Deeper than primes

[The Man]

Nice "year in review" there Little 10 Toes.

 

[The Man]

By using Standard Math please prove that a collection of points completely covers an endless line.

We have been over this time and time again Doron. The real number line (a line without ends mind you) is completely covered by the set of real numbers. It is a simple and trivial fact that any number (or point) being on the real number line that makes it a member of the set of real numbers just as any number that would not be in the set would not be on the line. Also as has already been explained to you the fact (both the line and set being continuous) that between any two points on that line (or numbers in that set) there is another point on that line (or number in that set), ensuring no gaps. Your entire conundrum seems to be centered around your own incompleteness. In that you seem to perceive that something (specifically a set) must have an end in order to be complete. Yet seem to attribute no such limitation to a line. However both can be complete, without ends and continuous. The only incompleteness seems to be in your notions and understanding of being continually, well, endless.

 

[laca]

Nice list Little 10 Toes

But you forgot to mention he doesn't get that 0.(9) = 1 (as a side-effect he makes up a nonexistent number, 0.(0)1 ) and that a line is made up of points

 

[Little 10 Toes]

I really started to develop OM only 7 years ago, during internet dialogs both with professional and non-professional persons.

Nope. You did it 30 years ago.

For example, please show me a totally accurate location in our physical realm, or totally non-accurate location in our physical realm.

You will not find them in the physical realm, but you will find them in the abstract realm of ideas.

Ok. "Totally accurate location in our physical realm". How about N 32° 3' 30" E 34° 45' 54"? According to what I see, I can get get gasoline close by. Is this true?

Complexity is developed by using both abstract and non-abstract realms, so from this comprehensive view, any result is based on the abstract AND the non-abstract, such that the abstract notions of today can become the non-abstract technology of tomorrow.

Too bad you still haven't defined "Complexity".

As for your designs that are based on Electricity, a better understanding of electrons may be used for better developments of Electric technology, and by QM we know that an electron is like a segment that is not entirely particle (local) and not entirely wave (non-local).

Now I'm no Quantium Mechanic, but I know that an electron is not a segment. From what I remember from high school, an electron is not "not entirely partical and not entirely wave".

Furthermore, concepts like Superposition, Uncertainty, Redundancy, Randomness, Locality, Non-locality, Finite, Infinite, Complexity, Serial, Parallel, etc… are all based on a one comprehensive model of the linkage between the non-local and local aspects of a one atomic state, where an atom is both existing AND empty (of any sub-things) thing.

Just so that you know it's a question, could you please define all those terms? And since you've used the term electron, I assume that when you say atomic/atom in this paragraph, you are referring to urelement.

New post

This is an abstraction that cannot actually be found in the physical world of segments, which are not totally local (like a point) AND not totally non-local (like an edgeless line).

And you say that you understand QM, but QM supports my argument about the physical realm, and not your naïve argument about totally accurate location like 44º35’25” North by 104º42’55” West that can be found in the physical realm.

You do not understand your own arguments, The Man.

Nope. You claimed that we could not give you an accurate point in the real world. You did not mention "the world of segments". I believe the term "backfill" is currently in use when you decide to make changes in your requests.

You are the one that claims the you can show that 44º35’25” North by 104º42’55” West is an exact 0-dim location in the physical world.

So this time please do that, I am still wating.

Pull up Google maps. I did. Found the location. Found another one too!

New post

1) I show that your community uses the word "curve" with relation to a line.

2) An open endeless line is also called "a straight curve" by your own community.

Now please answer to this question:

What is the difference between an endless open line (where only a 1-dim is considered) and an open line segment?

I also see that wikipedia uses the letters "e", "u", and "r". You use those same letters. You also miss the third sentence of the wikipedia entry of "Line (geometry)". "Lines are an idealisation of such objects and have no width or height at all and are usually considered to be infinitely long." No need to continue with infinite non-finite lines or infinite non-finite length.

By your question we can clearly see that you do not distinguish between a buiding-block (endless line) and a complex (a line segment, which is a line AND two points).

But a line segment is not a line.

Since I claim that a line isn't covered completely by points, and since you support standard Mathematics, which claims the opposite, then you claim that an endless line (= line , according to you) is completly covered by points. Please try again and provide standard Mathematics' proof about that subject.

Let's do the doron-tango. Step 1: doronshadmi makes a claim. Step 2: someone else doesn't believe that claim. Step 3: doronshadmi asks for everyone else to provide proof for their standings.

 

[Little 10 Toes]

Hey laca, I was away from this portion of the website for a while. Give me a break.

 

[The Man]

Hey laca, I was away from this portion of the website for a while. Give me a break.

Well certainly not an endless break L10T, since by OM we must be able to sick a 1 (or a fraction) after that endless end. Otherwise we are reasoning only locally that will entropify the universalrific frameworktivity of complexification leading to an asymmetricon un-incompletitude in non-edgelessality.

 

[doronshadmi]

In the same vain, my answer is that it is not an atomic property. Just as insightful, just as non-constructive as yours.

Since you cannot give an appropriate answer to the general question, perhaps you can deal with specifics:

• What is the dimensionality of a circle? How did you determine that?
• What is the dimensionality of a torus? How did you determine that?
• What is the dimensionality of a helix? How did you determine that?

After dealing with those, maybe you would reconsider the original question: What makes something n-dimensional?

Please provide your answer.

 

[The Man]

Please provide your answer.

You First

This is a thread you started to discuss your notions. If you wanted to discuss current mathematical notions you should have started one on the math sub-forum.


Oh wait you did and that was closed because you could not refrain from positing the philosophical aspects of your religious like notions.

So it is now your stage Doron, be a player (or is that too local for you?).

 

[doronshadmi]

We have been over this time and time again Doron. The real number line (a line without ends mind you) is completely covered by the set of real numbers.

EDIT:

No,

An endless line is a 1-D atom that has no edges and it is the minimal form of actual infinity.

A 0-D is the minimal form of actually finite.

A line segment is the intermediate result of actual infinity AND actually finite.

As a result no infinite extrapolation of segments is actual infinity AND no infinite interpolation is actually finite.

In other orders, there is an essential difference between the actual atomic states, and the potentiality of the complex, where a segment is some example of it.

Any collection is a complex, and no complex is an atomic state.

As a result any infinite collection is incomplete, because it is under infinite extrapolation or interpolation, such that no element of infinite extrapolation or interpolation of some collection is its final element.

Infinite extrapolation is the tendency of a complex result to be actual infinity.

Infinite interpolation is the tendency of a complex result to be actually finite.

Yet, there is a symmetry between infinite extrapolation and interpolation such that no one of them gets the actual state.

In that case the term “all” is wrong, because infinite collections are essentially incomplete (under tendency).

On the contrary, finite extrapolation or interpolation is the luck of tendency to get actual infinity or actually finite such that each one of the members of a non-empty collection is the final element of that collection.

In that case the term “all” is right, because finite collections are essentially complete (under non-tendency).

between any arbitrary pair of real numbers there is also a non-local number,where ≠ is its non-local property w.r.t any real number, where real numbers are local numbers (they have exact locations along the real-line).

For example: the real number PI is not the non-local number 3.141592653589793238462643383279502884197169399375...[base 10]

 

[doronshadmi]

You First

Already done at http://www.internationalskeptics.com/forums/showpost.php?p=5271296&postcount=6630.

 

[doronshadmi]

By your question we can clearly see that you do not distinguish between a buiding-block (endless line) and a complex (a line segment, which is a line AND two points).

But a line segment is not a line.

Right, it is a complex (an intermediate result between actual infinity and actually finite).

 

[jsfisher]

Already done at http://www.internationalskeptics.com/forums/showpost.php?p=5271296&postcount=6630.

Already done at http://www.internationalskeptics.com/forums/showpost.php?p=5271674&postcount=6637 and elsewhere.

Now, please address the follow-on questions in http://www.internationalskeptics.com/forums/showpost.php?p=5271674&postcount=6637.

 

[doronshadmi]

Already done at http://www.internationalskeptics.com/forums/showpost.php?p=5271674&postcount=6637 and elsewhere.

Now, please address the follow-on questions in http://www.internationalskeptics.com/forums/showpost.php?p=5271674&postcount=6637.

n-dimesional is related to one and only one dimesional value, and in this case any n-D is an atom.

Circle, torus or hellix are complex thikngs if more than a one dimesion is used to define them, exactly as the difference between an endless line (only 1-D) and a segment (1-D AND 0-D).

EDIT: Now please provide your proof that an endless (which is also edgeless) line is completely covered by points.

 

[doronshadmi]

Look at that:

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

http://en.wikipedia.org/wiki/Vector_(geometric)

In both cases, only complex things (1-D AND 0-D) are defined.

 

[jsfisher]

n-dimesional is related to one and only one dimesional value, and in this case any n-D is an atom.

Non-responsive to the questions.

Circle, torus or hellix are complex thikngs if more than a one dimesion is used to define them, exactly as the difference between an endless line (only 1-D) and a segment (1-D AND 0-D).

Non-responsive to the questions. So, what is the dimensionality of a circle, torus, and of a helix?

EDIT: Now please provide your proof that an endless (which is also edgeless) line is completely covered by points.

Off onto a completely different subject. Be that as it may, the onus is still yours to provide a proof. Your claim; your responsibility. You also need to explain how endless and/or edgeless lines differ from lines.

 

[doronshadmi]

Non-responsive to the questions.

This is exactly my accurate response.

Now please provide your response.

Off onto a completely different subject. Be that as it may, the onus is still yours to provide a proof. Your claim; your responsibility. You also need to explain how endless and/or edgeless lines differ from lines.

An edgeless line is a 1-D atom.

Now please show us how infinitely many segments, where each one of them has two edges, can fully cover an edgeless 1-D atom.

EDIT: You can also think about some arbitrary initial point along the 1-D atom, where each pair of points is located one to the left and one to the right w.r.t this initial point.

 

[Little 10 Toes]

By your question we can clearly see that you do not distinguish between a buiding-block (endless line) and a complex (a line segment, which is a line AND two points).

But a line segment is not a line.

Right, it is a complex (an intermediate result between actual infinity and actually finite).

Thank you for acknowledging that a line segment is a line and two points in your thinking.

Here's where you get into trouble. A line is endless. A line segment starts and stops at the two end points. Why don't you explain to us the difference between line and line segment?

 

[Sanelunatic]

I’ve only read the first and last score, or so, of pages; however, I think I am “up to speed” about what’s going on in this thread.

I have much to say, but let me start by first addressing a few points about your paper: “Zeno’s Achilles/Tortoise Race and Reconsiderations of Some Mathematical Paradigms.”

Let me start by saying your abstract is incomprehensible. It contains no small number of terms that have no meanings or different meanings outside of your paper, and as such the abstract is complete gibberish to anyone that has not already read the paper, defeating the purpose of having an abstract.

1: Zeno’s Achilles\Tortoise Race

Here you present 3 “cases” of Zeno’s Paradox, I will discuss them in order (even though each case has the same initial parameters, making the fact that we are considering 3 cases somewhat… bizarre)

Case A: Achilles speed = 10, Tortoise speed = 1, Tortoise head start = 10

In Case A Achilles wins, and rightfully so. While the length of the race is unspecified you correctly deduce that at the 2 second mark (actually he passes the tortoise at the 10/9 seconds mark, but for whatever reason you’re examining time here in discrete one second intervals) Achilles is ahead of the tortoise where he will remain indefinitely.

This is what actually happens in Zenos paradox. It is clear to anyone that Achilles actually wins the race and the business about having to catch up to the tortoise an infinite number of times thus taking infinite time is bunk. In fact, it was my understanding that the reason Zeno’s paradox was originally posed was not because it was a paradox in the true sense but because mathematical thinking of the day did not have a way to deal with such concepts as infinities, infinitesimals and limits.

Case B: Achilles speed = 10, Tortoise speed = 1, Tortoise head start = 10.

Here you present the classical version of the paradox. He must first run to 10 meters, then to 11, then to 11.1 then to 11.11 then to 11.111, the point being whenever he catches up to the tortoise the tortoise has used that time to advance slightly further and therefore Achilles NEVER catches up to the tortoise.

Here you claim the race continues “forever” because it takes an infinite number of steps to complete the race. However, I assume by “forever” that you mean it takes infinite time, and it is easy to see that this is not so. The first step takes 1 second, the next .1 the next .01 the next .001 and so forth for a total time of 1.111111… = 10/9. Whereupon Achilles catches the tortoise, and at any time greater than that it is easy to see that Achilles is past the tortoise.

It is here that you throw out Case C because as can be seen in Case B Achilles is always behind the tortoise. However you fail to consider what you learned from case A, that is to say Achilles starts out behind the tortoise and then ends up in front of the tortoise.

Consider the following. You present “Since Achilles position < Tortoise position is an invariant state of Case B…” as fact. Using your notation, this would imply that for loop k infinity Achilles position < Tortoise position. If this were to be true then

Tortoise position - Achilles position > 0

Which implies that there exists some number, lets call it e, e > 0 such that

Tortoise position - Achilles position = e

All that means is that there is some distance between Achilles and the Tortoise, that is to say that the distance between them is NOT zero.

Further, I assume you will accept that for some loop K n, loop K n+1 has Achilles closer to the Tortoise than he was at loop K n. As a consequence of this, any finite loop K n must have Achilles further from the tortoise than loop K infinity.

Lastly, you will note that if Dn represents the distance from Achilles to the Tortoise at loop K n, then D(n+1) = Dn/10.

Assuming you accept all of these things as logically true (and if you do not, let me know wherein the problem lies and I will attempt to clarify my reasoning), then we can draw the following conclusions.

Since e is a real number log(e) is defined and will give a real number. Let m denote the smallest integer strictly greater than |log(e)|. From this we can conclude that Dm < e. Now since for any nonzero e claimed to be the distance between the Tortoise and Achilles I can find an m such that at loop K m they are closer than that e, the only conclusion that can be drawn is that at k infinity

Tortoise position – Achilles position = 0

This is the classical mathematical way to resolve the paradox.

At this point in the paper we run into a problem. A severe one. The following terms are presented without explanation or definition and mean nothing to anyone other than the author: Local, Non-Local, existence, localities, locality linkage, linkage, dimension (it doesn’t seem to mean what it should).

On page 6 you give 3 definitions; however, believe me when I say these definitions are, as a whole, unsatisfactory and the ambiguity of these definitions makes the rest of the paper incomprehensible for the most part, so if you care to have me, at least, read the rest of this paper it is imperative that those terms get rigorous definitions that can be interpreted without any degree of ambiguity.

Further, here is a proof that a line is completely covered by points using the same technique.

Let x be a “spot” on the real number line that does not have a point on it. We must be able to define a distance between x and the nearest point to x, for if that distance were zero then we would find that there was a point on x! Let us call this distance e. Keep in mind that e is the distance between x and its closest neighbor, therefore e is the smallest possible distance between x and a point that is on the real number line. Let us denote this point, that is the point on the real number line closest to x as y. Making what we just typed equivalent to saying |y-x|= e

I’m assuming you accept the completeness of the reals (you should, its axiomatic), that is to say that if a,b are points on the real number line a+b and ab are also points on the real number line.

I’m also assuming I do not have to provide a proof that the integers and their inverses are also on the real number line, (the integers constructively, their inverses by axiom).

So, if y is a point on the real number line so to is y +(1/n) for any integer n. Now, since earlier we assumed that e was the smallest possible distance between x and a point on the real number line then clearly e < (1/n) for all integer n. However, no matter how we choose e =/= 0 we can find an integer n such that e > 1/n, therefore e = 0 and we find that there was a point on x all along.

 

[The Man]

EDIT:

No,

An endless line is a 1-D atom that has no edges and it is the minimal form of actual infinity.

A 0-D is the minimal form of actually finite.

A line segment is the intermediate result of actual infinity AND actually finite.

As a result no infinite extrapolation of segments is actual infinity AND no infinite interpolation is actually finite.

In other orders, there is an essential difference between the actual atomic states, and the potentiality of the complex, where a segment is some example of it.

Any collection is a complex, and no complex is an atomic state.

As a result any infinite collection is incomplete, because it is under infinite extrapolation or interpolation, such that no element of infinite extrapolation or interpolation of some collection is its final element.

Infinite extrapolation is the tendency of a complex result to be actual infinity.

Infinite interpolation is the tendency of a complex result to be actually finite.

Yet, there is a symmetry between infinite extrapolation and interpolation such that no one of them gets the actual state.

In that case the term “all” is wrong, because infinite collections are essentially incomplete (under tendency).

On the contrary, finite extrapolation or interpolation is the luck of tendency to get actual infinity or actually finite such that each one of the members of a non-empty collection is the final element of that collection.

In that case the term “all” is right, because finite collections are essentially complete (under non-tendency).

between any arbitrary pair of real numbers there is also a non-local number,where ≠ is its non-local property w.r.t any real number, where real numbers are local numbers (they have exact locations along the real-line).

For example: the real number PI is not the non-local number 3.141592653589793238462643383279502884197169399375...[base 10]

In case you need to be reminded of your question..

By using Standard Math please prove that a collection of points completely covers an endless line.

“By using Standard Math” was your requirement. Your ramblings above belie that requirement.


“the luck of tendency “? A new Doron catch phrase.

 

[laca]

First of all, welcome to the forum

I’ve only read the first and last score, or so, of pages; however, I think I am “up to speed” about what’s going on in this thread.

Yes, I think too.

I have much to say, but let me start by first addressing a few points about your paper: “Zeno’s Achilles/Tortoise Race and Reconsiderations of Some Mathematical Paradigms.”

Let me start by saying your abstract is incomprehensible. It contains no small number of terms that have no meanings or different meanings outside of your paper, and as such the abstract is complete gibberish to anyone that has not already read the paper, defeating the purpose of having an abstract.

1: Zeno’s Achilles\Tortoise Race

Here you present 3 “cases” of Zeno’s Paradox, I will discuss them in order (even though each case has the same initial parameters, making the fact that we are considering 3 cases somewhat… bizarre)

Case A: Achilles speed = 10, Tortoise speed = 1, Tortoise head start = 10

In Case A Achilles wins, and rightfully so. While the length of the race is unspecified you correctly deduce that at the 2 second mark (actually he passes the tortoise at the 10/9 seconds mark, but for whatever reason you’re examining time here in discrete one second intervals) Achilles is ahead of the tortoise where he will remain indefinitely.

This is what actually happens in Zenos paradox. It is clear to anyone that Achilles actually wins the race and the business about having to catch up to the tortoise an infinite number of times thus taking infinite time is bunk. In fact, it was my understanding that the reason Zeno’s paradox was originally posed was not because it was a paradox in the true sense but because mathematical thinking of the day did not have a way to deal with such concepts as infinities, infinitesimals and limits.

Case B: Achilles speed = 10, Tortoise speed = 1, Tortoise head start = 10.

Here you present the classical version of the paradox. He must first run to 10 meters, then to 11, then to 11.1 then to 11.11 then to 11.111, the point being whenever he catches up to the tortoise the tortoise has used that time to advance slightly further and therefore Achilles NEVER catches up to the tortoise.

Here you claim the race continues “forever” because it takes an infinite number of steps to complete the race. However, I assume by “forever” that you mean it takes infinite time, and it is easy to see that this is not so. The first step takes 1 second, the next .1 the next .01 the next .001 and so forth for a total time of 1.111111… = 10/9. Whereupon Achilles catches the tortoise, and at any time greater than that it is easy to see that Achilles is past the tortoise.

It is here that you throw out Case C because as can be seen in Case B Achilles is always behind the tortoise. However you fail to consider what you learned from case A, that is to say Achilles starts out behind the tortoise and then ends up in front of the tortoise.

Consider the following. You present “Since Achilles position < Tortoise position is an invariant state of Case B…” as fact. Using your notation, this would imply that for loop k infinity Achilles position < Tortoise position. If this were to be true then

Tortoise position - Achilles position > 0

Which implies that there exists some number, lets call it e, e > 0 such that

Tortoise position - Achilles position = e

All that means is that there is some distance between Achilles and the Tortoise, that is to say that the distance between them is NOT zero.

Further, I assume you will accept that for some loop K n, loop K n+1 has Achilles closer to the Tortoise than he was at loop K n. As a consequence of this, any finite loop K n must have Achilles further from the tortoise than loop K infinity.

Lastly, you will note that if Dn represents the distance from Achilles to the Tortoise at loop K n, then D(n+1) = Dn/10.

Assuming you accept all of these things as logically true (and if you do not, let me know wherein the problem lies and I will attempt to clarify my reasoning), then we can draw the following conclusions.

Since e is a real number log(e) is defined and will give a real number. Let m denote the smallest integer strictly greater than |log(e)|. From this we can conclude that Dm < e. Now since for any nonzero e claimed to be the distance between the Tortoise and Achilles I can find an m such that at loop K m they are closer than that e, the only conclusion that can be drawn is that at k infinity

Tortoise position – Achilles position = 0

This is the classical mathematical way to resolve the paradox.

Yes. However,

<doron mode> Phwah! That is local-only reasoning! You cannot get <insert random doron post here>! You don't have the skills to address this issue! Only I do! Everyone else is limited to local-only reasoning! I am teh powah!
</doron mode>

At this point in the paper we run into a problem. A severe one. The following terms are presented without explanation or definition and mean nothing to anyone other than the author: Local, Non-Local, existence, localities, locality linkage, linkage, dimension (it doesn’t seem to mean what it should).

Oh, that's just a subset of the subsets he's using: cardinality, entropy, atomic, ...

On page 6 you give 3 definitions; however, believe me when I say these definitions are, as a whole, unsatisfactory and the ambiguity of these definitions makes the rest of the paper incomprehensible for the most part, so if you care to have me, at least, read the rest of this paper it is imperative that those terms get rigorous definitions that can be interpreted without any degree of ambiguity.

Yeah, but don't hold your breath...

Further, here is a proof that a line is completely covered by points using the same technique.

Let x be a “spot” on the real number line that does not have a point on it. We must be able to define a distance between x and the nearest point to x, for if that distance were zero then we would find that there was a point on x! Let us call this distance e. Keep in mind that e is the distance between x and its closest neighbor, therefore e is the smallest possible distance between x and a point that is on the real number line. Let us denote this point, that is the point on the real number line closest to x as y. Making what we just typed equivalent to saying |y-x|= e

I’m assuming you accept the completeness of the reals (you should, its axiomatic), that is to say that if a,b are points on the real number line a+b and ab are also points on the real number line.

Uh-oh... You shouldn't assume anything about doron. He's non-local, for Jack's sake! He doesn't even acknowledge that 1 / 3 * 3 = 1! He also invented a number: 0.(0)1

I’m also assuming I do not have to provide a proof that the integers and their inverses are also on the real number line, (the integers constructively, their inverses by axiom).

Err... No, he most probably will ask you for proof.

So, if y is a point on the real number line so to is y +(1/n) for any integer n. Now, since earlier we assumed that e was the smallest possible distance between x and a point on the real number line then clearly e < (1/n) for all integer n. However, no matter how we choose e =/= 0 we can find an integer n such that e > 1/n, therefore e = 0 and we find that there was a point on x all along.

All in all, a good post. You just missed the 150+ pages to make you realize reason doesn't appeal to doron. He's not bound to our local-only reasoning, you see. He transcended reason a long time ago. He dwells in a non-local universe he made up. He gets to be the uber-emperor, so it's fun.