Deeper than primes

[doronshadmi]

EDIT:

Furthermore, Koch fractal can be found by using only non-locality (a single bended line):

Try to use only points (localities) in order to define Koch fractal, and you get no more than a single point (the line's bended levels are collapsed into a single point).

Koch fractal (finite or not) actually can be defined by using only a single line (non-locality), which is actually a 1-dim element over finite of infinitely many bended (scale) levels, which is not composed by segments AND no 0-dim is found along it (we do not need a point nor a segment in order to bend a line).


n= 1 to ∞

k = 0 to n-1

In general, we do not need k-dim elements or a collection of n-dim elements in order to define a bended n-dim element.

In that case the Koch fractal is some representation of the non-local atomic aspect, which is not the same as a Koch fractal that is defined by collections of n-dim elements or n-dim AND k-dim elements.

 

[zooterkin]

Furthermore, Koch fractal can be found by using only non-locality (a single banded line):

[qimg]http://farm3.static.flickr.com/2707/4408536446_ba7ba9f3d3_o.jpg [/qimg]

Try to use only points (localities) in order to define Koch fractal, and you get no more than a single point (the line's bended levels are collapsed into a single point).

On the contrary Koch fractal (finite or not) can be defined by using only a single line (non-locality), which is actually a 1-dim element over finite of infinitely many bended (scale) levels, which is not composed by segments AND no 0-dim is found along it (we do not need a point nor a segment in order to band a line).


n= 1 to ∞

k = 0 to n-1

In general, we do not need k-dim elements or a collection of n-dim elements in order to define a bended n-dim element.

I don't know what word you mean, but I think it's neither band nor bend, certainly not banded or bended.

You also do need to define where to apply the rules to generate the fractal pattern, which means you do define the point at which to do so.

 

[doronshadmi]

I don't know what word you mean, but I think it's neither band nor bend, certainly not banded or bended.

You also do need to define where to apply the rules to generate the fractal pattern, which means you do define the point at which to do so.

You are right, thank you, it has to be "bend".

Please refresh your screen and read it again.

 

[zooterkin]

You are right, thank you, it has to be "bend".

Please refresh your screen and read it again.

At least it is consistently wrong now.

 

[doronshadmi]

At least it is consistently wrong now.

It is "consistently wrong" for any one who closed under the concopt of Collection.

is possible (please ignore the illusion of movement) exactly because an infinite Koch fractal (composed or atomic) is based on ... +1+1+1+... form (of infinitely many scale levels) that has no sum.

 

[zooterkin]

It is "consistently wrong" for any one who closed under the concopt of Collection.

Can I quote you on that?

 

[doronshadmi]

sympathic,

 

[doronshadmi]

Can I quote you on that?

You do not need me, this is exactly your reasoning.

 

[The Man]

Wrong.

Complete refers to a non-composed thing (atomic) , where inclusion or exclusion refer to composed things, which are incomplete w.r.t to the non-composed.

Complete can refer to your “atomic”, but it also refers “to composed things” that are, well, complete (nothing is missing).

Again your notion does not distinguish between the complete (the atomic) and the incomplete (the non-atomic) where this distinction is based on the qualitative difference between the atomic and the composed.

Sure it does, it just doesn’t assume complete must be synonymous with your notion of “atomic”. Complete and incomplete are antonyms just as your “atomic” and “non-atomic” are. However both being antonyms does not make complete synonymous with your "atomic" or incomplete synonymous with your "non-atomic". Again the set {-15, 5} is complete as that set, but as the set of integers it is incomplete as most integers are missing from that set.

A point or a line are atomic exactly because no one of them is made by the other. The line is the simplest example of a non-local atomic quality and a point is the simplest example of a local atomic quality.

Again I’ve got no problems with your notion of a point being “atomic” (indivisible), but a line (or line segment) is divisible as line segments and that is the whole basis of the “Koch curve”.

By your quantitative-only notion you count them and conclude that the thing that includes them is non-atomic.

Nope as expressed above.

You are right about that, because that thing is called a collection, and a collection is a non-atomic thing.

Right about what? You ascribe some “quantitative-only notion” to me and call it “right”?

Since your abstraction ability can deal only with collections, you are unable to understand that the Non-local and Local atomic aspects are the minimal manifestations of the atomic state itself, which is beyond any manifestation (it is not Non-local AND not Local, or any other property that is used to define it).

Wait, so your “atomic state itself” is “beyond any manifestation” “that is used to define it” meaning it need not be “atomic” or a “state” as those are just your ‘manifestations’ that you like to think you use to “define it”. So thanks Doron for admitting that your “atomic state itself” is not bound by your notion of “atomic” defining what you think ‘manifests’ an “atomic state”.

Again, there is no such a thing like the entire collection of infinitely many things, exactly because the cardinality of infinitely many things (the 1+1+1+… form) has no sum, and it does not matter if things become smaller of bigger (in both cases biggest or smallest, do not hold).

You are still confusing (again I think deliberately) the number of members in a set with a sum of those members.

Simply wrong.

Nope, you’re simply unable to understand what was written.

If the sum between the limit and the partial sum equals to the partial sum you do not deal anymore with converge series, because you add two equal sizes to each other in order to reach the limit.

It is the difference between the limit and the partial sum that is itself a self similar infinite convergent series and in the case of the ½ common ratio series that remaining self similar convergent infinite series has a sum equal to the last element of the partial sum which is the difference between the partial sum and the limit (as already explained to you before).

Converge series do no have any two equal sizes along it in order to be considered as a converge series. The sizes of converge series must be different from each other no matter how infinitely many scales are involved.

Try actually reading the previous posts instead of just relating again your “two equal sizes” requirement that is just an assumption you have based on a finite number of elements.

Archimedes did not understand the real nature self similarity over infinitely many scale levels. This self-similarity is defined only if being smaller is an invariant property upon infinitely many scale levels, which permanently prevent the value of a given limit.

Obviously he understood it better then you.

This simple notion can easily be proven by researching, for example, Koch fractal.

Koch fractal is infinite only if there are infinitely many scale levels, where each scale level is represented by a given segment that is smaller than any arbitrary previous segment, for example:

[qimg]http://farm5.static.flickr.com/4001/4407014804_46d505b681_o.jpg[/qimg]

The Koch fractal is an example of infinite extrapolation\interpolation of scale levels of the form …+1+1+1+… that has no sum.


The self similarity over scales clearly shown also by the following diagram, where the values 1,2,4,8,16,32,… etc. are not reached, ad infinituum:

[qimg]http://farm5.static.flickr.com/4062/4405947817_0146693fb4_o.jpg[/qimg]

Again research is not just you imposing your assumptions of “no sum” on an infinite convergent series based on your (apparently deliberate) confusion between the number of elements in an infinite set (which is divergent) and the sum of those elements (that can be convergent) and your “two equal sizes” requirement based on the sum of a finite number of elements.

EDIT:

Furthermore, Koch fractal can be found by using only non-locality (a single bended line):

[qimg]http://farm3.static.flickr.com/2707/4408536446_ba7ba9f3d3_o.jpg [/qimg]

Try to use only points (localities) in order to define Koch fractal, and you get no more than a single point (the line's bended levels are collapsed into a single point).

Koch fractal (finite or not) actually can be defined by using only a single line (non-locality), which is actually a 1-dim element over finite of infinitely many bended (scale) levels, which is not composed by segments AND no 0-dim is found along it (we do not need a point nor a segment in order to bend a line).


n= 1 to ∞

k = 0 to n-1

In general, we do not need k-dim elements or a collection of n-dim elements in order to define a bended n-dim element.

In that case the Koch fractal is some representation of the non-local atomic aspect, which is not the same as a Koch fractal that is defined by collections of n-dim elements or n-dim AND k-dim elements.

http://en.wikipedia.org/wiki/Koch_fractal

Construction
One can imagine that the Koch curve was created by starting with an equilateral triangle, then recursively altering each line segment as follows:
1. divide the line segment into three segments of equal length.
2. draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.
3. remove the line segment that is the base of the triangle from step 2.
The Koch snowflake/star is generated using the same recursive process but starting with an equilateral triangle rather than a line segment. After doing this once for the Koch snowflake, the result is a shape similar to the Star of David.
The Koch curve is the limit approached as the above steps are followed over and over again.

Oh look “divide the line segment” which is defined by points into three segments, also defined by points. A “triangle that has the middle segment from step 1 as its base” both defined by points then “remove the line segment that is the base of the triangle from step 2” which is defined by the end points of that segment.

Keep going Doron and let us know when you find that line segment not defined by points that you can’t divide into “three segments of equal length”. If you do then you’re simply not talking about a “Koch curve”.

 

[doronshadmi]

Wait, so your “atomic state itself” is “beyond any manifestation” “that is used to define it” meaning it need not be “atomic” or a “state” as those are just your ‘manifestations’ that you like to think you use to “define it”. So thanks Doron for admitting that your “atomic state itself” is not bound by your notion of “atomic” defining what you think ‘manifests’ an “atomic state”.

Thank you for demonstrating again how you are unable to directly get that is beyond any definition (now you will say that "that is beyond any definition" is this thing, but it is not, exactly as saying or thinking "silence" is not the thing itself).

Again the set {-15, 5} is complete as that set, but as the set of integers it is incomplete as most integers are missing from that set.

I have noticed that you did not write "the set of ALL integers", which is a good start in order to get OM.

There is no such thing like infinite and complete class.

You still do not understand the following notion about completeness and incompleteness, for example:

A pointless circle is totally complete because is has no beginning AND no end (it has an atomic quality).

A circle with a single point along it defines a segment, and a closed segment is less complete than a pointless circle (completeness is a matter of Quality and not a matter of Quantity).

When more and more segments are used in order to define a circle, it becomes less and less complete, but as long as the amount of segments is finite, their added values have an accurate result known as Sum.

If an infinite amount of segments is used in order to define a circle, then also the accuracy of the result is lost and we do not have a Sum.

Since you notion is still closed under collections and classes, you are unable to get things from qualitative view and unable to understand the generalization of …+1+1+1+… form that stands at the basis of any infinite collection of distinct things (where this form is not limited to any particular class and therefore "missing" has no meaning at that abstraction level), which is beyond your partial and closed under collections and classes level).

You are still confusing (again I think deliberately) the number of members in a set with a sum of those members.

Your abstraction is simply still closed under collections and classes.

Try actually reading the previous posts instead of just relating again your “two equal sizes” requirement that is just an assumption you have based on a finite number of elements.

Simply wrong.

Two equal sizes are exactly two equal sizes, whether a size is finite or not.

If you deal with two equal sizes, say bye bye to the concept of Convergence.

Keep going Doron and let us know when you find that line segment not defined by points that you can’t divide into “three segments of equal length”. If you do then you’re simply not talking about a “Koch curve”.

Again The Man, keep going and let us know when you are able to get thing beyond the reasoning that is closed under collections and classes.

Some hint: You will not find it in Wikipedia, which is a tool that includes only already agreed knowledge about some given subject.

Again, I have noticed that you wrote "the set of integers" instead of "the set of ALL integers" which is a good start in order to get OM.

 

[doronshadmi]

Again I’ve got no problems with your notion of a point being “atomic” (indivisible), but a line (or line segment) is divisible as line segments and that is the whole basis of the “Koch curve”.

The whole basis of the “Koch curve” is exactly 1-dim that is bended over finite or infinitely many distinct scale levels, which are based on 1+1+1 (finite) or 1+1+1… (infinite) forms.

Since you are unable to get the qualitative difference between the atomic and the collection, let us do it in 3D.

You have a closed 3D string, with no 3D bead along it.

In that case we say that the beadless closed 3D string is the atomic base ground for any collection of 3D beads along it.

Any amount of 3D beads along the closed 3D string does not divide it into many 3D strings.

All they do is to measure it by giving a bead an agreed value, and then we add these values to each other and define some size to that closed string.

Change the size of the beads, and you get different results, which are related to the same closed string.

But in any given measurement, the measured string is not divided by the beads along it, and stays a one and only one 3D closed atomic string under any given measurement.

The same notion holds aslo if 2D or 1D are used.

 

[doronshadmi]

You are still confusing (again I think deliberately) the number of members in a set with a sum of those members.

No The Man, you simply do not get the general form that stands in the basis of (x/y)*(y/x).

For example if the considered case is (1/2^x), the general form is the result of the addition of the values, where each value is the result of (1/2^x)*(2^x/1), where (1/2^x) is the converges aspect and (2^x/1) is the diverges aspect of 1+1+1+... general form.

Again, your asymmetric view of this subject prevents from you to get the general form that stands at both converges or diverges series.

Maybe you can learn something form the great mathematician Poincaré (http://en.wikipedia.org/wiki/Henri_Poincaré), which understood geometric series better than Archimedes:

Poincaré was dismayed by Georg Cantor's theory of transfinite numbers, and referred to it as a "disease" from which mathematics would eventually be cured.

http://philmat.oxfordjournals.org/cgi/content/abstract/2/3/202

Poincaré's Conception of the Objectivity of Mathematics (Philosophia Mathematica 1994 2(3):202-227)

JANET FOLINA*

*Department of Philosophy, Macalester College, St Paul, Minnesota 55105, U. S. A.


There is a basic division in the philosophy of mathematics between realist, ‘platonist’ theories and anti-realist ‘constructivist’ theories. Platonism explains how mathematical truth is strongly objective, but it does this at the cost of invoking mind-independent mathematical objects. In contrast, constructivism avoids mind-independent mathematical objects, but the cost tends to be a weakened conception of mathematical truth. Neither alternative seems ideal.

The purpose of this paper is to show that in the philosophical writings of Henri Poincaré there is a coherent argument for an interesting position between the two traditional poles in the philosophy of mathematics. Relying on a semi-Kantian framework, Poincaré combines an epistemological and metaphysical constructivism with a more realist account of the nature of mathematical truth.

 

[zooterkin]

Still waiting.

You seem to have avoided this question:


Now, exactly what has ever been accomplished with OM?

 

[doronshadmi]

Still waiting.

http://www.internationalskeptics.com/forums/showpost.php?p=5674970&postcount=8882

 

[zooterkin]

http://www.internationalskeptics.com/forums/showpost.php?p=5674970&postcount=8882

Try again, that doesn't answer the question I'm asking.

What has been achieved with OM? (Not, what do you think it might do.)

 

[dafydd]

Try again, that doesn't answer the question I'm asking.

What has been achieved with OM? (Not, what do you think it might do.)

The only achievement so far is that it's given us all some good laughs.

 

[Apathia]

Try again, that doesn't answer the question I'm asking.

What has been achieved with OM? (Not, what do you think it might do.)

All of Doron's answers to this question so far have led me to conclude that the achievements or practical applications of OM (except for the conventional strictly quantitative aspects of number) are non-local in nature, qualitative and inspirational.

Though, I seem to remember Doron saying that a positive contribution was liberation from the evils of mathematics as it has developed since Euclid strayed from the path.

Another answer he has given is that you, yourself are an example of ./__ Interaction. An amazing achievement!

(Bottom line: what we're asking of him is some "local only," "deductive," profane expectation that doesn't pertain to OM.
Or the question we are asking is incoprehensible to him.)

 

[The Man]

Thank you for demonstrating again how you are unable to directly get that is beyond any definition (now you will say that "that is beyond any definition" is this thing, but it is not, exactly as saying or thinking "silence" is not the thing itself).

No I will simply say "that is beyond any definition" is just a cop out for you so you can pretend for things to mean anything you want them to.

I have noticed that you did not write "the set of ALL integers", which is a good start in order to get OM.

A mere oversight on my part as “the set of ALL integers” is clearly what I had intended.

There is no such thing like infinite and complete class.

You still do not understand the following notion about completeness and incompleteness, for example:

A pointless circle is totally complete because is has no beginning AND no end (it has an atomic quality).

A circle with a single point along it defines a segment, and a closed segment is less complete than a pointless circle (completeness is a matter of Quality and not a matter of Quantity).

When more and more segments are used in order to define a circle, it becomes less and less complete, but as long as the amount of segments is finite, their added values have an accurate result known as Sum.

If an infinite amount of segments is used in order to define a circle, then also the accuracy of the result is lost and we do not have a Sum.

Doron simply repeating your nonsense does not make it any less nonsense. If you find this nonsense makes you feel important, such that you can save our civilization from self-destruction. Then fine I’m happy for you, but don’t try to pawn it off as some new math, ethics or a combination of logic and ethics. It is just your fantasy.

Since you notion is still closed under collections and classes, you are unable to get things from qualitative view and unable to understand the generalization of …+1+1+1+… form that stands at the basis of any infinite collection of distinct things (where this form is not limited to any particular class and therefore "missing" has no meaning at that abstraction level), which is beyond your partial and closed under collections and classes level).

Again “1+1+1+…” is a divergent series not a convergent series and a “qualitative view” does not mean you simply get to deliberately confuse a divergent series with a convergent one (ok well maybe in your fantasies it does) .

Your abstraction is simply still closed under collections and classes.

Again abstraction does not mean you just making up whatever you want and deliberately confusing the number of elements with the sum of those elements (again except of course just in your fantasies it does).

Simply wrong.

So you won’t try acctualy reading the previous posts.

Two equal sizes are exactly two equal sizes, whether a size is finite or not.

If you deal with two equal sizes, say bye bye to the concept of Convergence.

Well since the sum of a convergent series is equal to a finite value, the only thing you would be saying “bye bye to” is your fantasies.

Again The Man, keep going and let us know when you are able to get thing beyond the reasoning that is closed under collections and classes.

Sure as soon as you find a line segment that can’t be divided on a “Koch curve” instead of just fantasizing about one.

Some hint: You will not find it in Wikipedia, which is a tool that includes only already agreed knowledge about some given subject.

You should try actually reading that too sometime, you will find much disagreement about a lot of issues. Just whom do you think makes all these ‘agreements’ about knowledge anyway? Since generally no one accepts or agrees with your fantasies I can understand how you need to construe the general conformity of actual knowledge with some nefarious agreement.

Again, I have noticed that you wrote "the set of integers" instead of "the set of ALL integers" which is a good start in order to get OM.

Oh so now just not writing “ALL” “is a good start in order to get OM”? I wonder what else you don’t want anyone to write in order to enforce your “get OM” doctrine?

The whole basis of the “Koch curve” is exactly 1-dim that is bended over finite or infinitely many distinct scale levels, which are based on 1+1+1 (finite) or 1+1+1… (infinite) forms.

Nope, try actually reading.

Since you are unable to get the qualitative difference between the atomic and the collection, let us do it in 3D.

You have a closed 3D string, with no 3D bead along it.

In that case we say that the beadless closed 3D string is the atomic base ground for any collection of 3D beads along it.

Any amount of 3D beads along the closed 3D string does not divide it into many 3D strings.

All they do is to measure it by giving a bead an agreed value, and then we add these values to each other and define some size to that closed string.

Change the size of the beads, and you get different results, which are related to the same closed string.

But in any given measurement, the measured string is not divided by the beads along it, and stays a one and only one 3D closed atomic string under any given measurement.

The same notion holds aslo if 2D or 1D are used.

Again simply repeating you string/bead analogy nonsense does not make it any less nonsense nor does adding the phrase “3D” to it. Who cares what you think your “string is not divided by”, line segments are still defined by points just as curves are (closed or not). Again it just shows how desperate you are to support you morbid fantasy of saving our civilization from self destruction. That you have to deliberately separate a line or curve from what define them, the points, just so you can combine them again into your “complex”.

No The Man, you simply do not get the general form that stands in the basis of (x/y)*(y/x).

For example if the considered case is (1/2^x), the general form is the result of the addition of the values, where each value is the result of (1/2^x)*(2^x/1), where (1/2^x) is the converges aspect and (2^x/1) is the diverges aspect of 1+1+1+... general form.

Again, your asymmetric view of this subject prevents from you to get the general form that stands at both converges or diverges series.

Again “1+1+1+... “ is not convergent which means it is not a “general form” of convergence. Once again “general from” does not mean you simply get to deliberately confuse a convergent series with a divergent series (again except of course just in your fantasies). Again your asymmetric fantasy view makes you think a divergent series is a “general form” of a convergent series.

Maybe you can learn something form the great mathematician Poincaré (http://en.wikipedia.org/wiki/Henri_Poincaré), which understood geometric series better than Archimedes:

I’m sure I can (and have already as have many others), I’m sure you could as well, but I doubt you actually will. What does one reference to Cantor have to do with a geometric series?

http://philmat.oxfordjournals.org/cgi/content/abstract/2/3/202

And you think this is a argument for what? You just making up whatever crap you want and often attributing it as someone else’s “understanding” as well as you deliverable confusing the number of members of a set with the sum of the members of that set in addition to your simply ignorance that a divergent series is simply not convergent (‘generally’ or otherwise).

 

[doronshadmi]

What does one reference to Cantor have to do with a geometric series?

The "sum" of an infinite collection.


In a diverges series each size is bigger than any arbitrary previous size.

In a converges series each size is smaller than any arbitrary previous size.

1+1+1+… is the general form of both cases, and it has no sum, if we deal with infinitely many sizes (diverges or converges, it does not matter).

 

[jsfisher]

The "sum" of an infinite collection.

Infinite sets don't exist in your universe, doron, remember? There can be no set of the integers (which necessarily would be all of them), there can be no set of the reals, there can be no infinite sets of any kind.