Scrutinizing Equality "=" more closely

[chikoka]

In set theory there is a construction called a cartesian product.This takes two sets and puts the elements in relation to each other. So if we have A={a;b} and B={c;d} then the cartesian product of these sets A X B would be all the orderd pairs that link all the elements in both sets together. A X B would be {(a,c);(a,d);(b,c);(b,d)}

as mentioned this is a way of relating two sets. So this cartesian product would also be a relation. Any subset of this cartesian product would also be a full blooded relation, so R= {(a,d);(b,d)} would also be a relation (made from a subset of A X B.

An equivalence relation is a particular subset of the cartesian product of a set with itself.
Now equivalence relations come in many flavours like this subset of the natural numbers with themselves. When we consider the ordered pair (2,4) we can see that it is equivalent to (1,2) in the sense that they give the same remainder when the first divides the second.More importantly the remainders both have the same causal profile which should be obvious .It is "0". The same "0" you shouldnt divide by ...etc.Another name for an equivalence relation is a congruence relation so all equilateral triangles are equivalent in an obvious way but only if you ignore other respects from shape (e.g. size)

now we come to the axiom of identity which expresses equivalence between things. This uses the equivalence relation known as equality, "=".

Let us scruitanize this relation more closely. We would like perhaps to say it represents equality in all respects or a total equality.But do total equivalence relations exist?

If two things were totaly equivalent then odering them would violate this totality because one would come before the other. How can (a,a) be total if the first "a" comes before the other one. The very definition of a relation is "orderd pair" .How can the two things we want to totaly equate , totaly equate when they differ in any respect like order?And how can they be related if they are not part of an oderd pair.

But their both from the same set you say.Its the exact same "a". Look at the word underlined there same. You have just invoked *another* equivalence relation that orders that set with that set.In order for that set to be that set it has to be part of an orderd pair that relates it with it-self.So no matter what we do ,we still keep running into order. At first we might be tempted because of this to say that the relation (A,A) through reverse induction *causes* the set "A" to be, but i think you could get away with removing time from this type of causality.

So we are forced to conclude that order (differentiantion) is a fundamental property in judging equivalence which makes total equivalence impossible. How can the two be totaly equivalent when they differ in respect of order.

Then as a last great attempt we could try this;

Since if two sets are a subset of each other then they are the same set ,then perhaps we could imagine "a preceding a" then "swap" them around and have "a preceding a".

My response would be if those two things are equivalent then your attempt doesnt work because you in effect havent swapped anything around. If you say that they not equivalent and that you have swaped the "a's" then i raise the point that what does;

"a preceding a" not equal to "a preceding a" ,

mean if not

A is not equal to A.

Whichever way you want to make sense of equivalence , total equivalence is a "married batchelor" .

[Russell Parr]

It appears you are mixing the 'law of identity' with 'causality' whereas each concept explains a slightly different aspect of reality. As an example of the law of identity, let's choose a word: Apple.

According to the law of identity, "The word 'Apple' at the end of the previous paragraph" is equivalent to "The word 'Apple' at the end of the previous paragraph".

Causality refers to the fundamental ever-changing nature of all things, whereas the law of identity helps to explain the nature of consciousness in relation to all things.

Try these links: Law of Identity and Causality

[chikoka]

"The word 'Apple' at the end of the previous paragraph" is equivalent to "The word 'Apple' at the end of the previous paragraph".

I'm not attacking the "Apple at the end of the previous paragraph". I'm attacking the term "is equivalent". Equivalence cannot make sense without order. Order itself implies differentiation.

[chikoka]

try this link

https://en.wikipedia.org/wiki/Equivalence_relation

[Russell Parr]

"The word 'Apple' at the end of the previous paragraph" is equivalent to "The word 'Apple' at the end of the previous paragraph".

I'm not attacking the "Apple at the end of the previous paragraph". I'm attacking the term "is equivalent". Equivalence cannot make sense without order. Order itself implies differentiation.

The law of identity is specifically in reference to "the word Apple at the end of the previous paragraph". When you refer to "is equivalent to" and speak of 'order', we are no longer talking about the law of identity alone. Instead, we're now delving into the nature of time and causality.

It would help to understand each concept individually first.

[chikoka]

"The word 'Apple' at the end of the previous paragraph" is equivalent to "The word 'Apple' at the end of the previous paragraph".

I'm not attacking the "Apple at the end of the previous paragraph". I'm attacking the term "is equivalent". Equivalence cannot make sense without order. Order itself implies differentiation.

The law of identity is specifically in reference to "the word Apple at the end of the previous paragraph". When you refer to "is equivalent to" and speak of 'order', we are no longer talking about the law of identity alone. Instead, we're now delving into the nature of time and causality.

It would help to understand each concept individually first.

I'd like to understand.
Tell me what that bold part means without equality being involved.

[chikoka]

Russel

I'm saying this not to offend you in any way, but you really should stop this "air of schooling" approach you have whenever we meet. Everythings fine untill "it would help to understand the concepts first".

If by enlightened we mean egoless then you are the most unenlightened person i have ever come accross.

try this link

#again_no_offense

[Russell Parr]

The act of equalizing is indeed a causal, time lapsing process. We cannot escape this fact. The law of identity is in reference to specific moments in time, that is, moments of identification. After such moments, representations of identification necessarily occur in a constantly changing, causal process.

I can't explain it much better than this. I would recommend pouring over those links I shared and try to see why, as a concept, it (the law of identity) is useful in a different way compared to the concept of causality.

Russel

I'm saying this not to offend you in any way, but you really should stop this "air of schooling" approach you have whenever we meet. Everythings fine untill "it would help to understand the concepts first".

If by enlightened we mean egoless then you are the most unenlightened person i have ever come accross.

try this link

#again_no_offense

Oh please :) No offense taken, nor is any intended from my side. I'm just pretty sure that I'm right, and vice versa, I'm sure.

I noticed that you responded 6 minutes after I posted those links, leading me to assume you hardly glanced at them. That led to me suggesting that you try to understand them better first as different concepts. I truly believe the error in your understand resides there.

[chikoka]

The act of equalizing is indeed a causal, time lapsing process.

I've never encountered a more wrong formulation as this.
"Equal" things by definition do not occur at different times.
Events for example. And everything is an event. An event of existing.

Please tell me what "the axiom of identity alone" is. And please dont do a walkabout.

Did you read the link i gave you

[chikoka]

I'm beginning to think you didnt read my post fully..or missed some points.

The whole post was to show that the equivalence relation "=" was incoherent.

A dog is made out of those particular things that always belong to a dog. E.g. 4 legs (i know) but not the name spot.
Simmilarly the only thing that belongs to the axiom of identity in all its forms is this; "=". You seem to identify it with variablles but by definition variables are not the same throughout.

[chikoka]

Russel[/b

I see what you dont understand about the OP.
An order does not always imply time.The order "a is left of b" does not need to be acted out i.e. in time. So in this sense there is no act of equalising. It might take time to state but this ordering in time is trivial and fundamentaly different from the oder "left of"

[Russell Parr]

"Equal" things by definition do not occur at different times.
Events for example. And everything is an event. An event of existing.

Yes, that is why I said the act of equalizing. Put another way: recalling a moment, a moment which represents a specific identification, occurs over time. To me, you are scrutinizing the act of equalizing instead of the law of identity itself, therefore accidentally confusing the two as one.

Let's try this: my niece glanced in a closet and saw a ghost. Upon further inspection, it turned out to be an old bed sheet.

Applying the law of identification: My niece identified a ghost. Even though she did so erroneously, the fact is, in that moment, she identified a ghost. There's nothing else to add to it.

Even the fact that recalling her identification necessarily happens later in time doesn't change the fact that her original identification occurred. Thus, my niece identifying a ghost = my niece identifying a ghost.

Did you read the link i gave you

I'm beginning to think you didnt read my post fully..or missed some points.

The whole post was to show that the equivalence relation "=" was incoherent.

I read some of that wiki, not all of it. I see where are you going with your math example, and I think your original post showed it clearly.

A dog is made out of those particular things that always belong to a dog. E.g. 4 legs (i know) but not the name spot.
Simmilarly the only thing that belongs to the axiom of identity in all its forms is this; "=". You seem to identify it with variablles but by definition variables are not the same throughout.

Variables do indeed change through time, but in the moment of identification, the variables are frozen in time; timeless, so to speak. The "=" refers to this.

I see what you dont understand about the OP.
An order does not always imply time.The order "a is left of b" does not need to be acted out i.e. in time. So in this sense there is no act of equalising. It might take time to state but this ordering in time is trivial and fundamentaly different from the oder "left of"

I still think that you are combining the "=" of the the law of identity with the truth that the act of identifying occurs over a time-lapsing process, thus causing confusion.

Anyway, I'll come back to this at another time. Perhaps someone else can chime-in in the meantime :)

[chikoka]

I still think that you are combining the "=" of the the law of identity with the truth that the act of identifying occurs over a time-lapsing process, thus causing confusion.

I'm seriously not. If i meant that i would be infact saying something equivalent to Saying the numbers from 1 to 10 backwards makes 10 smaller than 1.Thats a strawman, and it seems we've been making unintentional strawmen of each others posts.
i specificaly said this in the OP:

At first we might be tempted because of this to say that the relation (A,A) through reverse induction *causes* the set "A" to be, but i think you could get away with removing time from this type of causality.

I said equivalence implies order.Order implies differentiation. Like i said the time ordering is trivial wont you get it, the time ordering is not part of the definition of equivalence so mentioning the time ordering when expressing the equivalence is like saying the equivalence needs to be expressed in french say and not english.Language is just as irrelevant as time ordering becuase those two are not part of the definition of equivalence.

The ordering the OP reffers to is the ordering that is involved in the definition of equivalence..not in the defining of equivalence.

[Diebert van Rhijn]

But do total equivalence relations exist?

It's axiomatic to the system. That means you always assert equivalence before asserting anything else. Lets call it the reflexive axiom of equality.

In practice nothing is totally or absolutely equivalent to anything unless it would equal totality. The equivalence is a logical abstract only, like existence.

[chikoka]

But do total equivalence relations exist?

It's axiomatic to the system. That means you always assert equivalence before asserting anything else. Lets call it the reflexive axiom of equality.

In practice nothing is totally or absolutely equivalent to anything unless it would equal totality. The equivalence is a logical abstract only, like existence.

Nice.
If that is one of the axioms of "the system", then the system is inconsistent. That very axiom is there for one purpose only ;consitency in the system.
I (think) i have shown it fails at this.

In practice nothing is totally or absolutely equivalent to anything unless it would equal totality

What do you mean by this? Are you admiting that nothing can be absolutely equivalent to itself?

[chikoka]

"And i'm here for one thing Mr Anderson" to throw a spanner in the works"

:)

[chikoka]

Aah ! So they may be a way out-"says the never ending dialectic in my head".

Differentiation impies order but order may not imply differentiation.

We can imagine the order expressed in equality has "scale" zero or can be drawn as a point, just as how "0" expresses nothingness but still has properties.

#takes_spanner_back_out ...lol

But wait...

in order for the term orderscale to equal zero, this invokes another equivalence relation(between the order scale and zero), and for the orderscale of this new ER we need to invoke yet another ER for this second orderscale to equal zero as well and so on infinatley.

Is this regress vicious? Is there really a way out here?

maybe we can deny an infinite regress by saying all those orderscales are the same(another ER) and the orderscale of that samenesses ER is the same...

I think this second regress is also viscious....

So....i'm not too well versed on the difference between a virtuous and a viscious regress.

Perhaps someone could justify calling either (or both) of the above infinite regresses as virtuose and not viscious.

What do you think?

[chikoka]

Just been schooled : sorry...both are vicious

At every point in the regress we simply deffer judgement...i.e. equivalence is never reached.

[chikoka]

Just some notes on where i think you dont see what i mean Russel.


The word compare implies the number 2. If you compare a (1) thing to (2) itself. you also impy 2. The number 2 has parts , namely 1 and 1. These cannot be the same 1 otherwise the number 2 would be equal to 1.



I dont want to confuse you more , but the above exposition is Fractal

[Russell Parr]

I see you're fascinated by math, chikoka, and so am I at times. I enjoy learning about sacred geometry, fractal patterns in nature, and so on. Math does indeed have deep implications when it comes to the structure of reality. But the truth is, math doesn't have anything to do with the law of identity, as it is purely philosophical. In fact, the law of identity precedes mathematics; you cannot get to math without there first being the law of identity.

As far as equivalence goes, equivalence doesn't really occur in nature, as all things are different in some degree from one moment to the next. I think we can agree on this?

The law of identity, or, A=A (which is not a math equation; it's simply a symbolic representation of the axiom), is a philosophical law that explains the nature of consciousness, and it's role in reality. Using math to explain it: where 1 plus 2 equals 3, 1 is identified as 1 and nothing else. 2 is identified as 2, and so on. This even applies to "plus". Plus is identified as plus, equals as equals, and so on.

It's really that simple. The implications from there become more profound, but we must understand this first and foremost. It really can't go anywhere from here without having this well understood.

Don't take this as brow-beating. I am merely explaining something that you have badly misunderstood thus far. Hashtag not brow beating ;)

[Bobo]

Let's look at difference and call it a relation:

A != B

In this case "A" is different than "B", someone might object saying that they are both letters:

A != not-a-letter

You might say that "not-a-letter" is just a bunch of letters, it is not clear what not-a-letter would be beside the letters.

A != nothing

Nothing too is a word, probably what the person using this word is trying to convey is that "nothing" can have the meaning of not-something even if it is a word and thus something it means the contrary of that. So let's try to express that feeling:

A !=

And, error, we don't have an relation anymore which was a prerequisite to have a difference. Difference failed every test like equivalence, if we then say that ==!= have we said something that hasn't been said before?

[chikoka]

I see you're fascinated by math, chikoka, and so am I at times. I enjoy learning about sacred geometry, fractal patterns in nature, and so on. Math does indeed have deep implications when it comes to the structure of reality. But the truth is, math doesn't have anything to do with the law of identity, as it is purely philosophical. In fact, the law of identity precedes mathematics; you cannot get to math without there first being the law of identity.

As far as equivalence goes, equivalence doesn't really occur in nature, as all things are different in some degree from one moment to the next. I think we can agree on this?

The law of identity, or, A=A (which is not a math equation; it's simply a symbolic representation of the axiom), is a philosophical law that explains the nature of consciousness, and it's role in reality. Using math to explain it: where 1 plus 2 equals 3, 1 is identified as 1 and nothing else. 2 is identified as 2, and so on. This even applies to "plus". Plus is identified as plus, equals as equals, and so on.

It's really that simple. The implications from there become more profound, but we must understand this first and foremost. It really can't go anywhere from here without having this well understood.

Don't take this as brow-beating. I am merely explaining something that you have badly misunderstood thus far. Hashtag not brow beating ;)

Can this philosophical law be explained without using math?

[chikoka]

Let's look at difference and call it a relation:

A != B

In this case "A" is different than "B", someone might object saying that they are both letters:

A != not-a-letter

You might say that "not-a-letter" is just a bunch of letters, it is not clear what not-a-letter would be beside the letters.

A != nothing

Nothing too is a word, probably what the person using this word is trying to convey is that "nothing" can have the meaning of not-something even if it is a word and thus something it means the contrary of that. So let's try to express that feeling:

A !=

And, error, we don't have an relation anymore which was a prerequisite to have a difference. Difference failed every test like equivalence, if we then say that ==!= have we said something that hasn't been said before?

Hahahahaah :)

ALL THIS IS VALID REASONING!!!!!!!!!!!!!
you said it like it was a reducto absurdum proof that deifference cant be a relation. For there to be any difference there must first be a simmilarity (i.e. they are both being compared. If they share a simmilarity how can they be totaly different.

[Russell Parr]

A dog is a dog because it is identified as a dog.

If you think this is a literally a math equation, then there's nothing else I can tell you, other than: you've really got to work on your understanding on the basics of philosophy.

[Diebert van Rhijn]

If that is one of the axioms of "the system", then the system is inconsistent. That very axiom is there for one purpose only ;consitency in the system. I (think) i have shown it fails at this.

Obviously if you'd stop asserting the axion (hold it as true) a logical system will not function. That's why it's called axiom! So are you trying to prove a logical system does not function without its axioms to be true? The question you should ask is what remains after doing that, since you don't have a system anymore to organize your thoughts or definitions in. This holds true for specific logical systems as well for general mentation or observation.

In practice nothing is totally or absolutely equivalent to anything unless it would equal totality

What do you mean by this? Are you admiting that nothing can be absolutely equivalent to itself?

This is about the truth of change, which is also part of causality. There will always be differences between a thing and whatever it is one is comparing it to. So the equivalence is approximation, like everything else that is being observed. The abstract realm consist of idealized reality where the normal flux of reality is artificially suspended. But that is because you accept the approximation of the abstract, like you accept a drawing as representation only.