Deeper than primes

[zooterkin]

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.

"ends", Doron, not "edges".

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.

Is your problem that you can't see how to specify the adjacent points, where one segment ends and the next starts, given that there's an infinite number of other points between any two points?

 

[Skeptic]

He also invented a number: 0.(0)1

Cool!

 

[doronshadmi]

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?

I did.

A line (an edgeless line) is a 1-D atom.

A segment is a complex that is made of two kinds of atoms (1-D atom AND 0-D atom).

 

[doronshadmi]

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

Wrong assumption, any collection is incomplete with respect to actual infinity (that its minimal form is an edgeless line), or to actually finite (that its minimal form is a point).

According to my paper, there are two valid solutions to Zeno’s Achilles/Tortoise Race, which are Achilles wins OR Achilles does not win.

Since both cases are independent of each other, there is no paradox.

As for case b, Achilles does not win exactly because of the difference between the actual and the potential.

The actual is an atomic state (only one dimension is considered) where the potential is a complex state (more than a one dimension is considered).

Let as show it in details:

1. 0-D (a point) is the minimal form of actually finite (it is an atom).

2. 1-D (an edgeless line) is the minimal form of actual infinity (it is an atom).

3. A segment is an intermediate and complex result of 0-D AND 1-D.

4. No complex is an atom, and vice versa.

Let as use the following thought experiment:

a. Take the edgeless line and define an arbitrary point on it.

b. The result is a pair of left and right rays, where a ray has exactly one edge.

c. Locate infinitely many arbitrary pairs of points on the edgeless line, such that one point of each pair is located to the left of (a) point and the other point is located to the right of (a) point.

d. It is obvious that infinitely many points on the edgeless line do not change the fact of the permanent existence of left and right rays with respect to any collection of infinitely many points on an edgeless line.

e. According to (d) it is obvious that no collection of infinitely many points is equivalent to an edgeless line, or in other words, a collection of infinitely many points is a potential infinity w.r.t the actual infinity of an edgeless line.

f. According to (e) there is an infinite extrapolation between any collection of infinitely many points and an edgeless line.

g. The symmetrical state of an infinite extrapolation is an infinite interpolation, where no segment (a complex result of 1-D AND 0-D) is actually finite (it is not a point).

If you understand the difference between the actual (atomic) and the potential (complex), it is used like a key in order to understand my article.

1.111111… = 10/9

10/9[base 10] = 1.111...[base 10], which are two representations of the same non-local number, where a non-local number does not have an exact location along the real-line (it is used to measure potential infinity).

For more details, please look at http://www.internationalskeptics.com/forums/showpost.php?p=5246903&postcount=6333 .

Also you are invited to read:

http://www.scribd.com/doc/17039028/OMDP

http://www.scribd.com/doc/16542245/OMPT

http://www.scribd.com/doc/21954566/NXOR-XOR

http://www.scribd.com/doc/21954904/UP

 

[doronshadmi]

"ends", Doron, not "edges".


Is your problem that you can't see how to specify the adjacent points, where one segment ends and the next starts, given that there's an infinite number of other points between any two points?

No, you are not able to understand what is ≠ between any arbitrary pair of distinct locations along a line.

 

[zooterkin]

No, you are not able to understand what is ≠ between any arbitrary pair of distinct locations along a line.

Oh, good, so now you can specify adjacent points on a line. I'm glad of that, because you were having difficulty with it before.

 

[doronshadmi]

Oh, good, so now you can specify adjacent points on a line. I'm glad of that, because you were having difficulty with it before.

zooterkin,

I did not have any problem to a get complex, which is both 1-D AND 0-D.

You have a problem to get 1-D OR 0-D, where each one of them is an independent atom (not a complex).

 

[zooterkin]

W

According to my paper, there are two valid solutions to Zeno’s Achilles/Tortoise Race, which are Achilles wins OR Achilles does not win.

So, you resolve the paradox by not resolving it?

 

[laca]

So, you resolve the paradox by not resolving it?

In case he resolved all paradoxes within OM, nothing would remain. So yes, he's not avoiding contradictions, he builds on them.

 

[doronshadmi]

So, you resolve the paradox by not resolving it?

The two cases are valid and independent results (there is an OR connective between them).

Therefore there is no paradox.

 

[jsfisher]

The two cases are valid and independent results (there is an OR connective between them).

Therefore there is no paradox.

That word doesn't mean what you think it means.

 

[doronshadmi]

That word doesn't mean what you think it means.

The paradox of motion does not exist, at the moment that the difference between Non-locality\Locality, Atom\Complex and Actual\Potential is understood.

By this understanding the One\Many metaphysical problem (which is the core of Zeno's paradoxes) is solved.

Contemporary Mathematics does not even scratch Zeno's paradoxes, as clearly shown in http://www.internationalskeptics.com/forums/showpost.php?p=5263811&postcount=6530 (http://philsci-archive.pitt.edu/arc...ths_review_metaphysics_alba_papa_grimaldi.pdf ).

 

[doronshadmi]

Yahoo closed Geocities, without saving the web addresses.

As a result most of my articles and diagrams cannot be found anymore.

 

[zooterkin]

Verily, a tragedy to compare with the burning of the library at Alexandria.

 

[jsfisher]

The paradox of motion does not exist....

And the way you "prove" there is no paradox is by claiming two contradictory results are both tenable.

I say again, that word [paradox] doesn't mean what you think it means.

 

[doronshadmi]

And the way you "prove" there is no paradox is by claiming two contradictory results are both tenable.

I say again, that word [paradox] doesn't mean what you think it means.

Since the two cases are independent of each other, there is no paradox.

 

[laca]

Yahoo closed Geocities, without saving the web addresses.

As a result most of my articles and diagrams cannot be found anymore.

You mean they're non-local now? Aww... I'd ask if you had some backups, but why would you? That would be rational...

 

[doronshadmi]

You mean they're non-local now? Aww... I'd ask if you had some backups, but why would you? That would be rational...

All of the stuff is saved. The problem is that it cannot be seen anymore in the old posts of this forum. I'll ask the moderator to give me the ability to change the Geocities addresses to new addresses, where my stuff can be found.

 

[jsfisher]

Since the two cases are independent of each other, there is no paradox.

You might want to actually look that word up. You have analyzed the race in two different ways which you claim are both valid (they aren't, of course, but you'd never admit or even understand that), and you got contradictory results.

In what way are two seemingly valid but contradictory results not a paradox?

 

[Sanelunatic]

While I still have much to say, I will limit my post to talking about a few topics for the moment, and if discussion here is fruitful we can branch back out to other things. Furthermore, for the moment, I will drop my points about the real line and Zeno's paradox (for the moment, mind you!). I feel that neither of us will be able to convince the other of anything unless we both have a firm understanding of the other's ideas.

To that extent, I have two questions I would like you to answer when you get the chance.

Firstly;

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

Wrong assumption, any collection is incomplete with respect to actual infinity (that its minimal form is an edgeless line), or to actually finite (that its minimal form is a point).

When I say you should accept it, I mean there is a problem if you do not, a problem with mathematics.

The closure of the real number line under multiplication and addition is an axiom of mathematics.

For those not familiar with the term, an axiom is an idea that is accepted, without proof because it is "obviously true" or so goes the classical phrasing. There are (in classical algebra) 10 axioms.

They are, for those unfamiliar:

1) Closure: if x and y are in a set x + y and xy are also in the set
2) Commutativity of addition: x + y = y + x
3) Associativity of addition: (x + y) + z = x + (y + z)
4) Additive identity: There exists an element which we will denote 0 such that x + 0 = x
5) Additive inverse: There exists an element which we will denote -x such that x + -x = 0
6) Associativity of multiplication: (xy)z = x(yz)
7) Distributive law: x(y + z) = xy + xz
8) Commutativity of multiplication: xy = yx
9) Multiplicative Identity: There exists an element which we will denote 1 such that 1x = x
10) Multiplicative Inverse: There exists an element which we will denote x^-1 such that x^-1x = 1

Not all of these are absolutely necessary for math to make sense, particularly numbers 6+. By removing some of these we can obtain structures such as rings; however, "standard math" would accept all 10.

My first question is this:

Are there any of these axioms that you feel are either: untrue, true only in some cases or require any changes in order to be true. Additionally, do you feel that there are any axioms that should be added to this list?

This, for me at least, is the single most important question you can answer.

The second point I would like to address is your notion of an Atomic state. To quote your paper that I referenced in my last post:

Atomic state is an existing thing that has no sub-existing things...

You also state

In order to be expressed beyond the atomic state, we suggest at least two existing things that are linked to each other, without deriving from each other.

This seems to be a somewhat related concept to the standard notion of dimensionality and the basis of a set.

To express the notion loosely, a set B is a basis of a given set X provided that:

1) By combining, in various ways, the elements of B I can produce any element of X (and no elements not in X).

2) I can not, by combining in any way, form one element of B from any of the other elements of B.

Furthermore, if the set B has n elements, we refer to the set X as being n-dimensional.

This, of course, is not the formal mathematical definition (and if anyone is interested in seeing the formal definition, I suggest wikipedia); however, I feel this clearly communicates the idea.

My second question is this.

What do you think about the standard notion of dimensionality and the basis of a set with respect to your notions of atomic state and linkage? Do you feel that the two notions are essentially the same? Do you feel that one notion is lacking somehow? If so, how?

If you can answer, in part or in whole, those two questions it would greatly abet me in understanding your ideas.