a point.
QUOTE (->
| QUOTE |
| A point has no structure. A derivative is not a point. The derivative is related to the relationship of a point to adjacent points. You cannot take a derivative of a point. |
Again, within reality, things cannot shrink to zero. You have to subtract to actually get zero. When you subtract all of something, form is lost. There is not even a point.
I 100% agree that in physical reality there is neither infinitesimals, nor infinity, but somehow lately I have got the feeling that there is still a need to incorporate, tie the mathematical nothings and infinities into the physical world around us, perhaps via sub quantum substances and their phase transformations. That is what I am trying to express, how the ideal world of mathematics is connected to the real world of quantized matter and radiation. And, later, life and consciousness.
The idea about the form came from the fact that I was trying to imagine void- space without anything, without change- which seems indiscernible from absolutely ideal continuous solid. Then I thought but can this ideal solid still be particulated somehow, like have a lattice- e.g. tetrahedral -which is the simplest form of quantized volume, as triangle is of area.
Or, is there a possibility to imagine such solid and imaginary lattice it is there, but its not expressed. Then I thought is there in mathematics something which would behave so that if you reduce the size of the cell forming a lattice , something will be left even after You reduce the cell size to 0?
Surely You can not take a derivative of a point, but derivative exists at any point as a value y(xo) , and as functional dependence it exists everywhere it is defined, in each point as y(x).
To me it seemed, may be wrongly, that derivatives give answers about the FORM of curves and surfaces (surrounding volumes) even if there is no curve or surface - the information about the form is contained in derivative. The more complicated the form is, the more derivatives there exist. All derivatives together should tell all about the form in general= so 2 surfaces z1 (x,y) and z2(x,y) with all derivatives equal should have the same form, generally?
Now if we know only derivatives, we can create a space where only derivatives are present, not the form itself. Such a space would define all forms that can be created from these derivatives up to a constant, meaning this space would not tell where exactly relative to coordinate system chosen the surface is located. Which is kind of a principle of relativity- where ever You chose to place the surface , its form is independent from the choice of the beginning of coordinates.
But is seems i have not been able to make my point clearly.
As I pointed out before, if the "reduction in size to zero" also eliminates "information" about the object, then, well, it's not the same object any more, is it? Without the "information" about the object's structure, it can't be "reinflated" to its original shape.
Look at multiplication by zero as an information eraser.
Look at multiplication by zero as an information eraser.
Hej NoCleverName,
I really appreciate all comments in this attempt to define something that is escaping.
As I pointed out before, if the "reduction in size to zero" also eliminates "information" about the object, then, well, it's not the same object any more, is it? Without the "information" about the object's structure, it can't be "reinflated" to its original shape.
In first approximation classical/quantum physics may be yes. But not in mathematics.
As to 0, there are different orders of 0 if we look again at differential calculus.
if dx is the first order infinitesimal, than dx^2 is second order and so on. dx^2 is infinitely smaller than dx, although both are 0 in the end.
When dx->0, dx^2 ->0 via different path so once You have limit 0/0 You can sometimes solve the problem by differentiating the functions whose value is 0 in numerator and denominator.
Because of the different way each functions differences tend to 0 as higher order infinitesimals tend to 0, the limit 0/0 may have a finite convergent value, and is considered true limit of the initial sequence. So information about the ratio of a limit is hidden quite deep in the structure of numerator and denominator, and can be retrieved via differentiation even when both are 0. That is in my thinking analogue to a possibility to retrieve a form of something which has vanished.
When You integrate, You get infinities of a different order, in the same way.
In first approximation classical/quantum physics may be yes. But not in mathematics.
As to 0, there are different orders of 0 if we look again at differential calculus.
if dx is the first order infinitesimal, than dx^2 is second order and so on. dx^2 is infinitely smaller than dx, although both are 0 in the end.
When dx->0, dx^2 ->0 via different path so once You have limit 0/0 You can sometimes solve the problem by differentiating the functions whose value is 0 in numerator and denominator.
Because of the different way each functions differences tend to 0 as higher order infinitesimals tend to 0, the limit 0/0 may have a finite convergent value, and is considered true limit of the initial sequence. So information about the ratio of a limit is hidden quite deep in the structure of numerator and denominator, and can be retrieved via differentiation even when both are 0. That is in my thinking analogue to a possibility to retrieve a form of something which has vanished.
When You integrate, You get infinities of a different order, in the same way.
Look at multiplication by zero as an information eraser.
Now imagine You multiply infinity of order 1 with infinitesimal of order 1 = the answer will be infinity(1) * 0(1) = 1
If You have 2nd order of infinity multiplied by first order infinitesimal then
infinity (2)* 0(1) = infinity
If You have 1st order infinity and 2nd order infinitesimal:
infinity(1)*0(2) = 0
So while ALL infinitesimals in differential calculus must necessarily be 0 when compared to any FINITE value, they do not necessarily ERASE all information in all cases.
These are just few examples, I guess the whole subject is much more complicated.
I really appreciate all comments in this attempt to define something that is escaping.
QUOTE
As I pointed out before, if the "reduction in size to zero" also eliminates "information" about the object, then, well, it's not the same object any more, is it? Without the "information" about the object's structure, it can't be "reinflated" to its original shape.
In first approximation classical/quantum physics may be yes. But not in mathematics.
As to 0, there are different orders of 0 if we look again at differential calculus.
if dx is the first order infinitesimal, than dx^2 is second order and so on. dx^2 is infinitely smaller than dx, although both are 0 in the end.
When dx->0, dx^2 ->0 via different path so once You have limit 0/0 You can sometimes solve the problem by differentiating the functions whose value is 0 in numerator and denominator.
Because of the different way each functions differences tend to 0 as higher order infinitesimals tend to 0, the limit 0/0 may have a finite convergent value, and is considered true limit of the initial sequence. So information about the ratio of a limit is hidden quite deep in the structure of numerator and denominator, and can be retrieved via differentiation even when both are 0. That is in my thinking analogue to a possibility to retrieve a form of something which has vanished.
When You integrate, You get infinities of a different order, in the same way.
QUOTE (->
| QUOTE |
As I pointed out before, if the "reduction in size to zero" also eliminates "information" about the object, then, well, it's not the same object any more, is it? Without the "information" about the object's structure, it can't be "reinflated" to its original shape. |
In first approximation classical/quantum physics may be yes. But not in mathematics.
As to 0, there are different orders of 0 if we look again at differential calculus.
if dx is the first order infinitesimal, than dx^2 is second order and so on. dx^2 is infinitely smaller than dx, although both are 0 in the end.
When dx->0, dx^2 ->0 via different path so once You have limit 0/0 You can sometimes solve the problem by differentiating the functions whose value is 0 in numerator and denominator.
Because of the different way each functions differences tend to 0 as higher order infinitesimals tend to 0, the limit 0/0 may have a finite convergent value, and is considered true limit of the initial sequence. So information about the ratio of a limit is hidden quite deep in the structure of numerator and denominator, and can be retrieved via differentiation even when both are 0. That is in my thinking analogue to a possibility to retrieve a form of something which has vanished.
When You integrate, You get infinities of a different order, in the same way.
Look at multiplication by zero as an information eraser.
Now imagine You multiply infinity of order 1 with infinitesimal of order 1 = the answer will be infinity(1) * 0(1) = 1
If You have 2nd order of infinity multiplied by first order infinitesimal then
infinity (2)* 0(1) = infinity
If You have 1st order infinity and 2nd order infinitesimal:
infinity(1)*0(2) = 0
So while ALL infinitesimals in differential calculus must necessarily be 0 when compared to any FINITE value, they do not necessarily ERASE all information in all cases.
These are just few examples, I guess the whole subject is much more complicated.
QUOTE (Ivars+Nov 14 2007, 02:03 PM)
In first approximation classical/quantum physics may be yes. But not in mathematics.
Yes, in mathematics, too. I invite you to consider the work of Turing, Chaitin, etc., etc., before you dismiss the notion that information is not contained in mathematics.
Yes, in mathematics, too. I invite you to consider the work of Turing, Chaitin, etc., etc., before you dismiss the notion that information is not contained in mathematics.
Let me expand on that slightly.
Say a physical process is described by x^2 + 2. Now, if you differentiate, giving, 2x, then information has clearly been lost that cannot be recreated by the simple reversal of integration alone.
Taken as pure information, the equation x^2 + 2 has a certain bit-length (the ASCII characters). Differentiatian and re-integration are information transforms that results in an equation of lessor bit-length: x^2 --- which has less information. Similarly, the reduction of a geometric figure to a point will, in at least euclidian space, make a severe change in information content.
Say a physical process is described by x^2 + 2. Now, if you differentiate, giving, 2x, then information has clearly been lost that cannot be recreated by the simple reversal of integration alone.
Taken as pure information, the equation x^2 + 2 has a certain bit-length (the ASCII characters). Differentiatian and re-integration are information transforms that results in an equation of lessor bit-length: x^2 --- which has less information. Similarly, the reduction of a geometric figure to a point will, in at least euclidian space, make a severe change in information content.
QUOTE (NoCleverName+Nov 14 2007, 02:27 PM)
Let me expand on that slightly.
Say a physical process is described by x^2 + 2. Now, if you differentiate, giving, 2x, then information has clearly been lost that cannot be recreated by the simple reversal of integration alone.
hej NoCleverName,
Sure, I agree. If we do not have boundary conditions or some other equation where we could extract that 2, we loose information about the place of the process relative to the beginning of coordinate system where y=0, but we do not loose the basic information that this process was quadratic relative to variable x.
That is exactly what I am trying to say: Differentiation maintains the essence of process, but not the exact place . It keeps the FORM of a curve, surface, but not placement.
if we have y= (x-xo)^2 + 2 ,
then dy/dx = 2(x-xo) - we loose information about initial value of y(x=xo) =2
and after second differentiation :
d^2y/dx^2 = 2 we loose information also about xo, so our process is "freed" from coordinate system but still we can recover its basic quadratic dependence on x.
Which again leads to conclusion that we do not loose info about quadratic FORM of the process, we loose info about placement.
I guess that is basically why differential equations work at all- that we can recover the FORM of the process itself from its derivatives.
So again, we can have a space of ALL coordinate systems which may be placed wherever relative to each other- translationally at least - in such a space the FORM would be enough to represent all possible placements of the process by choosing the right coordinate system, fixing xo and y(xo).
To me this seems like a proper statement of relativity, so in essence:
The ability to recover the FORM of surface ( or essence of process) via its derivatives implies the relativity of translational transformations of coordinates.
What does c= const means then? It means that our space which is full of coordinate systems is such that whatever way you differentiate physical processes given in any of these coordinate systems, the first derivative of position by dt will never exceed c, and for light the second derivative will be 0.
So the question about relativity is to find such combination of all possible mathematical coordinate systems in which we can express physical laws which have this property where d/dt<=c, d/dt^2 for light = 0. Now because it will be mathematical construction, c will just be a number, ONE single number characterizing this POSSIBLE combination of spaces that exist in this space of ALL coordinate systems ( other number d/dt^2=0 is not very helpful as it may fit many combinations) which will correspond to the c we measure, so we will be able to find our local conversion factor between complicated mathematical POSSIBLE coordinate space NUMBER c and measured speed of light c.
Q.E.D
Say a physical process is described by x^2 + 2. Now, if you differentiate, giving, 2x, then information has clearly been lost that cannot be recreated by the simple reversal of integration alone.
hej NoCleverName,
Sure, I agree. If we do not have boundary conditions or some other equation where we could extract that 2, we loose information about the place of the process relative to the beginning of coordinate system where y=0, but we do not loose the basic information that this process was quadratic relative to variable x.
That is exactly what I am trying to say: Differentiation maintains the essence of process, but not the exact place . It keeps the FORM of a curve, surface, but not placement.
if we have y= (x-xo)^2 + 2 ,
then dy/dx = 2(x-xo) - we loose information about initial value of y(x=xo) =2
and after second differentiation :
d^2y/dx^2 = 2 we loose information also about xo, so our process is "freed" from coordinate system but still we can recover its basic quadratic dependence on x.
Which again leads to conclusion that we do not loose info about quadratic FORM of the process, we loose info about placement.
I guess that is basically why differential equations work at all- that we can recover the FORM of the process itself from its derivatives.
So again, we can have a space of ALL coordinate systems which may be placed wherever relative to each other- translationally at least - in such a space the FORM would be enough to represent all possible placements of the process by choosing the right coordinate system, fixing xo and y(xo).
To me this seems like a proper statement of relativity, so in essence:
The ability to recover the FORM of surface ( or essence of process) via its derivatives implies the relativity of translational transformations of coordinates.
What does c= const means then? It means that our space which is full of coordinate systems is such that whatever way you differentiate physical processes given in any of these coordinate systems, the first derivative of position by dt will never exceed c, and for light the second derivative will be 0.
So the question about relativity is to find such combination of all possible mathematical coordinate systems in which we can express physical laws which have this property where d/dt<=c, d/dt^2 for light = 0. Now because it will be mathematical construction, c will just be a number, ONE single number characterizing this POSSIBLE combination of spaces that exist in this space of ALL coordinate systems ( other number d/dt^2=0 is not very helpful as it may fit many combinations) which will correspond to the c we measure, so we will be able to find our local conversion factor between complicated mathematical POSSIBLE coordinate space NUMBER c and measured speed of light c.
Q.E.D
Differentiation does give you information about the form, but only if there are adjacent points, and therefore form. If the object goes to zero (though some inconceivable process) then there are no adjacent points, and no form.
That you somehow focus the derivitive on the point, and not on the point's relationship to its neighbors, is the crux of your apparent confusion. We can look carefully at how the derivitive is determined and see this (spare me for now, though).
The fact that the derivitive exists at a point is not the same as saying the point has the form. The point has no form. The point in relation to its neighbors has form, as expressed by the derivitive.
Now, you are saying that if you strip away all the adjacent points, there is information in the perviously taken derivitive that you conveniently remembered, but, unfortunately, that is not an inherent characteristic of the point. That is additional information you remembered, so is not pertinent to this discussion. That information is not contained in the point. If I get 15 points that were collapsed in this inconceivable way, I have no way of ever determining what form they had previously.
That you somehow focus the derivitive on the point, and not on the point's relationship to its neighbors, is the crux of your apparent confusion. We can look carefully at how the derivitive is determined and see this (spare me for now, though).
The fact that the derivitive exists at a point is not the same as saying the point has the form. The point has no form. The point in relation to its neighbors has form, as expressed by the derivitive.
Now, you are saying that if you strip away all the adjacent points, there is information in the perviously taken derivitive that you conveniently remembered, but, unfortunately, that is not an inherent characteristic of the point. That is additional information you remembered, so is not pertinent to this discussion. That information is not contained in the point. If I get 15 points that were collapsed in this inconceivable way, I have no way of ever determining what form they had previously.
MBG brings out an important, well, point. Calculus derived from limits speaks of mathematical properties around a collection of points as opposed to a property of a point itself. Without neighbors, points wouldn't have "slope", etc. So there would seem to be a difference between shrinking something to zero in the limit as opposed to actually shrinking it to zero.
I believe calculus derived from infinitesimals can preserve slope at a point, but that would not help you recreate an object's exact shape, just it's general form. But there would be a secondary difficulty of understanding what it means to have many points lying one on top of another in zero space.
Inherant in this discussion is the open question of whether or not physical reality is actually mathematical. So far, it seems that it is, but nobody has the slightest idea why that should be so. This would imply that math "exists" on its own quite apart from our imagination --- we aren't really making it up after all. Since having such a discussion would unleash the dogs of crankdom, let's just leave it lie.
I believe calculus derived from infinitesimals can preserve slope at a point, but that would not help you recreate an object's exact shape, just it's general form. But there would be a secondary difficulty of understanding what it means to have many points lying one on top of another in zero space.
Inherant in this discussion is the open question of whether or not physical reality is actually mathematical. So far, it seems that it is, but nobody has the slightest idea why that should be so. This would imply that math "exists" on its own quite apart from our imagination --- we aren't really making it up after all. Since having such a discussion would unleash the dogs of crankdom, let's just leave it lie.
QUOTE (NoCleverName+Nov 15 2007, 12:22 PM)
Inherent in this discussion is the open question of whether or not physical reality is actually mathematical. So far, it seems that it is, but nobody has the slightest idea why that should be so. This would imply that math "exists" on its own quite apart from our imagination --- we aren't really making it up after all. Since having such a discussion would unleash the dogs of crankdom, let's just leave it lie.
hej NoCleverName
I agree 100% with this - mathematics exists and is expressed partly via physical reality. It is nice that You can notice my basic idea which is behind these ramblings.
The challenge is to find the internal dynamical structure of mathematics to be able to understand its workings without being able to physically measure all of them, and then understand the connections between our perceived reality and that structure.
And discover/define a substance which could have such properties that structure of mathematics arises from the self dynamics of such substance in whatever space(s) it needs to perform them ( and how it leads also to physical , mental etc dynamics with time direction we observe/participate in).
Now we can take the blame.
As to derivatives around point and in point- I think we have more or less cleared the issue that without space there can not be a form, and I agree with MBG perfectly, but if there is a form in some space, it can be reconstructed in any space (as defined by coordinates) where its derivatives exist from reduced information contained in derivatives.
Which leads to a principle of relativity , in my opinion, or even broader principle than just certain types of coordinate systems- all systems that in principle preserve infinite number of derivatives and where in any of them dy/dx < c, and dy/dx (light ) = const, d^2y/dx^2 (light) = 0 are leading to special relativity, but not necessarily ALL physics possible.
if we add spaces where d^2y/dx^2(light) < 0, we are adding gravitation a la Einstein.
but what if we look for spaces where d^2y/dx^2(light) > 0? What spaces would fit? Of course it is more complex than this.
hej NoCleverName
I agree 100% with this - mathematics exists and is expressed partly via physical reality. It is nice that You can notice my basic idea which is behind these ramblings.
The challenge is to find the internal dynamical structure of mathematics to be able to understand its workings without being able to physically measure all of them, and then understand the connections between our perceived reality and that structure.
And discover/define a substance which could have such properties that structure of mathematics arises from the self dynamics of such substance in whatever space(s) it needs to perform them ( and how it leads also to physical , mental etc dynamics with time direction we observe/participate in).
Now we can take the blame.
As to derivatives around point and in point- I think we have more or less cleared the issue that without space there can not be a form, and I agree with MBG perfectly, but if there is a form in some space, it can be reconstructed in any space (as defined by coordinates) where its derivatives exist from reduced information contained in derivatives.
Which leads to a principle of relativity , in my opinion, or even broader principle than just certain types of coordinate systems- all systems that in principle preserve infinite number of derivatives and where in any of them dy/dx < c, and dy/dx (light ) = const, d^2y/dx^2 (light) = 0 are leading to special relativity, but not necessarily ALL physics possible.
if we add spaces where d^2y/dx^2(light) < 0, we are adding gravitation a la Einstein.
but what if we look for spaces where d^2y/dx^2(light) > 0? What spaces would fit? Of course it is more complex than this.
If we take a triangle and shrink it so that its side lengths approach zero, then then it remains a triangle unless it actually shrinks to become a point, ie with all vertices the same. If it ever did become a point, then all the geometric properties of the object become meaningless, eg the sum of its internal angles no longer equal pi.
QUOTE (Tim C+Nov 15 2007, 03:45 PM)
If we take a triangle and shrink it so that its side lengths approach zero, then then it remains a triangle unless it actually shrinks to become a point, ie with all vertices the same. If it ever did become a point, then all the geometric properties of the object become meaningless, eg the sum of its internal angles no longer equal pi.
hej Tim C,
No, I was suggesting that we shrink it so that AREA becomes 0. What happens to the edges and angles between them?
I guess these relations are preserved.
In a sense of infinitesimals, AREA is a second order infinitesimal, dx^2, while edge is first order infinitesimal, dx.
Obviously AREA proportional to dx^2 vanish compared to dx as dx tends to 0. Dx^2 is infinite times smaller than dx if You compare them arithmetically:
dx+dx^2 = infinitesimal edge + infinitesimal AREA = infinitesimal edge
But not the same happens if You compare them geometrically:
dx/dx^2 = 1/dx = infinity
what happens if we multiply this infinity with dx, which is 0?
infinity* dx = (1/dx)*dx = 1 in case of square, when AREA=dx^2
Since in triangle ( let us take equilateral triangle, with 3 equal sides, Length dx):
AREA = Sqrt(3)/4 *dx^2
The number left over after edge goes to 0 and AREA goes to 0 as second order infinitesimal:
dx/sgrt(3)/4*dx^2 = sgrt(3)/4 * 1/dx = sgrt (3)/4* infinity
If we multiply this by dx to get a FINITE value, we get sqrt(3)/4 as a number left over when triangle dissappears.
So every form can be characterized by a number that is related to its AREA/edge relation as edge-> 0 .
Now , if we see a number sgrt(3)/4 we can reconstruct equilateral triangle of any size from it, so form of equilateral triangle in plane is preserved in this number.
We can even recover angles as sgrt(3)/4= 2*sin (pi/3)
If You do the same with tetrahedron in 3D, You are left with sgrt(3) as a number characterizing AREA of tertrahedron and sqrt(2)/12 characterizing its Volume.
So I would not agree we can not recover its form-there are definitely not too many 3D shapes whose area are sgrt(3)* edge^2 and at the same time volume sgrt(2) /12 *edge^3. Perhaps regular tetrahedron is the only one.
hej Tim C,
No, I was suggesting that we shrink it so that AREA becomes 0. What happens to the edges and angles between them?
I guess these relations are preserved.
In a sense of infinitesimals, AREA is a second order infinitesimal, dx^2, while edge is first order infinitesimal, dx.
Obviously AREA proportional to dx^2 vanish compared to dx as dx tends to 0. Dx^2 is infinite times smaller than dx if You compare them arithmetically:
dx+dx^2 = infinitesimal edge + infinitesimal AREA = infinitesimal edge
But not the same happens if You compare them geometrically:
dx/dx^2 = 1/dx = infinity
what happens if we multiply this infinity with dx, which is 0?
infinity* dx = (1/dx)*dx = 1 in case of square, when AREA=dx^2
Since in triangle ( let us take equilateral triangle, with 3 equal sides, Length dx):
AREA = Sqrt(3)/4 *dx^2
The number left over after edge goes to 0 and AREA goes to 0 as second order infinitesimal:
dx/sgrt(3)/4*dx^2 = sgrt(3)/4 * 1/dx = sgrt (3)/4* infinity
If we multiply this by dx to get a FINITE value, we get sqrt(3)/4 as a number left over when triangle dissappears.
So every form can be characterized by a number that is related to its AREA/edge relation as edge-> 0 .
Now , if we see a number sgrt(3)/4 we can reconstruct equilateral triangle of any size from it, so form of equilateral triangle in plane is preserved in this number.
We can even recover angles as sgrt(3)/4= 2*sin (pi/3)
If You do the same with tetrahedron in 3D, You are left with sgrt(3) as a number characterizing AREA of tertrahedron and sqrt(2)/12 characterizing its Volume.
So I would not agree we can not recover its form-there are definitely not too many 3D shapes whose area are sgrt(3)* edge^2 and at the same time volume sgrt(2) /12 *edge^3. Perhaps regular tetrahedron is the only one.
QUOTE (Ivars+Nov 15 2007, 05:39 PM)
hej Tim C,
No, I was suggesting that we shrink it so that AREA becomes 0. What happens to the edges and angles between them?
I guess these relations are preserved.
Well, no one's going to accept a "guess".
Aside from mixing the apples of limits with the oranges of infinitesimals, the idea that an "area" could be zero while the enclosing boundaries non-zero would seem to violate the axiom that x * 0 = 0. It's going to take a lot more than hopeful arm-waving for this idea to pan out.
As far as the area being zero while the sides are not zero, I was reminded that this is true only for certain geometries (like euclidian). There is no way for this to happen in others. This would seem to contraindicate your hope for a physical analogy, as it's not clear physical geometry is strictly euclidian.
No, I was suggesting that we shrink it so that AREA becomes 0. What happens to the edges and angles between them?
I guess these relations are preserved.
Well, no one's going to accept a "guess".
Aside from mixing the apples of limits with the oranges of infinitesimals, the idea that an "area" could be zero while the enclosing boundaries non-zero would seem to violate the axiom that x * 0 = 0. It's going to take a lot more than hopeful arm-waving for this idea to pan out.
As far as the area being zero while the sides are not zero, I was reminded that this is true only for certain geometries (like euclidian). There is no way for this to happen in others. This would seem to contraindicate your hope for a physical analogy, as it's not clear physical geometry is strictly euclidian.
QUOTE (NoCleverName+Nov 15 2007, 07:08 PM)
Well, no one's going to accept a "guess".
Aside from mixing the apples of limits with the oranges of infinitesimals, the idea that an "area" could be zero while the enclosing boundaries non-zero would seem to violate the axiom that x * 0 = 0. It's going to take a lot more than hopeful arm-waving for this idea to pan out.
As far as the area being zero while the sides are not zero, I was reminded that this is true only for certain geometries (like euclidean). There is no way for this to happen in others. This would seem to contraindicate your hope for a physical analogy, as it's not clear physical geometry is strictly euclidean.
hej NoCleverName
1) I am still in Eulers time in analysis, I study it by reading his books as I find it the easiest way to understand it ( I have done it properly 20 years ago but did not understand very well, so I have some background-but Euler just kills those university guys).
What I like about Euler, he does not seem to have made many mistakes, does he? Although he never bothered with rigorous math. He is much simpler and clearer at least for me.
2)Now Euler does differentials quite like I did in the exercise with vanishing areas/edges.No limits, no rigor, but always correct results ( which I can not claim in my case, of course).
For him, 0*x=0 holds as long as x is finite. But he also says- divide both sides by 0 to get x=0/0 and than divide both sides by e.g. 4 to get x/4=0/0 and then put x=1 to get 1/4=0/0. From which he concludes that even if 0 is divided by 0, ratios must hold ( which we know is true when applying e.g. Lopitals rule). That does not contradict other usages of different infinity scales and infinitesimal scales as long as You stick to right principles consistently. Which he did , as his results show.
3)Of course the derivations are valid only for Euclidean space. For others they have to take into account all the ratios between differentials etc. which I have not looked at yet. May be it is true what You say about edge/area ratio there.
4)But that is also an interesting question, is physical space Euclidean principally, with other spaces representing further superpositions of activity on that background, or is it really fundamentally NOT. I do not think that has been resolved. Gravity may also be exposure of e.g Aether density gradients or other mechanisms of creating mass which happens on the background of Euclidean space. Otherwise, why would it be in mathematics?
Aside from mixing the apples of limits with the oranges of infinitesimals, the idea that an "area" could be zero while the enclosing boundaries non-zero would seem to violate the axiom that x * 0 = 0. It's going to take a lot more than hopeful arm-waving for this idea to pan out.
As far as the area being zero while the sides are not zero, I was reminded that this is true only for certain geometries (like euclidean). There is no way for this to happen in others. This would seem to contraindicate your hope for a physical analogy, as it's not clear physical geometry is strictly euclidean.
hej NoCleverName
1) I am still in Eulers time in analysis, I study it by reading his books as I find it the easiest way to understand it ( I have done it properly 20 years ago but did not understand very well, so I have some background-but Euler just kills those university guys).
What I like about Euler, he does not seem to have made many mistakes, does he? Although he never bothered with rigorous math. He is much simpler and clearer at least for me.
2)Now Euler does differentials quite like I did in the exercise with vanishing areas/edges.No limits, no rigor, but always correct results ( which I can not claim in my case, of course).
For him, 0*x=0 holds as long as x is finite. But he also says- divide both sides by 0 to get x=0/0 and than divide both sides by e.g. 4 to get x/4=0/0 and then put x=1 to get 1/4=0/0. From which he concludes that even if 0 is divided by 0, ratios must hold ( which we know is true when applying e.g. Lopitals rule). That does not contradict other usages of different infinity scales and infinitesimal scales as long as You stick to right principles consistently. Which he did , as his results show.
3)Of course the derivations are valid only for Euclidean space. For others they have to take into account all the ratios between differentials etc. which I have not looked at yet. May be it is true what You say about edge/area ratio there.
4)But that is also an interesting question, is physical space Euclidean principally, with other spaces representing further superpositions of activity on that background, or is it really fundamentally NOT. I do not think that has been resolved. Gravity may also be exposure of e.g Aether density gradients or other mechanisms of creating mass which happens on the background of Euclidean space. Otherwise, why would it be in mathematics?
QUOTE (Ivars+Nov 15 2007, 10:26 PM)
Although he never bothered with rigorous math.
Utter nonsense.
It's almost painful reading your posts. You just spout crap constantly.
Utter nonsense.
It's almost painful reading your posts. You just spout crap constantly.
QUOTE (Ivars+Nov 15 2007, 10:26 PM)
But he also says- divide both sides by 0 to get x=0/0 and than divide both sides by e.g. 4 to get x/4=0/0 and then put x=1 to get 1/4=0/0.
What book of his has him saying that? Which page? I'll check in the uni library tomorrow because I seriously doubt he said that.
QUOTE (Ivars+Nov 15 2007, 10:26 PM)
3)Of course the derivations are valid only for Euclidean space.
Complete crap.
There is absolutely nothing to do with the metric of a manifold in what you're talking about. Basic manipulation of the Reals is algebra, it's independent of manifolds. The metric given to a manifold is something completely different. The development of such rigorous notions in differential geometry didn't occur till more than a century after Euler died.
You truely are just talking BS. You consistently just make things up and hope that if you talk enough bollocks people will think you're onto something. You just continue to show the depth of your ignorance and your willingness to lie. By the looks of it you've not read any books about these topics, despite your claims. You just hope to get people like myself off your back if you say you are reading such books.
There is absolutely nothing to do with the metric of a manifold in what you're talking about. Basic manipulation of the Reals is algebra, it's independent of manifolds. The metric given to a manifold is something completely different. The development of such rigorous notions in differential geometry didn't occur till more than a century after Euler died.
You truely are just talking BS. You consistently just make things up and hope that if you talk enough bollocks people will think you're onto something. You just continue to show the depth of your ignorance and your willingness to lie. By the looks of it you've not read any books about these topics, despite your claims. You just hope to get people like myself off your back if you say you are reading such books.
hej AlphaNumeric,
AlphaNumeric: Utter nonsense.
Of course, I am not concrete enough;but the general idea is not mine:See Leonhard Euler in Wikipedia
Ivars:"But he also says- divide both sides by 0 to get x=0/0 and than divide both sides by e.g. 4 to get x/4=0/0 and then put x=1 to get 1/4=0/0."
Please have a look at : Euler, Foundations of Differential CalculusChapters 1 to 9, , translated by John D. Blanton, Springer, 2000, ISBN 0-387-98534-4.
Chapter 3: " On the infinite and the Infinitely Small.":
Please have a look at : Euler, Foundations of Differential CalculusChapters 1 to 9, , translated by John D. Blanton, Springer, 2000, ISBN 0-387-98534-4.
Chapter 3: " On the infinite and the Infinitely Small.":
Euler:Paragraph 84, page 51: The arithmetic ratio between any two zeros is an equality. This is not the case with a geometric ratio. We can easily see this from this geometric proportion 2:1=0:0, in which the fourth term is equal to 0, as is the third.From the nature of the proportion, it since the first term is twice the second, it is necessary that the third is twice the fourth.
Paragraph 85., page 51. : These things are very clear , even in ordinary arithmetic. Everyone knows that when zero is multiplied by any number, the product is zero and that n*0=0, so that n:1=0:0.. Hence, it is clear that any two zeros can be in geometric ratio, although form the perspective of arithmetic, the ratio is always of equals. Since between zeros any ratio is possible, in order to indicate this diversity we use different notations on purpose, especially when geometric ratio between two zeros is being investigated. In the calculus of infinitely small, we deal precisely with geometric ratios of infinitely small quantities.
Here is the book in Amazon.com: Foundations of Differential Calculus in Amazon.com
May be I am wrong, but it seems to me that Euler did not see it as purely algebraical manipulation. His differential/Integral calculus was essentially derived from and linked to use in geometry of curves and surfaces, as I guess were most of calculus in that time. About metrics and various spaces I did not quote Euler.
Sorry Ivars, but I'll have to call BS on the very idea that the area can become zero without losing the very concepts of edges and angles. If there are edges and angles, there must be area. I think that is axiomatic.
I don't think you can proceed with out rigorously proving there can be a closed shape with edges and angles with 0 area contained. You need to do it, or admit you can't.
For example, the simplest closed form is a triangle. Three points. But, there must always be smaller points. There is no limit. You can't say "these are the smallest possible points". There is no such thing.
@Tim C
I am maintaining that there is no such thing as a shrinking process that can cause a shape to actually become zero. You can calculate a limit, or such, but that is not a physical process. The object itself can never actually reach the limit. In order to actually become 0, some destructive process must occur.
I am maintaining that there is no such thing as a shrinking process that can cause a shape to actually become zero. You can calculate a limit, or such, but that is not a physical process. The object itself can never actually reach the limit. In order to actually become 0, some destructive process must occur.
hej MeBigGuy
I have to think about the first few claims more, as intuitively I am not ready to admit that, nor have rigorous proof at hand. I need more time.
I agree it is not physical process in usual sense that can lead to such results, but mathematical process of infinitesimals/infinities can.
Maybe there are "physical" processes beneath quantum level which behave exactly like mathematics- and in this case You are right- a phase transition must happen, so that usual physical object is destroyed in some sense, but its FORM remains as different phase of it and continues to behave as mathematics of infinitesimals of different orders prescribe- maintaining edges while losing AREA.
We can also say there is no infinity in Nature-it is not divisible infinitely nor is material Universe infinitel large - on other hand, we can never imagine the final number of any number sequence which would imply there is none bigger than it- so our mind can not do without infinity to understand the seemingly finite Nature- the conjecture is perhaps there is real infinity in Nature as well, which we just do not observe.
I'd guess that if there are quanta's of form in nature, then when something shrank to nothing, or to 1 quanta, there would be no form left.
Quantization does not preserve information. Information can be lost. (only exception is within the sampling theorm)
Tell me a number and give some time. May be a lot of time. For 3D we need 2 numbers. It would be easier for 2D. I hope that figure is equilateral, to make things easier.
No, zero volume is the only fact I can give you; it is the only property in euclidian space this object now has.
Maybe now you will understand what "loss of information" means.
QUOTE
Ivars:"Although he (Euler) never bothered with rigorous math."
QUOTE (->
| QUOTE |
| Ivars:"Although he (Euler) never bothered with rigorous math." |
AlphaNumeric: Utter nonsense.
Of course, I am not concrete enough;but the general idea is not mine:See Leonhard Euler in Wikipedia
QUOTE
The development of calculus was at the forefront of 18th century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigour,[20] his ideas led to many great advances.
QUOTE (->
| QUOTE |
| The development of calculus was at the forefront of 18th century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigour,[20] his ideas led to many great advances. |
Ivars:"But he also says- divide both sides by 0 to get x=0/0 and than divide both sides by e.g. 4 to get x/4=0/0 and then put x=1 to get 1/4=0/0."
QUOTE
AlphaNumeric:"What book of his has him saying that? Which page? I'll check in the uni library tomorrow because I seriously doubt he said that.Complete crap."
Please have a look at : Euler, Foundations of Differential CalculusChapters 1 to 9, , translated by John D. Blanton, Springer, 2000, ISBN 0-387-98534-4.
Chapter 3: " On the infinite and the Infinitely Small.":
QUOTE (->
| QUOTE |
| AlphaNumeric:"What book of his has him saying that? Which page? I'll check in the uni library tomorrow because I seriously doubt he said that.Complete crap." |
Please have a look at : Euler, Foundations of Differential CalculusChapters 1 to 9, , translated by John D. Blanton, Springer, 2000, ISBN 0-387-98534-4.
Chapter 3: " On the infinite and the Infinitely Small.":
Euler:Paragraph 84, page 51: The arithmetic ratio between any two zeros is an equality. This is not the case with a geometric ratio. We can easily see this from this geometric proportion 2:1=0:0, in which the fourth term is equal to 0, as is the third.From the nature of the proportion, it since the first term is twice the second, it is necessary that the third is twice the fourth.
Paragraph 85., page 51. : These things are very clear , even in ordinary arithmetic. Everyone knows that when zero is multiplied by any number, the product is zero and that n*0=0, so that n:1=0:0.. Hence, it is clear that any two zeros can be in geometric ratio, although form the perspective of arithmetic, the ratio is always of equals. Since between zeros any ratio is possible, in order to indicate this diversity we use different notations on purpose, especially when geometric ratio between two zeros is being investigated. In the calculus of infinitely small, we deal precisely with geometric ratios of infinitely small quantities.
Here is the book in Amazon.com: Foundations of Differential Calculus in Amazon.com
QUOTE
Basic manipulation of the Reals is algebra, it's independent of manifolds. The metric given to a manifold is something completely different. The development of such rigorous notions in differential geometry didn't occur till more than a century after Euler died.
May be I am wrong, but it seems to me that Euler did not see it as purely algebraical manipulation. His differential/Integral calculus was essentially derived from and linked to use in geometry of curves and surfaces, as I guess were most of calculus in that time. About metrics and various spaces I did not quote Euler.
QUOTE
No, I was suggesting that we shrink it so that AREA becomes 0. What happens to the edges and angles between them?
Sorry Ivars, but I'll have to call BS on the very idea that the area can become zero without losing the very concepts of edges and angles. If there are edges and angles, there must be area. I think that is axiomatic.
I don't think you can proceed with out rigorously proving there can be a closed shape with edges and angles with 0 area contained. You need to do it, or admit you can't.
For example, the simplest closed form is a triangle. Three points. But, there must always be smaller points. There is no limit. You can't say "these are the smallest possible points". There is no such thing.
@Tim C
I am maintaining that there is no such thing as a shrinking process that can cause a shape to actually become zero. You can calculate a limit, or such, but that is not a physical process. The object itself can never actually reach the limit. In order to actually become 0, some destructive process must occur.
QUOTE (meBigGuy+Nov 16 2007, 08:35 AM)
I am maintaining that there is no such thing as a shrinking process that can cause a shape to actually become zero. You can calculate a limit, or such, but that is not a physical process. The object itself can never actually reach the limit. In order to actually become 0, some destructive process must occur.
hej MeBigGuy
I have to think about the first few claims more, as intuitively I am not ready to admit that, nor have rigorous proof at hand. I need more time.
I agree it is not physical process in usual sense that can lead to such results, but mathematical process of infinitesimals/infinities can.
Maybe there are "physical" processes beneath quantum level which behave exactly like mathematics- and in this case You are right- a phase transition must happen, so that usual physical object is destroyed in some sense, but its FORM remains as different phase of it and continues to behave as mathematics of infinitesimals of different orders prescribe- maintaining edges while losing AREA.
We can also say there is no infinity in Nature-it is not divisible infinitely nor is material Universe infinitel large - on other hand, we can never imagine the final number of any number sequence which would imply there is none bigger than it- so our mind can not do without infinity to understand the seemingly finite Nature- the conjecture is perhaps there is real infinity in Nature as well, which we just do not observe.
QUOTE
We can also say there is no infinity in Nature-it is not divisible infinitely nor is material Universe infinitel large - on other hand, we can never imagine the final number of any number sequence which would imply there is none bigger than it- so our mind can not do without infinity to understand the seemingly finite Nature- the conjecture is perhaps there is real infinity in Nature as well, which we just do not observe.
I'd guess that if there are quanta's of form in nature, then when something shrank to nothing, or to 1 quanta, there would be no form left.
Quantization does not preserve information. Information can be lost. (only exception is within the sampling theorm)
QUOTE (meBigGuy+Nov 16 2007, 09:52 AM)
Quantization does not preserve information. Information can be lost. (only exception is within the sampling theorem)
hej MeBigGuy,
Sampling theorem!
But that should be enough. Once You measure some quanta and can guess /derive the filter characteristics, You are able to reconstruct infinitely divisible process that has been going behind them from data about quanta.
Quanta act as bandwidth filter here, the sub quantum processes when filtered, are the ones we observe as quantized Nature.
Next question is can the filters itself be formed from sub quantum processes in a way that leads to quantization of Nature at certain level- and why not? The only thing needed is some eternal process which would drive it.
So if there is a solution that gives both filter response function and observables, we can be quite sure that our guess about what happens on the other side of filter, its input, may be also correct.
Next- are there some unique formation of sub quantum processes that leads to self-resonant - meaning unavoidable- formation of quanta sooner or later which in turn behave as we observe? If yes, we can deconstruct it also backwards, and from all 500 zillion possible string theory equations choose the right one in the right time in the right place.
Yes Yes AlphaNumeric I do not know anything about string theory, but I have heard rumours there are problems with too many possible equations. And there should be many as otherwise there would be nothing to choose from, and Nature will become classical in a sence fully determined by convergent series which does not sound exciting either.
hej MeBigGuy,
Sampling theorem!
But that should be enough. Once You measure some quanta and can guess /derive the filter characteristics, You are able to reconstruct infinitely divisible process that has been going behind them from data about quanta.
Quanta act as bandwidth filter here, the sub quantum processes when filtered, are the ones we observe as quantized Nature.
Next question is can the filters itself be formed from sub quantum processes in a way that leads to quantization of Nature at certain level- and why not? The only thing needed is some eternal process which would drive it.
So if there is a solution that gives both filter response function and observables, we can be quite sure that our guess about what happens on the other side of filter, its input, may be also correct.
Next- are there some unique formation of sub quantum processes that leads to self-resonant - meaning unavoidable- formation of quanta sooner or later which in turn behave as we observe? If yes, we can deconstruct it also backwards, and from all 500 zillion possible string theory equations choose the right one in the right time in the right place.
Yes Yes AlphaNumeric I do not know anything about string theory, but I have heard rumours there are problems with too many possible equations. And there should be many as otherwise there would be nothing to choose from, and Nature will become classical in a sence fully determined by convergent series which does not sound exciting either.
Ivars, you are starting to wave your arms rather wildly.
I suggest you actually demonstrate a procedure to reconstruct the 3D geometric figure I now have in mind that presently has zero volume.
I suggest you actually demonstrate a procedure to reconstruct the 3D geometric figure I now have in mind that presently has zero volume.
QUOTE (NoCleverName+Nov 16 2007, 01:17 PM)
Ivars, you are starting to wave your arms rather wildly.
I suggest you actually demonstrate a procedure to reconstruct the 3D geometric figure I now have in mind that presently has zero volume.
hej NoCleverName
I think I do
Tell me a number and give some time. May be a lot of time. For 3D we need 2 numbers. It would be easier for 2D. I hope that figure is equilateral, to make things easier.
I suggest you actually demonstrate a procedure to reconstruct the 3D geometric figure I now have in mind that presently has zero volume.
hej NoCleverName
I think I do
Tell me a number and give some time. May be a lot of time. For 3D we need 2 numbers. It would be easier for 2D. I hope that figure is equilateral, to make things easier.
QUOTE (Ivars+Nov 16 2007, 01:49 PM)
Tell me a number and give some time. May be a lot of time. For 3D we need 2 numbers. It would be easier for 2D. I hope that figure is equilateral, to make things easier.
No, zero volume is the only fact I can give you; it is the only property in euclidian space this object now has.
Maybe now you will understand what "loss of information" means.
QUOTE (NoCleverName+Nov 16 2007, 02:02 PM)
No, zero volume is the only fact I can give you; it is the only property in euclidian space this object now has.
Maybe now you will understand what "loss of information" means.
Ok, it is tetrahedron.
That is the only thing I can reconstruct from so little information-luckily You mentioned Euclidian space, so I took the simplest thing that can exist there with volume.
Maybe now you will understand what "loss of information" means.
Ok, it is tetrahedron.
That is the only thing I can reconstruct from so little information-luckily You mentioned Euclidian space, so I took the simplest thing that can exist there with volume.
QUOTE (Ivars+Nov 16 2007, 08:46 AM)
While some of Euler's proofs are not acceptable by modern standards of mathematical rigour,
The notion of mathematical analysis proofs were not as well developed in Euler's day as they are now. This is why it's often a handicap to use original texts as something to learn from, because they don't have the hindsight that modern texts have.
Euler gave very good and rigorous proofs to a great many things, he didn't deliberately skim over proof. It's just we've now made concrete a lot of analysis and his methods are outdated.
The notion of mathematical analysis proofs were not as well developed in Euler's day as they are now. This is why it's often a handicap to use original texts as something to learn from, because they don't have the hindsight that modern texts have.
Euler gave very good and rigorous proofs to a great many things, he didn't deliberately skim over proof. It's just we've now made concrete a lot of analysis and his methods are outdated.
QUOTE (Euler+Nov 16 2007, 08:46 AM)
Paragraph 85., page 51. : These things are very clear , even in ordinary arithmetic. Everyone knows that when zero is multiplied by any number, the product is zero and that n*0=0, so that n:1=0:0.. Hence, it is clear that any two zeros can be in geometric ratio, although form the perspective of arithmetic, the ratio is always of equals. Since between zeros any ratio is possible, in order to indicate this diversity we use different notations on purpose, especially when geometric ratio between two zeros is being investigated. In the calculus of infinitely small, we deal precisely with geometric ratios of infinitely small quantities.
That does not say that 0/0 = 1/n. It says that just as 1*n = n, 0*n = 0, and so 1 and n can be in the same ratio as 0 and 0.
1*n = n and 1*0 = 0 do not imply that 0/0 = 1/n is valid. Multiplication and division are NOT completely symmetric, precisely because you cannot 'cancel zero's. No such number as 1/0 exists in the Reals.
[QUOTE=Ivars,Nov 16 2007, 08:46 AM]
May be I am wrong, but it seems to me that Euler did not see it as purely algebraical manipulation. His differential/Integral calculus was essentially derived from and linked to use in geometry of curves and surfaces, as I guess were most of calculus in that time. About metrics and various spaces I did not quote Euler./QUOTE]The theorems of derivatives and integrals are INDEPENDENT of manifolds. You can do calculus on a manifold, it's an essential notion to physicists, but manifolds are not required for calculus. Manifolds didn't become well developed in mathematics till people like Riemann developed Riemannian geometry. Hell, in Euler's day they didn't even realise there was anything other than Euclidean metrics. It wasn't until Gauss developed hyperbolic geometry this was realised.
At no point in the differentiation of something like f(x) = x² do you mention a metric. If you disagree, say where the metric is mentioned.
You are constantly wrong and like every other crank here, you don't like being told it.
1*n = n and 1*0 = 0 do not imply that 0/0 = 1/n is valid. Multiplication and division are NOT completely symmetric, precisely because you cannot 'cancel zero's. No such number as 1/0 exists in the Reals.
[QUOTE=Ivars,Nov 16 2007, 08:46 AM]
May be I am wrong, but it seems to me that Euler did not see it as purely algebraical manipulation. His differential/Integral calculus was essentially derived from and linked to use in geometry of curves and surfaces, as I guess were most of calculus in that time. About metrics and various spaces I did not quote Euler./QUOTE]The theorems of derivatives and integrals are INDEPENDENT of manifolds. You can do calculus on a manifold, it's an essential notion to physicists, but manifolds are not required for calculus. Manifolds didn't become well developed in mathematics till people like Riemann developed Riemannian geometry. Hell, in Euler's day they didn't even realise there was anything other than Euclidean metrics. It wasn't until Gauss developed hyperbolic geometry this was realised.
At no point in the differentiation of something like f(x) = x² do you mention a metric. If you disagree, say where the metric is mentioned.
You are constantly wrong and like every other crank here, you don't like being told it.
QUOTE (Ivars+Nov 16 2007, 02:38 PM)
Ok, it is tetrahedron.
That is the only thing I can reconstruct from so little information-luckily You mentioned Euclidian space, so I took the simplest thing that can exist there with volume.
That, I believe, might be a sphere. In any event, I hope now you can see how barren your idea is. Obviously you are reduced to guesswork and you can lay out no mathematical procedure for anyone else to use that will reconstruct the object.
Bzzzt! To Play Again, Insert another quarter!
That is the only thing I can reconstruct from so little information-luckily You mentioned Euclidian space, so I took the simplest thing that can exist there with volume.
That, I believe, might be a sphere. In any event, I hope now you can see how barren your idea is. Obviously you are reduced to guesswork and you can lay out no mathematical procedure for anyone else to use that will reconstruct the object.
Bzzzt! To Play Again, Insert another quarter!
hej AlphaNumeric,
Euler: Paragraph 85., page 51. : These things are very clear , even in ordinary arithmetic. Everyone knows that when zero is multiplied by any number, the product is zero and that n*0=0, so that n:1=0:0.. Hence, it is clear that any two zeros can be in geometric ratio, although form the perspective of arithmetic, the ratio is always of equals. Since between zeros any ratio is possible, in order to indicate this diversity we use different notations on purpose, especially when geometric ratio between two zeros is being investigated. In the calculus of infinitely small, we deal precisely with geometric ratios of infinitely small quantities.
We can easily see this from this geometric proportion 2:1=0:0
Honestly, I do not understand what You are trying to say. I do not see any difference between what I wrote, and what Euler wrote, except that he wrote n/1 =0/0 and I wrote 1/4 =0/0. You mean there is a difference if we write 4/1=0/0 or 1/4=0/0? That is ridiculous! Can You explain?
Honestly, I do not understand what You are trying to say. I do not see any difference between what I wrote, and what Euler wrote, except that he wrote n/1 =0/0 and I wrote 1/4 =0/0. You mean there is a difference if we write 4/1=0/0 or 1/4=0/0? That is ridiculous! Can You explain?
AlphaNumeric: At no point in the differentiation of something like f(x) = x² do you mention a metric. If you disagree, say where the metric is mentioned.
Of course, when You use linear equal differentials dx You imply something about the space?
QUOTE
For him(Euler), 0*x=0 holds as long as x is finite. But he also says- divide both sides by 0 to get x=0/0 and than divide both sides by e.g. 4 to get x/4=0/0 and then put x=1 to get 1/4=0/0.
QUOTE (->
| QUOTE |
| For him(Euler), 0*x=0 holds as long as x is finite. But he also says- divide both sides by 0 to get x=0/0 and than divide both sides by e.g. 4 to get x/4=0/0 and then put x=1 to get 1/4=0/0. |
Euler: Paragraph 85., page 51. : These things are very clear , even in ordinary arithmetic. Everyone knows that when zero is multiplied by any number, the product is zero and that n*0=0, so that n:1=0:0.. Hence, it is clear that any two zeros can be in geometric ratio, although form the perspective of arithmetic, the ratio is always of equals. Since between zeros any ratio is possible, in order to indicate this diversity we use different notations on purpose, especially when geometric ratio between two zeros is being investigated. In the calculus of infinitely small, we deal precisely with geometric ratios of infinitely small quantities.
We can easily see this from this geometric proportion 2:1=0:0
QUOTE
AlphaNumeric:That does not say that 0/0 = 1/n. It says that just as 1*n = n, 0*n = 0, and so 1 and n can be in the same ratio as 0 and 0.
1*n = n and 1*0 = 0 do not imply that 0/0 = 1/n is valid. Multiplication and division are NOT completely symmetric, precisely because you cannot 'cancel zero's. No such number as 1/0 exists in the Reals.
1*n = n and 1*0 = 0 do not imply that 0/0 = 1/n is valid. Multiplication and division are NOT completely symmetric, precisely because you cannot 'cancel zero's. No such number as 1/0 exists in the Reals.
Honestly, I do not understand what You are trying to say. I do not see any difference between what I wrote, and what Euler wrote, except that he wrote n/1 =0/0 and I wrote 1/4 =0/0. You mean there is a difference if we write 4/1=0/0 or 1/4=0/0? That is ridiculous! Can You explain?
QUOTE (->
| QUOTE |
| AlphaNumeric:That does not say that 0/0 = 1/n. It says that just as 1*n = n, 0*n = 0, and so 1 and n can be in the same ratio as 0 and 0. 1*n = n and 1*0 = 0 do not imply that 0/0 = 1/n is valid. Multiplication and division are NOT completely symmetric, precisely because you cannot 'cancel zero's. No such number as 1/0 exists in the Reals. |
Honestly, I do not understand what You are trying to say. I do not see any difference between what I wrote, and what Euler wrote, except that he wrote n/1 =0/0 and I wrote 1/4 =0/0. You mean there is a difference if we write 4/1=0/0 or 1/4=0/0? That is ridiculous! Can You explain?
AlphaNumeric: At no point in the differentiation of something like f(x) = x² do you mention a metric. If you disagree, say where the metric is mentioned.
Of course, when You use linear equal differentials dx You imply something about the space?
QUOTE (Ivars+Nov 17 2007, 05:38 PM)
Honestly, I do not understand what You are trying to say. I do not see any difference between what I wrote, and what Euler wrote, except that he wrote n/1 =0/0 and I wrote 1/4 =0/0. You mean there is a difference if we write 4/1=0/0 or 1/4=0/0? That is ridiculous! Can You explain?
No, he wrote that 1:n = 0:0. That is NOT the same as 1/n = 0/0. I can multiply by 0, I cannot divide!
1=2 is not true, but if I multiply both sides by 0, I get a true statement, 0*1 = 0*2. Euler is saying that if I have 2 entities in a ratio of 1:n, if I have zero amounts of both, I am still in the 1:n ratio, 0*1 : 0*n. That is NOT the same as saying 1/n = 0/0.
You are not quoting Euler, you are putting your own ignorant spin on it.
No way. You distort the sampling theorom beyond recognition.
read about aliasing.
No, he wrote that 1:n = 0:0. That is NOT the same as 1/n = 0/0. I can multiply by 0, I cannot divide!
1=2 is not true, but if I multiply both sides by 0, I get a true statement, 0*1 = 0*2. Euler is saying that if I have 2 entities in a ratio of 1:n, if I have zero amounts of both, I am still in the 1:n ratio, 0*1 : 0*n. That is NOT the same as saying 1/n = 0/0.
You are not quoting Euler, you are putting your own ignorant spin on it.
QUOTE (Ivars+Nov 17 2007, 05:38 PM)
Of course, when You use linear equal differentials dx You imply something about the space?
Linearity has nothing to do with it! You are not saying ANYTHING about the space.
If you have a function, f(x), and you change x by an amount dx, the function f(x) will change by an amount df(x). It's interesting to consider what the ratio of these changes is, df(x)/dx = (df/dx)(x).
If differential geometry, there's a new kind of derivative known as the covariant derivative, which including a 'connection' and which takes into account the properties of the space you are working in. HOWEVER, if you actually knew any differential geometry, you'd know that for a scalar function, which f(x) is by definition, the connection is zero! There is no difference between the covariant derivative and the usual derivative. So even in non-Eulclidean space, the derivative of f(x) is df/dx !
But you didn't know this, because you never know anything about the maths you try to talk about, you just hope that other people don't know it either and they assume you do.
Honestly, what purpose, other than to deceive, can you possibly have for doing such things? You don't have worthwhile discussions, you know that. The only people who engage you in any conversation other than to correct you are idiots like Mott.Carl whose even more ignorant than you (same goes for StevenA, Laidback etc) and if you aren't able to tell they are full of **** you're a lost cause!
Come on, explain to me how you hope to have a meaningful discussion when you continuely talk about topics noone in the discussion knows a thing about? You're not attempting to rederive mathematical results, you don't do any actual maths, so how do you hope to say something meaningful about non-Euclidean gometry and the like?
Have you managed to do ANY of those questions I asked you yet? Even as a matter of curiosity you should have at least had a go at one of them. Come on, let's see you actually dfo something you think you can do if you went to a book. Do one of those questions in the next 24 hours. That even gives you enough time to run to another forum and ask there! At least do something to demonstrate you can go from a question to an answer in something you try to BS about.
If you have a function, f(x), and you change x by an amount dx, the function f(x) will change by an amount df(x). It's interesting to consider what the ratio of these changes is, df(x)/dx = (df/dx)(x).
If differential geometry, there's a new kind of derivative known as the covariant derivative, which including a 'connection' and which takes into account the properties of the space you are working in. HOWEVER, if you actually knew any differential geometry, you'd know that for a scalar function, which f(x) is by definition, the connection is zero! There is no difference between the covariant derivative and the usual derivative. So even in non-Eulclidean space, the derivative of f(x) is df/dx !
But you didn't know this, because you never know anything about the maths you try to talk about, you just hope that other people don't know it either and they assume you do.
Honestly, what purpose, other than to deceive, can you possibly have for doing such things? You don't have worthwhile discussions, you know that. The only people who engage you in any conversation other than to correct you are idiots like Mott.Carl whose even more ignorant than you (same goes for StevenA, Laidback etc) and if you aren't able to tell they are full of **** you're a lost cause!
Come on, explain to me how you hope to have a meaningful discussion when you continuely talk about topics noone in the discussion knows a thing about? You're not attempting to rederive mathematical results, you don't do any actual maths, so how do you hope to say something meaningful about non-Euclidean gometry and the like?
Have you managed to do ANY of those questions I asked you yet? Even as a matter of curiosity you should have at least had a go at one of them. Come on, let's see you actually dfo something you think you can do if you went to a book. Do one of those questions in the next 24 hours. That even gives you enough time to run to another forum and ask there! At least do something to demonstrate you can go from a question to an answer in something you try to BS about.
QUOTE
But that should be enough. Once You measure some quanta and can guess /derive the filter characteristics
No way. You distort the sampling theorom beyond recognition.
read about aliasing.
QUOTE (AlphaNumeric+Nov 17 2007, 05:13 PM)
You are not quoting Euler, you are putting your own ignorant spin on it.
But you didn't know this, because you never know anything about the maths you try to talk about, you just hope that other people don't know it either and they assume you do.
I still do not understand- if ratio of 2 numbers is 1:n ->You imply it is not the same as division of 1 by n, or 1/n ? Why so?
Anyway, I found something I would like to read - non-standard analysis. I hoped there is something that is based on works of Leubniz, Euler, Newton and does not involve limits which appeared later. Sounds like that is the right way to study mathematics, as it does not deal with all that limit BS.
Non-standard analysis
Then I will be also able to solve Your tasks. I am curious about the solutions but I would never copy them or ask someone to do them for me - that is boring and brings no value.
y''+5y'+6y=e^3x
Obviously, one could look for solutions where Y contains e^3x as (e^3x)' = 3e^3x, (e^3x)''=9e^3x so equality could be possible.
So I wil try with y=C*e^3x
So 9Ce^3x+5C3e^3x+6Ce^3x=e^3x
or 30C e^3x=e^3x
so C=1/30 and Y=1/30*e^3x. That is one solution. There should be other.
I still do not understand- if ratio of 2 numbers is 1:n ->You imply it is not the same as division of 1 by n, or 1/n ? Why so?
Anyway, I found something I would like to read - non-standard analysis. I hoped there is something that is based on works of Leubniz, Euler, Newton and does not involve limits which appeared later. Sounds like that is the right way to study mathematics, as it does not deal with all that limit BS.
Non-standard analysis
Then I will be also able to solve Your tasks. I am curious about the solutions but I would never copy them or ask someone to do them for me - that is boring and brings no value.
y''+5y'+6y=e^3x
Obviously, one could look for solutions where Y contains e^3x as (e^3x)' = 3e^3x, (e^3x)''=9e^3x so equality could be possible.
So I wil try with y=C*e^3x
So 9Ce^3x+5C3e^3x+6Ce^3x=e^3x
or 30C e^3x=e^3x
so C=1/30 and Y=1/30*e^3x. That is one solution. There should be other.
If differential geometry, there's a new kind of derivative known as the covariant derivative, which including a 'connection' and which takes into account the properties of the space you are working in. HOWEVER, if you actually knew any differential geometry, you'd know that for a scalar function, which f(x) is by definition, the connection is zero! There is no difference between the covariant derivative and the usual derivative. So even in non-Eulclidean space, the derivative of f(x) is df/dx !
You may be really right there, I did not know, I was trying to figure out myself. That will take longer time to learn to be able to oppose what You say, so - OK.
My purpose on doing all this is to learn, while keeping things interesting and undecided. I value highly such people as Sylwester, StevenA, Laidback who have consistent ideas they try to work on, but, as me , they do not own the language to express them so that others would be able to read them. Which does not diminish the value of the ideas, but makes them very difficult to grasp or even discuss.
But you didn't know this, because you never know anything about the maths you try to talk about, you just hope that other people don't know it either and they assume you do.
QUOTE
No, he wrote that 1:n = 0:0. That is NOT the same as 1/n = 0/0. I can multiply by 0, I cannot divide!
I still do not understand- if ratio of 2 numbers is 1:n ->You imply it is not the same as division of 1 by n, or 1/n ? Why so?
Anyway, I found something I would like to read - non-standard analysis. I hoped there is something that is based on works of Leubniz, Euler, Newton and does not involve limits which appeared later. Sounds like that is the right way to study mathematics, as it does not deal with all that limit BS.
Non-standard analysis
Then I will be also able to solve Your tasks. I am curious about the solutions but I would never copy them or ask someone to do them for me - that is boring and brings no value.
y''+5y'+6y=e^3x
Obviously, one could look for solutions where Y contains e^3x as (e^3x)' = 3e^3x, (e^3x)''=9e^3x so equality could be possible.
So I wil try with y=C*e^3x
So 9Ce^3x+5C3e^3x+6Ce^3x=e^3x
or 30C e^3x=e^3x
so C=1/30 and Y=1/30*e^3x. That is one solution. There should be other.
QUOTE (->
| QUOTE |
| No, he wrote that 1:n = 0:0. That is NOT the same as 1/n = 0/0. I can multiply by 0, I cannot divide! |
I still do not understand- if ratio of 2 numbers is 1:n ->You imply it is not the same as division of 1 by n, or 1/n ? Why so?
Anyway, I found something I would like to read - non-standard analysis. I hoped there is something that is based on works of Leubniz, Euler, Newton and does not involve limits which appeared later. Sounds like that is the right way to study mathematics, as it does not deal with all that limit BS.
Non-standard analysis
Then I will be also able to solve Your tasks. I am curious about the solutions but I would never copy them or ask someone to do them for me - that is boring and brings no value.
y''+5y'+6y=e^3x
Obviously, one could look for solutions where Y contains e^3x as (e^3x)' = 3e^3x, (e^3x)''=9e^3x so equality could be possible.
So I wil try with y=C*e^3x
So 9Ce^3x+5C3e^3x+6Ce^3x=e^3x
or 30C e^3x=e^3x
so C=1/30 and Y=1/30*e^3x. That is one solution. There should be other.
If differential geometry, there's a new kind of derivative known as the covariant derivative, which including a 'connection' and which takes into account the properties of the space you are working in. HOWEVER, if you actually knew any differential geometry, you'd know that for a scalar function, which f(x) is by definition, the connection is zero! There is no difference between the covariant derivative and the usual derivative. So even in non-Eulclidean space, the derivative of f(x) is df/dx !
You may be really right there, I did not know, I was trying to figure out myself. That will take longer time to learn to be able to oppose what You say, so - OK.
My purpose on doing all this is to learn, while keeping things interesting and undecided. I value highly such people as Sylwester, StevenA, Laidback who have consistent ideas they try to work on, but, as me , they do not own the language to express them so that others would be able to read them. Which does not diminish the value of the ideas, but makes them very difficult to grasp or even discuss.
QUOTE (Ivars+Nov 18 2007, 03:56 PM)
I still do not understand- if ratio of 2 numbers is 1:n ->You imply it is not the same as division of 1 by n, or 1/n ? Why so?
No, that isn't what I'm implying.
If A and B are in a ratio of 1:n, then A*n = 1*B. B is n times larger than A. That can be expressed as B/A = n/1 PROVIDED A and B are non-zero. If they are zero, you cannot rearrange to say B/A = n, because you've taken a true statement, that 0=0 and divided by zero, which isn't allowed.
As I said, 0*1 = 0*2 is true. That statement divided by 0 isn't. Dividing by 0 does not guarentee that one statement implies another.
No, that isn't what I'm implying.
If A and B are in a ratio of 1:n, then A*n = 1*B. B is n times larger than A. That can be expressed as B/A = n/1 PROVIDED A and B are non-zero. If they are zero, you cannot rearrange to say B/A = n, because you've taken a true statement, that 0=0 and divided by zero, which isn't allowed.
As I said, 0*1 = 0*2 is true. That statement divided by 0 isn't. Dividing by 0 does not guarentee that one statement implies another.
QUOTE (Ivars+Nov 18 2007, 03:56 PM)
Anyway, I found something I would like to read - non-standard analysis. I hoped there is something that is based on works of Leubniz, Euler, Newton and does not involve limits which appeared later. Sounds like that is the right way to study mathematics, as it does not deal with all that limit BS.
I know what non-standard analysis is. I know a bit about it because I know how it violates standard analysis, which I know a great deal more about.
You know neither kind of analysis. You're trying to run before you can walk because you know that learning to walk (ie learning basic mathematics) is too hard for you so it's easier to skip to stuff less people know about and so less people will be able to expose your BS.
You know neither kind of analysis. You're trying to run before you can walk because you know that learning to walk (ie learning basic mathematics) is too hard for you so it's easier to skip to stuff less people know about and so less people will be able to expose your BS.
QUOTE (Ivars+Nov 18 2007, 03:56 PM)
That is one solution. There should be other.
No, there's actually 2 linearly independent solutions and the basic one you found.
This is stuff taught to school children, in high school. You'll find it in any textbook aimed at 17 year olds.
When given a differential equations L(y)(x) = f(x) you first find all solutions to L(y)(x) = 0 and then particular solutions to L(y)(x) = f(x). In the case of that equation, y=Ae^3x is a particular solution. There's two (since L is a 2nd order differential operator) linearly independent solutions which you haven't found (which basically means there's infinitely many solutions you've missed).
Solution :
L(y)(x) = y'' + 5y + 6y = e^3x
Consider y''+5y+6y = 0. Ansatz y = e^ax implies a²+5a+6=0 = (a+3)(a+2), so a=-3 or -2.
Therefore y(x) = Ae^(-2x) + B^(-3x) solves L(y)=0.
Particular solution to L(y)(x) = e^3x is y = Ce^(3x) => C = 1/30.
Therefore the full solution is y(x) = Ae^(-2x)+Be^(-3x)+(1/30)e^(3x). This is a solution for ANY values of A and B.
That is a question given to high school students (ages 16~18). You, despite wanting to learn about non-standard analysis (typically taught to maths graduates, ages 21+), are unable to do that.
This is stuff taught to school children, in high school. You'll find it in any textbook aimed at 17 year olds.
When given a differential equations L(y)(x) = f(x) you first find all solutions to L(y)(x) = 0 and then particular solutions to L(y)(x) = f(x). In the case of that equation, y=Ae^3x is a particular solution. There's two (since L is a 2nd order differential operator) linearly independent solutions which you haven't found (which basically means there's infinitely many solutions you've missed).
Solution :
L(y)(x) = y'' + 5y + 6y = e^3x
Consider y''+5y+6y = 0. Ansatz y = e^ax implies a²+5a+6=0 = (a+3)(a+2), so a=-3 or -2.
Therefore y(x) = Ae^(-2x) + B^(-3x) solves L(y)=0.
Particular solution to L(y)(x) = e^3x is y = Ce^(3x) => C = 1/30.
Therefore the full solution is y(x) = Ae^(-2x)+Be^(-3x)+(1/30)e^(3x). This is a solution for ANY values of A and B.
That is a question given to high school students (ages 16~18). You, despite wanting to learn about non-standard analysis (typically taught to maths graduates, ages 21+), are unable to do that.
QUOTE (Ivars+Nov 18 2007, 03:56 PM)
You may be really right there, I did not know, I was trying to figure out myself.
I am right. You'd know if you bothered to learn. And you aren't trying to figure this out for yourself, you're just hoping to BS long enough that some idiot (like Mott.Carl or Laidback) pats you on the back and praises you for how clever you must be to know about non-standard analysis or non-Euclidean manifolds. Except you aren't because you don't.
QUOTE (Ivars+Nov 18 2007, 03:56 PM)
My purpose on doing all this is to learn
See my previous comment.
QUOTE (Ivars+Nov 18 2007, 03:56 PM)
I value highly such people as Sylwester, StevenA, Laidback who have consistent ideas they try to work on
The blind praising the blind.
To restart with a new idea:
imagine the smallest thing we can have is a straight , unbending infinitesimal piece of a line. It has a direction, but has no magnitude, and is continuous , what ever it may mean.
When looked upon from one end it is a point, a discrete thing. So this element contains both discrete and continuous elements, being an infinitesimal line and a point at the same time.
Now since it is infinitesimal, let us denote its lenght as dx. Then, as we know, a triangle made from it will have area proportional to dx^2, so in the scale of dx it will be not present- infinitely smaller. So in that scale triangle will have no area, but 1 scale lower, infinity lower, there will be an area but edges will be infinitely large.
If we construct a tetrahedron form dx, its volume will be proportional to dx^3 in that scale where dx is infinitesimal, so 0. Volume will not exist in this scale , but 2 scales lower it will be infinitesimal volume dx^3.
So we can see that in any given scale we ,starting to builkd things NOT from points , but infinitesimal vectors, directions, when reducing things to infinitely small, will be left with form only, but vanishing area and volume.
The question is:
Given the method of constructing things from infinitesimal line segments which has direction, is there any surface which will have:
a) area in the same scale as dx
volume in the same scale as dx, or at least just one, not 2 scales lower
What would be the finite ? area/volume of such a thing.
To give some idea, if we had 2 infinite lines with lenght sgrt(2) , the area of a square in the scale based on sgrt(2) would be (sgrt(2)^2)=2 or finite.
However, area of a circle built on radius sgrt(2) would be pi* r^2= 2* pi- infinite( in terms of numbers needed to describe it).
An Area of infinitesimal equilateral triangle built on from dx will be 0, of a cube =0, volume 0 as well for both.
imagine the smallest thing we can have is a straight , unbending infinitesimal piece of a line. It has a direction, but has no magnitude, and is continuous , what ever it may mean.
When looked upon from one end it is a point, a discrete thing. So this element contains both discrete and continuous elements, being an infinitesimal line and a point at the same time.
Now since it is infinitesimal, let us denote its lenght as dx. Then, as we know, a triangle made from it will have area proportional to dx^2, so in the scale of dx it will be not present- infinitely smaller. So in that scale triangle will have no area, but 1 scale lower, infinity lower, there will be an area but edges will be infinitely large.
If we construct a tetrahedron form dx, its volume will be proportional to dx^3 in that scale where dx is infinitesimal, so 0. Volume will not exist in this scale , but 2 scales lower it will be infinitesimal volume dx^3.
So we can see that in any given scale we ,starting to builkd things NOT from points , but infinitesimal vectors, directions, when reducing things to infinitely small, will be left with form only, but vanishing area and volume.
The question is:
Given the method of constructing things from infinitesimal line segments which has direction, is there any surface which will have:
a) area in the same scale as dx
volume in the same scale as dx, or at least just one, not 2 scales lowerWhat would be the finite ? area/volume of such a thing.
To give some idea, if we had 2 infinite lines with lenght sgrt(2) , the area of a square in the scale based on sgrt(2) would be (sgrt(2)^2)=2 or finite.
However, area of a circle built on radius sgrt(2) would be pi* r^2= 2* pi- infinite( in terms of numbers needed to describe it).
An Area of infinitesimal equilateral triangle built on from dx will be 0, of a cube =0, volume 0 as well for both.
QUOTE (Ivars+Nov 7 2007, 05:19 PM)
Let us have a simplice of volume-tetrahedron. Its volume is some number. Let us have some process where we require this volume in limit to reach 0 ( the volume) - e.g differential dP/dV.
Question: what happens with the form? what will be the form of the 0 volume?
Tetrahedron, sphere, point (what is that?) etc?
Either it is a form IN SPACE or it is not. If you shrink it so that it doesn't occupy space AND it is still a tetrahedron you have extinguished its outer space only, but its inner space is still there.
A zero-size tetrahedron turns inside-out
http://www.miqel.com/fractals_math_pattern...h-platonic.html
because the definition of "zero-size tetrahedron" doesn't include -nor can it include- inner space of the subject of the exercize, the tetrahedron itself, only its relationship to the outside world.
Question: what happens with the form? what will be the form of the 0 volume?
Tetrahedron, sphere, point (what is that?) etc?
Either it is a form IN SPACE or it is not. If you shrink it so that it doesn't occupy space AND it is still a tetrahedron you have extinguished its outer space only, but its inner space is still there.
A zero-size tetrahedron turns inside-out
http://www.miqel.com/fractals_math_pattern...h-platonic.html
because the definition of "zero-size tetrahedron" doesn't include -nor can it include- inner space of the subject of the exercize, the tetrahedron itself, only its relationship to the outside world.
QUOTE (IAMoraes+Dec 24 2007, 06:55 PM)
Either it is a form IN SPACE or it is not. If you shrink it so that it doesn't occupy space AND it is still a tetrahedron you have extinguished its outer space only, but its inner space is still there.
A zero-size tetrahedron turns inside-out
http://www.miqel.com/fractals_math_pattern...h-platonic.html
because the definition of "zero-size tetrahedron" doesn't include -nor can it include- inner space of the subject of the exercise, the tetrahedron itself, only its relationship to the outside world.
hej IAM,
If we think of continuous as infinitely divisible, You are right-at any given scale, if volume is 0, there are scale where it is not 0 - most likely inside tetrahedron in this scale, but at infinitely smaller scales its outer shape and volume appears again, with a small caveat - it can be placed anywhere relative to the tetrahedron we had, so its smaller inner volume/outer form is free to wonder around in smaller scales while we have fixed its place in our scale.
Would that be the same as turning it inside out-even if it has volume 0, it has infinite internal scales which in outer space at our scale may look like a fractal? Because, if the smaller scales may wonder around in proportion to the size of scale , we would get a fractal. Normal fractal however, is a little boring, as it will lead to fractions of type 1/n - which is fine, but its placement in smaller scales will be limited, regular.
What i am looking for is a fractal which is more irregular, able to appear ALMOST anywhere but still maintaining some harmony-not totally chaotic. May be circle, may be spiral, may be ?
A zero-size tetrahedron turns inside-out
http://www.miqel.com/fractals_math_pattern...h-platonic.html
because the definition of "zero-size tetrahedron" doesn't include -nor can it include- inner space of the subject of the exercise, the tetrahedron itself, only its relationship to the outside world.
hej IAM,
If we think of continuous as infinitely divisible, You are right-at any given scale, if volume is 0, there are scale where it is not 0 - most likely inside tetrahedron in this scale, but at infinitely smaller scales its outer shape and volume appears again, with a small caveat - it can be placed anywhere relative to the tetrahedron we had, so its smaller inner volume/outer form is free to wonder around in smaller scales while we have fixed its place in our scale.
Would that be the same as turning it inside out-even if it has volume 0, it has infinite internal scales which in outer space at our scale may look like a fractal? Because, if the smaller scales may wonder around in proportion to the size of scale , we would get a fractal. Normal fractal however, is a little boring, as it will lead to fractions of type 1/n - which is fine, but its placement in smaller scales will be limited, regular.
What i am looking for is a fractal which is more irregular, able to appear ALMOST anywhere but still maintaining some harmony-not totally chaotic. May be circle, may be spiral, may be ?
QUOTE (Ivars+Dec 25 2007, 02:57 AM)
hej IAM,
If we think of continious as infinitelty divisible, You are right-at any given scale, if volume is 0, there are scale where it is not 0 - most likely inside tetrahedron in this scale, but at infinitely smaller scales its outer shape and volume appears again, with a small caveat - it can be placed anywhere relative to the tetragedron we had, so its smaller inner volume is free to wonder around in smaller scales while we have fixed its place in our scale.
Would that be the same as turning it inside out-even if it has volume 0, it has infinite internal scales which in outer space ar our scale may look like a fractal?
Yes, it would be the same thing. An object falling into the fourth dimension would get progressively smaller until it disappeared, it would get to "zero" size. From our point of view it would --for the same reason that a light beam perpendicular to a 2d plane would always be at the same angle from you if you moved inside that plane or not: it would be outside of your plane, and the exact data for triangulation would be mostly out of your reach. What would be left of the beam on your 2d world would be the (1 infinitesimal) of the intersection, and it would be incomprehensible to you that it is all around you and why it is.
Similarly, in "(0.9r=1)?" the (protonic? Photonic? You didn't tell me yet!) infinitesimal can not be gone because it is an intersection of planes. That is a whole infinitesimal for the previous thought exercize, though, and we are dealing with numbers below that now, so the level of uncertainty **below** "(0.9r=1)?" would necessarily be (half of an infinitesimal), which makes no sense at all except for the "1/2" which adds addressability to the incomprehensible.
Don't tell anyone...
(just saw your edit but it is 2:30 in the am. Will be back tomorrow.)
(I am aware that 0.9r is "only" an infinitesimal in base 10, but I am counting the FORM of the infinitesimal: it happens across all bases... in base n, (zero point ((n-1) recurring)) is always equal to 1. That is a property of division, thus a truth in the prime graph, thus inescapable.)
If we think of continious as infinitelty divisible, You are right-at any given scale, if volume is 0, there are scale where it is not 0 - most likely inside tetrahedron in this scale, but at infinitely smaller scales its outer shape and volume appears again, with a small caveat - it can be placed anywhere relative to the tetragedron we had, so its smaller inner volume is free to wonder around in smaller scales while we have fixed its place in our scale.
Would that be the same as turning it inside out-even if it has volume 0, it has infinite internal scales which in outer space ar our scale may look like a fractal?
Yes, it would be the same thing. An object falling into the fourth dimension would get progressively smaller until it disappeared, it would get to "zero" size. From our point of view it would --for the same reason that a light beam perpendicular to a 2d plane would always be at the same angle from you if you moved inside that plane or not: it would be outside of your plane, and the exact data for triangulation would be mostly out of your reach. What would be left of the beam on your 2d world would be the (1 infinitesimal) of the intersection, and it would be incomprehensible to you that it is all around you and why it is.
Similarly, in "(0.9r=1)?" the (protonic? Photonic? You didn't tell me yet!) infinitesimal can not be gone because it is an intersection of planes. That is a whole infinitesimal for the previous thought exercize, though, and we are dealing with numbers below that now, so the level of uncertainty **below** "(0.9r=1)?" would necessarily be (half of an infinitesimal), which makes no sense at all except for the "1/2" which adds addressability to the incomprehensible.
Don't tell anyone...
(just saw your edit but it is 2:30 in the am. Will be back tomorrow.)
(I am aware that 0.9r is "only" an infinitesimal in base 10, but I am counting the FORM of the infinitesimal: it happens across all bases... in base n, (zero point ((n-1) recurring)) is always equal to 1. That is a property of division, thus a truth in the prime graph, thus inescapable.)
hej IAM
So there are 5 dimensions measurable in our scale for any structure:
3D-location
Inner
Outer
so that inner +outer = 4th dimension, which is imaginary
Inner dimension would be related to ways to divide the structure infinitely, outer- the way this subdimension interacts with our scale via lower , smaller scales. So one of them or both can be fractional.
Time is related to this information exchange speed, so it is local, and scaleable, while inertia is related to content , complexity of infinite division inside structure - again in relation to our scale.
I like 1/2 as well as connector between scales, but have not been able yet to derive it somehow. I think it is connected with Reynolds number of infinitesimal flows, it is not just a fraction, as it contains infinite imaginary components as well.
So there are 5 dimensions measurable in our scale for any structure:
3D-location
Inner
Outer
so that inner +outer = 4th dimension, which is imaginary
Inner dimension would be related to ways to divide the structure infinitely, outer- the way this subdimension interacts with our scale via lower , smaller scales. So one of them or both can be fractional.
Time is related to this information exchange speed, so it is local, and scaleable, while inertia is related to content , complexity of infinite division inside structure - again in relation to our scale.
I like 1/2 as well as connector between scales, but have not been able yet to derive it somehow. I think it is connected with Reynolds number of infinitesimal flows, it is not just a fraction, as it contains infinite imaginary components as well.
I still maintain that nothing can actually shrink to zero area. While the concept of the limit of the area of a shrinking shape might be zero, that is different from shrinking to zero.
I still also maintain that a single point can have no form.
I also maintain that connected points enclosing an area cannot enclose zero area.
@Alphanumeric,
Am I wrong on any of this? Those all seem obvious to me, but could be wrong.
I still also maintain that a single point can have no form.
I also maintain that connected points enclosing an area cannot enclose zero area.
@Alphanumeric,
Am I wrong on any of this? Those all seem obvious to me, but could be wrong.