A hidden assumption

[becomingagodo]

doronshadmi, if I was you I would learn how to do normal mathematics first. Then everybody will be able to understand you.

http://www.amazon.com/Course-Pure-M...=sr_1_1?ie=UTF8&s=books&qid=1205709326&sr=1-1
Try this and this
http://www.amazon.com/Complex-Analy...=sr_1_2?ie=UTF8&s=books&qid=1205709402&sr=1-2

Seriously, if you don't do this then no one will understand you and your ideas will die. Also, I would drop the idear of Cantor being wrong, it is blasphemy.

 

[Apathia]

A probable reason Doron Shadmi has been ignored, for good reason:
http://www.nytimes.com/library/national/science/020999sci-math-crank.html

 

[jsfisher]

Please show this universal principle by using traditional Math, because after all you claim that my system is no more than giving different names to well known mathematical concepts.

Not true. I said some of what you wrote was just renaming stuff and making up new definitions for existing terminology. I also gave you full credit for gibberish. Lets not forget the gibberish.

Your gibberish, by the way, is very persuasive. And I mean that sincerely, with as much sarcasm as I can possibly muster.

 

[MattusMaximus]

Any way in my system the real-line is not a set of infinitely many distinct elements, as defined by the traditional point of view, but it is an atom or more accurately, a non-local atom.

The magnitude of the non-local atom is bigger than the magnitude of any non-finite collection, and I show it by using logic.

Urrrmmm... how about actually defining what the hell an "atom" is in your system, much less a "non-local" (or "local" for that matter) one?

I'd always thought that real mathematicians - you know, the ones who publish in those professional journals you seem to avoid - tended to define new terms when presenting them for the first time.

Woo... woo... woobitty-woo...

 

[MattusMaximus]

In order to get this stuff, please ask yourself a simple question, which is:

How the multitude (or the multiple) is defined?

Wow... he obviously knows what he's talking about... because he's YELLING!!!

 

[doronshadmi]

Are you trying to say that True and False cannot be compared without understanding mathematical symmetry?

I am trying to say this:

Total Symmetry is a non-local atom, understood as unity.

Total Asymmetry is a local atom, understood as isolation.

Nothing is comparable (and therefore not researchable) at total unity state.

Nothing is comparable (and therefore not researchable) at total isolation state.

The comparable (and therefore the researchable) is possible only in the case of total symmetry\total Asymmetry relation.

The multiple is the outcome of total Symmetry\total Asymmetry relation.

Logic (which is a researchable thing) is the outcome of this relation, and not vice versa.

Non-Locality is addressed here in terms of NXOR.

Locality is addressed here in terms of XOR.

Any other logical state is the outcome of NXOR\XOR relation.

No multiple thing has the magnitude of total Symmetry, or in other words, it is permanently incomplete (cannot reach total symmetry's magnitude and therefore permanently open) when compared to total symmetry's magnitude.

The non-finite of traditional Math is understood only in terms of the multitude (infinitely many …).

In that case it is no more than a potential infinity (an incomplete and open infinity) when compared with the complete magnitude of total symmetry (total unity, known also as the non-local ur-element real-line, which is a non-local atom).

If you disagree with me, than please define a researchable thing that is not a total Aymmetry\total Asymmetry relation.

As in you cannot tell the difference between 1 and 0 if you do not understand that 1=1 and 0=0 and 1=/=0?

No I cannot tell the difference between 1 and 0 if Symmetry is not used as the common basis (non-locality) that enables the comparison between at least two isolated (local) Asymmetries.

No pair exists without Symmetry\Asymmetry relation, and True\False pair is a particular case of this relation.

If you disagree with me, than please than please show how True\False are comparable without a common basis (what I call Symmetry, which is invariant under exchange).

Urrrmmm... how about actually defining what the hell an "atom" is in your system, much less a "non-local" (or "local" for that matter) one?

I'd always thought that real mathematicians - you know, the ones who publish in those professional journals you seem to avoid - tended to define new terms when presenting them for the first time.

Well, this is exactly what I tried to do, by addressing it as NXOR\XOR relation in order to help the reader to get Asymmetry\Asymmetry relation's notion.

If you have a better way to do it, I'll be thankful.

This is exactly the reason why I share my ideas with you.

Here is the relevant part of my first post (an edited version of it, which is the result of our dialog):

NXOR:

NXOR (if Symmetry is addresed logically) is the logical connective that enables us to define the relations among abstract/non-abstract elements.

Ths notion is not defined in any book of modern Mathematics, because Modern Mathematics uses it as an hidden assumption.

NXOR can be recognized as a property called memory, which enables us to connect things and research their relations.

Our natural ability to connect between objects (notated by ↔ and known as map, or function) is a hidden assumption of the current formal language.

It has to be understood that in order to define even a 1↔1 map, we need not less than a XOR outcome (notated as 1) and a NXOR outcome (notated as ↔).

Map is a connection (a NXOR outcome) that enables us to define the relations between more than a 1 XOR outcome, and we cannot go beyond cardinal 1 without ↔ between 1,1 .

The number 2 is actually the relations of our own memory as ↔ (as a function) between 1 object XOR 1 object.

So the multiple (or the multitude) is a NXOR\XOR outcome.

 

[doronshadmi]

doronshadmi, if I was you I would learn how to do normal mathematics first. Then everybody will be able to understand you.

http://www.amazon.com/Course-Pure-M...=sr_1_1?ie=UTF8&s=books&qid=1205709326&sr=1-1
Try this and this
http://www.amazon.com/Complex-Analy...=sr_1_2?ie=UTF8&s=books&qid=1205709402&sr=1-2

Seriously, if you don't do this then no one will understand you and your ideas will die. Also, I would drop the idear of Cantor being wrong, it is blasphemy.

Thank you for this reply. My work is based on a very careful learning of the fundamental mathematical notions (and their formalization methods) of the past 2500 years, with a special care of the mathematical development of the 19th and 20th centuries.

The main persons of the modern time are (in my opinion) Peano, Cantor, Dedekind, Frege, Russell and Hilbert, that their fundamental notions shaped the paradigm of Modern mathematics.

If we read Cantor's notions about the non-finite, we define that he mixed between religion and scientific reasoning when he developed his transfinite system. As a result he missed the notion of non-locality (by saying that it belongs to God) instead of using it as a part of his mathematical research.

By mixing between religion and scientific reasoning he made, in my opinion, a mistake.

If "pure" mathematics is a kind of a religion, than to say that Cantor is wrong is indeed blasphemy.

Urrrmmm... how about actually defining what the hell an "atom" is in your system, much less a "non-local" (or "local" for that matter) one?

I did my best in order to define non-local and local atoms in http://www.geocities.com/complementa...y/NXOR-XOR.pdf , page 8

If you have a better idea how to formalize it, it will be great.

 

[jsfisher]

I am trying to say this:

Total Symmetry is a non-local atom, understood as unity.

Let's seen, you are defining "total symmetry" using an undefined term, "non-local atom". Well done. Then again, if it really is understood as unity, why not just call it that?

Total Asymmetry is a local atom, understood as isolation.

Silly me. I would have expected some sort of (dare I say it) symmetry between the meanings of "total symmetry" and "total asymmetry". Instead, I seen one is unity and the other, isolation. Or, did you mean something different for "unity" than the common mathematical meaning?

Nothing is comparable (and therefore not researchable) at total unity state.

Ok, now you have reverted to full gibberish. "Total unity state" sounds more like a Zen philosophy than anything mathematical. Also, you seem to have a private definition for "researchable." Care to share with the rest of us?

By the way, do you realize that much of Set Theory does not involve comparisons? Set membership, for example, doesn't involve comparisons.

<snip>
If you disagree with me, than please define a researchable thing that is not a total Aymmetry\total Asymmetry relation.

Your definitions are incomplete, your concepts are nonsense, so what you ask has no meaning.

<more snip>
Well, this is exactly what I tried to do, by addressing it as NXOR\XOR relation in order to help the reader to get Asymmetry\Asymmetry relation's notion.

If you have a better way to do it, I'll be thankful.

This is exactly the reason why I share my ideas with you.

Oh, there's a whole body of Mathematics knowledge that presents a far better way to do it. It seems to be fairly self-consistent, too, with well-defined terms. You should try it sometime.

 

[doronshadmi]

By the way, do you realize that much of Set Theory does not involve comparisons? Set membership, for example, doesn't involve comparisons.

Axioms are mutually independent self-evident statements that their outcomes are consistent with each other.

Try to define the above without using any comparison.

Try to define anything that its cardinality > 1 without using any comparison.

So, as I said, you are just going to use new names for all things to make it sound more important. You have added no value, only obfuscation.



OR membership? OR membership? You aren't following your own rules. You said in your paper:



You invented these rules; you should at least stick to them. There's no OR, AND, NAND, etc., just XOR and it's negative.

You were the one that said the in and out relations were minimally required for the concept of Membership, and that the concept of set grew from the XOR and NXOR outcomes of the in and out relations. I have provided exactly this information, so where's the set?

By the way, care to explain what, exactly, you meant by "related to the NXOR outcome but...logically not a part of it" in plain English? You seem to just smash the two together and mark all the outcomes as T. Why not F? Why not some of each?

Ok, I see that my pervious post didn't help much, so let's try it this way:

Let X be a domain.

Now let us use 2-valued logic in order to research it.

By using 2-velued logic, there can be at least 2 states that are researchable.

These states are: in X , out X.

Let x be a place holder of some relation with X.

Let 0 represent no relation.

Let 1 represent a relation.

In out
0 0 --> In this case x is both not in and not out of X
0 1 --> In this case x is out of X
1 0 --> In this case x is in X
1 1 --> In this case x is both in and out of X

Let us use Symmetry in order to reduce x to its minimal state.

In this case the difference between 1 and 0 is ignored and we have Symmetry invariance:

In out
0 0 --> In is the same as out.
1 1 --> In is the same as out.


Let us use Asymmetry in order to reduce x to its minimal state.

In this case the difference between 1 and 0 is not ignored and we have Asymmetry variance:

In out
0 1 --> In is not the same as out.
1 0 --> In is not the same as out.

Symmetry alone is total unity where nothing is comparable (and therefore it is not researchable).

Asymmetry alone is total isolation where nothing is comparable (and therefore it is not researchable).

By carefully examine these results, we realized that that these results cannot be researched and defined unless Symmetry and Asymmetry complement each other into a one realm which is simultaneously unified\isolated realm.


In that case 2-valued logic or any other system that its cardinality > 1, cannot be but the outcome of Symmetry\Asymmetry simultaneous relations.

So the multiple is at least, which is Hilbert's organic paradigm of the mathematical science and its logical reasoning:

Z operation Y = X

We can ask: what value is expressed here?

The traditional approach will say that the value's exact representation is the description that can be reduced into a single cardinality so the value is X.

I say that this is wrong and the value's exact representation is not less than:

Z operation Y = X

You may say: but what if the operation does not change the value of Z or Y?

In that case Z=X or Y=X but then there is no operation here but tautology, where some arbitrary representation ("X" in this case) is commonly accepted by different mathematicians in order to avoid confusion.

So different things (mathematicians in this case) must agree about some common thing (a single representation) in order to communicate between them.

So the exact system is not less than the organic one:

Y   Z
 \ /
  X

which gives the exact answer to this question:

What is the minimal condition that enables different things to communicate?

If you disagree with me then you actually say that since a value is only the common, then you and me are actually the same thing, but then the same thing cannot disagree on anything. Since this is not the case then all is left is the organic approach which is Symmetry\Asymmetry outcome.

 

[ben m]

In out
0 0 --> In this case x is both not in and not out of X
0 1 --> In this case x is out of X
1 0 --> In this case x is in X
1 1 --> In this case x is both in and out of X

OK, fine, you've defined a state table. No problem there. Obviously you're using "in" and "out" in a fairly abstract way---you could write an equally-interesting state table with "tall" and "thin". 0 0 --> in this case x is both not-tall and not-thin, and so on.

In out
0 0 --> In is the same as out.
1 1 --> In is the same as out.

In out
0 1 --> In is not the same as out.
1 0 --> In is not the same as out.

OK, you've divided your state table into two categories. Some x fall into category 1 ("00 or 11") and some fall into category 2 ("01 or 10").

Symmetry alone is total unity where nothing is comparable (and therefore it is not researchable).

Asymmetry alone is total isolation where nothing is comparable (and therefore it is not researchable).

By carefully examine these results, we realized that that these results cannot be researched and defined unless Symmetry and Asymmetry complement each other into a one realm which is simultaneously unified\isolated realm.

... and now you're descending into an abyss. "Symmetry alone is total unity?" No it's not. Formally, all we know about "symmetry" from the above is "symmetry is the 00 or 11 loci in-out space". "therefore it is/isn't researchable?" What the heck? "Asymmetry alone is total isolation"? Um, "01 or 10 loci in in-out space" doesn't sound like total isolation, it sounds like two out of the four points in a four-point space. You haven't proven any of your statements, you haven't defined them, you haven't even described what you're getting at.

If you want to stick to group theory, with the normal definitions of "in = a member of group X" and "out = not a member of group X"---well, you can prove that there are no x which are both in X and out of X. There are no x which are both not-in X and not-out of X. So all x are in the "asymmetry" loci and there are no x in the "symmetry" loci. That's the most interesting thing you can possibly say about it.

 

[doronshadmi]

OK, fine, you've defined a state table. No problem there. Obviously you're using "in" and "out" in a fairly abstract way---you could write an equally-interesting state table with "tall" and "thin". 0 0 --> in this case x is both not-tall and not-thin, and so on.



OK, you've divided your state table into two categories. Some x fall into category 1 ("00 or 11") and some fall into category 2 ("01 or 10").



... and now you're descending into an abyss. "Symmetry alone is total unity?" No it's not. Formally, all we know about "symmetry" from the above is "symmetry is the 00 or 11 loci in-out space". "therefore it is/isn't researchable?" What the heck? "Asymmetry alone is total isolation"? Um, "01 or 10 loci in in-out space" doesn't sound like total isolation, it sounds like two out of the four points in a four-point space. You haven't proven any of your statements, you haven't defined them, you haven't even described what you're getting at.

If you want to stick to group theory, with the normal definitions of "in = a member of group X" and "out = not a member of group X"---well, you can prove that there are no x which are both in X and out of X. There are no x which are both not-in X and not-out of X. So all x are in the "asymmetry" loci and there are no x in the "symmetry" loci. That's the most interesting thing you can possibly say about it.

Hi bem,

Thank you for your fine critics.

Please look at the notion behind the notations.

The notion is this:

'0 0' or '1 1' means that in,out the same.

'0 1 or ' 1 0' means that in,out not the same.

Now try to compare between in,out without any common thing.

In this case '0' is totally isolated of '1' and nothing can be compared and concluded.

I call this state total asymmetry.

Now try to compare between in,out where all you have is a common thing (sameness).

In this case '0' and '1' is the same and nothing can be compared and concluded.

I call this state total symmetry.

If you have any idea how to formally address this notion, then you are more than wellcome.


When I realized this, I defined the researchable as the outcome of total symmetry\total asymmetry synthesis.

 

[Gate2501]

Hi bem,

Thank you for your fine critics.

Please look at the notion behind the notations.

The notion is this:

'0 0' or '1 1' means that in,out the same.

'0 1 or ' 1 0' means that in,out not the same.

Now try to compare between in,out without any common thing.

In this case '0' is totally isolated of '1' and nothing can be compared and concluded.

I call this state total asymmetry.

Now try to compare between in,out where all you have is a common thing (sameness).

In this case '0' and '1' is the same and nothing can be compared and concluded.

I call this state total symmetry.

If you have any idea how to formally address this notion, then you are more than wellcome.


When I realized this, I defined the researchable as the outcome of total symmetry\total asymmetry synthesis.

I would venture to theorize that the psychosis on display here is of the variety Faeces Vespadelus finlaysoni.

 

[Richard Masters]

Hi bem,

Thank you for your fine critics.

Please look at the notion behind the notations.

The notion is this:

'0 0' or '1 1' means that in,out the same.

'0 1 or ' 1 0' means that in,out not the same.

Now try to compare between in,out without any common thing.

In this case '0' is totally isolated of '1' and nothing can be compared and concluded.

I call this state total asymmetry.

Now try to compare between in,out where all you have is a common thing (sameness).

In this case '0' and '1' is the same and nothing can be compared and concluded.

I call this state total symmetry.

If you have any idea how to formally address this notion, then you are more than wellcome.


When I realized this, I defined the researchable as the outcome of total symmetry\total asymmetry synthesis.

I've read this material before. Are you plagiarizing?

PDF

It seems to me that you are discussing philosophy and not math. If I understand you correctly, you are basically saying two-value logic is not enough to describe itself.

 

[jsfisher]

By the way, do you realize that much of Set Theory does not involve comparisons? Set membership, for example, doesn't involve comparisons.

Axioms are mutually independent self-evident statements that their outcomes are consistent with each other.

How did we get to axioms? By the way, axioms do not have to be self-evident, and mutual independence isn't always evident (as was the case, for example with the Peano axioms, if I recall correctly). The subject is 'set membership'. I chose 'set membership' specifically because you said it required comparisons. It doesn't.

Try to define the above without using any comparison.

Easy. Membership (with respect to a given set) is a one-variable function with domain of all possible admissible set elements and range of either "is a member" or "is not a member". Something either "is" or "isn't" a set member. No comparison required.

 

[jsfisher]

doronshadmi, did you once write this:

any point is not in the continuum, but an event that breaks the continuum

If yes, do you stand by that statement?

Do you also stand by this statement:

By the standard notion a set is determined by its members, which is the same as saying that the union of all members of X is set X.

Is it also your contention that in the standard notion of a set, the empty set is a member of itself?

 

[doronshadmi]

How did we get to axioms? By the way, axioms do not have to be self-evident,

By "self-evident" I mean that an axiom does not need a proof, but any proof is based on it (on its self-evident).

... and mutual independence isn't always evident (as was the case, for example with the Peano axioms, if I recall correctly).

If not all Peano axioms are mutually-independent, than there exists at least one statement that depends on another statement. But then it is not an axiom but some statement that is deduced from some axiom (where the axiom itself is a self-evident true that does not need any proof).

In other words, axioms must be mutually-independent in order to be considered as axioms.

If Peano axioms are not
The subject is 'set membership'. I chose 'set membership' specifically because you said it required comparisons. It doesn't.

... Membership (with respect to a given set) is a one-variable function with domain of all possible admissible set elements and range of either "is a member" or "is not a member". Something either "is" or "isn't" a set member. No comparison required.

A domain is a common-property that each member shares with any other member, so you cannot avoid element\common-property comparison in order to define if the given element is or isn't a set(domain) member.

 

[doronshadmi]

doronshadmi, did you once write this:

any point is not in the continuum, but an event that breaks the continuum

If yes, do you stand by that statement?

Thank you for reading an old stuff of mine.

In my system cotinuum is a non-local atom (NXOR) and a point is a local atom (XOR).

In that case the continuum (NXOR) is not made of (it is not a collection of) infinitely many elements (XOR outcomes) exactly as the NXOR connective is logically not the XOR connective.

Do you also stand by this statement:

By the standard notion a set is determined by its members, which is the same as saying that the union of all members of X is set X.

Is it also your contention that in the standard notion of a set, the empty set is a member of itself?

No, also by the standard notion the empty set is not a member of itself, and this is exactly why |{}|=0 and not 1.

By the standard notion the container (notated as "{" "}") is used as a hidden assumption.

In my system it is logically defined as the container (the non-local atom) that is not content (a local atom)

Some claims that in my system {_}_ is a member of some container.

My answer is: in my system a member is an element that can be simultaneously in exactly one relation with a given domain.

In that case __ of {_}_ is the exact way to represent a container (no container is fully in or fully out of some domain, and therefore cannot be considered as a content of another container).

 

[jsfisher]

No, also by the standard notion the empty set is not a member of itself, and this is exactly why |{}|=0 and not 1.

Great! So statements you made in that PDF you referenced a few times in this thread are flatly wrong.

I just want to be clear on that.

If you know the statements are wrong, why not correct them?

 

[Zep]

...is DS still going on about this at length and not made any sense yet??

Obsessive or what!

 

[Gravy]

Please look at the notion behind the notations.

Is the notion, "Illness can cause people to believe and promote absurdities?" Am I close?