Deeper than primes - Continuation

[epix]

The expression "r=0 ≠ r>0" is always resulted by T as its truth value, or in other words, it is a tautology (In logic, a tautology (from the Greek word ταυτολογία) is a formula which is true in every possible interpretation (http://en.wikipedia.org/wiki/Tautology_(logic))).

If one does not agree about it, he has to provide a False result to "r=0 ≠ r>0" expression.

Even Wikipedia shows that you are wrong. Here is again an example of tautology:
Let @ be some logical connective. Then

(a<b @ c<d) is True
(a<b @ c>d) is True
(a>b @ c<d) is True
(a>b @ c>d) is True

is tautology.

Your expression "r=0 ≠ r>0" lacks the necessary premise options for tautology to exist, but the main problem is that ≠ is not a logical connective. The inequality r>0 is just the only true (metrics) alternative to r≠0 and vice versa when r is a radius of a circle - the expression is not tautology, as opposed to "belongs AND doesn't belong". Remember?

Also Circumradius (does not related to an obvious center) ≠ Radius (does related to an obvious center), and also this case is a tautology.

What is "an obvious center?"
Look again at the definition of radius applied to regular polygon,
http://www.mathopenref.com/polygonradius.html
and then reconsider your inequality Radius≠Circumradius by looking at this:
http://etc.usf.edu/clipart/43400/43444/5c_43444_lg.gif
Then digest the fact that "circumradius" derives from the word "circumscribe." In both cases the magnitude of r is the same for the regular polygon and the circumscribed circle. Polygons aside, still, your initial reference to the radius was a case of insufficient definition, coz radius can apply to sphere as well as circle.

 

[epix]

[email protected] is available to any one, who really wishes to communicate with Dr. K. Ghosh about his work.

Why don't you communicate with Dr. K. Ghosh regarding the conjecture that you predictably skipped? He will surely help you to prove or disprove it. That gives us the opportunity to see if he would use again the non-standard descriptive means. Here it is once again:

Conjecture:

There is no prime number p such as that

p = a²+b² = c³+d³
There used to be a web site devoted to prime numbers where folks emailed related material, which was first reviewed by the webmaster mathematician. If I sent that conjecture there, then it wouldn't be very likely published, coz it is way too easy to prove or disprove - it doesn't pose much of a challenge. That means Dr. Ghosh shouldn't trip over it. Just tell him that it was a long time ago when you tinkered with number theory and if he could help you out.

 

[doronshadmi]

Look at these expression:

"but the main problem is that ≠ is not a logical connective."

"The inequality r>0 is just the only true (metrics) alternative to r≠0 and vice versa when r is a radius of a circle - the expression is not tautology,"

These expressions are wrong (they get things only in terms of context-dependent frameworks) because r is any value, which is not restricted to metric space or logical interpretation. By using that Cross-contexts generality the expression "r=0 ≠ r>0" is always true.

Any attempt to restrict "r=0 ≠ r>0" only to logical framework, misses the main principle of being Tautology.

Tautology at an actual universal state is a form(ula) which is true in every possible interpretation, whether the interpretation is done logically, arithmetically, or whatever.

Once again, context-dependent-only skill(ers) can't get the actuality of Tautology, which is not less than Cross-contexts.

An expression like:

"(In logic, a tautology (from the Greek word ταυτολογία) is a formula which is true in every possible interpretation ( http://en.wikipedia.org/wiki/Tautology_(logic) )" is not Cross-contexts and therefore it is not an actual Tautology.

As for the question: "What is "an obvious center?" the answer is found in ( http://en.wikipedia.org/wiki/Radius ):

"If the object does not have an obvious center, the term may refer to its circumradius,", and this http://www.mathopenref.com/polygonradius.html is a case of circumradius.

Look at this expression:

"your initial reference to the radius was a case of insufficient definition, coz radius can apply to sphere as well as circle"

This is wrong since r>0 is not restricted to any dimensional space > 0.

Once again the context-dependent-only reasoning airs its fragmented view.

 

[epix]

These expressions are wrong (they get things only in terms of context-dependent frameworks) because r is any value, which is not restricted to metric space or logical interpretation. By using that Cross-contexts generality the expression "r=0 ≠ r>0" is always true.

Your effort to weasel out of your erroneous statements brings in more contradictions, such as "r is any value." The left side of your doronian inequality clearly shows that r doesn't have any value, coz r=0 in there. If you switch to a particular case where r=0 and r=5, where r is the last letter of "number, then "r=0 ≠ r>0" becomes 0≠5, which is inequality, not tautology.

Any attempt to restrict "r=0 ≠ r>0" only to logical framework, misses the main principle of being Tautology.

You burned you dinner once again. The principle of tautology is an unnecessary redundancy. That means unnecessary redundancy where "unnecessary" is a redundant adjective. There is nothing redundant about 0≠5, is there?

Your attempt to come up with special math syntax, like "r=0 ≠ r>0" just reflects upon your overall confusion regarding relations. The same goes for your double-speak which is supposed to describe the result of some uncommon deep thought.

Tautology at an actual universal state is a form(ula) which is true in every possible interpretation, whether the interpretation is done logically, arithmetically, or whatever.

That's what you wish for, but tautology is applicable in well-defined domains and none of them includes arithmetics. Your effort to change that hasn't started that well, as everyone can see.

Once again, context-dependent-only skill(ers) can't get the actuality of Tautology, which is not less than Cross-contexts.

In
An expression like:

"(In logic, a tautology (from the Greek word ταυτολογία) is a formula which is true in every possible interpretation ( http://en.wikipedia.org/wiki/Tautology_(logic) )" is not Cross-contexts and therefore it is not an actual Tautology.

That's too bad that "Cross contexts" is not an "actual" tautology, which is "not less than Cross-contexts." Since the reference to tautology in Wiki never mentions any cross-contexts, it's apparent that the relation between tautology and cross-contexts is pure Doronetics.

 

[The Man]

Any attempt to restrict "r=0 ≠ r>0" only to logical framework, misses the main principle of being Tautology.

Well since a Tautology, in logic, is a proposition that is always true regardless of the truth values of its elements (the negation of a contradiction) it is obviously and deliberately you who " misses the main principle of being Tautology" in that regard.

In rhetoric a Tautology is exemplified by the repetition of meaning, as "r=0" has a different meaning than " r>0" you have missed the "main principle of being Tautology" in that regard as well.

As it is now clear that you are simply not restricting your "Tautology" to being any kind of, well, Tautology you continue on the previously noted "(totally packed)" with BS trend.

If it was your intent to demonstrate that your failure could cross contexts in the " main principle of being Tautology", you have succeeded.


By all means please Doron, do tell us what in you mind is "the main principle of being Tautology."? Simply making a true statement does not make it a tautology, in rhetoric or logic.

As already made clear to you many times before just because "r" can, or does, approach "0" does not restrict "r" from being "0", as in r ≥ 0 . Again a limit can be a member of the set it limits, though it does not have to be. Do please learn something Doron.


Oh just a little FYI, if you are formulating a response to this post then I again recommend that you address it and attribute quotes correctly, otherwise I simply will not read or reply to it.

 

[doronshadmi]

Context-dependent skill(ers) can't get concepts, unless they are restricted to some domain, for example:

"Well since a Tautology, in logic," means that Tautology is defined only if it is restricted (in this case) to Logic, or in other words, it can't be defined in terms of Cross-contexts principle.

Since "r=0" has a different meaning than " r>0" then r simultaneously ≥ 0 (which is a line) only if r is at and not at r=0 (where r=0 is a point) location.

Please look at these context-dependent restrictions:

"As already made clear to you many times before just because "r" can, or does, approach "0" does not restrict "r" from being "0", as in r ≥ 0 . Again a limit can be a member of the set it limits, though it does not have to be."


Case 1: at most one element:

If r only = 0, then r is a point.

If r simultaneously ≥ 0, the r is a line.


Case 2: at least two elements:

if r > OR = 0, then r is two different points, or a point and a line (the expression "r=0 ≠ r>0" is Tautology among at least two elements).


In both cases r is a point (can be only in one location) OR a line (can be in more than one location).


Points only approach a given limit (whether it belongs OR does not belong to a given set).

Lines can actually reach a given limit (whether it belongs OR does not belong to a given set).


Please look at the restrictions of context-dependent-only skill(er):

"Oh just a little FYI, if you are formulating a response to this post then I again recommend that you address it and attribute quotes correctly, otherwise I simply will not read or reply to it."


Once again the limited (formal or informal) reasoning of Context-dependent-only skill(ers) is strictly demonstrated.

------------

The expression "0≠5" is always true, no matter what context is used.

Please look at this expression:

"Since the reference to tautology in Wiki never mentions any cross-contexts, it's apparent that the relation between tautology and cross-contexts is pure Doronetics."

Wiki describes only already agreed marital, so relation between tautology and cross-contexts (which is a paradigm-shift of the already agreed marital) can't be found in Wiki.

 

[epix]

Oh just a little FYI, if you are formulating a response to this post then I again recommend that you address it and attribute quotes correctly, otherwise I simply will not read or reply to it.

Doronetics and tautology are very closely related. No matter what kind of options you take to instruct Doron, the outcome will be the same.

 

[doronshadmi]

Pi is a dimensionless size, because it is not impacted by the sizes of all existing circles.

Zero circumference or diameter do not allow the existence of Pi as a dimensionless size.

 

[epix]

Pi is a collection of two letters. They are the 19th and the 20th letters in a text. What text?

Oh, Lord! What have I done that would deserve punisment like that, for the Heavens know that no mortal soul of the world can match Your omniscience.

In that case I shall send an angel to assist your non-omniscient mind . . .

Pi is a dimensionless size, because it is not impacted by the sizes of all existing circles.

Zero circumference or diameter do not allow the existence of Pi as a dimensionless size.

(Let's see... 19+20=39, and there are 39 books in The Old Testament.)
The text is a part of the Bible - it is in The Old Testament, my Lord.

You must put more rigor into your effort, my son, to please my heart.

rolleyes: It looks like a brute force search to me . . .)
I have analytically arrived at the answer, my Lord.

Please, reveal it to Me.

With his own spear you pierced his head
when his warriors stormed out to scatter us,
gloating as though about to devour
the wretched who were in hiding.
Habakkuk 3:14

I'm done, Doron. You can fly away. Thanks for the assistence. Btw, if you see an Atheist temple on your way back, let me know. I feel like converting.

 

[Little 10 Toes]

Pi is a dimensionless size, because it is not impacted by the sizes of all existing circles.

Zero circumference or diameter do not allow the existence of Pi as a dimensionless size.

Who ever said pi is a) dimensionless, b) a size, or c) a dimensionless size?

Pi is a ratio between a circle's circumference and it's diameter.

 

[The Man]

Doronetics and tautology are very closely related. No matter what kind of options you take to instruct Doron, the outcome will be the same.

Indeed, but at least I gave him the chance and the choice. What he does with it is up to him alone.

 

[The Man]

Pi is a dimensionless size, because it is not impacted by the sizes of all existing circles.

Zero circumference or diameter do not allow the existence of Pi as a dimensionless size.

Who ever said pi is a) dimensionless, b) a size, or c) a dimensionless size?

Pi is a ratio between a circle's circumference and it's diameter.

Technically Pi is a dimensionless quantity, whatever Doron actually means by “size” let alone “dimensionless size” is really irrelevant as it will vary whenever it suits him.


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

 

[Little 10 Toes]

Technically Pi is a dimensionless quantity, whatever Doron actually means by “size” let alone “dimensionless size” is really irrelevant as it will vary whenever it suits him.


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

Well, I was using pi^WP for my info. So neener.

 

[epix]

Technically Pi is a dimensionless quantity, whatever Doron actually means by “size” let alone “dimensionless size” is really irrelevant as it will vary whenever it suits him.

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

Wiki quote:

In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and in everyday life (such as in counting). Numerous well-known quantities, such as π, e, and φ, are dimensionless.

LOL.
A little distraction would also raise the question why a quantity without dimension has been assigned a dimension of 1. Just stick with the initial part of the definition and don't assign any dimension to it. Right?

If pi is a dimensionless quantity, then what is number pi?

I guess it has to be a dimensionless number.

And what is a dimensionless number?

A number representing a property of a physical system, but not measured on a scale of physical units (as of time, mass, or distance).

That's why we don't see around expressions, like pi miles/h, pi mV, and so on.

The Wikipedia writers usually never do any explaining by example for those who know a little about a given subject but are not completely ignorant. Doron, as usually, sees things differently:

Pi is a dimensionless size, because it is not impacted by the sizes of all existing circles.

 

[doronshadmi]

Pi has a physical size as distance along the real line but it is dimensionless in terms of an invariant ratio among all circles.

In other words, Pi is Cross-contexts w.r.t all circles' curvature degrees, and without such Cross-contexts property circles with different curvature degrees can't be gathered into a one class.

Yet being gathered into a one class does not mean that all in terms of finite cardinality is equivalent to all in terms of transfinite cardinality, simply because the expression "+1" is an invariant Cross-contexts property w.r.t all infinite collections, exactly as Pi is Cross-contexts w.r.t all circles' curvature degrees.

In other words, the incompleteness of infinite collections is their "+1" expression (in terms of cardinality) as an invariant Cross-contexts property.

 

[epix]

Pi has a physical size as distance along the real line but it is dimensionless in terms of an invariant ratio among all circles.

In other words,

pi is a constant ratio when associated with circles.

Pi is Cross-contexts w.r.t all circles' curvature degrees, and without such Cross-contexts property circles with different curvature degrees can't be gathered into a one class.

Yet being gathered into a one class does not mean that all in terms of finite cardinality is equivalent to all in terms of transfinite cardinality, simply because the expression "+1" is an invariant Cross-contexts property w.r.t all infinite collections, exactly as Pi is Cross-contexts w.r.t all circles' curvature degrees.

In other words, the incompleteness of infinite collections is their "+1" expression (in terms of cardinality) as an invariant Cross-contexts property.

If you think so . . .

 

[The Man]

Wiki quote:

LOL.
A little distraction would also raise the question why a quantity without dimension has been assigned a dimension of 1. Just stick with the initial part of the definition and don't assign any dimension to it. Right?

From the first referance in that article….

^ "1.8 (1.6) quantity of dimension one dimensionless quantity". International vocabulary of metrology — Basic and general concepts and associated terms (VIM). ISO. 2008. http://www.iso.org/sites/JCGM/VIM/JCGM_200e_FILES/MAIN_JCGM_200e/01_e.html#L_1_8. Retrieved 2011-03-22.

1.8 (1.6)
quantity of dimension one
dimensionless quantity
quantity for which all the exponents of the factors corresponding to the base quantities in its quantity dimension are zero
NOTE 1 The term “dimensionless quantity” is commonly used and is kept here for historical reasons. It stems from the fact that all exponents are zero in the symbolic representation of the dimension for such quantities. The term “quantity of dimension one” reflects the convention in which the symbolic representation of the dimension for such quantities is the symbol 1 (see ISO 31 0:1992, 2.2.6).
NOTE 2 The measurement units and values of quantities of dimension one are numbers, but such quantities convey more information than a number.
NOTE 3 Some quantities of dimension one are defined as the ratios of two quantities of the same kind.
EXAMPLES Plane angle, solid angle, refractive index, relative permeability, mass fraction, friction factor, Mach number.
NOTE 4 Numbers of entities are quantities of dimension one.
EXAMPLES Number of turns in a coil, number of molecules in a given sample, degeneracy of the energy levels of a quantum system.

Actually the initial part of the definition (as you quoted) does say “or quantity of dimension one”

If pi is a dimensionless quantity, then what is number pi?


I guess it has to be a dimensionless number.

And what is a dimensionless number?

A “quantity of dimension one”

That's why we don't see around expressions, like pi miles/h, pi mV, and so on.

The Wikipedia writers usually never do any explaining by example for those who know a little about a given subject but are not completely ignorant.

While that may be the case in some instances, it is not (see the later examples cited below) in this and reference as well as additional links are usually provided in most cases.



http://en.wikipedia.org/wiki/Dimensionless_number#Example

Example
The power consumption of a stirrer with a given shape is a function of the density and the viscosity of the fluid to be stirred, the size of the stirrer given by its diameter, and the speed of the stirrer. Therefore, we have n = 5 variables representing our example.
Those n = 5 variables are built up from k = 3 dimensions which are:
• Length: L (m)
• Time: T (s)
• Mass: M (kg).
According to the π-theorem, the n = 5 variables can be reduced by the k = 3 dimensions to form p = n − k = 5 − 3 = 2 independent dimensionless numbers which are, in case of the stirrer:
• Reynolds number (a dimensionless number describing the fluid flow regime)
• Power number (describing the stirrer and also involves the density of the fluid)

ExamplesConsider this example: Sarah says, "Out of every 10 apples I gather, 1 is rotten.". The rotten-to-gathered ratio is (1 apple) / (10 apples) = 0.1 = 10%, which is a dimensionless quantity.

Another more typical example in physics and engineering is the measure of plane angles. An angle is measured as the ratio of the length of a circle's arc subtended by an angle whose vertex is the centre of the circle to some other length. The ratio, length divided by length, is dimensionless. When using radians as the unit, the length that is compared is the length of the radius of the circle. When using degree as the units, the arc's length is compared to 1/360 of the circumference of the circle.

In the case of the dimensionless quantity π, being the ratio of a circle's circumference to its diameter, the number would be constant regardless of what unit is used to measure a circle's circumference and diameter (eg. centimetres, miles, light-years, etc), as long as the same unit is used for both.

See also.

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

Doron, as usually, sees things differently:

Well that will probably always be the case.

 

[The Man]

Pi has a physical size as distance along the real line but it is dimensionless in terms of an invariant ratio among all circles.

Really a “physical size”, what “size”? Keeping in mind that 100cm has a smaller “physical size” than 10m

In other words, Pi is Cross-contexts w.r.t all circles' curvature degrees, and without such Cross-contexts property circles with different curvature degrees can't be gathered into a one class.

In the same words as before unless you are using the word “circles” in some other context there is simply and still nothing that crosses any context in your purported “Cross-contexts”. Learn the meaning of the word context at least, please Doron.

Yet being gathered into a one class does not mean that all in terms of finite cardinality is equivalent to all in terms of transfinite cardinality, simply because the expression "+1" is an invariant Cross-contexts property w.r.t all infinite collections, exactly as Pi is Cross-contexts w.r.t all circles' curvature degrees.

In other words, the incompleteness of infinite collections is their "+1" expression (in terms of cardinality) as an invariant Cross-contexts property.

OK, who ordered this word salad?

 

[doronshadmi]

A typical response of Context-dependent-only skill(er):

"Really a “physical size”, what “size”? Keeping in mind that 100cm has a smaller “physical size” than 10m"

He\she can't comprehend Pi as a Cross-contexts value among all circles, where curvature's value > 0 AND < ∞ is context-dependent among all circles.

 

[zooterkin]

A typical response of Context-dependent-only skill(er):

"Really a “physical size”, what “size”? Keeping in mind that 100cm has a smaller “physical size” than 10m"

He\she can't comprehend the concept of Distance, unless it is defined by some particular value.

What units do you measure pi in? Are you familiar with what a ratio is?