I could not find my old thread where the puzzling question was that if volume is reduced to 0, form or area reduces infinitely slower, so there should be a moment when volume is already 0, but area not ( and of course edge not as well , it lasts longer than area).
So here is new development of that old idea:
Given the dimensionality of space, at each dimension x there exists a smallest indivisible which can not be divided further. Further division can happen only by reducing its dimensionality, i.e. x.
Since we are used to work in 3D space, the smallest spatial indivisible could be found as a thing when dimensionality of space is x=3 which still has 3D shape and area but 0 volume in 3D- it would have volume in space with less than 3D , but not in 3D. As such, since its volume in 3D is 0, it is not further divisible spatially, but it can be divided further only by reducing space dimensionality.
The reasoning why this should be possible is based on fact that if You reduce small volume of space in 3D to infinitesimal, its volume ( since it is a cube of edge, roughly) reduces to 0 infinitely faster than its area, which in turn reduces to zero infinitely faster than edge. So at each dimensionality, there has to be an smallest indivisible just on border between previous and next smaller dimensionality.
QUOTE (Ivars+Jul 3 2008, 05:33 PM)
Given the dimensionality of space, at each dimension x there exists a smallest indivisible which can not be divided further. Further division can happen only by reducing its dimensionality, i.e. x.
This is wrong. It's basically the same proof as there always being a number between any two different numbers, as I proved in the 0.9r thread.
Suppose you have a non-zero volume V in N dimensions so V>0. I halve the extension of the volume in all N directions, V -> V/(2^N) > 0.
Therefore, for EVERY volume larger than zero, there's always a smaller volume which is also bigger than zero.
You obviously learnt nothing since you last posted your BS.
This is wrong. It's basically the same proof as there always being a number between any two different numbers, as I proved in the 0.9r thread.
Suppose you have a non-zero volume V in N dimensions so V>0. I halve the extension of the volume in all N directions, V -> V/(2^N) > 0.
Therefore, for EVERY volume larger than zero, there's always a smaller volume which is also bigger than zero.
You obviously learnt nothing since you last posted your BS.
QUOTE (Ivars+Jul 3 2008, 04:33 PM)
I could not find my old thread where the puzzling question was that if volume is reduced to 0, form or area reduces infinitely slower, so there should be a moment when volume is already 0, but area not ( and of course edge not as well , it lasts longer than area).
So here is new development of that old idea:
Given the dimensionality of space, at each dimension x there exists a smallest indivisible which can not be divided further. Further division can happen only by reducing its dimensionality, i.e. x.
Since we are used to work in 3D space, the smallest spatial indivisible could be found as a thing when dimensionality of space is x=3 which still has 3D shape and area but 0 volume in 3D- it would have volume in space with less than 3D , but not in 3D. As such, since its volume in 3D is 0, it is not further divisible spatially, but it can be divided further only by reducing space dimensionality.
The reasoning why this should be possible is based on fact that if You reduce small volume of space in 3D to infinitesimal, its volume ( since it is a cube of edge, roughly) reduces to 0 infinitely faster than its area, which in turn reduces to zero infinitely faster than edge. So at each dimensionality, there has to be an smallest indivisible just on border between previous and next smaller dimensionality.
A dimension is defined by a property that can vary - if a dimension defines nothing that can be varied, then the dimension has no influence on anything and effectively doesn't exist. (Multiplying a dimension by 0 doesn't scale a dimension to a point but instead destroys the existance of the dimension)
If we look at the possible volume defined by a space, it's comprised of the number of unique combinations of traits for every dimension describing it and if the properties defined by each dimension are entirely independent of each other (orthogonal) and an object within this space could possess any combination of all possible properties for each dimension, then the volume is a multiplication of the number of unique states that can exist for each dimension (so this is simply say that, for example, if we cars can be of 3 types - sports car, luxury car and truck and each of these could be in one of 5 different colors, then the total "space", which could be described as a 2-D surface would be 3*5=15 possible locations, but if for example, no trucks can be red, then we only have 14 possible locations in this space and the dimensions are not entirely independent/orthogonal to each other)
In the case of a 3-D space with widths (or quantities of unique states/traits for the property describing that dimension) that are all equal to each other, then if these all allow n possible states in each dimension to exist, then we have n^3 possible states in the volume describing the possible objects.
In order to alter the dimensionality of a space, the properties describing each dimension must become entangled and no longer remain orthogonal/independent of each other, and if we had, for example 3 dimensions of space initially, but the properties of one dimension were gradually copied to another dimension such these 2 dimensions became identical to each and by knowing the properties of either dimension, we could determine the property of the other, then the space would be compressed from 3 dimensions to 2 dimensions as one of the dimensions became irrelevant and non-existant to the 3-D space and we could see this similar to a compression of a dimension into "nothing" (though compression is not a very good description as we have the destruction of a dimension instead as its simply been overwritten and removed from existance - none of the information it contained it recoverable and saying that it arose from the initial space is a misnoemer as there is nothing to correlate the versions of these spaces together).
If something contains no volume in 3 dimensions, then I'd interprete this to mean it has no width/expanse and no free parameters in that space and would act as a 0-D point, with the existance of 3 dimensions only being appended as an outside context to it. If an object could only exist in 1 of 2 states, then it could at most be interpreted to exist in a single dimension (with 2 states) and if it could exist in 1 of 3 possible states, there wouldn't appear to be a manner to describe this in terms of 2 entirely independent dimensions, as the area should be a product of the number of states in each and 3 is a prime number, though we could potentially draw this similar to a polar/rectangular form in which we have a parameter of distance and angle - when the distane property was 0, the angle would not alter the position, so if we had distances of 0 or 1 and angles of 0 and 90 degs, then there would be only 3 unique positions generated by these 4 states (the distance 0 at 0 and 90 degs would lie at the same point in this space). As you progress to an object encompassing a volume of 4 states, then you have the possibility for 2 independent dimensions with binary properties and 5 through 7 would likely appeared mapped as objects between 2 and 3 dimensions and with a volume of 8, we have a space that could be defined by 3 orthogonal binary attributes.
If we had some states that were less common or frequent, then there could also potentially exist intermediate configurations between these dimensional forms (it would be interesting to trace through, from an informational perspective, what classes of alterations to the probabilities of some states existing would make the space appear to warp into different forms - for example, if you had an even distribution of 8 states, then this could naturally appear as 3 independent binary properties, but a biasing of a single state of these 8 into being more or less common could then construct the perception of an origin or orientation to each dimension and interactions between dimensions - if we biased the probabilities of 4 states, the space should appear to configure itself into 2 parallel planes with a single dimension that had a specific orientation and the other 2 dimesions being independent). I'm not certain off hand how altering the probabilities of 3 or 5 (should be symetrical for either 3 or 5) of 8 possible states would appear to bias spacial properties within that space, but you get the general idea (it would likely appear similar to the 2 plane configuration with a specific original - it's interesting to see a similarity here with number theory regarding defining all numbers as sums of primes (http://primes.utm.edu/notes/conjectures/) - there might be a way to extend upon this concept in some manner by creating something similar to imaginary/complex numbers or vectors, in which we're working with quantities in higher dimensions and find there are naturally spaces described by sums of products of primes that correlate with them ... something interesting to think about anyway).
So here is new development of that old idea:
Given the dimensionality of space, at each dimension x there exists a smallest indivisible which can not be divided further. Further division can happen only by reducing its dimensionality, i.e. x.
Since we are used to work in 3D space, the smallest spatial indivisible could be found as a thing when dimensionality of space is x=3 which still has 3D shape and area but 0 volume in 3D- it would have volume in space with less than 3D , but not in 3D. As such, since its volume in 3D is 0, it is not further divisible spatially, but it can be divided further only by reducing space dimensionality.
The reasoning why this should be possible is based on fact that if You reduce small volume of space in 3D to infinitesimal, its volume ( since it is a cube of edge, roughly) reduces to 0 infinitely faster than its area, which in turn reduces to zero infinitely faster than edge. So at each dimensionality, there has to be an smallest indivisible just on border between previous and next smaller dimensionality.
A dimension is defined by a property that can vary - if a dimension defines nothing that can be varied, then the dimension has no influence on anything and effectively doesn't exist. (Multiplying a dimension by 0 doesn't scale a dimension to a point but instead destroys the existance of the dimension)
If we look at the possible volume defined by a space, it's comprised of the number of unique combinations of traits for every dimension describing it and if the properties defined by each dimension are entirely independent of each other (orthogonal) and an object within this space could possess any combination of all possible properties for each dimension, then the volume is a multiplication of the number of unique states that can exist for each dimension (so this is simply say that, for example, if we cars can be of 3 types - sports car, luxury car and truck and each of these could be in one of 5 different colors, then the total "space", which could be described as a 2-D surface would be 3*5=15 possible locations, but if for example, no trucks can be red, then we only have 14 possible locations in this space and the dimensions are not entirely independent/orthogonal to each other)
In the case of a 3-D space with widths (or quantities of unique states/traits for the property describing that dimension) that are all equal to each other, then if these all allow n possible states in each dimension to exist, then we have n^3 possible states in the volume describing the possible objects.
In order to alter the dimensionality of a space, the properties describing each dimension must become entangled and no longer remain orthogonal/independent of each other, and if we had, for example 3 dimensions of space initially, but the properties of one dimension were gradually copied to another dimension such these 2 dimensions became identical to each and by knowing the properties of either dimension, we could determine the property of the other, then the space would be compressed from 3 dimensions to 2 dimensions as one of the dimensions became irrelevant and non-existant to the 3-D space and we could see this similar to a compression of a dimension into "nothing" (though compression is not a very good description as we have the destruction of a dimension instead as its simply been overwritten and removed from existance - none of the information it contained it recoverable and saying that it arose from the initial space is a misnoemer as there is nothing to correlate the versions of these spaces together).
If something contains no volume in 3 dimensions, then I'd interprete this to mean it has no width/expanse and no free parameters in that space and would act as a 0-D point, with the existance of 3 dimensions only being appended as an outside context to it. If an object could only exist in 1 of 2 states, then it could at most be interpreted to exist in a single dimension (with 2 states) and if it could exist in 1 of 3 possible states, there wouldn't appear to be a manner to describe this in terms of 2 entirely independent dimensions, as the area should be a product of the number of states in each and 3 is a prime number, though we could potentially draw this similar to a polar/rectangular form in which we have a parameter of distance and angle - when the distane property was 0, the angle would not alter the position, so if we had distances of 0 or 1 and angles of 0 and 90 degs, then there would be only 3 unique positions generated by these 4 states (the distance 0 at 0 and 90 degs would lie at the same point in this space). As you progress to an object encompassing a volume of 4 states, then you have the possibility for 2 independent dimensions with binary properties and 5 through 7 would likely appeared mapped as objects between 2 and 3 dimensions and with a volume of 8, we have a space that could be defined by 3 orthogonal binary attributes.
If we had some states that were less common or frequent, then there could also potentially exist intermediate configurations between these dimensional forms (it would be interesting to trace through, from an informational perspective, what classes of alterations to the probabilities of some states existing would make the space appear to warp into different forms - for example, if you had an even distribution of 8 states, then this could naturally appear as 3 independent binary properties, but a biasing of a single state of these 8 into being more or less common could then construct the perception of an origin or orientation to each dimension and interactions between dimensions - if we biased the probabilities of 4 states, the space should appear to configure itself into 2 parallel planes with a single dimension that had a specific orientation and the other 2 dimesions being independent). I'm not certain off hand how altering the probabilities of 3 or 5 (should be symetrical for either 3 or 5) of 8 possible states would appear to bias spacial properties within that space, but you get the general idea (it would likely appear similar to the 2 plane configuration with a specific original - it's interesting to see a similarity here with number theory regarding defining all numbers as sums of primes (http://primes.utm.edu/notes/conjectures/) - there might be a way to extend upon this concept in some manner by creating something similar to imaginary/complex numbers or vectors, in which we're working with quantities in higher dimensions and find there are naturally spaces described by sums of products of primes that correlate with them ... something interesting to think about anyway).
QUOTE (AlphaNumeric+Jul 3 2008, 06:19 PM)
This is wrong. It's basically the same proof as there always being a number between any two different numbers, as I proved in the 0.9r thread.
Suppose you have a non-zero volume V in N dimensions so V>0. I halve the extension of the volume in all N directions, V -> V/(2^N) > 0.
Therefore, for EVERY volume larger than zero, there's always a smaller volume which is also bigger than zero.
You obviously learnt nothing since you last posted your BS.
You're missing the fact that a dimension is defined by some change in a property. You can scale the virtual size to anything you want as long as some change in a property is possible to define the dimension. So the only impossible "scaling" of a dimension is where it goes from the point of being defined by some property to the point of being defined by nothing.
Of course if we're referring to a real physical space measured by quanta then the (informationally) smallest dimension could be defined by a choice of 1 of 2 possible states, there would be nothing smaller that could exist than this unless we're working with statistical models (which are abstract).
Suppose you have a non-zero volume V in N dimensions so V>0. I halve the extension of the volume in all N directions, V -> V/(2^N) > 0.
Therefore, for EVERY volume larger than zero, there's always a smaller volume which is also bigger than zero.
You obviously learnt nothing since you last posted your BS.
You're missing the fact that a dimension is defined by some change in a property. You can scale the virtual size to anything you want as long as some change in a property is possible to define the dimension. So the only impossible "scaling" of a dimension is where it goes from the point of being defined by some property to the point of being defined by nothing.
Of course if we're referring to a real physical space measured by quanta then the (informationally) smallest dimension could be defined by a choice of 1 of 2 possible states, there would be nothing smaller that could exist than this unless we're working with statistical models (which are abstract).
QUOTE (StevenA+Jul 4 2008, 05:34 AM)
You're missing the fact that a dimension is defined by some change in a property. You can scale the virtual size to anything you want as long as some change in a property is possible to define the dimension. So the only impossible "scaling" of a dimension is where it goes from the point of being defined by some property to the point of being defined by nothing.
The dimension of an object is defined by the smallest number of variables required to uniquely define a point of the object. Not by "by some change in a property". Is this another case of you making up meanings of words again? I wouldn't be suprised if you 'translate' this post as "StevenA is the greatest" over and over again. Your ability to delude yourself is that much.
For instance a 2-sphere is 2 dimensional because you only need a longtitude and a latitude. A 2-ball is 3 dimensional because you need a radial distance too.
A volume in 3 dimensions needs 3 values to define a point within it. Scaling it's lengths by any positive real number doesn't alter that. So the dimensionality is unchanged.
The dimension of an object is defined by the smallest number of variables required to uniquely define a point of the object. Not by "by some change in a property". Is this another case of you making up meanings of words again? I wouldn't be suprised if you 'translate' this post as "StevenA is the greatest" over and over again. Your ability to delude yourself is that much.
For instance a 2-sphere is 2 dimensional because you only need a longtitude and a latitude. A 2-ball is 3 dimensional because you need a radial distance too.
A volume in 3 dimensions needs 3 values to define a point within it. Scaling it's lengths by any positive real number doesn't alter that. So the dimensionality is unchanged.
QUOTE (StevenA+Jul 4 2008, 05:34 AM)
Of course if we're referring to a real physical space measured by quanta then the (informationally) smallest dimension could be defined by a choice of 1 of 2 possible states, there would be nothing smaller that could exist than this unless we're working with statistical models (which are abstract).
You're like NeoNo.1. You'll put your foot in it and then think you're going for something complicated as a 'punch line' and all you do is prove how ignorant you are.
QUOTE (AlphaNumeric+Jul 4 2008, 05:22 AM)
The dimension of an object is defined by the smallest number of variables required to uniquely define a point of the object. Not by "by some change in a property".
(...)
There's no need to continue as you began with a contradiction and continued with it. The "variable" you refer to defines a changeable property. How those potential states are sliced/factored into multiple attributes is arbitrary and not immediately relevant to defining the volume of the space (in other words I could construct the volume by addition or multiplication, but that's separate from the fact that more than a single point exists within it, in order that at least one dimension exists to describe the volume/potential solutions of the space). If, for example, a space contains 17 possible states, we could factor this into 2 dimensions as one state with binary properties and another as having 9 possible values, with a single combination being denied (17=2*9-1), or we could describe it as a single dimension of 17 points or as a 5 dimension binary hypercube with only one valid solution in one of the planes (17=2*2*2*2+1=2^4+1). You can always describe a space as being linear by enumerating, in a single dimension, all possible states it could contain.
A variable encompasses multiple solutions, otherwise it would be a constant.
(...)
There's no need to continue as you began with a contradiction and continued with it. The "variable" you refer to defines a changeable property. How those potential states are sliced/factored into multiple attributes is arbitrary and not immediately relevant to defining the volume of the space (in other words I could construct the volume by addition or multiplication, but that's separate from the fact that more than a single point exists within it, in order that at least one dimension exists to describe the volume/potential solutions of the space). If, for example, a space contains 17 possible states, we could factor this into 2 dimensions as one state with binary properties and another as having 9 possible values, with a single combination being denied (17=2*9-1), or we could describe it as a single dimension of 17 points or as a 5 dimension binary hypercube with only one valid solution in one of the planes (17=2*2*2*2+1=2^4+1). You can always describe a space as being linear by enumerating, in a single dimension, all possible states it could contain.
A variable encompasses multiple solutions, otherwise it would be a constant.
QUOTE (AlphaNumeric+)
Is this another case of you making up meanings of words again? I wouldn't be suprised if you 'translate' this post as "StevenA is the greatest" over and over again. Your ability to delude yourself is that much.
Ok, let's see (once again) if this is a matter of your misunderstanding of concepts or of me misusing terminology. Again, we can look at a simple dictionary definition to see which context is most applicable:
http://dictionary.reference.com/browse/variable
I see various definitions like this:
"1. apt or liable to vary or change; changeable: variable weather; variable moods."
Score 1 for the change of a property definition
2. capable of being varied or changed; alterable: a variable time limit for completion of a book.
Score 2 for the change of a property definition
3. inconstant; fickle: a variable lover.
Score 3 for the change of a property definition
4. having much variation or diversity.
Score 4 for the change of a property definition (this immediately implies a dimension populated with possible solutions)
5. Biology. deviating from the usual type, as a species or a specific character.
Score 5 for the change of a property definition (this also implies a dimension populated with possible solutions with the inclusion of an origin as it refers to a deviation from a nominal)
6. Astronomy. (of a star) changing in brightness.
Score 6 for the change of a property definition.
7. Meteorology. (of wind) tending to change in direction.
Score 7 for the change of a property definition.
8. Mathematics. having the nature or characteristics of a variable.
–noun 9. something that may or does vary; a variable feature or factor.
Score 8 for the change of a property definition.
10. Mathematics, Computers. a. a quantity or function that may assume any given value or set of values.
b. a symbol that represents this.
Score 9 for the change of a property definition.
11. Logic. (in the functional calculus) a symbol for an unspecified member of a class of things or statements. Compare bound variable, free variable.
Score 10 for the change of a property definition.
etc...
If you'd read and thought about my post above, you'd recognize that I also indicated how higher dimensional spaces can be perceived to be derived from these possible states in properties.
Notice that entirely orthogonal dimensions should not interact and not be perceivable as existing within the same space - if one property could truly be changed arbitrarily without any possible influence on another dimension then any mapping of them into a common space should be as meaningless and arbitrary as constructing a 2 dimensional space relating the number of visible clouds today to yesterday first drawn winning lotto number (I guess there would have to be some potentially miniscule interaction, as these two quantities exist within the same universe and could not be entirely unrelated, but you get the point).
The previous paragraph implies that physical spaces are described by a prime number of states that can't be factored into independent dimensions - quite interesting
(Though we could have spaces that contained a prime number of points that was close to some multidimensional product and closely mimicked the properties of that space)
Notice that if we had a variable that could be 1 of 4 possible values and no outside context to differentiate between these, then designating this as existing within a single dimension is as arbitrary as saying it's a 2x2 space containing four vertices of a square.
Ok, let's see (once again) if this is a matter of your misunderstanding of concepts or of me misusing terminology. Again, we can look at a simple dictionary definition to see which context is most applicable:
http://dictionary.reference.com/browse/variable
I see various definitions like this:
"1. apt or liable to vary or change; changeable: variable weather; variable moods."
Score 1 for the change of a property definition
2. capable of being varied or changed; alterable: a variable time limit for completion of a book.
Score 2 for the change of a property definition
3. inconstant; fickle: a variable lover.
Score 3 for the change of a property definition
4. having much variation or diversity.
Score 4 for the change of a property definition (this immediately implies a dimension populated with possible solutions)
5. Biology. deviating from the usual type, as a species or a specific character.
Score 5 for the change of a property definition (this also implies a dimension populated with possible solutions with the inclusion of an origin as it refers to a deviation from a nominal)
6. Astronomy. (of a star) changing in brightness.
Score 6 for the change of a property definition.
7. Meteorology. (of wind) tending to change in direction.
Score 7 for the change of a property definition.
8. Mathematics. having the nature or characteristics of a variable.
–noun 9. something that may or does vary; a variable feature or factor.
Score 8 for the change of a property definition.
10. Mathematics, Computers. a. a quantity or function that may assume any given value or set of values.
b. a symbol that represents this.
Score 9 for the change of a property definition.
11. Logic. (in the functional calculus) a symbol for an unspecified member of a class of things or statements. Compare bound variable, free variable.
Score 10 for the change of a property definition.
etc...
If you'd read and thought about my post above, you'd recognize that I also indicated how higher dimensional spaces can be perceived to be derived from these possible states in properties.
Notice that entirely orthogonal dimensions should not interact and not be perceivable as existing within the same space - if one property could truly be changed arbitrarily without any possible influence on another dimension then any mapping of them into a common space should be as meaningless and arbitrary as constructing a 2 dimensional space relating the number of visible clouds today to yesterday first drawn winning lotto number (I guess there would have to be some potentially miniscule interaction, as these two quantities exist within the same universe and could not be entirely unrelated, but you get the point).
The previous paragraph implies that physical spaces are described by a prime number of states that can't be factored into independent dimensions - quite interesting
(Though we could have spaces that contained a prime number of points that was close to some multidimensional product and closely mimicked the properties of that space)Notice that if we had a variable that could be 1 of 4 possible values and no outside context to differentiate between these, then designating this as existing within a single dimension is as arbitrary as saying it's a 2x2 space containing four vertices of a square.
QUOTE (StevenA+Jul 4 2008, 07:42 AM)
There's no need to continue as you began with a contradiction and continued with it. The "variable" you refer to defines a changeable property.
A variable encompasses multiple solutions, otherwise it would be a constant.
I define 'dimension' as used by mathematicians and you address my usage of 'variable'.
Besides, your definition is wrong. A variable is something you vary. The number of solutions is something else. For instance f(x) = x. x is a variable but there's only one solution to f(x) = 5.
A variable encompasses multiple solutions, otherwise it would be a constant.
I define 'dimension' as used by mathematicians and you address my usage of 'variable'.
Besides, your definition is wrong. A variable is something you vary. The number of solutions is something else. For instance f(x) = x. x is a variable but there's only one solution to f(x) = 5.
QUOTE (StevenA+Jul 4 2008, 07:42 AM)
http://dictionary.reference.com/browse/variable
Once again, Steven doesn't know how to stick with the context! It's a thread about maths so he goes to a non-maths dictionary! 
QUOTE (StevenA+Jul 4 2008, 07:42 AM)
Notice that entirely orthogonal dimensions should not interact and not be perceivable as existing within the same space - if one property could truly be changed arbitrarily without any possible influence on another dimension then any mapping of them into a common space should be as meaningless and arbitrary as constructing a 2 dimensional space relating the number of visible clouds today to yesterday first drawn winning lotto number (I guess there would have to be some potentially miniscule interaction, as these two quantities exist within the same universe and could not be entirely unrelated, but you get the point).
No, only certain systems lead to independent dynamics in orthogonal directions. For instance, in electromagnetism, the motion of a charged particle through a magnetic field creates a force orthogonal to it's motion and the magnetic field.
Besides, you talk about something completely different to the context of this thread. And you misuse terminology. What you mean is 'independent systems'. The use of dimensions, orthogonality and scaling I use in my posts are not those usages of the words. But since you don't understand what the words mean you don't understand the context.
I'm certain if I gave you a vector calculus question you'd be unable to do it. Want to prove me wrong?
Besides, you talk about something completely different to the context of this thread. And you misuse terminology. What you mean is 'independent systems'. The use of dimensions, orthogonality and scaling I use in my posts are not those usages of the words. But since you don't understand what the words mean you don't understand the context.
I'm certain if I gave you a vector calculus question you'd be unable to do it. Want to prove me wrong?
QUOTE (StevenA+Jul 4 2008, 07:42 AM)
Notice that if we had a variable that could be 1 of 4 possible values and no outside context to differentiate between these, then designating this as existing within a single dimension is as arbitrary as saying it's a 2x2 space containing four vertices of a square.
Wow, it's look you took the number 4 and thought "I'll think of a shape with 4 vertices!".
Shame it doesn't demonstrate you understand geometry or vector calculus. Want to put your maths where your mouth is? Considering you continue to avoid posting here I'd guess you're unable to admit you cannot.
What do you say Steven? It takes you more time to type out your lengthy BS than it would to answer say questions 10, 11 or 12. But we all know you won't.
Shame it doesn't demonstrate you understand geometry or vector calculus. Want to put your maths where your mouth is? Considering you continue to avoid posting here I'd guess you're unable to admit you cannot.
What do you say Steven? It takes you more time to type out your lengthy BS than it would to answer say questions 10, 11 or 12. But we all know you won't.
QUOTE (AlphaNumeric+)
I define 'dimension' as used by mathematicians and you address my usage of 'variable'.
Don't try to blame mathematicians here. A good mathematician is capable of carrying on conversations about mathematics in many contexts. You should have already recognized from my post the context within which I was using the terms.
Don't try to blame mathematicians here. A good mathematician is capable of carrying on conversations about mathematics in many contexts. You should have already recognized from my post the context within which I was using the terms.
QUOTE (AlphaNumeric+)
Besides, your definition is wrong. A variable is something you vary. The number of solutions is something else. For instance f(x) = x. x is a variable but there's only one solution to f(x) = 5.
5 has no meaning outside a space/dimension that contains it - 5 what? In this case you're referring to a dimension f(x) for which you selected a point 5 out of whatever possible solutions f(x) could have possessed without the selection of 5 (so the possible solutions of f(x) supplied the context of the dimension for which the selected point 5, was chosen, but without the context of f(x)=5, 5 itself doesn't reside within some specific space and it's actually the potential range of f(x) that defines the dimension and not 5).
5 has no meaning outside a space/dimension that contains it - 5 what? In this case you're referring to a dimension f(x) for which you selected a point 5 out of whatever possible solutions f(x) could have possessed without the selection of 5 (so the possible solutions of f(x) supplied the context of the dimension for which the selected point 5, was chosen, but without the context of f(x)=5, 5 itself doesn't reside within some specific space and it's actually the potential range of f(x) that defines the dimension and not 5).
QUOTE (AlphaNumeric+)
Once again, Steven doesn't know how to stick with the context! It's a thread about maths so he goes to a non-maths dictionary!
Ok, show me one of these mathematics dictionaries that defines a variable somehow differently ... (I'll be waiting).
Also consider that if we really did try to make mathematical terms into something incompatible with common expressions in a language, that would only serve to detach the mathematical meanings from the common usages of the terms (yes, there are always mathematical contexts that provide additional information, as I stated above and you should have enough familiarity with mathematical terms to recognize this)
Ok, show me one of these mathematics dictionaries that defines a variable somehow differently ... (I'll be waiting).
Also consider that if we really did try to make mathematical terms into something incompatible with common expressions in a language, that would only serve to detach the mathematical meanings from the common usages of the terms (yes, there are always mathematical contexts that provide additional information, as I stated above and you should have enough familiarity with mathematical terms to recognize this)
QUOTE (AlphaNumeric+)
No, only certain systems lead to independent dynamics in orthogonal directions.
And what would the properties of those "certain" systems be?
And what would the properties of those "certain" systems be?
QUOTE (AlphaNumeric+)
For instance, in electromagnetism, the motion of a charged particle through a magnetic field creates a force orthogonal to it's motion and the magnetic field.
The from my prior statement, there should be some combinations of these motions, forces and charges that are impossible to construct and they do not exist independent of each other (the space defined by these properties would not be symmetrically factorable - not immediately that that the force is dependent upon the charge, motion and the field, and we could, for example, interprete this as saying that motion, for example, is not independent of the charge and the field - hence the space of these properties does not exist as an orthogonal 3-D space of motion (ignoring that this is not defined in a single dimension), charge and field).
The from my prior statement, there should be some combinations of these motions, forces and charges that are impossible to construct and they do not exist independent of each other (the space defined by these properties would not be symmetrically factorable - not immediately that that the force is dependent upon the charge, motion and the field, and we could, for example, interprete this as saying that motion, for example, is not independent of the charge and the field - hence the space of these properties does not exist as an orthogonal 3-D space of motion (ignoring that this is not defined in a single dimension), charge and field).
QUOTE (AlphaNumeric+)
Besides, you talk about something completely different to the context of this thread.
No, I began by referencing how a dimishing volume for space transforms the possible dimensional forms in which a space could exist (go back read my comments about trying to factor volumes into independent dimensions).
No, I began by referencing how a dimishing volume for space transforms the possible dimensional forms in which a space could exist (go back read my comments about trying to factor volumes into independent dimensions).
QUOTE (AlphaNumeric+)
And you misuse terminology.
Within the limited context that you understand, most certainly.
Within the limited context that you understand, most certainly.
QUOTE (AlphaNumeric+)
What you mean is 'independent systems'. The use of dimensions, orthogonality and scaling I use in my posts are not those usages of the words. But since you don't understand what the words mean you don't understand the context.
The problem is that if I used your definitions and calculations, then I'd simply get the same results as you do. I'm not interested in rehashing basic geometry and mathematics, nor do I believe Ivars is. We're both trying to look deeper into the mechanics of how spaces are constructed (I tend to take a more discrete and digital approach than Ivars, but who knows if, in the limit, these structures don't approach the same thing).
The problem is that if I used your definitions and calculations, then I'd simply get the same results as you do. I'm not interested in rehashing basic geometry and mathematics, nor do I believe Ivars is. We're both trying to look deeper into the mechanics of how spaces are constructed (I tend to take a more discrete and digital approach than Ivars, but who knows if, in the limit, these structures don't approach the same thing).
QUOTE (AlphaNumeric+)
I'm certain if I gave you a vector calculus question you'd be unable to do it. Want to prove me wrong?
You want me to ask you to optimize some logic equations for me?
Ultimately, what's more fundamental - mathematics or logic? Mathematics is built upon logic and if you don't have the logic right, then your mathematics is simply going to inherit the problems.
You want me to ask you to optimize some logic equations for me?
Ultimately, what's more fundamental - mathematics or logic? Mathematics is built upon logic and if you don't have the logic right, then your mathematics is simply going to inherit the problems.
QUOTE (AlphaNumeric+)
Wow, it's look you took the number 4 and thought "I'll think of a shape with 4 vertices!".
The question is over what physical properties these vertices represent as to whether or not their interpreted as existing in different physical forms.
For example, if I showed you some random pictures of rooms in a house, if you wanted to construct a model of the house within some common spacial framework, then you'd have to find common spacial features to extract and link together between the pictures.
So the apparent spacial properties are defined by the paths and interconnections between points in the space (which could, for example, make a space appear locally 3 dimensional if we had, for example, positive and negative motions available in 3 dimensions at some point, whereas the entire volume of space outside this could have a different form of interconnections)
The question is over what physical properties these vertices represent as to whether or not their interpreted as existing in different physical forms.
For example, if I showed you some random pictures of rooms in a house, if you wanted to construct a model of the house within some common spacial framework, then you'd have to find common spacial features to extract and link together between the pictures.
So the apparent spacial properties are defined by the paths and interconnections between points in the space (which could, for example, make a space appear locally 3 dimensional if we had, for example, positive and negative motions available in 3 dimensions at some point, whereas the entire volume of space outside this could have a different form of interconnections)
QUOTE (AlphaNumeric+)
...
etc.
etc.
To specify some of the ideas more precisely, if we had, for example 3 properties A, B and C for which each property was simply a binary attribute and an object within this space could have any of 8 possible combinations of these attributes (we could list them as 1 and 0 and simply write all 3 bit binary combinations 000,001,010,011,100,101,110 and 111 to enumerate all possible spaces that an object could reside at).
Now in this case, there is nothing inherent to the space that determines any physical asymmetries between these attributes and any interactions/restriction for them would beed to exist as additional information appended to this space. In other words, the space itself does not cause these attributes to interact in any manner and we could just as easily state that there are 8 independent locations/objects (0 to 7 in octal, for example) within within this space and none of them are related or interact with any of the locations/objects. We could also factor these 8 objects/locations into a 2-D description as 8=4*2 and say alternately that it's a 2 dimensional space with 2 properties/attributes for each object within the space and one such property/attribute exists in one of 4 states, whereas the other is binary.
Notice that all 3 of these interpretations are equally valid (or invalid) except for the single case in which we specified it as a single dimension with 8 positions because each position is entirely independent of the others and so no multidimensional features would appear present in the properties of these objects.
But, on the other hand if we had two spaces - one defined in terms of multidimensional attributes of the objects within them, such as this example with 3 binary attributes, and then we had an alternate "true"/physical space in which they existed that only allowed 7 of these possible combinations to exist, then an apparent interation between this abstract properties would occur because we could no longer specify each attribute/property of the object independently of the others and, for example, if one attribute was associated with a position and the other an acceleration, then the mismatch between these two spaces (the abstract/virtual and the actual) would bias these objects into appearing that an acceleration was dependent upon a position and hence a field of force would appear to exist in the actual space, when compared to our (compressed in some respects, though approximate) interpretation of it.
And again, notice that you specified the dimensionality of a space as being the fewer number of dimensions required to specify the space but that in reality, as long as we have even a single dimension, we could compress all discrete locations within a higher dimensional space into a single dimension. The assumption you make is that there is no such manner to do this with positions defined by real numbers as the dimensions are infinitely subdivisible and the problem becomes one of ordering multiple infinite quantities, but the reality is that no such space can physically exist (because we don't have infinite precision for samples of position), nor is the logic implied by it deterministic without specifying a precise manner in which interactions occur between dimensions (as a side note: irrational numbers for distances in a Euclidean are just windowed approximations of events over time and not precise logical results measured at any moment of time - I could go into more detail, but it's redundant to have multiple infinite/unbounded quantities in a computation as a single one is "large enough" to fit the rest without anyone noticing the difference, if x, y and z all approach infinity, then so would a volume described by x*y*z and we can simply take a single long thread and wrap it around to fill up the space instead of using a model with an infinite number of infinitely wide surfaces comprised of infinite length threads and then trying to figure out how to connect them all together into a single connected space where we can trace connections between objects within them - and if the volume of that space happened to actually be something like x*y*z-1, then we'd automatically have a manner in which the assumed independent dimensions defining it could appear to interact within that space and appear to create "laws" of interactions between dimensions, though I think that form is still too symmetrical and possibly something like x*(y*(z-1)-1)-1 could fit the bill better as each dimension would have unique properties and remain detectably independent of the others, yet they would all appear to interact within a space defined by them as dimensions, relative to an assumed abstract space in which the volume should have been x*y*z (or a product of 2 of these for a 2 dimensional representation).
Now in this case, there is nothing inherent to the space that determines any physical asymmetries between these attributes and any interactions/restriction for them would beed to exist as additional information appended to this space. In other words, the space itself does not cause these attributes to interact in any manner and we could just as easily state that there are 8 independent locations/objects (0 to 7 in octal, for example) within within this space and none of them are related or interact with any of the locations/objects. We could also factor these 8 objects/locations into a 2-D description as 8=4*2 and say alternately that it's a 2 dimensional space with 2 properties/attributes for each object within the space and one such property/attribute exists in one of 4 states, whereas the other is binary.
Notice that all 3 of these interpretations are equally valid (or invalid) except for the single case in which we specified it as a single dimension with 8 positions because each position is entirely independent of the others and so no multidimensional features would appear present in the properties of these objects.
But, on the other hand if we had two spaces - one defined in terms of multidimensional attributes of the objects within them, such as this example with 3 binary attributes, and then we had an alternate "true"/physical space in which they existed that only allowed 7 of these possible combinations to exist, then an apparent interation between this abstract properties would occur because we could no longer specify each attribute/property of the object independently of the others and, for example, if one attribute was associated with a position and the other an acceleration, then the mismatch between these two spaces (the abstract/virtual and the actual) would bias these objects into appearing that an acceleration was dependent upon a position and hence a field of force would appear to exist in the actual space, when compared to our (compressed in some respects, though approximate) interpretation of it.
And again, notice that you specified the dimensionality of a space as being the fewer number of dimensions required to specify the space but that in reality, as long as we have even a single dimension, we could compress all discrete locations within a higher dimensional space into a single dimension. The assumption you make is that there is no such manner to do this with positions defined by real numbers as the dimensions are infinitely subdivisible and the problem becomes one of ordering multiple infinite quantities, but the reality is that no such space can physically exist (because we don't have infinite precision for samples of position), nor is the logic implied by it deterministic without specifying a precise manner in which interactions occur between dimensions (as a side note: irrational numbers for distances in a Euclidean are just windowed approximations of events over time and not precise logical results measured at any moment of time - I could go into more detail, but it's redundant to have multiple infinite/unbounded quantities in a computation as a single one is "large enough" to fit the rest without anyone noticing the difference, if x, y and z all approach infinity, then so would a volume described by x*y*z and we can simply take a single long thread and wrap it around to fill up the space instead of using a model with an infinite number of infinitely wide surfaces comprised of infinite length threads and then trying to figure out how to connect them all together into a single connected space where we can trace connections between objects within them - and if the volume of that space happened to actually be something like x*y*z-1, then we'd automatically have a manner in which the assumed independent dimensions defining it could appear to interact within that space and appear to create "laws" of interactions between dimensions, though I think that form is still too symmetrical and possibly something like x*(y*(z-1)-1)-1 could fit the bill better as each dimension would have unique properties and remain detectably independent of the others, yet they would all appear to interact within a space defined by them as dimensions, relative to an assumed abstract space in which the volume should have been x*y*z (or a product of 2 of these for a 2 dimensional representation).
Hej StevenA
Thanks for jumping in to keep idea alive:)
The thought of the order in which infinitesimal volume disappears - first edge, then dimension, leads to a notion of continuos space dimensionalities (which in language of AN number of variables would mean non-integer number of variables, which should not be impossible, since there are non-itnteger number of terms in sums, non-integer iteration of functions, non-integer Euler characteristics (or measures) of polytopes etc.) where integer ones are just very important special cases. If there are continuos space dimensionalities y, then there can be also imaginary
and negative.
We may assume in the beginning these continuos dimensionalities to be real number, but very soon ( e.g. via derivatives or tetration) some special cases will start to pop out, perhaps in the end leading to some prime polytype structures in space being more important than others.
This leads to natural (?) interpretation of exponentiation as hypercube of
edge x and dimension y= x^y.
Particularly interesting seems case where y<1 in which case it seems to me space
will become disconnected, or ? I have not read anything about this yet.
Which in turn allows to consider case x^x but here both x have
DIFFERENT geometrical meaning- one is edge, another SPACE dimension.
The speed of growth or decline of hypervolume x^x depends DIFFERENTLY on each x, so the derivative of the function makes them distinquishable. For some
reason, if the edge of such hypercube is e, (so e^x) all its
derivatives are e^x.
If we take partial derivatives of x^y, then they are:
y*x^y-1 - this is derivative by length of edge x, dimensionality y=constant
and
x^y*ln(y) - this is derivative by dimension y, edge x= constant.
If we now replace y with x, these derivatives are equal with oppossite signs at x= 1/e. The value of hypervolume is (1/e)^(1/e)=0,69... At x=e,at hypervolume e^e=15,.. the derivative by edge length gets smaller than derivative by dimension, so if edge is smaller than e AND dimension is smaller than e, then volume disapppear first by reducing edge in dimension x, than dimension x itself, as I suggested in first post. If edge is bigger than e AND dimension is bigger than e, then increase of hypervolume happens first by increase in DIMENSION x , and then by increase of EDGE to x in THAT dimension x.
Understanding case x^y and exponentiation in general in dimensional
terms should allow to interpret tetration x^x^x^ ........
geometrically, and what infinite tetration of e^pi/2 = +-i might mean
geometrically and physically.
Of interest are also other geometrical figures in X dimensional space,
like sphere, and its volume/area values as function of radius.
Thanks for jumping in to keep idea alive:)
The thought of the order in which infinitesimal volume disappears - first edge, then dimension, leads to a notion of continuos space dimensionalities (which in language of AN number of variables would mean non-integer number of variables, which should not be impossible, since there are non-itnteger number of terms in sums, non-integer iteration of functions, non-integer Euler characteristics (or measures) of polytopes etc.) where integer ones are just very important special cases. If there are continuos space dimensionalities y, then there can be also imaginary
and negative.
We may assume in the beginning these continuos dimensionalities to be real number, but very soon ( e.g. via derivatives or tetration) some special cases will start to pop out, perhaps in the end leading to some prime polytype structures in space being more important than others.
This leads to natural (?) interpretation of exponentiation as hypercube of
edge x and dimension y= x^y.
Particularly interesting seems case where y<1 in which case it seems to me space
will become disconnected, or ? I have not read anything about this yet.
Which in turn allows to consider case x^x but here both x have
DIFFERENT geometrical meaning- one is edge, another SPACE dimension.
The speed of growth or decline of hypervolume x^x depends DIFFERENTLY on each x, so the derivative of the function makes them distinquishable. For some
reason, if the edge of such hypercube is e, (so e^x) all its
derivatives are e^x.
If we take partial derivatives of x^y, then they are:
y*x^y-1 - this is derivative by length of edge x, dimensionality y=constant
and
x^y*ln(y) - this is derivative by dimension y, edge x= constant.
If we now replace y with x, these derivatives are equal with oppossite signs at x= 1/e. The value of hypervolume is (1/e)^(1/e)=0,69... At x=e,at hypervolume e^e=15,.. the derivative by edge length gets smaller than derivative by dimension, so if edge is smaller than e AND dimension is smaller than e, then volume disapppear first by reducing edge in dimension x, than dimension x itself, as I suggested in first post. If edge is bigger than e AND dimension is bigger than e, then increase of hypervolume happens first by increase in DIMENSION x , and then by increase of EDGE to x in THAT dimension x.
Understanding case x^y and exponentiation in general in dimensional
terms should allow to interpret tetration x^x^x^ ........
geometrically, and what infinite tetration of e^pi/2 = +-i might mean
geometrically and physically.
Of interest are also other geometrical figures in X dimensional space,
like sphere, and its volume/area values as function of radius.
QUOTE (StevenA+Jul 4 2008, 08:45 AM)
Don't try to blame mathematicians here. A good mathematician is capable of carrying on conversations about mathematics in many contexts. You should have already recognized from my post the context within which I was using the terms.
And a rational person is capable of not using words he doesn't know the meaning of.
And a rational person is capable of not using words he doesn't know the meaning of.
QUOTE (StevenA+Jul 4 2008, 08:45 AM)
5 has no meaning outside a space/dimension that contains it - 5 what?
Wrong. The concept of 5 is well defined. It's the property shared by 5 kilograms, 5 potatos, 5 light bulbs, 5 thoughts, 5 lots of 3 squirells etc etc.
QUOTE (StevenA+Jul 4 2008, 08:45 AM)
Ok, show me one of these mathematics dictionaries that defines a variable somehow differently ... (I'll be waiting).
I was talking about dimensions. Didn't you follow what I said?
Do you know what a basis is?
Do you know what a basis is?
QUOTE (StevenA+Jul 4 2008, 08:45 AM)
Also consider that if we really did try to make mathematical terms into something incompatible with common expressions in a language, that would only serve to detach the mathematical meanings from the common usages of the terms (yes, there are always mathematical contexts that provide additional information, as I stated above and you should have enough familiarity with mathematical terms to recognize this)
So you're basically saying that mathematicians should never use words which already have meanings?
So to you a 'field' is a large area of grass? A ring is a circular object? A set is where badgers live?
How dare people use preexisting words for their own purposes!
So to you a 'field' is a large area of grass? A ring is a circular object? A set is where badgers live?
How dare people use preexisting words for their own purposes!
QUOTE (StevenA+Jul 4 2008, 08:45 AM)
And what would the properties of those "certain" systems be?
Counter examples to your claim involve things like Maxwell's equations. Despite being expressible in Cartesian coordinates, the equations for a particular direction involve other directions.
Your requirement is that a set of n equations in n variables (F_i)(x1,...,xn) = 0 reduce to (F_j)(x_j)=0, so they are actually the product of n 1d decoupled systems. Pretty much all physics is more complicated than that.
Your requirement is that a set of n equations in n variables (F_i)(x1,...,xn) = 0 reduce to (F_j)(x_j)=0, so they are actually the product of n 1d decoupled systems. Pretty much all physics is more complicated than that.
QUOTE (StevenA+Jul 4 2008, 08:45 AM)
Within the limited context that you understand, most certainly.
In other words, since I cannot predict the personal interpretation you put on words, because you don't understand the technical meaning, I'm at fault?
QUOTE (StevenA+Jul 4 2008, 08:45 AM)
The problem is that if I used your definitions and calculations, then I'd simply get the same results as you do.
It's almost as if consistent terminology allows people to understand each other's work.
But I doubt you could do the results I do. Actually, I know you couldn't do the results I do.
But I doubt you could do the results I do. Actually, I know you couldn't do the results I do.
QUOTE (StevenA+Jul 4 2008, 08:45 AM)
We're both trying to look deeper into the mechanics of how spaces are constructed (I tend to take a more discrete and digital approach than Ivars, but who knows if, in the limit, these structures don't approach the same thing)..
Except neither of you know any geometry.
QUOTE (StevenA+Jul 4 2008, 08:45 AM)
You want me to ask you to optimize some logic equations for me?
I'm not claiming to know those things though. Real logic is a particular branch of mathematics. You're always claiming to be competant at various things. Funny how you can never show it. How many times have I shown I can do the physics and maths I claim I'm competant at? Plenty.
It took you more time to write out that post than I would have taken to answer one of the questions I linked to. As usual, the crank goes for whining about why he won't answer, but could, rather than actually answering.
Your silence speaks volume.
Tell me, if you're solving so many things in physics and maths, why are you a nobody whining on an internet forum about how you've got all the answers to problems mathematicians and physicists don't even know exist? Could it be you're full of BS? Even with an offer of $1000 for minimal amounts of work, you refuse. Because you know you'd be the one paying out $1000. Come on, if you think you're right, take up my challenge. Chicken!
It took you more time to write out that post than I would have taken to answer one of the questions I linked to. As usual, the crank goes for whining about why he won't answer, but could, rather than actually answering.
Your silence speaks volume.
Tell me, if you're solving so many things in physics and maths, why are you a nobody whining on an internet forum about how you've got all the answers to problems mathematicians and physicists don't even know exist? Could it be you're full of BS?
Even with an offer of $1000 for minimal amounts of work, you refuse. Because you know you'd be the one paying out $1000. Come on, if you think you're right, take up my challenge. Chicken!
QUOTE (Ivars+Jul 4 2008, 05:44 PM)
Hej StevenA
Thanks for jumping in to keep idea alive:)
The thought of the order in which infinitesimal volume disappears - first dimension, then edge, leads to a notion of continuos space dimensionalities (which in language of AN number of variables would mean non-integer number of variables, which should not be impossible, since there are non-itnteger number of terms in sums, non-integer iteration of functions, non-integer Euler characteristics (or measures) of polytopes etc.) where integer ones are just very important special cases. If there are continuos space dimensionalities y, then there can be also imaginary
and negative.
Within a space where every position can possess a unique property independent of other locations within it and we could effectively fill every point within this space with "random" information, independently of considering the values at other locations.
But if such a space is not possible and there are restrictions upon the values that a location can contain that are dependent upon values in other locations, then the actual number of combinations of values that could exist within this space is less than an idealized product of the possible values each dimension could contain independently and an equivalent dimensionality is considered to be lower.
Here's a link referring to fractal dimensions that you might find interesting, though it would be additionally interesting to consider whether or not there's a way to construct a linear interpolation between these spaces (there may not be, other than as statistical structures or with spaces with a potentially unbounded number of dimensions). But anyway here's a link to the idea of fractal dimensions http://en.wikipedia.org/wiki/Fractal_dimension.
Thanks for jumping in to keep idea alive:)
The thought of the order in which infinitesimal volume disappears - first dimension, then edge, leads to a notion of continuos space dimensionalities (which in language of AN number of variables would mean non-integer number of variables, which should not be impossible, since there are non-itnteger number of terms in sums, non-integer iteration of functions, non-integer Euler characteristics (or measures) of polytopes etc.) where integer ones are just very important special cases. If there are continuos space dimensionalities y, then there can be also imaginary
and negative.
Within a space where every position can possess a unique property independent of other locations within it and we could effectively fill every point within this space with "random" information, independently of considering the values at other locations.
But if such a space is not possible and there are restrictions upon the values that a location can contain that are dependent upon values in other locations, then the actual number of combinations of values that could exist within this space is less than an idealized product of the possible values each dimension could contain independently and an equivalent dimensionality is considered to be lower.
Here's a link referring to fractal dimensions that you might find interesting, though it would be additionally interesting to consider whether or not there's a way to construct a linear interpolation between these spaces (there may not be, other than as statistical structures or with spaces with a potentially unbounded number of dimensions). But anyway here's a link to the idea of fractal dimensions http://en.wikipedia.org/wiki/Fractal_dimension.
Hi Steven,
No, I think it is not fractal dimension which should be used in this context, but topological or some equivalent measure that is possible to extend to rational and real numbers. So it is more linked to:
Fractional Calculus
which is actually real number iterates of differentiation/integration operations and involves gamma function ( factorial!).
Since integer differentation reduces dimension in exponent by 1, while integration increases,
d/dx (x^n) = n*x^(n-1)
it is reasonable to assume that any noninteger topological dimension would be accesable via fractional differentiation/integration.
But I have just started to read on this topic. More later:)
No, I think it is not fractal dimension which should be used in this context, but topological or some equivalent measure that is possible to extend to rational and real numbers. So it is more linked to:
Fractional Calculus
which is actually real number iterates of differentiation/integration operations and involves gamma function ( factorial!).
Since integer differentation reduces dimension in exponent by 1, while integration increases,
d/dx (x^n) = n*x^(n-1)
it is reasonable to assume that any noninteger topological dimension would be accesable via fractional differentiation/integration.
But I have just started to read on this topic. More later:)
Word processors should not be used the way food processors are.
That is to say word salad is not a good thing to make.
That is to say word salad is not a good thing to make.
QUOTE (Ivars+Jul 4 2008, 07:41 PM)
Hi Steven,
No, I think it is not fractal dimension which should be used in this context, but topological or some equivalent measure that is possible to extend to rational and real numbers. So it is more linked to:
Fractional Calculus
which is actually real number iterates of differentiation/integration operations and involves gamma function ( factorial!).
Since integer differentation reduces dimension in exponent by 1, while integration increases,
d/dx (x^n) = n*x^(n-1)
it is reasonable to assume that any noninteger topological dimension would be accesable via fractional differentiation/integration.
But I have just started to read on this topic. More later:)
Yes, fractals are only one form of possible topologies for a space.
The comment regarding fractional calculus is good, and it would be very interesting to study what geometric forms could appear to naturally arise from them (great suggestion actually, that could be one of those ideas that sticks in my head for a while!
), though beyond this, if we're really trying to be general, there would appear to be some implicit assumptions made about these that might not truely be restrictions.
If we look at the volume of a space, with an identical width for all dimensions of n and and integer, m. number of dimensions then for a purely orthogonal/rectangular set of dimensions the volume of this space would be n^m. We could assume that for fractional dimensions of size x, the volume should then be n^x instead, but notice that for fractional exponents, most of these values would be irrational, except for specific cases of n which had integer roots (for example 8^(4/3)=16), but this is still made under the assumption that such spaces all had the same width, n. For an orthogonal/rectangular space with varied width dimensions (which I think should be the case if dimensions exist due to a distinct width/wavelength associated with them), then the volume of the space would more generally be a product of all the widths (n0*n1*n2*...), but again, from my previous comments, if a volume/space is constructed in this manner, it treats the dimensions as being entirely unrelated to each other and requires additional information outside the abstract definition of this to provide those dimensions with interrelated properties ... if we instead had this abstract volume, defined by products or non-prime volumes mapped to a smaller volume, then interactions between dimensions naturally arises.
So, for example, one manner to analyze the properties of such mismatched volumes of spaces is to construct a mapping between them and have every location within the smaller space mapped to virtual/abstract properties in the larger orthogonal space (described by independent attributes).
As a quick example, if we had 2 possible locations in space, with each location containing one of 2 possible objects, and we might describe these as, for example, positive and negative charges in positions A and B of space, then the entire possible state of all of these would be one of the four combinations for positions A and B:
AB
--
-+
+-
++
So the "solutions" to this description reside within one of 4 possible states, but if we denied the exist of one of these combinations, such as ++, then we'd only see one of the remaining 3 combinations:
AB
--
-+
+-
Notice that if we have a negative charge in either position, the alternate position can be either positive or negative and no bias in the other state occurs (so the apparent interaction of a negative charge at either location would actually appear as neutral with respect to the other location), but if either location is measured to be positive, then the alternate position can be know to be negative and hence that space would not be detectable as existing independently of the positive and we'd really perceive the existance of a positive as being in one of two states as either the A or B version of the positive (assuming something outside this allowed for the difference between these forms to be detected).
Consider also that if only a single element could change at a time, then we'd actually have a linear space that could be described as containing the positions {+-,--,-+} or alternately {-+,--,+-}. Notice that without some manner to treat A and B differently, there's nothing that would supply any preferential interpretion of the direction of this space and the -+ and +- elements would likely appear superimposed to an observer (Notice that if we'd left the existance of the ++ valid in this space, then +s and -s would also appear symmetrically distributed in this space as well). Any, of there's nothing outside this space that allows for the -+ and +- states to appear biased and distinguishable from each other, then the entire space collapses into binary attribute of whether or not a single + was present (it's the ratio of 2 of 3 spaces having this as true versus the 1 of 3 spaces for which it's false that allows these two classes to be differentiated, but the -+ and +- states are both equally significant and hence non distinguishable from within the context of this space itself).
This could lead to a requirement for a construction of space in which the volumes contained by various elements remain relatively prime to other attributes within this space (otherwise we can compress out the symmetries into a smaller space). And this again leads to the interesting correlation with Goldbach's conjecture that all numbers beyond a trivial minimum are the sum of two primes (we could interprete this, in the context I gave above, as meaning that all volumes of spaces are numbers because the space could only be seen to exist if it both 1) defined a constrast between two states and 2) that the volumes of each state were relatively prime - though there's almost certainly more to consider, but that's enough for now
No, I think it is not fractal dimension which should be used in this context, but topological or some equivalent measure that is possible to extend to rational and real numbers. So it is more linked to:
Fractional Calculus
which is actually real number iterates of differentiation/integration operations and involves gamma function ( factorial!).
Since integer differentation reduces dimension in exponent by 1, while integration increases,
d/dx (x^n) = n*x^(n-1)
it is reasonable to assume that any noninteger topological dimension would be accesable via fractional differentiation/integration.
But I have just started to read on this topic. More later:)
Yes, fractals are only one form of possible topologies for a space.
The comment regarding fractional calculus is good, and it would be very interesting to study what geometric forms could appear to naturally arise from them (great suggestion actually, that could be one of those ideas that sticks in my head for a while!
), though beyond this, if we're really trying to be general, there would appear to be some implicit assumptions made about these that might not truely be restrictions.If we look at the volume of a space, with an identical width for all dimensions of n and and integer, m. number of dimensions then for a purely orthogonal/rectangular set of dimensions the volume of this space would be n^m. We could assume that for fractional dimensions of size x, the volume should then be n^x instead, but notice that for fractional exponents, most of these values would be irrational, except for specific cases of n which had integer roots (for example 8^(4/3)=16), but this is still made under the assumption that such spaces all had the same width, n. For an orthogonal/rectangular space with varied width dimensions (which I think should be the case if dimensions exist due to a distinct width/wavelength associated with them), then the volume of the space would more generally be a product of all the widths (n0*n1*n2*...), but again, from my previous comments, if a volume/space is constructed in this manner, it treats the dimensions as being entirely unrelated to each other and requires additional information outside the abstract definition of this to provide those dimensions with interrelated properties ... if we instead had this abstract volume, defined by products or non-prime volumes mapped to a smaller volume, then interactions between dimensions naturally arises.
So, for example, one manner to analyze the properties of such mismatched volumes of spaces is to construct a mapping between them and have every location within the smaller space mapped to virtual/abstract properties in the larger orthogonal space (described by independent attributes).
As a quick example, if we had 2 possible locations in space, with each location containing one of 2 possible objects, and we might describe these as, for example, positive and negative charges in positions A and B of space, then the entire possible state of all of these would be one of the four combinations for positions A and B:
AB
--
-+
+-
++
So the "solutions" to this description reside within one of 4 possible states, but if we denied the exist of one of these combinations, such as ++, then we'd only see one of the remaining 3 combinations:
AB
--
-+
+-
Notice that if we have a negative charge in either position, the alternate position can be either positive or negative and no bias in the other state occurs (so the apparent interaction of a negative charge at either location would actually appear as neutral with respect to the other location), but if either location is measured to be positive, then the alternate position can be know to be negative and hence that space would not be detectable as existing independently of the positive and we'd really perceive the existance of a positive as being in one of two states as either the A or B version of the positive (assuming something outside this allowed for the difference between these forms to be detected).
Consider also that if only a single element could change at a time, then we'd actually have a linear space that could be described as containing the positions {+-,--,-+} or alternately {-+,--,+-}. Notice that without some manner to treat A and B differently, there's nothing that would supply any preferential interpretion of the direction of this space and the -+ and +- elements would likely appear superimposed to an observer (Notice that if we'd left the existance of the ++ valid in this space, then +s and -s would also appear symmetrically distributed in this space as well). Any, of there's nothing outside this space that allows for the -+ and +- states to appear biased and distinguishable from each other, then the entire space collapses into binary attribute of whether or not a single + was present (it's the ratio of 2 of 3 spaces having this as true versus the 1 of 3 spaces for which it's false that allows these two classes to be differentiated, but the -+ and +- states are both equally significant and hence non distinguishable from within the context of this space itself).
This could lead to a requirement for a construction of space in which the volumes contained by various elements remain relatively prime to other attributes within this space (otherwise we can compress out the symmetries into a smaller space). And this again leads to the interesting correlation with Goldbach's conjecture that all numbers beyond a trivial minimum are the sum of two primes (we could interprete this, in the context I gave above, as meaning that all volumes of spaces are numbers because the space could only be seen to exist if it both 1) defined a constrast between two states and 2) that the volumes of each state were relatively prime - though there's almost certainly more to consider, but that's enough for now
Hi Steven
And here comes the connection of hypervolumes of previous posts to tetration and other higher hyperoperations:
When looking at x^y as hypercube in dimension y with edge x, I have
made implicit assumption that the ANGLE in which each edge of such
hypercube creates with another is 90 degrees in any dimension y, just
by analogy, or Pi/2, as in general definitions of hypercube.
However, this angle may be another,may be negative, may be imaginary.
This leads to interpretation of real etc. extensions of hyperoperation number z in x[z]y as angle between such edges x of a
hypercube in dimension y either directly, or via some exponential
relation like log (pi/2)^n/log (pi/2) = n for integer z, and z*log
pi/2/log (pi/2) =z for all other. , or n= log (e^n*I*pi/2)/log I , z=
log (e^z*I*pi/2)/log I.
As for imaginary, negative angles and other imaginary things like
points, edges etc. it seems obvious that such interpretation of
tetration and higher operations and its extension for real/imaginary/negative number of
operations leads to/is connected with projective geometry in its
original complex form, as developed by Lambert, Gauss, Stoudt, Cayley,
Klein. It seems to me that real numbers as foundations of geometry
fail here (in extensions of tetration to real , imaginary, negative
numbers) , which is of no big surprise since their foundation are
based on physical convenience but not abstract proofs (since it is
unprovable so far) , so alternative models are possible as well.
But since I only conjectured this yesterday, there are many things to
read, particularly about origins of non-euclidian geometry . Just to
summarize:
In expression where x is on which hyperoperation z acts, and y is how many times it does ( e.g. x^x= x[4]2, x^a= x[3]a, x*x=x[2]2, x+x= x[1]2)
x[z]y
x- edge of hypervolume in y dimensions
y- dimensions of hypervolume with edge x
z- is related to varying angle between edges x of hypervolume in y
dimensions, perhaps via logarithm or some trigonometric function. A
meaningful definition should recover usual angles pi/2 in case of
hypercube in integer dimensions.
The geometric interpretation of hyperoperations in general and their real,
complex extensions is related to projective geometry in imaginary (complex) form
extended to non-integer, negative and imaginary dimensions.
To give a more concrete test example:
I[I]I is a hypervolume with edge I in I dimensions with angle between imaginary edges (e^(I*I*pi/2) = e^(-pi/2)
I[-I]I is a hypervolume with edge I in I dimensions with angle between imaginary edges e^pi/2. In I dimensional space, such angle might have a meaning.
It also seems that his hyperangle conveyed via z is a composite one, so related to twisting of hypervolume x[z]y of dimension y. This is probably possible to check if some volumes in ordinary integer spaces of twisted hypercubes are known.
And here comes the connection of hypervolumes of previous posts to tetration and other higher hyperoperations:
When looking at x^y as hypercube in dimension y with edge x, I have
made implicit assumption that the ANGLE in which each edge of such
hypercube creates with another is 90 degrees in any dimension y, just
by analogy, or Pi/2, as in general definitions of hypercube.
However, this angle may be another,may be negative, may be imaginary.
This leads to interpretation of real etc. extensions of hyperoperation number z in x[z]y as angle between such edges x of a
hypercube in dimension y either directly, or via some exponential
relation like log (pi/2)^n/log (pi/2) = n for integer z, and z*log
pi/2/log (pi/2) =z for all other. , or n= log (e^n*I*pi/2)/log I , z=
log (e^z*I*pi/2)/log I.
As for imaginary, negative angles and other imaginary things like
points, edges etc. it seems obvious that such interpretation of
tetration and higher operations and its extension for real/imaginary/negative number of
operations leads to/is connected with projective geometry in its
original complex form, as developed by Lambert, Gauss, Stoudt, Cayley,
Klein. It seems to me that real numbers as foundations of geometry
fail here (in extensions of tetration to real , imaginary, negative
numbers) , which is of no big surprise since their foundation are
based on physical convenience but not abstract proofs (since it is
unprovable so far) , so alternative models are possible as well.
But since I only conjectured this yesterday, there are many things to
read, particularly about origins of non-euclidian geometry . Just to
summarize:
In expression where x is on which hyperoperation z acts, and y is how many times it does ( e.g. x^x= x[4]2, x^a= x[3]a, x*x=x[2]2, x+x= x[1]2)
x[z]y
x- edge of hypervolume in y dimensions
y- dimensions of hypervolume with edge x
z- is related to varying angle between edges x of hypervolume in y
dimensions, perhaps via logarithm or some trigonometric function. A
meaningful definition should recover usual angles pi/2 in case of
hypercube in integer dimensions.
The geometric interpretation of hyperoperations in general and their real,
complex extensions is related to projective geometry in imaginary (complex) form
extended to non-integer, negative and imaginary dimensions.
To give a more concrete test example:
I[I]I is a hypervolume with edge I in I dimensions with angle between imaginary edges (e^(I*I*pi/2) = e^(-pi/2)
I[-I]I is a hypervolume with edge I in I dimensions with angle between imaginary edges e^pi/2. In I dimensional space, such angle might have a meaning.
It also seems that his hyperangle conveyed via z is a composite one, so related to twisting of hypervolume x[z]y of dimension y. This is probably possible to check if some volumes in ordinary integer spaces of twisted hypercubes are known.
hi Steven
This twisting I mentioned previosly is a link between space and information storage:
We can use these twisting or rotation properties of hyperoperations to
get a glimpse what in x[z]y y dimensional space is:
We know that ( i may be mistaken in these analogies):
x^x=x[3]x=x[4]2
So according to my conventions, in 2 dimensional space, this is edge x
TWISTED 4 times by pi/2, or 2pi, or a TWISTED line, related to
torsion of line.
In x dimensional space, it is edge x twisted 3 times via pi/2. So one
twist is removed from line and attributed to space itself.
Hopefully this can be extended to lower operations like multiplication
and addition and zeration to see their geometrical meaning. Negation
as inverse twists and imagination as rotation (?) of edges will
follow. Rotations ( imaginations) will lead in case of straight
angles to a kind of Hilbert space filling curve with edge x in n or
other number of dimensions.
This twisting I mentioned previosly is a link between space and information storage:
We can use these twisting or rotation properties of hyperoperations to
get a glimpse what in x[z]y y dimensional space is:
We know that ( i may be mistaken in these analogies):
x^x=x[3]x=x[4]2
So according to my conventions, in 2 dimensional space, this is edge x
TWISTED 4 times by pi/2, or 2pi, or a TWISTED line, related to
torsion of line.
In x dimensional space, it is edge x twisted 3 times via pi/2. So one
twist is removed from line and attributed to space itself.
Hopefully this can be extended to lower operations like multiplication
and addition and zeration to see their geometrical meaning. Negation
as inverse twists and imagination as rotation (?) of edges will
follow. Rotations ( imaginations) will lead in case of straight
angles to a kind of Hilbert space filling curve with edge x in n or
other number of dimensions.
QUOTE (StevenA+Jul 5 2008, 12:14 AM)
This could lead to a requirement for a construction of space in which the volumes contained by various elements remain relatively prime to other attributes within this space (otherwise we can compress out the symmetries into a smaller space). And this again leads to the interesting correlation with Goldbach's conjecture that all numbers beyond a trivial minimum are the sum of two primes (we could interprete this, in the context I gave above, as meaning that all volumes of spaces are numbers because the space could only be seen to exist if it both 1) defined a constrast between two states and 2) that the volumes of each state were relatively prime - though there's almost certainly more to consider, but that's enough for now
The word processor must be set to chop to make this word salad.
The word processor must be set to chop to make this word salad.
Hello Ivars,
There's a fundamental issue/problem of symmetry I see with a volume of space being interpreted as something of the form n^m (unless potentially n and m are not integers) and I think I had another slight epiphany regarding how randomness appears in the context of a physical space.
Notice that n^m implies m orthongonal dimensions of width n. Now if the properties of each dimension exist independently/orthogonally to each other, then this would be a natural manner to describe the space, but recognize that this independence of dimensions implies they do not actually interact within this space and this space is just a generalization of any possible (effectively randomly selected) relationship between those dimensions.
For example, if we had a checkerboard and had two independent sources select x and y coordinates to place a marker on, over time, this board would collect markers on each space and with an idealized random correlation (which can also be interpreted as no correlation) between these coordinates, the space ultimately fills uniformly and ideally approaches a "density" of selection for each location within this grid of 1/n^2 where n is "width" of each dimension.
So this uniform/orthogonal space has no features within it and the areas/significance of all cell/locations within it are identical - we could select any random point within it and it would be just as distinguishable in properties as any other point within it.
Now in this example we could imagine some short term properties might be interpreted to exist because we're selecting individual samples of this space over time and so when plotted against time, the grid no longer appears uniform as each moment of time (as well as finite windows of time) a specific point within it is preferentially treated, but notice that there would be no manner to construct any provable "laws of physics" regarding this selection as no such correlations would exist over time (though recognize that making predictions itself implies a non-uniform treatment in the relationships between these cells and hence the predictions themselves could be seen similar to such laws).
So imagine a universe in which at any moment you simply observed a random object (with a "random" collection of attributes) each moment. If we attempted to plot what form of space would be apparent, there would, in the long run, be no correlations between attributes and every dimension for an attribute would appear unrelated and orthogonal to every other dimension - the space would be perfectly flat and uniform in the long run.
Now consider that though some elements of this may appear present in actual physical laws, we can still feel rather confident that when we drop an apple, it will fall and though the possibility that at some point an apple may not fall, there's an extreme bias toward one outcome versus the other and this bias implies that we effectively have a large bias between the volumes containing these two possible results (notice that the space of this has been biased by the question or manner of observation and classification of the two results - we could construct a virtual space much closer correlated with a 50/50 distribution of a binary attribute by restructing the test to be whether or not the apple happened to fall faster or slower than some nominal estimate - in that case, the resulting binary "space" we're measuring is much closer to a random space and we can similarly interprete this as meaning that by intelligent observation and estimation we've constructed an inverse to the biases of physical laws upon a space and reached a point where we're sampling "randomness" and this space appears flat and "undistorted" and the dimension describing points in time of my sampling are independent of the binary result of whether or not the apple fell faster or slower than the estimated nominal that I predicted).
You can see this process very similar to physics/the body/brain etc. forming in a subconscious manner imposing specific properties upon something fundamentally random/energetic (though you can draw the line in different places, there remains an unknown source that provides randomness)
Notice that generally variables are effectively unknown and have random features and don't contribute to a structure but are simply the "paint" you fill it with:
There's a fundamental issue/problem of symmetry I see with a volume of space being interpreted as something of the form n^m (unless potentially n and m are not integers) and I think I had another slight epiphany regarding how randomness appears in the context of a physical space.
Notice that n^m implies m orthongonal dimensions of width n. Now if the properties of each dimension exist independently/orthogonally to each other, then this would be a natural manner to describe the space, but recognize that this independence of dimensions implies they do not actually interact within this space and this space is just a generalization of any possible (effectively randomly selected) relationship between those dimensions.
For example, if we had a checkerboard and had two independent sources select x and y coordinates to place a marker on, over time, this board would collect markers on each space and with an idealized random correlation (which can also be interpreted as no correlation) between these coordinates, the space ultimately fills uniformly and ideally approaches a "density" of selection for each location within this grid of 1/n^2 where n is "width" of each dimension.
So this uniform/orthogonal space has no features within it and the areas/significance of all cell/locations within it are identical - we could select any random point within it and it would be just as distinguishable in properties as any other point within it.
Now in this example we could imagine some short term properties might be interpreted to exist because we're selecting individual samples of this space over time and so when plotted against time, the grid no longer appears uniform as each moment of time (as well as finite windows of time) a specific point within it is preferentially treated, but notice that there would be no manner to construct any provable "laws of physics" regarding this selection as no such correlations would exist over time (though recognize that making predictions itself implies a non-uniform treatment in the relationships between these cells and hence the predictions themselves could be seen similar to such laws).
So imagine a universe in which at any moment you simply observed a random object (with a "random" collection of attributes) each moment. If we attempted to plot what form of space would be apparent, there would, in the long run, be no correlations between attributes and every dimension for an attribute would appear unrelated and orthogonal to every other dimension - the space would be perfectly flat and uniform in the long run.
Now consider that though some elements of this may appear present in actual physical laws, we can still feel rather confident that when we drop an apple, it will fall and though the possibility that at some point an apple may not fall, there's an extreme bias toward one outcome versus the other and this bias implies that we effectively have a large bias between the volumes containing these two possible results (notice that the space of this has been biased by the question or manner of observation and classification of the two results - we could construct a virtual space much closer correlated with a 50/50 distribution of a binary attribute by restructing the test to be whether or not the apple happened to fall faster or slower than some nominal estimate - in that case, the resulting binary "space" we're measuring is much closer to a random space and we can similarly interprete this as meaning that by intelligent observation and estimation we've constructed an inverse to the biases of physical laws upon a space and reached a point where we're sampling "randomness" and this space appears flat and "undistorted" and the dimension describing points in time of my sampling are independent of the binary result of whether or not the apple fell faster or slower than the estimated nominal that I predicted).
You can see this process very similar to physics/the body/brain etc. forming in a subconscious manner imposing specific properties upon something fundamentally random/energetic (though you can draw the line in different places, there remains an unknown source that provides randomness)
Notice that generally variables are effectively unknown and have random features and don't contribute to a structure but are simply the "paint" you fill it with:
QUOTE (Ivars+)
We can use these twisting or rotation properties of hyperoperations to
get a glimpse what in x[z]y y dimensional space is:
We know that ( i may be mistaken in these analogies):
x^x=x[3]x=x[4]2
So according to my conventions, in 2 dimensional space, this is edge x
TWISTED 4 times by pi/2, or 2pi, or a TWISTED line, related to
torsion of line.
In x dimensional space, it is edge x twisted 3 times via pi/2. So one
twist is removed from line and attributed to space itself.
Notice that you're taking a flat/uniform/abstract representation for x^x (the fact that you're using variables makes the content of this space non-discrete and nonspecific, so it's a very general form of abstraction) and then attempting to map it to the structures x[3]x and x[4]2 and though you've given descriptions of what these quantities are, I'm not familiar with this representation. Maybe a couple concrete examples of how they're constructed would work.
How do you compute, 2[3]4, for example. I need to know the precise algorithm in order to see how these spaces are constructed.
But still, notice that you began with an abstract virtual and uncorrelated/random space of x dimensions with widths of x (very little of any distinct features to isolate any spacial properties of it as being specific - it's highly symmetric) and then you impose specific ratios of attributes upon it by substituting in constants and the resulting interpretations of these new fixed spaces arise from your imposing non-symmetric features to it, of which the significant attributes are quantities and I believe there's a manner to even describe your interpretation of the geometric interrelationships between these numbers as a number itself (as a simple example to demonstrate the possibility of this, we could, for example, write a computer program to compute such interpretations of spaces and interprete the instructions for this program and input as a number itself and the spaces your describe become numbers within this context - though notice that defining the instruction set could appear to exist as something beyond this, but if that instruction set was also defined in terms of geometric objects with numeric interpretations, then the entire computation could be seen as isolating a subset of the fractal space that this "seed" structure, defining the basic operations of such a geometric processor, that could evolve into this - in other words, specific programs would be the same as selecting specific points within this space, much like searching a fractal landscape for an object of a specific form).
Notice that in order to create an actual physical space, the dimensions must not be entirely unrelated within it (which could similarly be seen as being perfectly correlated and indistinguishable) or there would be no ability to construct the equivalent of laws that were able to connect them into interacting within a single universe.
So if we have a space that might initially appear as n^m, this is simply the mental abstraction of such a space that can potentially describe any combination of the n varieties of attributes in m dimensions - but notice again that this ability to describe any of them also means that it doesn't specifically describe anything and in fact we could reorder dimensions without influencing the properties of this space, so there's nothing that keeps any of these specific dimensions as actually remaining specific and distinct.
Now notice that instead if we have an asymmetrical volume described by, for example 3 dimensions of widths 2, 3 and 5, then the volume of this space would be 30 (if the dimensions were truly unrelated) and at this point we can inherently determine specifically what forms of properties exist within each of these dimensions as (given the assumption that it's 3 dimension - which could be seen to arise as information from outside this volume of 30 as we could interprete it to represent various other dimensional forms as a single dimension of 30 or a 2 space of 5x6 size etc.), and the dimensions are inherently represented by binary, ternary and quinary (?) properties and this is determined by factorizing the volume of a space into independent dimensions.
Notice that objective reality is defined in terms of shared quantities - the specific order of objects with a shared class between observers isn't inherently common, but the fact that quantities of both of them can be compared between observers is what determines common features in this landscape (for example, the color orange is determined by ratios of quantities of photons possessing specific wavelengths that are measured by quantities of distance or time, a forest is recognized by quantities of trees and smells are ratios of quantities of chemical interactions determined by atoms which interact via. discrete quantities of electrical charges - you can rearrange the specific ordering of individual events for an observer and the experience can be perceived differently, but objective reality between observers is constructed by taking a window of time of such events and extract various commonly measured quantities embedded within these aggregate features over time) - so two people don't see the same photon because it's a discrete event that can't be broken up into finer quantities, but two observers seeing similar ratios of photons from two objects can determine that one object is brighter than the other or supplies a greater force etc.).
If we consider what a truly objective universe would appear as, it would not possess any specific conscious qualities to it - red photons would not be specifically red (as truly I can't see what red looks like to someone else) but would instead be defined by a specific "object" within this objective view the universe, and in order to locate it in a precise and unique manner, I'd need a collection of all possible common elements within this objective universe and this would effectively be the basic alphabet for communication and further, it's need these to each remain unique and not be swapped in interpretation and this requires a space to store them and appears to require that this space actually exist as a single dimension (there would be no purpose to describing these symbols in any higher dimensional form because they, in themselves do not have any specific significance or meaning, yet, until related into more complex structures) - and because they each need to remain distinct and not swappable with any other element, then a precise ordering within this dimension is required and we could effectively enumerate them (in any order) as A, B, C etc. or 1, 2, 3 ... but recognize that the numeric ordering implies numeric features, which do not (yet) exist in this example because quantities of these occur over time and collections of them describe aggregate properties of objects in a common objective space.
But though these symbols have no specific numeric interpretation, the fact that they must exist in a specific linear ordering in a dimension automatically imposes a numeric manner of referencing them and this, in itself, can construct rules by which some are found relative to others. (Notice that such a view creates an objective universe that could be viewed from diverse subjective conscious qualities and that the conscious qualities of some of these traits could be swapped/altered in many ways, though the quantity of these would need to match so that a 1 to 1 perception of physical properties remained in common)
Now if we instead constructed a space similar to the form n^n, but required that each dimension of this represent a different quantity, then we'd have a volume of 1*2*3*4*...*n, or n! instead of n^n.
Notice that n! and n^n are closely related http://mathworld.wolfram.com/StirlingsApproximation.html, and that, for large n, log(n!)~=log(n^n), so you could interprete this to mean that the apparent number of dimensions, of constant width, required to describe the volume of these two spaces would appear very similar, but the n! space has non-random properties inherent in it that the n^n interpretation doesn't (you might see this similar to a mismatch between a virtual/mental space described by abstract properties versus an actual space of objects required to be defined by specific and unique properties).
An interesting approximation, more accurate that Stirling's is this one, given by Gosper:
n!~=((2n+1/3)*pi)^.5 * n^n * e^-n
Notice that this approximation could be seen as a scaleing factor to the n^n space, if we rewrote it like this:
n!~=n^n * scaling
scaling=(2pi*(n+1/6))^.5 * e^-n
Notice that the term e^-n implies a recursive scaling by 1/e or ~.368 as every additional dimension is added. This could be interpreted in a geometric context similar to a half life (notice this implies a randomness) to correlations between spaces as each new layer is added, and if we remove this term and look at what's happening before this decay in correlation between dimensions, we have:
scaling(excluding decay of time in correlations)=2pi*(n+1/6)^.5
We cancel the square root by squaring:
scaling(excluding decay of time in correlations)^2=2pi*(n+1/6)
And this has an interesting correlation with a complete cycle of rotation (2pi) for every new dimension added, though we begin with a 1/6th rotation (with many interesting geometric correlations here).
Anyway, those are just some potentially interesting ideas to consider. (Alternately we could possibly make some correlations with a gaussian form here as we have an e^-n term as well as sqrt(pi) term)
I started typing up a large reply (as usual), but they take so long and usually go off on some tangents, so here's a short and quick version
get a glimpse what in x[z]y y dimensional space is:
We know that ( i may be mistaken in these analogies):
x^x=x[3]x=x[4]2
So according to my conventions, in 2 dimensional space, this is edge x
TWISTED 4 times by pi/2, or 2pi, or a TWISTED line, related to
torsion of line.
In x dimensional space, it is edge x twisted 3 times via pi/2. So one
twist is removed from line and attributed to space itself.
Notice that you're taking a flat/uniform/abstract representation for x^x (the fact that you're using variables makes the content of this space non-discrete and nonspecific, so it's a very general form of abstraction) and then attempting to map it to the structures x[3]x and x[4]2 and though you've given descriptions of what these quantities are, I'm not familiar with this representation. Maybe a couple concrete examples of how they're constructed would work.
How do you compute, 2[3]4, for example. I need to know the precise algorithm in order to see how these spaces are constructed.
But still, notice that you began with an abstract virtual and uncorrelated/random space of x dimensions with widths of x (very little of any distinct features to isolate any spacial properties of it as being specific - it's highly symmetric) and then you impose specific ratios of attributes upon it by substituting in constants and the resulting interpretations of these new fixed spaces arise from your imposing non-symmetric features to it, of which the significant attributes are quantities and I believe there's a manner to even describe your interpretation of the geometric interrelationships between these numbers as a number itself (as a simple example to demonstrate the possibility of this, we could, for example, write a computer program to compute such interpretations of spaces and interprete the instructions for this program and input as a number itself and the spaces your describe become numbers within this context - though notice that defining the instruction set could appear to exist as something beyond this, but if that instruction set was also defined in terms of geometric objects with numeric interpretations, then the entire computation could be seen as isolating a subset of the fractal space that this "seed" structure, defining the basic operations of such a geometric processor, that could evolve into this - in other words, specific programs would be the same as selecting specific points within this space, much like searching a fractal landscape for an object of a specific form).
Notice that in order to create an actual physical space, the dimensions must not be entirely unrelated within it (which could similarly be seen as being perfectly correlated and indistinguishable) or there would be no ability to construct the equivalent of laws that were able to connect them into interacting within a single universe.
So if we have a space that might initially appear as n^m, this is simply the mental abstraction of such a space that can potentially describe any combination of the n varieties of attributes in m dimensions - but notice again that this ability to describe any of them also means that it doesn't specifically describe anything and in fact we could reorder dimensions without influencing the properties of this space, so there's nothing that keeps any of these specific dimensions as actually remaining specific and distinct.
Now notice that instead if we have an asymmetrical volume described by, for example 3 dimensions of widths 2, 3 and 5, then the volume of this space would be 30 (if the dimensions were truly unrelated) and at this point we can inherently determine specifically what forms of properties exist within each of these dimensions as (given the assumption that it's 3 dimension - which could be seen to arise as information from outside this volume of 30 as we could interprete it to represent various other dimensional forms as a single dimension of 30 or a 2 space of 5x6 size etc.), and the dimensions are inherently represented by binary, ternary and quinary (?) properties and this is determined by factorizing the volume of a space into independent dimensions.
Notice that objective reality is defined in terms of shared quantities - the specific order of objects with a shared class between observers isn't inherently common, but the fact that quantities of both of them can be compared between observers is what determines common features in this landscape (for example, the color orange is determined by ratios of quantities of photons possessing specific wavelengths that are measured by quantities of distance or time, a forest is recognized by quantities of trees and smells are ratios of quantities of chemical interactions determined by atoms which interact via. discrete quantities of electrical charges - you can rearrange the specific ordering of individual events for an observer and the experience can be perceived differently, but objective reality between observers is constructed by taking a window of time of such events and extract various commonly measured quantities embedded within these aggregate features over time) - so two people don't see the same photon because it's a discrete event that can't be broken up into finer quantities, but two observers seeing similar ratios of photons from two objects can determine that one object is brighter than the other or supplies a greater force etc.).
If we consider what a truly objective universe would appear as, it would not possess any specific conscious qualities to it - red photons would not be specifically red (as truly I can't see what red looks like to someone else) but would instead be defined by a specific "object" within this objective view the universe, and in order to locate it in a precise and unique manner, I'd need a collection of all possible common elements within this objective universe and this would effectively be the basic alphabet for communication and further, it's need these to each remain unique and not be swapped in interpretation and this requires a space to store them and appears to require that this space actually exist as a single dimension (there would be no purpose to describing these symbols in any higher dimensional form because they, in themselves do not have any specific significance or meaning, yet, until related into more complex structures) - and because they each need to remain distinct and not swappable with any other element, then a precise ordering within this dimension is required and we could effectively enumerate them (in any order) as A, B, C etc. or 1, 2, 3 ... but recognize that the numeric ordering implies numeric features, which do not (yet) exist in this example because quantities of these occur over time and collections of them describe aggregate properties of objects in a common objective space.
But though these symbols have no specific numeric interpretation, the fact that they must exist in a specific linear ordering in a dimension automatically imposes a numeric manner of referencing them and this, in itself, can construct rules by which some are found relative to others. (Notice that such a view creates an objective universe that could be viewed from diverse subjective conscious qualities and that the conscious qualities of some of these traits could be swapped/altered in many ways, though the quantity of these would need to match so that a 1 to 1 perception of physical properties remained in common)
Now if we instead constructed a space similar to the form n^n, but required that each dimension of this represent a different quantity, then we'd have a volume of 1*2*3*4*...*n, or n! instead of n^n.
Notice that n! and n^n are closely related http://mathworld.wolfram.com/StirlingsApproximation.html, and that, for large n, log(n!)~=log(n^n), so you could interprete this to mean that the apparent number of dimensions, of constant width, required to describe the volume of these two spaces would appear very similar, but the n! space has non-random properties inherent in it that the n^n interpretation doesn't (you might see this similar to a mismatch between a virtual/mental space described by abstract properties versus an actual space of objects required to be defined by specific and unique properties).
An interesting approximation, more accurate that Stirling's is this one, given by Gosper:
n!~=((2n+1/3)*pi)^.5 * n^n * e^-n
Notice that this approximation could be seen as a scaleing factor to the n^n space, if we rewrote it like this:
n!~=n^n * scaling
scaling=(2pi*(n+1/6))^.5 * e^-n
Notice that the term e^-n implies a recursive scaling by 1/e or ~.368 as every additional dimension is added. This could be interpreted in a geometric context similar to a half life (notice this implies a randomness) to correlations between spaces as each new layer is added, and if we remove this term and look at what's happening before this decay in correlation between dimensions, we have:
scaling(excluding decay of time in correlations)=2pi*(n+1/6)^.5
We cancel the square root by squaring:
scaling(excluding decay of time in correlations)^2=2pi*(n+1/6)
And this has an interesting correlation with a complete cycle of rotation (2pi) for every new dimension added, though we begin with a 1/6th rotation (with many interesting geometric correlations here).
Anyway, those are just some potentially interesting ideas to consider. (Alternately we could possibly make some correlations with a gaussian form here as we have an e^-n term as well as sqrt(pi) term)
QUOTE (buttershug+)
The word processor must be set to chop to make this word salad.
So your mind twists post into word salads ... interesting.
So your mind twists post into word salads ... interesting.
Here's another way to look at this with regard to the potential differences in abstract/mental versus real/perceivable volumes of space:
If we have a space with a volume of n units, and each unit contained, for example either "nothing" (or empty space) or a proton, there could be considered to be 2^n possible configurations for this space - this is the abstract/mental model.
Now consider that if we swap two units of empty space or two units of protons, the space, from this abstract perspective is identical and we remain at one of those 2^n possible configurations and the same is true for swapping units of "empty" space. (Notice that both the "empty" space and the protons can actually move within their respective fields, to the extent that such motions remain undetectable, or alternately we could say that such motions are meaningless/impossible and two protons, for example, could not swap places within this space - the only way to allow such protons to swap spaces would be to treat each proton as if it were not identical and that proton A moved to the location where proton B was and visa versa, but notice that this space requires a memory to construct and is effectively larger in information content and has a form that's not exponential but factorial)
If we have a space in which, otherwise indistinguishable, objects can appear to move then we're appending an individual context to each that makes it unique and allows us (relative to an observed history of that space) to distinguish between them and no longer is a proton, for example, described solely by its physical properties identical to other protons, but it has now become proton A or proton B etc.
In this case, if we have a space of n units, and we have m protons within it, but each is uniquely designated, then we have a larger possible number of configurations as the ordering of protons is considered significant and, for example, the existance of 3 protons, A, B and C would allow for these to placed in one of 6 possible ways as ABC, BAC, ACB, etc.
If we consider the empty spaces within this to be indistinguishable, then we do not have such a context for units of empty space and the volume of space can grow much larger without providing the information complexity that matter does within a smaller volume.
To compute the total number of possible configurations of m distinct objects contained within a space of n units, with the remaining n-m units of empty space being considered indistinguishable from each other (and motions within this empty space being considered irrelevant as individual units of it aren't differentiable), we can compute the form of this equation to be a product of two terms - one term indicates the number of possible manners in which the matter could be reorganized internally, within the same volume of space, and the other indicates the spacial locations at which these units of matter reside within space (so one could be interpreted as an inherent complexity to mass and the other an inherent complexity to space position/motion of that mass - which is effectively just described by the locations of space, where matter, is not).
For some background here on the mathematics http://en.wikipedia.org/wiki/Combinatorics
#configurations of mass with m distinct units=m!
We can define s as the number of units of empty space as:
s=n-m
And we can see this simply as selectively placing s units within a space of n, of which the first unit of space could be placed at one of n locations, and the second at one of n-1 locations (there would no ability for space to be twice as dense, for example, using this model) and the the next at n-2 etc. until we place n-m units and then we divide by the number of possible reorderings of s, because these reordering of space would not be detectable and we get:
#configurations of space=n!/s!
Which can alternately/conversely be interpreted to be the possible spacial forms this mass could reside within.
And we get the total number of possible states which considering possible permutations for mass above by multiplying:
#total states=m!*n!/s!
Notice that if we ignore the term detailing properties of mass, and look at n>s (which would be the case if any matter existed), we can cancel the division and construct this as a product of purely independent dimensions, and for example if n=7 and s=4, then we'd have:
#states_excluding_matter(n=7,s=4)=(7*6*5*4*3*2*1)/(4*3*2*1)
#states_excluding_matter(n=7,s=4)=(7*6*5) * (4*3*2*1)/(4*3*2*1)
And the second term cancels and we simply have:
#states_excluding_matter(n=7,s=4)=(7*6*5)
Which could be interpreted as a 3 dimensional space with dimensional widths of 7, 6 and 5.
Notice that for a large n (large volume of space) and small m (few distinguishable units of mass) the terms left remaining would simply be a product of m terms, near n and with large n, the relative difference in terms of these dimensions would be small.
For example, if n=100 and m=3, (s=100-3)=97 then we have:
#total states(n=100,s=97)=(3*2*1) * (100*99*98)
Notice that this value is approximately 99^3, and this space could appear to closely resemble a 3 dimensional with uniform dimensions, though it would contain 3 units of mass, A, B and C, which could appear to exist in one of 6 states (would could, for example, interprete these as pointing in a bipolar direction in one of those 3 dimensions and have the aggregate model appear to closely mimic a single unit of matter, with a field oriented in a direction of either +x, -x, +y, -y or +z and -z and the size of this space would be approximately 99x99x99, though in reality it would be 100x99x98, though for any such space with 3 units of small this approximation would be, if we assign j=n-1, we have (j+1)*(j)*(j-1)=j^3-j, or in our above case it would actually be 99^3-99 - and that's where infinitesimal calculus takes a wrong turn and treats this physically similar to the form 99^3 and end up seeing a flat and uniform space then can't determine how asymmetries arise! and there are some other analogies we can show as well for the (misconstruction) of real numbers).
If we have a space with a volume of n units, and each unit contained, for example either "nothing" (or empty space) or a proton, there could be considered to be 2^n possible configurations for this space - this is the abstract/mental model.
Now consider that if we swap two units of empty space or two units of protons, the space, from this abstract perspective is identical and we remain at one of those 2^n possible configurations and the same is true for swapping units of "empty" space. (Notice that both the "empty" space and the protons can actually move within their respective fields, to the extent that such motions remain undetectable, or alternately we could say that such motions are meaningless/impossible and two protons, for example, could not swap places within this space - the only way to allow such protons to swap spaces would be to treat each proton as if it were not identical and that proton A moved to the location where proton B was and visa versa, but notice that this space requires a memory to construct and is effectively larger in information content and has a form that's not exponential but factorial)
If we have a space in which, otherwise indistinguishable, objects can appear to move then we're appending an individual context to each that makes it unique and allows us (relative to an observed history of that space) to distinguish between them and no longer is a proton, for example, described solely by its physical properties identical to other protons, but it has now become proton A or proton B etc.
In this case, if we have a space of n units, and we have m protons within it, but each is uniquely designated, then we have a larger possible number of configurations as the ordering of protons is considered significant and, for example, the existance of 3 protons, A, B and C would allow for these to placed in one of 6 possible ways as ABC, BAC, ACB, etc.
If we consider the empty spaces within this to be indistinguishable, then we do not have such a context for units of empty space and the volume of space can grow much larger without providing the information complexity that matter does within a smaller volume.
To compute the total number of possible configurations of m distinct objects contained within a space of n units, with the remaining n-m units of empty space being considered indistinguishable from each other (and motions within this empty space being considered irrelevant as individual units of it aren't differentiable), we can compute the form of this equation to be a product of two terms - one term indicates the number of possible manners in which the matter could be reorganized internally, within the same volume of space, and the other indicates the spacial locations at which these units of matter reside within space (so one could be interpreted as an inherent complexity to mass and the other an inherent complexity to space position/motion of that mass - which is effectively just described by the locations of space, where matter, is not).
For some background here on the mathematics http://en.wikipedia.org/wiki/Combinatorics
#configurations of mass with m distinct units=m!
We can define s as the number of units of empty space as:
s=n-m
And we can see this simply as selectively placing s units within a space of n, of which the first unit of space could be placed at one of n locations, and the second at one of n-1 locations (there would no ability for space to be twice as dense, for example, using this model) and the the next at n-2 etc. until we place n-m units and then we divide by the number of possible reorderings of s, because these reordering of space would not be detectable and we get:
#configurations of space=n!/s!
Which can alternately/conversely be interpreted to be the possible spacial forms this mass could reside within.
And we get the total number of possible states which considering possible permutations for mass above by multiplying:
#total states=m!*n!/s!
Notice that if we ignore the term detailing properties of mass, and look at n>s (which would be the case if any matter existed), we can cancel the division and construct this as a product of purely independent dimensions, and for example if n=7 and s=4, then we'd have:
#states_excluding_matter(n=7,s=4)=(7*6*5*4*3*2*1)/(4*3*2*1)
#states_excluding_matter(n=7,s=4)=(7*6*5) * (4*3*2*1)/(4*3*2*1)
And the second term cancels and we simply have:
#states_excluding_matter(n=7,s=4)=(7*6*5)
Which could be interpreted as a 3 dimensional space with dimensional widths of 7, 6 and 5.
Notice that for a large n (large volume of space) and small m (few distinguishable units of mass) the terms left remaining would simply be a product of m terms, near n and with large n, the relative difference in terms of these dimensions would be small.
For example, if n=100 and m=3, (s=100-3)=97 then we have:
#total states(n=100,s=97)=(3*2*1) * (100*99*98)
Notice that this value is approximately 99^3, and this space could appear to closely resemble a 3 dimensional with uniform dimensions, though it would contain 3 units of mass, A, B and C, which could appear to exist in one of 6 states (would could, for example, interprete these as pointing in a bipolar direction in one of those 3 dimensions and have the aggregate model appear to closely mimic a single unit of matter, with a field oriented in a direction of either +x, -x, +y, -y or +z and -z and the size of this space would be approximately 99x99x99, though in reality it would be 100x99x98, though for any such space with 3 units of small this approximation would be, if we assign j=n-1, we have (j+1)*(j)*(j-1)=j^3-j, or in our above case it would actually be 99^3-99 - and that's where infinitesimal calculus takes a wrong turn and treats this physically similar to the form 99^3 and end up seeing a flat and uniform space then can't determine how asymmetries arise!
and there are some other analogies we can show as well for the (misconstruction) of real numbers).
hi Steven
I was thinking about impossibility to have a generalization of logarithm for higher hyperoperations ( I have a thread where I generalized it, only to find out it results in simplest case in a function that has 2 values 0 and 1 simultaneously).
So, why not to add an additional ( and probably infinitely many) degree of freedom INTERNAL to real, fractional, integer numbers? Than any number would have much more capacity.
The idea is like a torsion (twist) of a curve in 3D. In 2 D curve has only one parameter at each point, curvature. in 1D, even that is missing.
But if a line ( e.g. real number line, or any number) would have internal, or , since it is not present in 2D, imaginary torsion/twist in plane and 1D, than, also numbers representing such twist would have extra degree of freedom.
Also, such twisted intervals of number lines would of course interact with each other, so that in mathematical space, there would be interesting things happening.
Returning to generalized logarithms, that would mean that function f(x) = {0;1} twists (oscillates) between 0 and 1 . The speed of these oscillations ( period in twist space) would than be the related to hyper logarithm which has property :
3_log(a^ b )=3_log(a)+3_log( b ) as compared to usual
log(a^b )= b*log(a).
If such 3_log would exist, it would facilitate computation as did logarithms in 16 th-20th centuries.
And of course, such twisting degrees of freedom can be easily nested, in principle- like a rope that consists of thinner and thinner plaits.
There are no empty spaces, and each of spatial places is different from other by the things that exist in it and are nearby.
I was thinking about impossibility to have a generalization of logarithm for higher hyperoperations ( I have a thread where I generalized it, only to find out it results in simplest case in a function that has 2 values 0 and 1 simultaneously).
So, why not to add an additional ( and probably infinitely many) degree of freedom INTERNAL to real, fractional, integer numbers? Than any number would have much more capacity.
The idea is like a torsion (twist) of a curve in 3D. In 2 D curve has only one parameter at each point, curvature. in 1D, even that is missing.
But if a line ( e.g. real number line, or any number) would have internal, or , since it is not present in 2D, imaginary torsion/twist in plane and 1D, than, also numbers representing such twist would have extra degree of freedom.
Also, such twisted intervals of number lines would of course interact with each other, so that in mathematical space, there would be interesting things happening.
Returning to generalized logarithms, that would mean that function f(x) = {0;1} twists (oscillates) between 0 and 1 . The speed of these oscillations ( period in twist space) would than be the related to hyper logarithm which has property :
3_log(a^ b )=3_log(a)+3_log( b ) as compared to usual
log(a^b )= b*log(a).
If such 3_log would exist, it would facilitate computation as did logarithms in 16 th-20th centuries.
And of course, such twisting degrees of freedom can be easily nested, in principle- like a rope that consists of thinner and thinner plaits.
There are no empty spaces, and each of spatial places is different from other by the things that exist in it and are nearby.
I started typing up a large reply (as usual), but they take so long and usually go off on some tangents, so here's a short and quick version
QUOTE (Ivars+)
hi Steven
I was thinking about impossibility to have a generalization of logarithm for higher hyperoperations ( I have a thread where I generalized it, only to find out it results in simplest case in a function that has 2 values 0 and 1 simultaneously).
Yes, notice that most logarithms give irrational results and these can be interpreted as being impossible to construct physically in some instantaineous form (though instead the irrational results can potentially be interpreted as statistical properties over time of a non-stationary object)
I was thinking about impossibility to have a generalization of logarithm for higher hyperoperations ( I have a thread where I generalized it, only to find out it results in simplest case in a function that has 2 values 0 and 1 simultaneously).
Yes, notice that most logarithms give irrational results and these can be interpreted as being impossible to construct physically in some instantaineous form (though instead the irrational results can potentially be interpreted as statistical properties over time of a non-stationary object)
QUOTE (Ivars+)
So, why not to add an additional ( and probably infinitely many) degree of freedom INTERNAL to real, fractional, integer numbers? Than any number would have much more capacity.
Yes, real numbers are effectively lists of elements that can be interpreted as existing within arbitrarily complex dimensional frameworks. The real issue would be over how these values appear to interact within this space and what properties these interactions would appear to give.
Yes, real numbers are effectively lists of elements that can be interpreted as existing within arbitrarily complex dimensional frameworks. The real issue would be over how these values appear to interact within this space and what properties these interactions would appear to give.
QUOTE (Ivars+)
The idea is like a torsion (twist) of a curve in 3D. In 2 D curve has only one parameter at each point, curvature. in 1D, even that is missing.
I think many numeric quantities generated in calculus are more accurately described as higher dimensional objects with only the dimensional features encompassing the largest volume of space being retained (but of course throwing away this lower dimensional information can cause many problems, of which 1) the operation can't be reversed without indeterminism arising and 2) the actual result may be physically impossible to construct as the infinitesimal framework required to create it has been removed - likely most irrational results are of this form)
I think many numeric quantities generated in calculus are more accurately described as higher dimensional objects with only the dimensional features encompassing the largest volume of space being retained (but of course throwing away this lower dimensional information can cause many problems, of which 1) the operation can't be reversed without indeterminism arising and 2) the actual result may be physically impossible to construct as the infinitesimal framework required to create it has been removed - likely most irrational results are of this form)
QUOTE (Ivars+)
But if a line ( e.g. real number line, or any number) would have internal, or , since it is not present in 2D, imaginary torsion/twist in plane and 1D, than, also numbers representing such twist would have extra degree of freedom.
In this case the results could more generally be described in terms of a set of quantized/integer vectors or as a multivariable/multidimensional function mapping discrete quantities to discrete quantities. So you should have a precise description available in a form similar to {i,j,k}=f(l,m,n), where these elements are all integers and the function description is also effectively quantized from a finite set of possible operations, though you could have the dimensions of the domain and range of this be different, in which case you're mapping a space described by dimensions of f,l,m and n into a space described by i,j and k, for example (so the above relationship {i,j,k}=f(l,m,n) could be interpreted as a compression of a 4 dimensional space into a 3 dimensional space, if we assume the function itself could be described numerically).
In this case the results could more generally be described in terms of a set of quantized/integer vectors or as a multivariable/multidimensional function mapping discrete quantities to discrete quantities. So you should have a precise description available in a form similar to {i,j,k}=f(l,m,n), where these elements are all integers and the function description is also effectively quantized from a finite set of possible operations, though you could have the dimensions of the domain and range of this be different, in which case you're mapping a space described by dimensions of f,l,m and n into a space described by i,j and k, for example (so the above relationship {i,j,k}=f(l,m,n) could be interpreted as a compression of a 4 dimensional space into a 3 dimensional space, if we assume the function itself could be described numerically).
QUOTE (Ivars+)
Also, such twisted intervals of number lines would of course interact with each other, so that in mathematical space, there would be interesting things happening.
Yes, most any non-linear transformation, where we do not have a 1 to 1 mapping possible leads to complexity and appears fundamentally rooted in a problem of being irreversible. This irreversibility though leads to another problem in which you can't determine precisely which input space created the output space (and this might be physically interpreted in the context of a superposition of virtual input spaces - as a quick example, if we have some line "curving" in one dimension and we assume an ability to transform the properties of this curve into some other dimension, then the other dimension must effectively be at least as "dense" in its ability to represent those locations as the initial dimension was, otherwise you end up mapping multiple possible input points to the same output location and then you lose an ability to prove or trace through specifically which mapping gave rise to a specific output. So if the output space is required to be consistantly, at least as "dense" in representations as the input space, then the could at most remain a density equal to the input space, though if we treated all locations within all dimensions as having a uniform density, then we could never return a result back to an original dimension because it would also be required to be more dense in representations than it is - a logical paradox and attempting to iterate such functions either causes the information to appear to become compressed in space within this, or be grow ever more chaotically interleaved over time, or expand toward infinity - for example, saying that y=f(x), where f is a non-linear function, implies that y is required to represent results relative to x in a non-uniform manner over its dimension and if we iterated this, once again as y=f(f(x)) we can potentially square the required range of variations in density, though the specific alignment of these non-linearities within that dimension determine the maximum growth possible)
Yes, most any non-linear transformation, where we do not have a 1 to 1 mapping possible leads to complexity and appears fundamentally rooted in a problem of being irreversible. This irreversibility though leads to another problem in which you can't determine precisely which input space created the output space (and this might be physically interpreted in the context of a superposition of virtual input spaces - as a quick example, if we have some line "curving" in one dimension and we assume an ability to transform the properties of this curve into some other dimension, then the other dimension must effectively be at least as "dense" in its ability to represent those locations as the initial dimension was, otherwise you end up mapping multiple possible input points to the same output location and then you lose an ability to prove or trace through specifically which mapping gave rise to a specific output. So if the output space is required to be consistantly, at least as "dense" in representations as the input space, then the could at most remain a density equal to the input space, though if we treated all locations within all dimensions as having a uniform density, then we could never return a result back to an original dimension because it would also be required to be more dense in representations than it is - a logical paradox and attempting to iterate such functions either causes the information to appear to become compressed in space within this, or be grow ever more chaotically interleaved over time, or expand toward infinity - for example, saying that y=f(x), where f is a non-linear function, implies that y is required to represent results relative to x in a non-uniform manner over its dimension and if we iterated this, once again as y=f(f(x)) we can potentially square the required range of variations in density, though the specific alignment of these non-linearities within that dimension determine the maximum growth possible)
QUOTE (Ivars+)
Returning to generalized logarithms, that would mean that function f(x) = {0;1} twists (oscillates) between 0 and 1 . The speed of these oscillations ( period in twist space) would than be the related to hyper logarithm which has property :
3_log(a^ b )=3_log(a)+3_log( b ) as compared to usual
log(a^b )= b*log(a).
If such 3_log would exist, it would facilitate computation as did logarithms in 16 th-20th centuries.
I've tried to mess around with the ratio log(3)/log(2) or log_2(3)~=1.58496 a few times before, trying to find some potentially juicy physical correlations, but not with much luck - though, something that bothers me is that there should similar influences for other logarithmic processes, but I can imagine smaller integers to be more influencial. I'm still trying to determine natural manners in which these should interact so that there's less guesswork involved (the best results have come from rigorously tracing through some concepts, though the source of those concepts is semi-random - but there's still nothing like precise logic and mathematics to derive specific results).
3_log(a^ b )=3_log(a)+3_log( b ) as compared to usual
log(a^b )= b*log(a).
If such 3_log would exist, it would facilitate computation as did logarithms in 16 th-20th centuries.
I've tried to mess around with the ratio log(3)/log(2) or log_2(3)~=1.58496 a few times before, trying to find some potentially juicy physical correlations, but not with much luck - though, something that bothers me is that there should similar influences for other logarithmic processes, but I can imagine smaller integers to be more influencial. I'm still trying to determine natural manners in which these should interact so that there's less guesswork involved (the best results have come from rigorously tracing through some concepts, though the source of those concepts is semi-random
- but there's still nothing like precise logic and mathematics to derive specific results).QUOTE (Ivars+)
And of course, such twisting degrees of freedom can be easily nested, in principle- like a rope that consists of thinner and thinner plaits.
It sounds like we're thinking along similar lines here, though this is usually the point where my ability to take steps intuitively and reach the right answers falters and I think my only resort is simulation. (I'm rather certain there are some correlations here with other structures commonly encountered in number theory, which could be seen as nice on one hand in that there are potentially outside resources to use in characterizing properties of such space/objects, but on the other hand that's not too good because it would also tend to imply that you'd get stuck at the same point others have ...)
It sounds like we're thinking along similar lines here, though this is usually the point where my ability to take steps intuitively and reach the right answers falters and I think my only resort is simulation. (I'm rather certain there are some correlations here with other structures commonly encountered in number theory, which could be seen as nice on one hand in that there are potentially outside resources to use in characterizing properties of such space/objects, but on the other hand that's not too good because it would also tend to imply that you'd get stuck at the same point others have ...)
QUOTE (Ivars+)
There are no empty spaces, and each of spatial places is different from other by the things that exist in it and are nearby.
I'd agree that there should be nothing as empty space. Every location in space should possess a manner in which it's uniquely referenced, otherwise you have a problem of those spaces appearing to be superimposed or folded together properties. Also the fact that the space is defined in properties by neighboring connections allows the interpretation to be extended to a space able to support the concept of constant velocity motion within it. Sounds good to me
I'd agree that there should be nothing as empty space. Every location in space should possess a manner in which it's uniquely referenced, otherwise you have a problem of those spaces appearing to be superimposed or folded together properties. Also the fact that the space is defined in properties by neighboring connections allows the interpretation to be extended to a space able to support the concept of constant velocity motion within it. Sounds good to me
Hi Steven,
That's better.Structure and limits in size helps. Same for me, many times.
That's better.Structure and limits in size helps. Same for me, many times.
Yes, real numbers are effectively lists of elements that can be interpreted as existing within arbitrarily complex dimensional frameworks. The real issue would be over how these values appear to interact within this space and what properties these interactions would appear to give.
Real numbers try to tell what is the structure inside them/linked to them.
Or rather, multivalued functions,but that does not help much if there is nothing more inside these values . function mapping exponetation to a set of 2 values f(x^y)={0;1} does not give much help, but if there are phase relations between 0;1 in some other dimension- no problem.
If I am not mistaken , in early projective geometry ( geometry of position) continuity was associated with existance of IMAGINARY points in space so as it would represent all roots of algebraic equations. This sounds logical to require such continuity requirement. The notion of continuity of real number line was introduced later in geometry and, as it happens, is not provable (continuity hypothesis).
Or rather, multivalued functions,but that does not help much if there is nothing more inside these values . function mapping exponetation to a set of 2 values f(x^y)={0;1} does not give much help, but if there are phase relations between 0;1 in some other dimension- no problem.
If I am not mistaken , in early projective geometry ( geometry of position) continuity was associated with existance of IMAGINARY points in space so as it would represent all roots of algebraic equations. This sounds logical to require such continuity requirement. The notion of continuity of real number line was introduced later in geometry and, as it happens, is not provable (continuity hypothesis).
I've tried to mess around with the ratio log(3)/log(2) or log_2(3)~=1.58496 a few times before, trying to find some potentially juicy physical correlations, but not with much luck
No , You misunderstood me, I am not talking about logarithm of 2 in base 3, I am talking about next level, more effective analytic logarithm which would make exponentiation replaceable by addition directly. I have noted such logarithm 3_log as opposed to normal log . Such logarithm would be also related to exponential factorial n^(((n-1)^((n-2)........(.2^1) of bigger 3_factorial 2^(3^(4....((n-1)^n) and its extension, some new Gamma function.
See? In the end, c is just a number, determined by mathematical properties of space how it can be.
QUOTE
I started typing up a large reply (as usual), but they take so long and usually go off on some tangents, so here's a short and quick version
That's better.Structure and limits in size helps. Same for me, many times.
QUOTE (->
| QUOTE |
| I started typing up a large reply (as usual), but they take so long and usually go off on some tangents, so here's a short and quick version |
That's better.Structure and limits in size helps. Same for me, many times.
Yes, real numbers are effectively lists of elements that can be interpreted as existing within arbitrarily complex dimensional frameworks. The real issue would be over how these values appear to interact within this space and what properties these interactions would appear to give.
Real numbers try to tell what is the structure inside them/linked to them.
QUOTE
In this case the results could more generally be described in terms of a set of quantized/integer vectors or as a multivariable/multidimensional function mapping discrete quantities to discrete quantities.
Or rather, multivalued functions,but that does not help much if there is nothing more inside these values . function mapping exponetation to a set of 2 values f(x^y)={0;1} does not give much help, but if there are phase relations between 0;1 in some other dimension- no problem.
If I am not mistaken , in early projective geometry ( geometry of position) continuity was associated with existance of IMAGINARY points in space so as it would represent all roots of algebraic equations. This sounds logical to require such continuity requirement. The notion of continuity of real number line was introduced later in geometry and, as it happens, is not provable (continuity hypothesis).
QUOTE (->
| QUOTE |
| In this case the results could more generally be described in terms of a set of quantized/integer vectors or as a multivariable/multidimensional function mapping discrete quantities to discrete quantities. |
Or rather, multivalued functions,but that does not help much if there is nothing more inside these values . function mapping exponetation to a set of 2 values f(x^y)={0;1} does not give much help, but if there are phase relations between 0;1 in some other dimension- no problem.
If I am not mistaken , in early projective geometry ( geometry of position) continuity was associated with existance of IMAGINARY points in space so as it would represent all roots of algebraic equations. This sounds logical to require such continuity requirement. The notion of continuity of real number line was introduced later in geometry and, as it happens, is not provable (continuity hypothesis).
I've tried to mess around with the ratio log(3)/log(2) or log_2(3)~=1.58496 a few times before, trying to find some potentially juicy physical correlations, but not with much luck
No , You misunderstood me, I am not talking about logarithm of 2 in base 3, I am talking about next level, more effective analytic logarithm which would make exponentiation replaceable by addition directly. I have noted such logarithm 3_log as opposed to normal log . Such logarithm would be also related to exponential factorial n^(((n-1)^((n-2)........(.2^1) of bigger 3_factorial 2^(3^(4....((n-1)^n) and its extension, some new Gamma function.
QUOTE
I'd agree that there should be nothing as empty space. Every location in space should possess a manner in which it's uniquely referenced, otherwise you have a problem of those spaces appearing to be superimposed or folded together properties. Also the fact that the space is defined in properties by neighboring connections allows the interpretation to be extended to a space able to support the concept of constant velocity motion within it. Sounds good to me
See? In the end, c is just a number, determined by mathematical properties of space how it can be.
QUOTE:
"I'd agree that there should be nothing as empty space. Every location in space should possess a manner in which it's uniquely referenced, otherwise you have a problem of those spaces appearing to be superimposed or folded together properties. Also the fact that the space is defined in properties by neighboring connections allows the interpretation to be extended to a space able to support the concept of constant velocity motion within it. Sounds good to me
See? In the end, c is just a number, determined by mathematical properties of space how it can be."
Except for the need of a "free room" - a "place" - "imaginary space" - where calculations can be executed before they are being placed in physical space.
Once an information has been placed in 3D it is probably impossible to get rid of that information in 3D, once an informational bit has taken the shape and size of a dimensionality, the destiny of said informational bit is to be adjusted to the neighboring in such a way that there is no physical void - and because physical space is constantly being "feeded" by new (and smaller) informational bits - the achieving of such a fit is an ever ongoing struggle -
By not having such "free room" - imaginary space, it would be impossible to have ANY dynamic. And terefore it is pure logic that there must exist a smallest dimension in physical space. And it is pure logic that the "interior" of smallest physical dimensions cannot be made up of 3D structures and it is most likely a math-expreession dealing with 3 interrelated qualities by itsself leads to irreversability.
"I'd agree that there should be nothing as empty space. Every location in space should possess a manner in which it's uniquely referenced, otherwise you have a problem of those spaces appearing to be superimposed or folded together properties. Also the fact that the space is defined in properties by neighboring connections allows the interpretation to be extended to a space able to support the concept of constant velocity motion within it. Sounds good to me
See? In the end, c is just a number, determined by mathematical properties of space how it can be."
Except for the need of a "free room" - a "place" - "imaginary space" - where calculations can be executed before they are being placed in physical space.
Once an information has been placed in 3D it is probably impossible to get rid of that information in 3D, once an informational bit has taken the shape and size of a dimensionality, the destiny of said informational bit is to be adjusted to the neighboring in such a way that there is no physical void - and because physical space is constantly being "feeded" by new (and smaller) informational bits - the achieving of such a fit is an ever ongoing struggle -
By not having such "free room" - imaginary space, it would be impossible to have ANY dynamic. And terefore it is pure logic that there must exist a smallest dimension in physical space. And it is pure logic that the "interior" of smallest physical dimensions cannot be made up of 3D structures and it is most likely a math-expreession dealing with 3 interrelated qualities by itsself leads to irreversability.
Hi Steven, bukh
I studied what i could for 2 weeks and understood little, got lost almost totally but also understood that most books on whatever start with the same- real vector spaces , and then go on developing the ideas. Most who write today about quaternions or not seem not to have studied Hamiltons Elements of quaternions since he gives there much more than just vectors to be used for 3D rotations in visualization. He develops quaternion mathematics- functions, series, calculus, links to geometry, etc. It is not mentioned anywhere in new books even about quaternions. Even first explanator of quaternions Tait did not use most of that. But that is another story to be looked at.
Briefly, if we compare number types geometrically with straight line elements:
Scalars are length of line, or ordinary numbers. They also may measure the translational motion of one end into other.
Vectors are directed ordinary numbers, obtain by inverse summation : A-B . e.g 7-5 is a directed number 2 and is a vector. While -2 is the same vector in opposite direction. Normal imaginary unit have been related to these.
Than , by same logic we must (?) have numbers that are obtained by inverse multiplication, i.e. division. It seems quaternions are these, as Grassmann or Clifford defined his algebra from existence multiplicative inverse, of which there are only 4 known. For some reason, it involves imaginary space of 3D and is related to rotations in space. What is rotated in imaginary space around origin is ordinary imaginary number I- point on imaginary 2-sphere- which is the same as I for normal complex numbers, but generally not the same as pure quaternion axis i,j,k - none of them. x^2+1=0 has infinitely many solutions- +- I -is -antipodal points on this imaginary 2-sphere in pure quaternion space.
Then, next we may have numbers obtained by inverse exponentiation, or log. They may be twisted (?) numbers. Or what else could we do to an imaginary line segment (I have not figured out how and why) . e.g. Because octonions are non-associative, as is not exponentiation usually ( x^y is not y^x usually). Twisted numbers may oscillate?
Then, we move to 4th operation, or tetration. and think about numbers obtained by inverse tetration. What numbers we may obtain by inverse tetration? The inversion of infinite tetration is self root. Self root may denote looping numbers. But for self root to be inverse of tetration, we need infinite tetrations. In the finite tetrations, we get inverses as so called nth - superroots (hyperroots) and superlogarithms. We may need another set of numbers or we may be able to survive with combinations of existing. Basically what we can add to number as imaginary line after translating, directing, rotating, twisting it is what? It may be further internal twists of the imaginary line? if that is related to sedenions, what could it mean geometrically and why it is not anymore divisor algebra?
The reason I mention this is that it seems to me that numbers have internal structure, which may well be imaginary since we have no idea about it when we look at number line or measure lengths relative to scales set by other lengths when oriented parallel to what we want to measure. This structure only reveals itself under proper operations. So number line looks (perhaps) as a rope which has internal degrees of freedom, brought out under fast enough operations applied enough or proper amount of times. This imaginary structure at least at integer operation numbers may be related to imaginary numbers and their geometric developments into quaternions, octonions, sedenions etc. ...
I studied what i could for 2 weeks and understood little, got lost almost totally but also understood that most books on whatever start with the same- real vector spaces , and then go on developing the ideas. Most who write today about quaternions or not seem not to have studied Hamiltons Elements of quaternions since he gives there much more than just vectors to be used for 3D rotations in visualization. He develops quaternion mathematics- functions, series, calculus, links to geometry, etc. It is not mentioned anywhere in new books even about quaternions. Even first explanator of quaternions Tait did not use most of that. But that is another story to be looked at.
Briefly, if we compare number types geometrically with straight line elements:
Scalars are length of line, or ordinary numbers. They also may measure the translational motion of one end into other.
Vectors are directed ordinary numbers, obtain by inverse summation : A-B . e.g 7-5 is a directed number 2 and is a vector. While -2 is the same vector in opposite direction. Normal imaginary unit have been related to these.
Than , by same logic we must (?) have numbers that are obtained by inverse multiplication, i.e. division. It seems quaternions are these, as Grassmann or Clifford defined his algebra from existence multiplicative inverse, of which there are only 4 known. For some reason, it involves imaginary space of 3D and is related to rotations in space. What is rotated in imaginary space around origin is ordinary imaginary number I- point on imaginary 2-sphere- which is the same as I for normal complex numbers, but generally not the same as pure quaternion axis i,j,k - none of them. x^2+1=0 has infinitely many solutions- +- I -is -antipodal points on this imaginary 2-sphere in pure quaternion space.
Then, next we may have numbers obtained by inverse exponentiation, or log. They may be twisted (?) numbers. Or what else could we do to an imaginary line segment (I have not figured out how and why) . e.g. Because octonions are non-associative, as is not exponentiation usually ( x^y is not y^x usually). Twisted numbers may oscillate?
Then, we move to 4th operation, or tetration. and think about numbers obtained by inverse tetration. What numbers we may obtain by inverse tetration? The inversion of infinite tetration is self root. Self root may denote looping numbers. But for self root to be inverse of tetration, we need infinite tetrations. In the finite tetrations, we get inverses as so called nth - superroots (hyperroots) and superlogarithms. We may need another set of numbers or we may be able to survive with combinations of existing. Basically what we can add to number as imaginary line after translating, directing, rotating, twisting it is what? It may be further internal twists of the imaginary line? if that is related to sedenions, what could it mean geometrically and why it is not anymore divisor algebra?
The reason I mention this is that it seems to me that numbers have internal structure, which may well be imaginary since we have no idea about it when we look at number line or measure lengths relative to scales set by other lengths when oriented parallel to what we want to measure. This structure only reveals itself under proper operations. So number line looks (perhaps) as a rope which has internal degrees of freedom, brought out under fast enough operations applied enough or proper amount of times. This imaginary structure at least at integer operation numbers may be related to imaginary numbers and their geometric developments into quaternions, octonions, sedenions etc. ...
QUOTE (Ivars+)
Briefly, if we compare number types geometrically with straight line elements:
Scalars are length of line, or ordinary numbers. They also may measure the translational motion of one end into other.
Vectors are directed ordinary numbers, obtain by inverse summation : A-B . e.g 7-5 is a directed number 2 and is a vector. While -2 is the same vector in opposite direction. Normal imaginary unit have been related to these.
Than , by same logic we must (?) have numbers that are obtained by inverse multiplication, i.e. division. It seems quaternions are these, as Grassmann or Clifford defined his algebra from existence multiplicative inverse, of which there are only 4 known.
Consider that if we begin with integers then multiplication (with finite limits to number of iterations) yields integers, but the inverse can yield rational numbers (just like addition of positive integers only yields positive integers, whereas subtraction creates negative numbers), which can actually be interpreted as vectors, if we ignore reals.
Notice that though we interprete 1/2 as a single number in terms of reals, it's actually an incomplete division with 2 inputs, divide(1,2) or can be seen as a collection of the elements {/,1,2}, whereas the reduction of something like 4/2=2 returns to a 1-D result.
Also notice that though we consider 6/3=4/2, physically these quantities aren't necessarily identical (notice that we're making comparisons along a 1-D project of 2-D values and only effectively comparing for a common angle from the origin, but not measuring the distance from it) as 6 units divided into groups of 3 units each, though constructing 2 such groups is composed of 3 units in each group, where 4 divided into groups of 2 units leaves two groups of 2 each. So though we might say that 6/3=2 and 4/2=2, in one case the 2 can be divided by 3, whereas in the other case the 2 can be divided by 2.
So division is taking a 2 dimensional quantity and effectively only measuring the angle to the origin, with the reduced fraction being the first point at which this slope encounters, though the actual point constructing this slope can have different physical properties. (A more informationally complete representation would pair the result of a division as both the angle and the distance)
Scalars are length of line, or ordinary numbers. They also may measure the translational motion of one end into other.
Vectors are directed ordinary numbers, obtain by inverse summation : A-B . e.g 7-5 is a directed number 2 and is a vector. While -2 is the same vector in opposite direction. Normal imaginary unit have been related to these.
Than , by same logic we must (?) have numbers that are obtained by inverse multiplication, i.e. division. It seems quaternions are these, as Grassmann or Clifford defined his algebra from existence multiplicative inverse, of which there are only 4 known.
Consider that if we begin with integers then multiplication (with finite limits to number of iterations) yields integers, but the inverse can yield rational numbers (just like addition of positive integers only yields positive integers, whereas subtraction creates negative numbers), which can actually be interpreted as vectors, if we ignore reals.
Notice that though we interprete 1/2 as a single number in terms of reals, it's actually an incomplete division with 2 inputs, divide(1,2) or can be seen as a collection of the elements {/,1,2}, whereas the reduction of something like 4/2=2 returns to a 1-D result.
Also notice that though we consider 6/3=4/2, physically these quantities aren't necessarily identical (notice that we're making comparisons along a 1-D project of 2-D values and only effectively comparing for a common angle from the origin, but not measuring the distance from it) as 6 units divided into groups of 3 units each, though constructing 2 such groups is composed of 3 units in each group, where 4 divided into groups of 2 units leaves two groups of 2 each. So though we might say that 6/3=2 and 4/2=2, in one case the 2 can be divided by 3, whereas in the other case the 2 can be divided by 2.
So division is taking a 2 dimensional quantity and effectively only measuring the angle to the origin, with the reduced fraction being the first point at which this slope encounters, though the actual point constructing this slope can have different physical properties. (A more informationally complete representation would pair the result of a division as both the angle and the distance)
QUOTE (Ivars+)
For some reason, it involves imaginary space of 3D and is related to rotations in space. What is rotated in imaginary space around origin is ordinary imaginary number I- point on imaginary 2-sphere- which is the same as I for normal complex numbers, but generally not the same as pure quaternion axis i,j,k - none of them. x^2+1=0 has infinitely many solutions- +- I -is -antipodal points on this imaginary 2-sphere in pure quaternion space.
Sounds similar to the problem with rational numbers as well in that you have an infinite number of possible solutions for n and m, if n/m=j/k as we have 1/2=2/4=3/6=4/8=...
It's interesting to consider how this could affect the inverse, multiplication, which could be seen similar to the identity n*(m/m)=n, which doesn't appear to have any similar problems with quantization but notice that for multiplication of integers n and m where m>1, we have n*m>n/m and we can see an equivalent compression ratio between these two spaces as one grows larger by a factor of m and the other shirnks by a factor of m and so multiplication allows for a growth of the representation space, whereas division shrinks it and effectively leaves some information regarding the division undetermined (for a division we have a potentially infinite number of multiples of the inputs that could describe the possible points in the "input space" or domain, whereas with multiplication there are fewer, though considering something like 4*6=2*12, there's still information that can be lost - of course, whether or not any of this is relevant depends upon the specific traits a result is intended to describe, and it would not inherently describe anything beyond those results, similar to the fact that taking a limit in calculus removes information regarding infinitesimals).
Sounds similar to the problem with rational numbers as well in that you have an infinite number of possible solutions for n and m, if n/m=j/k as we have 1/2=2/4=3/6=4/8=...
It's interesting to consider how this could affect the inverse, multiplication, which could be seen similar to the identity n*(m/m)=n, which doesn't appear to have any similar problems with quantization but notice that for multiplication of integers n and m where m>1, we have n*m>n/m and we can see an equivalent compression ratio between these two spaces as one grows larger by a factor of m and the other shirnks by a factor of m and so multiplication allows for a growth of the representation space, whereas division shrinks it and effectively leaves some information regarding the division undetermined (for a division we have a potentially infinite number of multiples of the inputs that could describe the possible points in the "input space" or domain, whereas with multiplication there are fewer, though considering something like 4*6=2*12, there's still information that can be lost - of course, whether or not any of this is relevant depends upon the specific traits a result is intended to describe, and it would not inherently describe anything beyond those results, similar to the fact that taking a limit in calculus removes information regarding infinitesimals).
QUOTE (Ivars+)
Then, next we may have numbers obtained by inverse exponentiation, or log.
Again, something to look at is the equivalent compression ratio of spaces, or more accurately how the relative non-linearities in the slope of each interact - if a segment of a slope of a function is unity, then it's inverse can potentially be constructed without information loss (at least at that point), whereas for a function whose slope is less than 1, you have a compression of "space" and for quantized spaces that implies multiple values of the input can be mapped to multiple values of the output and the reverse mapping isn't necessarily possible without appending additional information, for a slope approaching zero, the amount of information required to invert the operation (which is required to "prove" it did a specific mapping) could grow potentially infinitely larger than the range of the function (this could be a mechanism by which the addition of a complex dimension could appear to arise).
Notice that though we might imagine space to be compressible, if we assume such a compression of a slope, or distance in space is possible and treat the new space as if it had properties identical to the original space, then we've effectively constructed something that can compress everything to a point (if we look at the typical interpretation taken by many in calculus), yet supposedly the linear properties of space that constructed it should still be present, such as, for a>0 and b>0 then a*b>0. This is a paradox in calculus and you can interprete it in the context of a compress of space being required to forever retain a linear ordering (which would typically quite naturally be assumed to exist in an alternate dimension, but it's actually derived from the requirement that in order to proof something specific has been compressed to a point, information still needs to be retained about specifically what dimension or linear ordering of elements, was placed into that point, otherwise nothing specific at all created that point ... here's another way of looking at the problem/issue - if you have a collection of objects, defined by some property, compressed to a point in space, then the spacial locations of these objects, defined by this property, need to become identical. If the objects are defined by a volume of space, for which not all positions are identical, then in order to retain an ability for those objects to remain at the same point after compression, they can no longer be defined by a volume of space, as there is no longer any volume to define them, and this paradox implies that the objects would need to possess an attribute not associated with a range of spacial positions, but instead with an attribute that would be retained as to whether or not they would later exist as a member of such a point feature. The alternative is that such objects could never be compressed to a point, but that a(n observer) scaling of that space could be altered and make those objects become indistiguishable in terms of spacial position).
Again, something to look at is the equivalent compression ratio of spaces, or more accurately how the relative non-linearities in the slope of each interact - if a segment of a slope of a function is unity, then it's inverse can potentially be constructed without information loss (at least at that point), whereas for a function whose slope is less than 1, you have a compression of "space" and for quantized spaces that implies multiple values of the input can be mapped to multiple values of the output and the reverse mapping isn't necessarily possible without appending additional information, for a slope approaching zero, the amount of information required to invert the operation (which is required to "prove" it did a specific mapping) could grow potentially infinitely larger than the range of the function (this could be a mechanism by which the addition of a complex dimension could appear to arise).
Notice that though we might imagine space to be compressible, if we assume such a compression of a slope, or distance in space is possible and treat the new space as if it had properties identical to the original space, then we've effectively constructed something that can compress everything to a point (if we look at the typical interpretation taken by many in calculus), yet supposedly the linear properties of space that constructed it should still be present, such as, for a>0 and b>0 then a*b>0. This is a paradox in calculus and you can interprete it in the context of a compress of space being required to forever retain a linear ordering (which would typically quite naturally be assumed to exist in an alternate dimension, but it's actually derived from the requirement that in order to proof something specific has been compressed to a point, information still needs to be retained about specifically what dimension or linear ordering of elements, was placed into that point, otherwise nothing specific at all created that point ... here's another way of looking at the problem/issue - if you have a collection of objects, defined by some property, compressed to a point in space, then the spacial locations of these objects, defined by this property, need to become identical. If the objects are defined by a volume of space, for which not all positions are identical, then in order to retain an ability for those objects to remain at the same point after compression, they can no longer be defined by a volume of space, as there is no longer any volume to define them, and this paradox implies that the objects would need to possess an attribute not associated with a range of spacial positions, but instead with an attribute that would be retained as to whether or not they would later exist as a member of such a point feature. The alternative is that such objects could never be compressed to a point, but that a(n observer) scaling of that space could be altered and make those objects become indistiguishable in terms of spacial position).
QUOTE (Ivars+)
They may be twisted (?) numbers. Or what else could we do to an imaginary line segment (I have not figured out how and why) . e.g. Because octonions are non-associative, as is not exponentiation usually ( x^y is not y^x usually). Twisted numbers may oscillate?
If they aren't associative, this implies the space is non-linear and we can't treat the rotations in an additive fashion (consider that rotations are defined by cycles that are not infinitely extensible - physical objects don't typically appear to have distinguishable properties dependent upon the number of complete cycles of rotations they've undergone, yet such a property must exist, even if it's only subjective, in order that we could count such rotations. Basically, if you can observe something being rotated 360 degrees, the rotation is only determined by a subset of the total information - obviously some of this information could be interpreted as a component of time).
If you think that something could potentially oscillate, then it would take to perform this. If the two or more states are actually indeterminant and effectively identical in some perceived attribute, then the states would simply appear merged to an identical state (and for my models, I don't think quantity if relevant to this state - you could have any number of states compressed to the appearance of a single state and the observable feature should simple be that the state is true/exists etc. If it's detected as multiple states in some manner, then there's an element of time counting repetitions of the same state and the same state is being observed within different contexts).
If they aren't associative, this implies the space is non-linear and we can't treat the rotations in an additive fashion (consider that rotations are defined by cycles that are not infinitely extensible - physical objects don't typically appear to have distinguishable properties dependent upon the number of complete cycles of rotations they've undergone, yet such a property must exist, even if it's only subjective, in order that we could count such rotations. Basically, if you can observe something being rotated 360 degrees, the rotation is only determined by a subset of the total information - obviously some of this information could be interpreted as a component of time).
If you think that something could potentially oscillate, then it would take to perform this. If the two or more states are actually indeterminant and effectively identical in some perceived attribute, then the states would simply appear merged to an identical state (and for my models, I don't think quantity if relevant to this state - you could have any number of states compressed to the appearance of a single state and the observable feature should simple be that the state is true/exists etc. If it's detected as multiple states in some manner, then there's an element of time counting repetitions of the same state and the same state is being observed within different contexts).
QUOTE (Ivars+)
Then, we move to 4th operation, or tetration. and think about numbers obtained by inverse tetration.
To me, the most likely relevance of fast growing functions would be for a space with evolutionary properties in which a class of operation that is capable of effectively reproducing itself faster (which requires a manner to measure equivalent time between such classes of operations), then an operation that "reproduced" faster would be the most prevalent (though there are also issues regarding information theory as to what properties of such a space could be informative - a vast crystalline structure with identical features would only appear capable of conveying information regarding the width in various dimensions and information regarding the "seed" properties of it, but not much else, unless additional information was injected into this structure somehow).
To me, the most likely relevance of fast growing functions would be for a space with evolutionary properties in which a class of operation that is capable of effectively reproducing itself faster (which requires a manner to measure equivalent time between such classes of operations), then an operation that "reproduced" faster would be the most prevalent (though there are also issues regarding information theory as to what properties of such a space could be informative - a vast crystalline structure with identical features would only appear capable of conveying information regarding the width in various dimensions and information regarding the "seed" properties of it, but not much else, unless additional information was injected into this structure somehow).
QUOTE (Ivars+)
What numbers we may obtain by inverse tetration? The inversion of infinite tetration is self root. Self root may denote looping numbers. But for self root to be inverse of tetration, we need infinite tetrations.
It's interesting that you mention this because I was trying to extrapole some upon how learning/intelligence could operate and how ever more complex concepts could be built upon those "roots" (pun somewhat intended :)) and the issue for me is how can you wrap up all these layers into a single recursive object that could extrapolated out to any potential complexity of computation ... if every new "layer" is described in terms of new or novel features present in the properties of the previous transformation, then you're effectively stuck counting finite numbers, but if it can be wrapped into a single entirely self contained structure, then it could grow infinitely complex with a finite description - seems like there's a paradox there though and I'm not certain if it's possible to have a finite description of something like that (though it's interesting to consider that some descriptions that use functions that grow as fast as possible could become somewhat optimal in describing a maximum number of properties within some finite description .... I'm just pondering stuff out loud here and I recognize these are very abstract ideas)
It's interesting that you mention this because I was trying to extrapole some upon how learning/intelligence could operate and how ever more complex concepts could be built upon those "roots" (pun somewhat intended :)) and the issue for me is how can you wrap up all these layers into a single recursive object that could extrapolated out to any potential complexity of computation ... if every new "layer" is described in terms of new or novel features present in the properties of the previous transformation, then you're effectively stuck counting finite numbers, but if it can be wrapped into a single entirely self contained structure, then it could grow infinitely complex with a finite description - seems like there's a paradox there though and I'm not certain if it's possible to have a finite description of something like that (though it's interesting to consider that some descriptions that use functions that grow as fast as possible could become somewhat optimal in describing a maximum number of properties within some finite description .... I'm just pondering stuff out loud here and I recognize these are very abstract ideas)
QUOTE (Ivars+)
In the finite tetrations, we get inverses as so called nth - superroots (hyperroots) and superlogarithms. We may need another set of numbers or we may be able to survive with combinations of existing.
From the track record, you'll likely need a new set of numbers :D There appears to be an issue here with non-linearity (all these functions are non-linear) and information theory - effectively every one of these adds a non-linearity that needs to be described as it effectively provides a manner to interleave elements within a dimension.
If we look at all possible computations that result in a single dimensional result, we have a resulting number line that for some specific set of inputs a<b<c<d<... results in various combinations of them being output, such as b<c<a<d<.... This really all you need to describe any function in 1 dimension and we can extend upon this concept to more than 1 dimension as well in that the coordinates of all the outputs in a single dimension need to be similarly orderable (notice that for any computation in which we have transformed information that lies precisely at the same position within a dimension, we potentially have an indeterminant inverse mapping, if multiple input points can all be mapped to the same output - effectively we've changed the input space in that case, which can be seen as a separate issue).
Now notice as well that every coordinate of an output object in a dimension can be described as a function of it's position in its input positions for each dimension (if you don't need to input some input dimensions to generate an output dimension then these spaces can potentially be detached and the descriptions of the dimensionalities of the relationship between these two spaces can become arbitrary as it's truly the interactions that define whether or not 2 dimensions exist within the same space - so that's an interesting observation that spaces are defined by interactions of dimensions and not simply the assumed existance of a dimension, which could be seen similar to the idea of parallel, non-interacting universes).
So, if for example, an x coordinate of a space can only be described using all of, for example, x, y and z dimensions in some other space, then we can see those spaces interacting via. a non-linearity and constructing an object with features inherited by the properties of each of the input dimensions, notice that if we do not have a non-linear combination of dimensions, but instead simply a linear combination of the inputs, then the input dimensions are all symettrically related to the output and indistinguishable from each other, so one potentially significant feature of functions with higher order features is that they could potentially represent orthogonal forms of non-linear relationships that could be interpreted as existing within different physical dimensions. (For example, x, 1/x and x^2 are all non-linear relative to each other, but a dimension constructed with properties of x^2 could appear to be a non-linear interaction of two linear spaces in x as we have x^2=x*x, or similarly if we make observations from a dimension related to x and observe a dimension containing only a constant, the properties can appear to mimic 1/x as we're making measurements of a constant c, using units of x and seeing relationships following c/x. We don't necessarily need a large number of different non-linear "seed" operations to be considered to all be orthogonal to each other if we simply consider that such function mappings along a dimension are only significant, from the perspective of a quantized space, when a linear ordering is altered (or a chronological sequence of them, if we want to interprete this as a space through which motion occurs)).
So if we have some values along a dimension/axis such that a<b<c there are only, for example, six possible unique "non-linear" functions that could be generated to derive various combinations of b<a<c, a<c<b, etc. You can actually reduce the complexity of this somewhat by breaking all these combinations up into a few "kernel" operations, or similar to factoring as either a "swap" opera
From the track record, you'll likely need a new set of numbers :D There appears to be an issue here with non-linearity (all these functions are non-linear) and information theory - effectively every one of these adds a non-linearity that needs to be described as it effectively provides a manner to interleave elements within a dimension.
If we look at all possible computations that result in a single dimensional result, we have a resulting number line that for some specific set of inputs a<b<c<d<... results in various combinations of them being output, such as b<c<a<d<.... This really all you need to describe any function in 1 dimension and we can extend upon this concept to more than 1 dimension as well in that the coordinates of all the outputs in a single dimension need to be similarly orderable (notice that for any computation in which we have transformed information that lies precisely at the same position within a dimension, we potentially have an indeterminant inverse mapping, if multiple input points can all be mapped to the same output - effectively we've changed the input space in that case, which can be seen as a separate issue).
Now notice as well that every coordinate of an output object in a dimension can be described as a function of it's position in its input positions for each dimension (if you don't need to input some input dimensions to generate an output dimension then these spaces can potentially be detached and the descriptions of the dimensionalities of the relationship between these two spaces can become arbitrary as it's truly the interactions that define whether or not 2 dimensions exist within the same space - so that's an interesting observation that spaces are defined by interactions of dimensions and not simply the assumed existance of a dimension, which could be seen similar to the idea of parallel, non-interacting universes).
So, if for example, an x coordinate of a space can only be described using all of, for example, x, y and z dimensions in some other space, then we can see those spaces interacting via. a non-linearity and constructing an object with features inherited by the properties of each of the input dimensions, notice that if we do not have a non-linear combination of dimensions, but instead simply a linear combination of the inputs, then the input dimensions are all symettrically related to the output and indistinguishable from each other, so one potentially significant feature of functions with higher order features is that they could potentially represent orthogonal forms of non-linear relationships that could be interpreted as existing within different physical dimensions. (For example, x, 1/x and x^2 are all non-linear relative to each other, but a dimension constructed with properties of x^2 could appear to be a non-linear interaction of two linear spaces in x as we have x^2=x*x, or similarly if we make observations from a dimension related to x and observe a dimension containing only a constant, the properties can appear to mimic 1/x as we're making measurements of a constant c, using units of x and seeing relationships following c/x. We don't necessarily need a large number of different non-linear "seed" operations to be considered to all be orthogonal to each other if we simply consider that such function mappings along a dimension are only significant, from the perspective of a quantized space, when a linear ordering is altered (or a chronological sequence of them, if we want to interprete this as a space through which motion occurs)).
So if we have some values along a dimension/axis such that a<b<c there are only, for example, six possible unique "non-linear" functions that could be generated to derive various combinations of b<a<c, a<c<b, etc. You can actually reduce the complexity of this somewhat by breaking all these combinations up into a few "kernel" operations, or similar to factoring as either a "swap" opera