Deeper than primes

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ddt said:
this can only mean that there is a third object z
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No, there are a point (which is an object), a line segment (which is an object), a single relation (which is not and object) or several relations connected by AND w.r.t a single object.
 
ddt said:
Very good observation, jsfisher! I see my prediction

was quite spot-on. :D

I see doron tries to deny your conclusion and your example. It's clear he simply doesn't get the point of your argument. I'll try another approach. It's even constructivist, i.e., I won't be using argument by contradiction :).

Let's first see what it should mean that "an object x is distinguished from another object y by relation R". IMNSHO, this can only mean that there is a third object z such that
(x, z) in R and not (y, z) in R or not (x, z) in R and (y, z) in R​
in other words: the question "with which objects is z related by R" is answered differently for x and y.

Let's take two different points A and B. Then draw the circle with centre A and radius the length of the line segment AB. Likewise, draw the circle with centre B and the same radius. These two circles have two intersection points; call them C and D.

With four (known) points, we can make 2^16 different relations between these points. I just define two different ones now:

R1 := { (A, C) }
R2 := { (B, C) }​

Now it's clear that R1 distinguishes A from B, as (A, C) is in R1 and (B, C) is not. Likewise, R2 distinguishes A from B.

As we have now two relations that distinguish A from B (and I could define many more), A is not local. QED.

Yes, it's clear this definition of "local" is nonsensical. It's also again clear that doron doesn't understand the basest and simplest mathematical arguments.
Click to expand...

In order to get the difference between the points in your particular example, you have used a non-local element (a line segment, in this case) that is both = and ≠ to any given point.

Without it each point is totally isolated of any other point in your particular example, and nothing can be concluded about the differences between them.

In other words, you have used non-locality in order to define the difference between locals.

Again:

(A,C) is not less than:

((A) AND (a line segment that is both A AND not-A(called C)))

OR

((C) AND (a line segment that is both C AND not-C(called A)))
 
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doronshadmi said:
In order to get the difference between the points in your particular example, you have used a non-local element (a line segment, in this case) that is both = and ≠ to any given point.
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Nonsense. I didn't use the line segment for anything other than to get its length and plot out a circle with my compass. A revered technique since the Ancients to Gauss' construction of the regular 17-sided polygon.

I could also show you that any point is non-local to any line or line segment with your ludicrous definition.

doronshadmi said:
Without it each point is totally isolated of any other point in your particular example, and nothing can be concluded about the differences between them.
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The whole point of geometry (in this case, planimetry) is to have both points and lines. Without the lines, the points are pointless (pun intended).
 
doronshadmi said:
ddt said:
this can only mean that there is a third object z
Click to expand...
No, there are a point (which is an object), a line segment (which is an object), a single relation (which is not and object) or several relations connected by AND w.r.t a single object.
Click to expand...
This sentence was about interpreting your definition in real mathematical terms - not about the particular case of points and lines. Look up what a "relation" is in math.

Or heed the advice of Skeptic, similar to what I already expressed many moons ago in this post.
 
doronshadmi said:
There is an OR connective between any pair of relatrions between a point and a line...
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The logical connector in my example was AND, but so what? It just means I can establish additional relations using already known relations in boolean combinations.

Your definition requires there be exactly one relation between two objects. Clearly, you definition is completely bogus since there are always more than one.. It doesn't mean what you think it means.

Try again.
 
ddt said:
Look up what a "relation" is in math.
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Relation (which is not an element) is between an element to itself or not to itself.

Please show a relation which is not what was written above.
 
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jsfisher said:
The logical connector in my example was AND again.
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jsfisher said:
Relation 1: The point is not on the line.
Relation 2: The point is not equal to the line.
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In that case there is no problem to reduce it to . ≠ __ (where not on = not equal in the case that __ is observed through . )

Try to do it in the case that . is observed through __ , where __ ≠ and = . , for example:

x = __
y = .


x is non-local w.r.t y if:

x < and = y (example: _. )

x < and > y (example: _._ )

x = and > y (example: ._ )

You still do not get the notion of "y is obsereved through x"
 
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I have changed definition 5 to:


Definition 5:
An object that is distinguished from another object by simultaneously more than a one relation, is called Non-local.

Only a non-local thing (a relation or a non-local element) can define the difference between local elements.

Without it, each local element is totally isolated and the difference cannot be defined, for example:

{a,b,c} is actually {a≠b,a≠c,b≠c}, where in this case the non-local ≠ is used.
 
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doronshadmi said:
In that case there is no problem to reduce it to . ≠ __ (where not on = not equal in the case that __ is observed through . )
Click to expand...


Your continued fascination with dots, dashes, and other symbols as substitutes for rational statements does nothing to salvage your failed definition.

Remember? This is the definition under fire:

Definition 4: An object that is distinguished from another object by exactly one relation, is called Local.​

Also, please note, that if a point is local with respect to a line, then that line must be local with respect to that point. It's your definition. Live with its consequences.
 
doronshadmi said:
I have changed definition 5 to:


Definition 5:
An object that is distinguished from another object by simultaneously more than a one relation, is called Non-local.
Click to expand...


You need to define "distinguished" first. The word "simultaneously" isn't necessary.
 
jsfisher said:
Your continued fascination with dots, dashes, and other symbols as substitutes for rational statements does nothing to salvage your failed definition.

Remember? This is the definition under fire:

Definition 4: An object that is distinguished from another object by exactly one relation, is called Local.​

Also, please note, that if a point is local with respect to a line, then that line must be local with respect to that point. It's your definition. Live with its consequences.
Click to expand...

In order to get Definition 4 you have to get y through x.

x = .

y = __ or .


y is observed through x:

If relations <, = , >, ≠ are used, then x cannot be simultaneously in more than a one relation with y.

If you disagree with me than please show how x is simultaneously in more than a one relation with y.

EDIT:

jsfisher said:
Also, please note, that if a point is local with respect to a line, then that line must be local with respect to that point. It's your definition. Live with its consequences.
Click to expand...

It depends on the observation's point of view, as can be seen in pages 3-5 of http://www.geocities.com/complementarytheory/UR.pdf .
 
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doronshadmi said:
In order to get Definition 4 you have to get y through x.
Click to expand...

No, in order for you to get Definition 4 you need to pay attention to what you wrote. It says nothing about "y through x". It deals solely with two objects and relations between them.

Stop making up stuff that isn't in the definition.

...
If you disagree with me than please show how x is simultaneously in more than a one relation with y.
Click to expand...

(Irrelevant word removed.) Already did...I stopped at three.
 
doronshadmi said:
In order to get Definition 4 you have to get y through x.

x = .

y = __ or .


y is observed through x:

If relations <, = , >, ≠ are used, then x cannot be simultaneously in more than a one relation with y.

If you disagree with me than please show how x is simultaneously in more than a one relation with y.
Click to expand...

The ‘relations’ < (less then) as well as > (greater then) both infer ≠ (not equal to) as the typical exclusive variants of those ‘relations’. Which is why the indications of <= (less then and equal to) as well as >= (greater then and equal to) are used to denote the inclusive variants. Since greater then as well as less then can only be inclusive or exclusive, two of the relations you have given (< as well as >) must always include one or the other of the two relation given (= or ≠) or are ‘relations’ that are inherently and simultaneously at least one of your other given ‘relations’.
 
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jsfisher said:
No, in order for you to get Definition 4 you need to pay attention to what you wrote. It says nothing about "y through x". It deals solely with two objects and relations between them.

Stop making up stuff that isn't in the definition.



(Irrelevant word removed.) Already did...I stopped at three.
Click to expand...

Ok,

Maybe these are clearer:

Definition 4:

If object x can be distinguished from object y by using a one relation, then x is called Local.


Definition 5:

If object x cannot be distinguished from object y unless more than a one relation is simultaneously used, then x is called Non-Local.
 
doronshadmi said:
Ok,

Maybe these are clearer:

Definition 4:

If object x can be distinguished from object y by using a one relation, then x is called Local.
Click to expand...

So, since a line can be distinguished from a point by the relation "isn't the same as", a line is therefore local.

Definition 5:

If object x cannot be distinguished from object y unless more than a one relation is simultaneously used, then x is called Non-Local.
Click to expand...

You switched things around then. Before nothing was local and everything was non-local. Now, with these improved definitions it's the other way around.

Is that really what you meant to do?
 
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