Felling the axiom of identity

[chikoka]

Hi all

Just wanted to bump these ideas with you before i go .
Zimbabwes life expectancy is only 29 and i just got there , so any time...:^)

Consider an upward opening parabola that crosses the x axis at 2 points.
The roots of the equation of the parabola will be the x values at which it crosses.
If the equation is such that only the tip of the parabola touches the x axis it is considered
to still have 2 roots but that the roots are equal.

Point being ; an upward opening parabola always has 2 roots.

Now what would the roots of an upward opening parabola be if its turning point is situated at y=2?
this would seem to imply that the parabola has no roots since it does not cross the x axis.

But to say so would be to err because it is the case that the parabola actualy does cross the x axis and any upward opening parabola will too, no matter if the fulcrum is at y greater than 0.
Only that it crosses at imaginary x values.2 of them to be exact.

Moral of the story:

The parabola whose turning point is at y > 0 can cross the x axis while it looks to all appearances as if it doesnt.

Isnt that how the axiom of identity is framed?
All thought all things have to submit to the axiom of identity.
Could it be the case that it seems so obvious , just as the parabola with a turning point at y=2 "obviously" does not cross the x axis and yet just as the parabola actualy does have roots (points it crosses the x axis) there could actually be "things" that are not "things".

Violating the axiom of identity.

[chikoka]

I know these ideas are shaky , but could you guys try to build and at the same time criticize it where you see it needs it.
Not to stick to your side of the fence just because historically you've always been there if you see what could build or take apart the op.

[Kelly Jones]

There's no challenge to the law of identity there. Your definition was that an upward opening parabola always has two roots if the vertex is at y≤0; and if positioned at y=0, the two roots are identical. But if the vertex is at y>0, then there aren't any roots, because the vertex doesn't touch the x axis. If the definition was that all upward opening parabolae always have roots, then the definition of roots would be something other than where the parabola crosses the x axis.

[chikoka]

There's no challenge to the law of identity there. Your definition was that an upward opening parabola always has two roots if the vertex is at y≤0; and if positioned at y=0, the two roots are identical. But if the vertex is at y>0, then there aren't any roots, because the vertex doesn't touch the x axis. If the definition was that all upward opening parabolae always have roots, then the definition of roots would be something other than where the parabola crosses the x axis.

That was not my definition. The definition is that all upward opening parabolars do infact have 2 roots.
What definition of root is there other than where the parabolar crosses the x axis?
Thats the definition.
Imaginary numbers are just as objectivly "real" as the "real" numbers.

This is all about complex numbers.

If there can be a value for two simmilar numbers that when multiplied give a negative one, now called an imaginary number , however that works , why cant u have a thing that is not a thing ,now called an imaginary thing?

[Kelly Jones]

I'm guessing that the equation establishing the position of roots will be unchanged for any position of the vertex on the y axis. It doesn't mean that the identity of the parabola changes essentially, when the vertex crosses the y axis at greater than 0. It just means the roots are more like potentials, what would be the case if the vertex crossed at less than 0.

[chikoka]

The parabola example is just an elaborate version of the square root of - 1 , one i gave later on.
I don't know why i didn't just give that one (probably Alzheimer :)) before, so lets stick to that shall we.

[chikoka]

Just so you know , i have no access to the net over the weekend, so you'll just have to wait for monday for my replies.
The debate doesnt realy need me though and you can all carry on.
I'm here for a bit right now though:

Sat Jan 22, 2011 12:24 am
according to theabsolute

or

Fri Jan 21, 2011 16:28 in zimbabwe

[cousinbasil]

chikoka is talking about complex numbers of the form a + bi, where i is defined as the square root of -1.

[cousinbasil]

Leonhard Euler is a towering figure in mathematical physics. He is the "e" which is the base of the natural logarithm ln. Euler brought the use of so-called "imaginary" numbers into mainstream mathematics. Imaginary numbers of the form bi, where i is defined as the square root of -1, are of course no more imaginary than regular numbers, which are now called "real" numbers to distinguish them from the imaginary numbers.

When combined in the form a + bi, where a and b are real numbers, we get what is called a complex number. The complex numbers are often depicted on the ["Argand"] plane, since every complex number has two components. The real numbers we are used to are then regarded as a subset of the complex numbers where b has been set to zero.

It is when b is not set to zero that the magic happens! Quantum mechanics could not be expressed properly without considering the imaginary part of complex numbers. Analog to digital conversions also rely on the properties of i.

Since this is a thread about identity, I will provide a link to the wikipedia treatment of Euler's Identity, which loses some of its intrinsic beauty when rendered in bbcode.

[Dan Rowden]

Anyone who thinks they can violate the law of identity automatically indicates that they have no idea what it even means.

[cousinbasil]

I think chikoka was just being playful with mathematical identities.

A=A, while apparently significant philosophically when taken as a logical statement or [identity] axiom, is in math what is known as a trivial relation, for it offers no further insight into the nature of A. Furthermore, it is equally valid if replace A with B or anything else: B=B. B=B conveys precisely what A=A conveys; neither provide additional information. A trivial relation ends where it begins and covers no ground, as it were.

Euler's identity, on the other hand, is not trivial:
e^i(pi) + 1 = 0

It is a special case of Euler's formula:
e^ix = cos x + i sin x

That the square root of -1 is denoted by i (for "imaginary") is an historical artifact of its long road to acceptance by the mathematical community, the term "imaginary" being at first pejorative.

Euler's formula leads immediately to many simple trigonometric identities and formulas; without it, these same formulas might be difficult if not impossible to prove. All mechanical engineering depends critically on the trigonometric relations. Without Euler's relations, no modern science. (I tend to think ideas know their own time, that is, without Euler himself, then somebody else or perhaps several somebodies would have made these discoveries.)

In other words, A=A can be regarded as trivial in mathematics, although, of course, an implicit part of every proof. That it is implicit and not stated usually is evidence of its triviality.

Given that it was widely known that the square of any number is a positive number, it was decidedly nontrivial to postulate the existence of i.

Once the existence of i has been accepted, it is again no trivial matter to view a natural number as a special case of the set of complex numbers. i completes algebra, as chikoka has noted.

Therefore A=A becomes:

A=A +iB where B is set to zero (B = 0).

In Quantum Mechanics, state vectors are considered complex quantities. If QM operators were to operate on just the real component, the result would be a purely real result. Unfortunately, it would not correspond to anything. Instead, operators operate on complex vectors which have both a real and imaginary component. The result is complex and does not resemble the result of just using the real component alone. When the (complex) result is obtained, the imaginary part is ignored. It turns out the real part does correspond to the desired real-world value, whether it be momentum position, or whatever.

This is an astounding result. It is as if nature itself is complex. The simple postulation of an imaginary component to what we normally see as ordinary numbers makes QM the most exact theory of the physical sciences.

note: edited for grammar

[chikoka]

Anyone who thinks they can violate the law of identity automatically indicates that they have no idea what it even means.

Just as the greeks could argue that anyone who can think of a negative number indicates that they have no idea what a number means.

Please give more fully fleshed arguments as i sort of know what you are going to say and i'm already preparing answers for them.
Just want to here you say them first.

[Diebert van Rhijn]

just as the parabola with a turning point at y=2 "obviously" does not cross the x axis and yet just as the parabola actualy does have roots (points it crosses the x axis) there could actually be "things" that are not "things".

Aren't you here just equalling "things" with the collection of "real numbers", lets say R? What prevents me to say "things" are a collection of all real and imaginary numbers combined, lets say C?

[chikoka]

To pre-empt the married batchelor example:

It is possible for 2 different objects to occupy the same space ...
only if it happens at different times.
It is also possible for 2 different objects to occupy the same space at the
same time... only if it happens at different points along a fifth dimension.
and so on adding an additional dimension each time.

Now each of these examples seem to violate the axiom of identity or at least
an equivalent axiom untill we add and additional dimension to solve the paradox.

Each time there is a way to break the axiom of identity for n dimensions by utilizing
n+1 dimensions.

There is therefore no *absolute* law of identity because we can always violate it by adding
one more dimension and i use the term dimension in as abstract a manner as i can.

There is always a different angle from which to view an objects identity that will violate
that identity.

There is also the case that human language and so conceptualisation was not made to be able
to deal with imaginary things.
This is analogous to the case of richards paradox where one form of language_conceptualisation
; the english language is innapropriate to deal with the issues of set theory and a whole
new language was built to deal with its innadequacies.

Quote
---
In order to introduce one of the thorny issues, let’s consider the set of
all those numbers which can be easily described, say in fewer then twenty
English words. This leads to something called Richard’s Paradox. The set

{x : x is a number which can be described
in fewer than twenty English words}

must be finite since there are only finitely many English words. Now, there
are infinitely many counting numbers (i.e., the natural numbers) and so there
must be some counting number (in fact infinitely many of them) which are not
in our set. So there is a smallest counting number which is not in the set. This
number can be uniquely described as “the smallest counting number which
cannot be described in fewer than twenty English words”. Count them—14
words. So the number must be in the set. But it can’t be in the set. That’s

a contradiction. What is wrong here?

Our naive intuition about sets is wrong here. Not every collection of
numbers with a description is a set. In fact it would be better to stay away
from using languages like English to describe sets. Our first task will be to
build a new language for describing sets, one in which such contradictions
cannot arise.
----



This language still leads to paradoxies too and the search is on for a better language to
describe set theory.

We may not have invented yet the terminology needed to deal with imaginary things since the
one we use leads to paradoxies (think married batchelor) just as the english language_conceptualisation
platform lead to richards paradox in set theory.

This takes me to the parable of the tar baby where the rabit could not free itself for the
tar object and every time it tried to free itself by using a limb to push it away that limb
only got stuck and every effort or strugle doomed him further and he could not get away to the
all the rest that was not the tar baby,

This is simmilar of our attempts to use the englishlanguage_conceptualisation platform to escape
from the axiom_of_identity-tarbaby and reach the rest of reality.
Each thought or contemplation of this axiom (using the englishlanguage_conceptualisation platform)
gets you more and more stuck (convinced) to it just as the rabbit got more and more trapped
the harder he tried to escape into the world.

Our task is now to find a language in which to describe all that is real.

[cousinbasil]

Our first task will be to
build a new language for describing sets, one in which such contradictions
cannot arise.

The first requirement of formal set theory is that it be consistent, that its axioms do not lead to logical contradictions. This would be a requirement of any conceivable formal set theory. As I understand it, Gödel-Rosser demonstrates that any consistent set theory must include statements which can neither be proved nor disproved using the axioms of the theory. Logically, one must assume that at least some of these statements have to be true. That is, they cannot all be assumed to be false; likewise, they cannot all be assumed to be true. If either assumption is made, then the notion of proof itself is rendered moot.

Simply put, consistency implies incompleteness.

I think Gödel actually went a step further, although trying to follow his paper is maddening to a relative set-theory layperson such as myself. I believe he demonstrated that it is possible to construct a statement within a consistent formal set theory that one knows to be true yet cannot prove using only the axioms of the formal system.

[cousinbasil]

Here's a nice link to some quick thoughts of Kurt Gödel from Hao Wang’s biography Reflections on Kurt Gödel, (MIT Press, 1987).

[chikoka]

just as the parabola with a turning point at y=2 "obviously" does not cross the x axis and yet just as the parabola actualy does have roots (points it crosses the x axis) there could actually be "things" that are not "things".

Aren't you here just equalling "things" with the collection of "real numbers", lets say R? What prevents me to say "things" are a collection of all real and imaginary numbers combined, lets say C?

I'm saying while what seems an impossibility to our understanding leads to very pragmatically useful results such as the notion of "i" why cant we find the same sort of thing in philosophy where we have some things that seem to be an impossibility to us such as imaginary things could be true non the less.

[Cahoot]

just as the parabola with a turning point at y=2 "obviously" does not cross the x axis and yet just as the parabola actualy does have roots (points it crosses the x axis) there could actually be "things" that are not "things".

Aren't you here just equalling "things" with the collection of "real numbers", lets say R? What prevents me to say "things" are a collection of all real and imaginary numbers combined, lets say C?

I'm saying while what seems an impossibility to our understanding leads to very pragmatically useful results such as the notion of "i" why cant we find the same sort of thing in philosophy where we have some things that seem to be an impossibility to us such as imaginary things could be true non the less.

Well, we do. We have the view of realized self-perfected natural mind of infinite potentiality. Truth abides in each thing, each thing is a manifestation from and of infinite potentiality, each manifestation is shaped by conditions (including the condition of perception), and conditions are bound to the limitations of relative existence. Understanding the nature of limitations is to understand the nature of truth.

[chikoka]

Our first task will be to
build a new language for describing sets, one in which such contradictions
cannot arise.

The first requirement of formal set theory is that it be consistent, that its axioms do not lead to logical contradictions. This would be a requirement of any conceivable formal set theory. As I understand it, Gödel-Rosser demonstrates that any consistent set theory must include statements which can neither be proved nor disproved using the axioms of the theory. Logically, one must assume that at least some of these statements have to be true. That is, they cannot all be assumed to be false; likewise, they cannot all be assumed to be true. If either assumption is made, then the notion of proof itself is rendered moot.

Simply put, consistency implies incompleteness.

I think Gödel actually went a step further, although trying to follow his paper is maddening to a relative set-theory layperson such as myself. I believe he demonstrated that it is possible to construct a statement within a consistent formal set theory that one knows to be true yet cannot prove using only the axioms of the formal system.

One of the limitations of the present zermelo-fraenklo axioms of set theory is that the contiuum hypothesis, which is a statement about the ordinal nature of the set of infinities,is undescidable under them.
I've read that all we need is a new axiom but that sounds suspiciously circular to me .
Dont know.

[paco]

Anyone who thinks they can violate the law of identity automatically indicates that they have no idea what it even means.

Amen.

[chikoka]

Anyone who thinks they can violate the law of identity automatically indicates that they have no idea what it even means.

Amen.

...and you say this by.....faith? visa vi the "amen".

Or could you find where my logic is weak.

Its not that i don't expect you to BTW.

[paco]

Hi all

Just wanted to bump these ideas with you before i go .
Zimbabwes life expectancy is only 29 and i just got there , so any time...:^)

Consider an upward opening parabola that crosses the x axis at 2 points.
The roots of the equation of the parabola will be the x values at which it crosses.
If the equation is such that only the tip of the parabola touches the x axis it is considered
to still have 2 roots but that the roots are equal.

Point being ; an upward opening parabola always has 2 roots.

Now what would the roots of an upward opening parabola be if its turning point is situated at y=2?
this would seem to imply that the parabola has no roots since it does not cross the x axis.

But to say so would be to err because it is the case that the parabola actualy does cross the x axis and any upward opening parabola will too, no matter if the fulcrum is at y greater than 0.
Only that it crosses at imaginary x values.2 of them to be exact.

Moral of the story:

The parabola whose turning point is at y > 0 can cross the x axis while it looks to all appearances as if it doesnt.

Isnt that how the axiom of identity is framed?
All thought all things have to submit to the axiom of identity.
Could it be the case that it seems so obvious , just as the parabola with a turning point at y=2 "obviously" does not cross the x axis and yet just as the parabola actualy does have roots (points it crosses the x axis) there could actually be "things" that are not "things".

Violating the axiom of identity.

Chikoka,

The law of identity applies to all men. Hence, women are subversital creatures.

[paco]

I think chikoka was just being playful with mathematical identities.

A=A, while apparently significant philosophically when taken as a logical statement or [identity] axiom, is in math what is known as a trivial relation, for it offers no further insight into the nature of A. Furthermore, it is equally valid if replace A with B or anything else: B=B. B=B conveys precisely what A=A conveys; neither provide additional information. A trivial relation ends where it begins and covers no ground, as it were.

Euler's identity, on the other hand, is not trivial:
e^i(pi) + 1 = 0

It is a special case of Euler's formula:
e^ix = cos x + i sin x

That the square root of -1 is denoted by i (for "imaginary") is an historical artifact of its long road to acceptance by the mathematical community, the term "imaginary" being at first pejorative.

Euler's formula leads immediately to many simple trigonometric identities and formulas; without it, these same formulas might be difficult if not impossible to prove. All mechanical engineering depends critically on the trigonometric relations. Without Euler's relations, no modern science. (I tend to think ideas know their own time, that is, without Euler himself, then somebody else or perhaps several somebodies would have made these discoveries.)

In other words, A=A can be regarded as trivial in mathematics, although, of course, an implicit part of every proof. That it is implicit and not stated usually is evidence of its triviality.

Given that it was widely known that the square of any number is a positive number, it was decidedly nontrivial to postulate the existence of i.

Once the existence of i has been accepted, it is again no trivial matter to view a natural number as a special case of the set of complex numbers. i completes algebra, as chikoka has noted.

Therefore A=A becomes:

A=A +iB where B is set to zero (B = 0).

In Quantum Mechanics, state vectors are considered complex quantities. If QM operators were to operate on just the real component, the result would be a purely real result. Unfortunately, it would not correspond to anything. Instead, operators operate on complex vectors which have both a real and imaginary component. The result is complex and does not resemble the result of just using the real component alone. When the (complex) result is obtained, the imaginary part is ignored. It turns out the real part does correspond to the desired real-world value, whether it be momentum position, or whatever.

This is an astounding result. It is as if nature itself is complex. The simple postulation of an imaginary component to what we normally see as ordinary numbers makes QM the most exact theory of the physical sciences.

note: edited for grammar

White

[paco]

To pre-empt the married batchelor example:

It is possible for 2 different objects to occupy the same space ...
only if it happens at different times.
It is also possible for 2 different objects to occupy the same space at the
same time... only if it happens at different points along a fifth dimension.
and so on adding an additional dimension each time.

Now each of these examples seem to violate the axiom of identity or at least
an equivalent axiom untill we add and additional dimension to solve the paradox.

Each time there is a way to break the axiom of identity for n dimensions by utilizing
n+1 dimensions.

There is therefore no *absolute* law of identity because we can always violate it by adding
one more dimension and i use the term dimension in as abstract a manner as i can.

There is always a different angle from which to view an objects identity that will violate
that identity.

There is also the case that human language and so conceptualisation was not made to be able
to deal with imaginary things.
This is analogous to the case of richards paradox where one form of language_conceptualisation
; the english language is innapropriate to deal with the issues of set theory and a whole
new language was built to deal with its innadequacies.

Quote
---
In order to introduce one of the thorny issues, let’s consider the set of
all those numbers which can be easily described, say in fewer then twenty
English words. This leads to something called Richard’s Paradox. The set

{x : x is a number which can be described
in fewer than twenty English words}

must be finite since there are only finitely many English words. Now, there
are infinitely many counting numbers (i.e., the natural numbers) and so there
must be some counting number (in fact infinitely many of them) which are not
in our set. So there is a smallest counting number which is not in the set. This
number can be uniquely described as “the smallest counting number which
cannot be described in fewer than twenty English words”. Count them—14
words. So the number must be in the set. But it can’t be in the set. That’s

a contradiction. What is wrong here?

Our naive intuition about sets is wrong here. Not every collection of
numbers with a description is a set. In fact it would be better to stay away
from using languages like English to describe sets. Our first task will be to
build a new language for describing sets, one in which such contradictions
cannot arise.
----



This language still leads to paradoxies too and the search is on for a better language to
describe set theory.

We may not have invented yet the terminology needed to deal with imaginary things since the
one we use leads to paradoxies (think married batchelor) just as the english language_conceptualisation
platform lead to richards paradox in set theory.

This takes me to the parable of the tar baby where the rabit could not free itself for the
tar object and every time it tried to free itself by using a limb to push it away that limb
only got stuck and every effort or strugle doomed him further and he could not get away to the
all the rest that was not the tar baby,

This is simmilar of our attempts to use the englishlanguage_conceptualisation platform to escape
from the axiom_of_identity-tarbaby and reach the rest of reality.
Each thought or contemplation of this axiom (using the englishlanguage_conceptualisation platform)
gets you more and more stuck (convinced) to it just as the rabbit got more and more trapped
the harder he tried to escape into the world.

Our task is now to find a language in which to describe all that is real.

Statis quo.

[paco]

Anyone who thinks they can violate the law of identity automatically indicates that they have no idea what it even means.

Dan Rowdan,