The Foundations of a Non-Naive Mathematics

[arildno]

I with you on that one. I would prefer mysticism to religion any day of the week. But look at how many times people in history have looked silly for trying to mix mysticism/religion with science; for example, Kepler tried to explain the orbits of the planets in terms of known 3D polyhedrons of the time because they were "perfect" or "divine". That's garbage. It was only after he let go of the idea of "perfect" shapes that considered using elliptical orbits and sucessfully formulated his laws.

Perhaps the most blatant example of misplaced mysticism may be found in the philosopher Hegel.
He became so enthused by the idea that he'd found an overarching system(which "integrated" religion/morality/society/science/nature) that his "philosophy of nature" is just about the worst crank writing imaginable.

However, his earlier, less connected works (in which he didn't suffer from his unifying obsession to the same degree as later on) does contain bits an pieces of rather poignant and subtle social analysis.

If he had had less than the crank nature in him, he might well have developed a truly interesting social theory (placing him in the company of Max Weber, and others)

 

[sparkster]

Perhaps the most blatant example of misplaced mysticism may be found in the philosopher Hegel.
He became so enthused by the idea that he'd found an overarching system(which "integrated" religion/morality/society/science/nature) that his "philosophy of nature" is just about the worst crank writing imaginable.

However, his earlier, less connected works (in which he didn't suffer from his unifying obsession to the same degree as later on) does contain bits an pieces of rather poignant and subtle social analysis.

If he had had less than the crank nature in him, he might well have developed a truly interesting social theory (placing him in the company of Max Weber, and others)

Yes, Hegel is a good example. My most favorite of his statements is when he claimed that the scientists were wasting their time searching for heavenly bodies--if they would just study philosophy they would realize that since 7 is the number of perfection, there can only be 7 and no more.

Incidentally, this brings me back to Lama/Dorian/Organic/etc. In E.T. Bells' Men of Mathematics, he is discussing the immortal Gauss, and what Gauss had to say about people like Hegel. He states that "those who wish to peck away at the foundations of mathematics would do well to sharpen their dull beaks on some hard mathematics first."

 

[arildno]

Or another of Gauss' immortal comments on Hegel:
"Der Mann ist verrückt"

 

[e(ho0n3]

LOL. I'm sure all matheticians have that to say about each other.

 

[matt grime]

So instead of being able to give a tautology you've now had to remove it from your definition. Does this mean you're going to remove all your pointless posts where you refused to give a tautology and just reposted the same silly definition.

Now, if you could just remove direction from your new definition, which you've not defined then you're getting somewhere. Note, it appears that you're presuming the real numbers exist, but what are they in your system? You've just presumed they exist when they don't necessarily since they are just a construct in our system that you reject. Also note that you've admitted something we've been telling you for a while that you've ignored: that the organic numbers which you state are BASED on N, cannot therefore be more basic than them since you;ve used N in its definition.

 

[Lama]

So instead of being able to give a tautology you've now had to remove it from your definition

On the contrary, I added two statements, each one for each definition, see for yourself in: http://www.geocities.com/complementarytheory/My-first-axioms.pdf.


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


Dear persons,



Thank you very much for your open-hearted replies.

1) I do not start every single concept from scratch, so words (for example) like, finite, and scale have their standard meaning.

2) Lowest bound or Highest bound are simply the edge values of a non-empty ordered set.

3) Scale factor is determined by the ratio of any R member to the entire R members.

As I wrote to ex-xian, I cannot do the whole work alone to convince someone about my point of view, if a person choose not to move from his spot, where things looks fundamentally different.

In short, you cannot stand in some place and say: “Please convince me to move to your place by describing what do you see from your place”.

I can do my best to describe (define) what I see, but the best advice that I can say is:
Please come to my place and see it by yourself, and only then we can argue about our different interpretation about what we see from this place.

Again, no one of you did this simple step, which is: To come to my place and see things by using his own eyes.

And this is the reason why I said that I have found that the most persons here acting like full_time_job bodyguards.

It is very important to check any new idea before we air our view about it, but it cannot be done if we are full_time_job bodyguards, which means, we do nothing to really see something by our own eyes, and instead we want that the mountain will come to us, instead of us to come and at least first see the mountain by using our own eyes.

And when we see the mountain by using our own eyes, then and only then we are in a position to decide if we want to clime on it, or not.

You have to understand that if you do not let yourself to see things by using your own eyes, we cannot move further in our dialog.


In short, I need your active participation on order to develop the dialog between us.


I hope that I explained my point of view about the dialog between you and me.


Please reply your remarks and insights.



Yours,

Lama

 

[matt grime]

1) is patently false since you reject the proper definition of cardinality for example, so at best you occasionally use the proper definition of words, however, none of the words you've used in this item have a formal standard mathematical meaning that we can apply here with any certainty that doesn't require you to accept standard mathematics first. And even then "scale" is not well-understood.

2) Lowest and Highest imply an ordering, you've not proven any such exists in your system: what is the highest and lowest bound of the circle? See, you need to actually have the Real numbers as we know them already extant, and that is a contradictory position for you to adopt.

3) makes no sense (how do you define the ratio of an element of that set to the set? and that is a set yo'uve not even bothered to define)

and since you now say that you've not removed the requirement to be defined by tautology you muist still have some proposition in mind, so for the, what, 10th time what is that proposition?

 

[Lama]

what is the highest and lowest bound of the circle.

x1 and y1 are R members.

x1 and x2 are its lowest and highest bounds where x1 < x2, and its length is |x2-x1|*xs, where xs is any R member.

Matt why do you ignore my request for help that can be seen in my previous post?

 

[matt grime]

you could also clarify what you mean by "indivisible" since I can "divide" the interval [0,2] into [0,1] and (1,2].

As it is your grasp of set theory is looking even shakier, and I bet I can make all your statements apply to Q or C, or P(3) if I felt like it, since they are so ill-defined. Of course, you are still presuming the existence in your model of something that you've not shown to exist. Your definitions are becoming increasingly circular: your set of reals is the set of points and intervals somehow, presumably, yet you can't define a point or interval without using the properties of your reals (order and equality, it is for instance not part of your axioms that this set even has an ordering, yet you use it).

 

[matt grime]

Where did you set of reals obtain its ordering? Why is length of an interval not well defined? What is R in your system? What is the tautology that defines x1 and y1? were did the abs value operator come from?

 

[Lama]

you could also clarify what you mean by "indivisible" since I can "divide" the interval [0,2] into [0,1] and (1,2].

I clearly and simply clarify what I mean by "indivisible".

In my system [0,1] and [1,2] are two {.}_AND_{._.} indivisible elements where [0,2] is also {.}_AND_{._.} indivisible element.

Also I have a different interpretation to (r1,r2].

Again you cannot see (understand) my framework in terms of your framework,
and if you can't grasp this, then we cannot communicate.

 

[matt grime]

It is not that I cannot understand your terms, but that you haven't defined them. incidentally the correct mathematical term is connected, not indivisible.

You have still not given this tautology that you say exists.
A note for everyone: shall we refuse to comment or in anyway communicate in this thread until that simple request has been met?

Let us absolutely make clear what is required:

something is alleged to be defined (only) by a tautology. That thing is a set. A tautology is a proposition. Please give the (an) example of this (or any other) tautology that defines this (or any other) set.

Until you can do that you have no need to post anything else, Doron.

 

[Lama]

Matt, please read this again and pay attention to * and * propositions.

Tautology:
x implies x (An example: suppose Paul is not lying. Whoever is not lying, is telling the truth Therefore, Paul is telling the truth) http://en.wikipedia.org/wiki/Tautology.
(tautology is also known as the opposite of a contradiction).

(EDIT: instead of the above definition, I change Tautology to: The identity of a thing to itself.

It means that in my framework we do not need 'if, then' to define a Tautology)


Set:
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.

Multiset:
A set-like object in which order is ignored, but multiplicity is explicitly significant.

Singleton set:
A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set.

Urelement:(no internal parts)
An urelement contains no elements, belongs to some set, and is not identical with the empty set http://mathworld.wolfram.com/Urelement.html.

A definition for a point:
A singleton set p that can be defined only by tautology* ('='), where p has no internal parts.

A definition for an interval (segment):
A singleton set s that can be defined by tautology* ('=') and ('<' or '>'), where s has no internal parts.

(Sign '<' means that we look at the segment from left to the right.
Sign '>' means that we look at the segment from right to the left.
When both '<' , '>' are used then we have a directionless segment.)

By the definition of a segment we get {._.}, which is the indivisible singleton set that exists between any two {.}.
Now we have the minimal building-blocks that allow us to define the standard R members.

(edit:

*A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.

*A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.)

The axiom of independency:
p and s cannot be defined by each other.

By the above axiom {.} and {._.} are independed building blocks.

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Now please tell us dear Matt why * and * are not tautological propositions?

(Logical statements in which the conclusions are equivalent to the premises)

incidentally the correct mathematical term is connected, not indivisible.

No, {._.} or {.} are indivisible exactly as I say.

If you want to force your point of view on my framework then do not west your time, beacuse it will nor work and you will stay in your spot.

That thing is a set

No Matt, that thing is a content of a set, which its existence cleary and simply defined by an axiom and astatement for each case of '.' or '_' .

 

[matt grime]

They are not even propositions, Doron.

 

[Lama]

Ok Matt since you say that you understand my first two axioms, then please help me and write their logical propositions (statmants), because I do not know what do you mean when you say 'propositions' in this case.

Thank you.

 

[matt grime]

A proposition is a statement of the form If X then Y where X and Y are some mathematical statements to which we can assign truth values.

Example:

If an integer, n, is divisible by 4, then n is even.

We start from the antecedent (n is divisible by 4) and come to the conclusion (n is even).

This proposition is true.

A proposition here would be:

If P is a point then p is defined only by a tautology.

Note, that that in itself is NOT the tautology that defines p, if it were you'd have a circular argument, which is not the tautology you require since it does not tell us what p is (ie offer any indication p even exists). You cannot rely on the definition of p to define p like that, since it does not prove that any such "p" exists. You must therefore offer another model of some such p, and by doing so you will contradict the minimality that it is only defined by (this alleged) tautology as you will be defining it in some other way. Sorry, but as we've been telling you your definition is not consistent.



In short, you cannot define things by tautologies like this, and you cannot prove that any such p exists since it contradicts its own definition. That is not to say it cannot exist, but that we cannot know it exists.

It makes someone wonder what you thought you were doing? An ungenerous person might think you'd been displaying ignorance of things you claim to understand. After all, it's you who's been posting the wikipedia definitions, did you not understand them?

And I didn't say I understood your two axioms, I said I was happy to accept them, and asked you to provide examples and models of things satisfying those axioms. You didn't, and, it appears, can't.


An example of a tautology (in mathematics)


If A then (if A then A)

which is equivalen to (not(A)) or ((not(A)) or A), which is equivalent to (not(A))orA, which is always true irrespective of the truth of A.

note that this of course requires you to have boolean logic and excluded middle and so, some things you reject, so actually you are once again possibly misusing a mathematical term, since your tautologies, may not be our tautologies, even after you've made a propositional form out of it. More contradictions again.

 

[Lama]

The best I can do is:

If a content of a set is a singleton and a urelement and has no directions, then it is a point.

If a content of a set is a singleton and a urelement and also has directions, then it is a segment.

I understans tautology in a different way than you.

For me tautology is not if,then proposition but a self avident existence of a thing to itself for example:

{}={}, {.}={.}, {._.}={._.}, {__}={__}

This is the reason why I call '=' 'only by tautology'.

 

[matt grime]

And you have removed the requirement of some tautology then. Which is good, because that was wrong.

Now, you just need to explain what a direction is.

It is still not clear of you've "defined R" and now define points and intervals in terms of R, or if you are defining points and intervals and then will define what R is.

Which of those two orders is it?

It appears, since you are talking about direction, and equality and ordering (<,> etc) that R must exist in your alleged system, so what is R?

 

[matt grime]

So, your next requirement is to tell us all what YOUR reals are. Until you do that we can do nothing.

 

[Lama]

Ok, let us write is again:

If a content of a set is a singleton and a urelement and has no directions, then it is a point.

If a content of a set is a singleton and a urelement and also has directions, then it is a segment.


(more detailed explanation of the first two definitions:

Let us examine these first two definitions by using the symmetry concept:

1) {.} content is the most symmetrical (the most "tight" on itself) content of a non-empty set.

It means that the direction concept does not exist yet and '.' can be defined only by '=' (tautology), which is the identity of '.' to itself.

2) {._.} content is the first content that "breaks" the most "tight" symmetry of {.} content, and now in addition to '=' by tautology (which is the identity of '._.' to itself) we have for the first time an existing direction '<' left-right, '>' right-left and also '<>' no-direction, which is different from the most "tight" non-empty element '.'

In short, by these two first definitions we get the different non-empty and indivisible contents '.'(a point) or '_'(a segment) .

In short, in both definitions (of {.} or {._.}) the conclusion cannot be different from the premise (mathworld.wolfram.com/Tautology.html)

As you see, in my framework '<','>' symbols have a deeper meaning then 'order'.

Actually, in order to talk about 'order' we first need a 'direction'.

 

[matt grime]

For me tautology is not if,then proposition but a self avident existence of a thing to itself for example:

a tautology is not required to be a proposition, merely a statement that is true irrespective of the truth values of the elements in it.

A=>A is a tautology

A=>B is not

A<=>A is not

(A=>(B=>C) => (A=>C) is i think a tautology.

it is not a self evident truth.

so stop saying tautology and pointing to the mathematical definition as if it is the one you are using. everything is in some sense equivalent to itself (tautologous in your new fangled interpretation) so it is impossible to say what is only defined by a tautology, since we cannot define it in any other way. that attempt at a definition is illogical.

Now, what is R in your system?

 

[matt grime]

I almost give up: what you've shown is that the definition of something implies itself. that is always true and not that useful directly. of course, your use of the word definition is different from everyone elses.


Now, what is R? in you system of course.

 

[Lama]

Since you missed it then here it is again:

If a content of a set is a singleton and a urelement and has no directions, then it is a point.

If a content of a set is a singleton and a urelement and also has directions, then it is a segment.


(more detailed explanation of the first two definitions:

Let us examine these first two definitions by using the symmetry concept:

1) {.} content is the most symmetrical (the most "tight" on itself) content of a non-empty set.

It means that the direction concept does not exist yet and '.' can be defined only by '=' (tautology), which is the identity of '.' to itself.

2) {._.} content is the first content that "breaks" the most "tight" symmetry of {.} content, and now in addition to '=' by tautology (which is the identity of '._.' to itself) we have for the first time an existing direction '<' left-right, '>' right-left and also '<>' no-direction, which is different from the most "tight" non-empty element '.'

In short, by these two first definitions we get the different non-empty and indivisible contents '.'(a point) or '_'(a segment) .

In short, in both definitions (of {.} or {._.}) the conclusion cannot be different from the premise (mathworld.wolfram.com/Tautology.html)

As you see, in my framework '<','>' symbols have a deeper meaning then 'order'.

Actually, in order to talk about 'order' we first need a 'direction'.

Also by my system a=a is a tautology, and we do not need 'if,then' to define it.

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

Do you have any remarks before we visit R?

(I have changed Tautology to: The identity of a thing to itself)

 

[Lama]

Matt, there are some points that we have to check before we examine R by my framework:



1) In an included-middle reasoning contradiction is not used because any two opposites are simultaneously preventing/defining their middle domain.

2) A thing is defined by its identity to itself, and we do not need the 'if, then' proposition (as we do in an excluded-middle reasoning) in order to define the existence of something.

3) By this reasoning we distinguish between a true statement and a tautology, which by included-middle reasoning is simpler and stronger then any existence that must an 'if, then' to exist.

4) The included-middle interpretation of a Tautology is circular only if we look at it from an excluded-middle reasoning. But then we must realize that we see and understand things which are not from an included-middle point of view.

If you stay in standard point of view, then you cannot understand my new framework (and in this sentence I used the standard 'if, then' reasoning)

Included-middle reasoning is the logic of mutual communication between opposites (and in this sentence I used the self identity of a thing to itself)

In short, I gave just now a simple demonstration that I understand very well the standard point of view.

Can you do something which demonstrates that you are able to write something from an included-middle logical reasoning?

If you cannot do it at this stage, then your mathematical skills will not help you to understand my system (and here I used again the 'if, then' reasoning).

An included-middle reasoning can be understood only by an included-middle reasoning (and here I used an included-middle reasoning).

Some examples:

By my system 4 not= 2+2.

By my system 4=4 and 2+2=2+2.

Now can say: But you can do nothing with these two trivial and circular equations, right?

My answer is: In included-middle reasoning we can do very interesting things because 4 not= 2+2, for example:

For example, let us represent the variations of cardinals(*) 2,3,4:

Let Redundancy be more then one copy of the same value can be found.

Let Uncertainty be more than one unique value can be found.

Let XOR be #

Let a=0,b=1,c=2,d=3 then we get:

    b  b                                        
    #  #                                        
   {a, a,  {a, b}                               
    .  .    .  .                                
    |  |    |  |                                
    |__|_   |__|                                
    |       |                                   
                                                
    {x,x}  {{x},x}                              
                                                
                                                
                                 
                                                
                                                
     c  c  c                                    
     #  #  #                                    
     b  b  b          b  b                      
     #  #  #          #  #                      
    {a, a, a,}       {a, a, c}       {a, b, b}  
     .  .  .          .  .  .         .  .  .   
     |  |  |          |  |  |         |  |  |   
     |  |  |          |__|_ |         |__|_ |   
     |  |  |          |     |         |     |   
     |__|__|_         |_____|         |_____|   
     |                |               |         
     |                |               |         
    {{x,x,x}         {{x,x},x}       {{x},x},x}

              
                [COLOR=Red][B]Uncertainty[/B][/COLOR]
  <-[B][COLOR=Blue]Redundancy[/COLOR][/B]->^
    d  d  d  d  |
    #  #  #  #  |
    c  c  c  c  |
    #  #  #  #  |
    b  b  b  b  |
    #  #  #  #  |
   {a, a, a, a} V   {a, b, c, d}
    .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |
    |  |  |  |       |__|  |  |
    |  |  |  |       |     |  | <--(Standard Math language uses only 
    |  |  |  |       |_____|  |     this no-redundancy_
    |  |  |  |       |        |     no-uncertainty_symmetry)
    |__|__|__|_      |________|
    |                |
    ={x,x,x,x}       ={{{{x},x},x},x}



============>>>

                [COLOR=Red][B]Uncertainty[/B][/COLOR]
  <-[B][COLOR=Blue]Redundancy[/COLOR][/B]->^
    d  d  d  d  |          d  d             d  d
    #  #  #  #  |          #  #             #  #        
    c  c  c  c  |          c  c             c  c
    #  #  #  #  |          #  #             #  #   
    b  b  b  b  |    b  b  b  b             b  b       b  b  b  b
    #  #  #  #  |    #  #  #  #             #  #       #  #  #  #   
   {a, a, a, a} V   {a, a, a, a}     {a, b, a, a}     {a, a, a, a}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |  |  |  |       |__|_ |  |       |__|  |  |       |__|_ |__|_
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |__|__|__|_      |_____|__|_      |_____|__|_      |_____|____
    |                |                |                |
    {x,x,x,x}        {{x,x},x,x}      {{{x},x},x,x}    {{x,x},{x,x}}     
 
                                      c  c  c
                                      #  #  #      
          b  b                        b  b  b          b  b
          #  #                        #  #  #          #  #         
   {a, b, a, a}     {a, b, a, b}     {a, a, a, d}     {a, a, c, d}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |__|  |__|_      |__|  |__|       |  |  |  |       |__|_ |  |
    |     |          |     |          |  |  |  |       |     |  |
    |     |          |     |          |__|__|_ |       |_____|  |
    |     |          |     |          |        |       |        |
    |_____|____      |_____|____      |________|       |________|
    |                |                |                |
    {{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x}      {{{x,x},x},x}
 
    a, b, c, d}
    .  .  .  .
    |  |  |  |
    |__|  |  |
    |     |  | <--(Standard Math language uses only this
    |_____|  |     no-redundancy_no-uncertainty_symmetry)
    |        |
    |________|
    |    
    {{{{x},x},x},x}

Also please pay attantion that the last form is the standard R members 0,1,2,3:

 0 = .

 1 = 0[COLOR=Blue]______1[/COLOR]

 2 = 0[COLOR=DarkRed]____________2[/COLOR]  

 3 = 0[COLOR=Green]___________________3[/COLOR]

And the standrard [B]R[/B] is nothing but the above 2-D representation 
in a 1-D representation:

0[COLOR=Blue]______1[/COLOR][COLOR=DarkRed]______2[/COLOR][COLOR=Green]______3[/COLOR]

And because no R member is both Multiset_AND_Set, I call it: The "shadow" of my new number system.

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

(*) Please pay attention that we are not talking about the natural numbers 2,3,4 but the cardinals 2,3,4.

It means that our Organic Natural Numbers are actually a general representation of information-trees, where any finite quantity of names of R members can be described by them, for example:

Instead of a=0,b=1,c=2,d=3 we can use a=0,b=.5,c=3,d=pi and then we use the same information-trees above.


I called these general information-trees 'Organic Natural Numbers' because:

1) These information-trees of cardinals are always having a structure, which is based on N members.

2) They can be used as natural (not forced) and general representation for any interaction between complementary states, which simultaneously preventing/defining their middle domain.

3) Because no R member is divisible by my system, it has its own organic (complete) unique and independent self existence.

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



Matt, please reply your comments to this post before we continue, thank you.

 

[sparkster]

Since you're trying to redefine another word, in this case "tautology," it's better to just take it out of your definition.

1) I do not start every single concept from scratch, so words (for example) like, finite, and scale have their standard meaning.

If "finite" has it's standard mathematical meaning, then you are assuming the existence of the natural numbers. Here's Rudin's definition of a finite set

For any positive integer (natural number) n, let J_n be the set whose elements are the integers 1, 2, ..., n; let J be the set consisting of all positive integers (or natural numbers)...A is finite if A has the same cardinality of J_n for some n

Again, if you're using finite as a concept in your definitions, you must accept the standard definitions for the natural numbers (and also cardinality). If you do not concede this, then you cannot use the concept of finite, or else define it yourself.

By accepting the natural numbers, you must either construct them yourself, or accept the standard construction. This in turn forces you to accept standard ZF set theory. Here are the axioms:


1. The empty set is a set as is every member of a set.

2. If X is a set and, for each x in X, P(x) is a proposition, then
{x in X: P(x)} is a set.

3. If X is a set and Y is a set, so is {X, Y}.

4. If X is a set, {z: z in x for some x in X} is a set.
This set the “union of X”.

5. If X is a nonempty set, {z: zin x for each x in X} is a set.
This set is the “intersection of X”

6. If X is a set, {z: z is a subset of X} is a set.
This set is called the power set of X.

7. The set N of all natural numbers is a set.

8. No set is a member of itself.

9. If X is a set and Y is a set, so is X cross Y.

I left off the axiom of choice, which is often included.


Are you prepared to concede all this to have your real numbers? If not, you must provide your own construction of the naturals.

 

[matt grime]

My simple comment is that you evidently do not understand the distinction between equivalent and equal, but heck, you're not very well educated mathematically, so whose fault is that?

99% of your last post is nonsensical unless you actually get round to defining all of the terms you use so that the rest of the world might know what you're doing. You should also put a huge disclaimer saying:

contrary to what I said earlier, when I use words I do not in fact use them with their proper well understood meanings.

You're talking about something you've labelled R butnot actually saying what it is. AGAIN.

You also don't seem to understand that, whilst you say your interpretation of "tautology" is different from that in formal mathematics, you cannot then cite its definition! And there is a difference between statement being tautologous, and it being true, since everything satisfies its own definition, then including this "tautology" is unnecessary.
It reminds me very much of your confusion over the Collatz conjecture, that you thought you'd figured out.

 

[Lama]

Matt,

My simple comment is that you evidently do not understand the distinction between equivalent and equal, but heck, you're not very well educated mathematically, so whose fault is that?

I know exactly the difference between:

_____
_____     AND    ______   NOTATIONS
_____            ______

The first one is stronger then the other.

By standard Math, if internal structural properties are omitted and only quantity remains, then and only then (and because of this trivial, again not abstract but trivial approach) there is a difference between 'identical' and 'equal'.

But you see, in my framework a '=' notation has one and only one meaning, which is the identity of a thing to itself, and the reason is:

Any given element in my system is unique and cannot be represented by any other element but itself.

And by this fundamental approach, my system is sensitive to any information change between elements, and can use these differences to make richer and deeper Math.

You are still standing in your spot and observe my system from your standard point of view, and I am talling you again, it will not work.

You're talking about something you've labelled R butnot actually saying what it is. AGAIN.

I alreadry crearly and simply defined my number system, and it is very easy to see how R members are the "shadows" of my number system.

Here it is again, and if you still standing in your spot, you will not understand it:


Tautology:
The identity of a thing to itself.

Set:
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.

Multiset:
A set-like object in which order is ignored, but multiplicity is explicitly significant.

Singleton set:
A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set.

Urelement:(no internal parts)
An urelement contains no elements, belongs to some set, and is not identical with the empty set http://mathworld.wolfram.com/Urelement.html.

A definition for a point:
A singleton set p that can be defined only by tautology*('='), where p has no internal parts.

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

Now let us move to the next step in order to define what is a number in my system.

First let us examine a well-known relation between mathematical objects and their representations.

=>> is ‘represented by’

|{}|=>>0 ; |{{}}|=>>|{0}|=>>1 ; |{{},{{}}}|=>>|{0,{0}}|=>>|{0,1}|=>>2 ;

|{{},{{},{{}}}}|=>>|{0,{0,{0}}}|=>>|{0,1,2}|=>>3 ; …

A definition for an interval (segment):
A singleton set s that can be defined by tautology* ('=') and ('<' or '>'), where s has no internal parts.

(Sign '<' means that we look at the segment from left to the right.
Sign '>' means that we look at the segment from right to the left.
When both '<' , '>' are used then we have a directionless segment.)

By the definition of a segment we get {._.}, which is the indivisible singleton set that exists between any two {.}.
Now we have the minimal building-blocks that allows us to define the standard R members.

(more detailed explanation of the first two definitions:

Let us examine these first two definitions by using the symmetry concept:

1) {.} content is the most symmetrical (the most "tight" on itself) content of a non-empty set.

It means that the direction concept does not exist yet and '.' can be defined only by '=' (tautology), which is the identity of '.' to itself.

2) {._.} content is the first content that "breaks" the most "tight" symmetry of {.} content, and now in addition to '=' by tautology (which is the identity of '._.' to itself) we have for the first time an existing direction '<' left-right, '>' right-left and also '<>' no-direction, which is different from the most "tight" non-empty element '.'

In short, by these two first definitions we get the different non-empty and indivisible contents '.'(a point) or '_'(a segment) .

*A statement for a point:
A point is an indivisible finite content of a non-empty set that has no directions.

*A statement for a segment:
A segment is an indivisible finite content of a non-empty set that also has directions.

(In standard Math we had to write:

A point: If a content of a set is a singleton and a urelement and has no directions, then it is a point.

A segment: If a content of a set is a singleton and a urelement and also has directions, then it is a segment.

But in this framework A=A is a tautology, and we do not need an ‘if, then’ proposition in order to define it) )



The axiom of independency:
p and s cannot be defined by each other.

By the above axiom {.} and {._.} are independed building blocks.

The axiom of complementarity:
p and s are simultaneously preventing/defining their middle domain (please look at http://www.geocities.com/complementarytheory/CompLogic.pdf to understand the Included-Middle reasoning).

By the above axiom we define the basic property of the middle domain between {.} and {._.}

The axiom of minimal structure:
Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.

The above axiom allows us to:

1) To define the internal structure of standard R members.
2) To define the internal structures of my new number system.

The axiom of duality(*):
Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by |{}| or s) of the entire collection.

The above axiom allows us to construct a collection of R members and also a collection of my new number system.

First, let us see how we use my method to construct a collection of R members.


R members are constructed like this:

1) First let us examine how we represent a number by my system:

=>> is ‘represented by’

a) |{}|=>>0

b) There is 1-1 and onto between ‘0’ and the left point of {._.} and we get {‘0’_.}

c) |{{}}|=>>|{0}|=>>1

e) There is 1-1 and onto between ‘1’ and the right point of {._.} and we get {‘0’_’1’}

In short, {.} is the initial place of R collection, which is represented by ‘0’, where {‘0’_.} is the initial place of the second place of R collection, which is represented by ‘1’, and we get our first two must-have building-blocks of R collection.


2) When we get {‘0’_’1’} we have our two must-have numbers, which are ‘0’ and _’1’.

Be aware that ‘0’ is the representation of {.} where ‘1’ is the representation of {._.}.


3) If we get {.}_AND_{._.}, then and only then we have the minimal must-have information to construct the entire R collection because:

a) We have ‘0’ AND _’1’ that give us the to basic scale factors 0 and _1.

b) We also have our initial domain _1, which standing in the basis of any arbitrary scale factor that is determined by the ratio between the initial domain _1 and another segment that is smaller or bigger than the initial domain _1 , for example:

 0 = .

 1 = 0[COLOR=Blue]______1[/COLOR]

 2 = 0[COLOR=DarkRed]____________2[/COLOR]  

 3 = 0[COLOR=Green]___________________3[/COLOR]

.5 = 0[COLOR=Red]__.5[/COLOR]    

pi = 0[COLOR=Magenta]______________________pi[/COLOR]

The negative numbers are the left mirror image of the above numbers.


There is no division in my number system because both {.} and {._.} are indivisible by definition.

In short, any segment is an independent element, that clearly can be shown in the above 2-D representation.

If we use a 1-D representation, we get the standard Real-line representation, but then we can understand that division is only an illusion of an overlap of independent elements when they are put on top of each other in a 1-D representation, for example:

0[COLOR=Red]__.5[/COLOR] [COLOR=Blue]__1[/COLOR][COLOR=DarkRed]_____2[/COLOR][COLOR=Green]_____3[/COLOR][COLOR=Magenta]__pi[/COLOR]

(*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations.


Since in my system nothing is divisible, then '/' stands for a ratio between at least any given two (indivisible) numbers.


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

 

[Lama]

If "finite" has it's standard mathematical meaning, then you are assuming the existence of the natural numbers. Here's Rudin's definition of a finite set

Let us stop here (before we continue to my new number system) to get your remarks.

I already gave you my answer to 'finite', which is:

The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels.

Then you asked me what is lowest and highest bounds, and my answer is:

The ends of some given element, where beyond them it cannot be found.

This last definition (and its opposite) is for {},{.},{._.} and {__} sets.

 

[CrankFan]

No matter how many times you repost your incomprehensible crap it's still incomprehensible crap.

 

[arildno]

No matter how many times you repost your incomprehensible crap it's still incomprehensible crap.

What's really sad is that he's been into this crap for over 20 years, according to some of his more lucid statements..

 

[matt grime]

So, you've got something that is your version of R, though you've not shown that it is a field, that it has a metric (though you've used it, preversely), that it is complete, etc... of course I understand that this is not necessary because your R is not the R as understood by the rest of humanity, just as those aren't naturals as we know them, tautology has a new meaning, as does =, symmetry you use without explanation, and I'm very impressed that you've introduced another spurious and unexplained property (tight). also, you're presuming that your set R has a direction given by an order, something else you've not shown to be true, nor that there is anything in those intervals, or that pi even exists in that system as a deducible fact from the axioms.

your axioms are rather clever in that part of them is that anything you choose to exist exists without proof.

 

[Lama]

arildno and CrankFan,

No matter how many times you will look at my work from your spot of reasoning, you will not be able to understand it, unless you come to my spot.

If you do not want to do it or cannot do it, then do not west your time in this thread, because you will see nothing, even after 20 years.

 

[arildno]

arildno and CrankFan,

No matter how many times you will look at my work from your spot of reasoning, you will not be able to understand it, unless you come to my spot.

If you do not want to do it or cannot do it, then do not west your time in this thread, because you will see nothing, even after 20 years.

Please explain what a "spot of reasoning" is.
And, mind you, no silly analogies this time.

 

[Lama]

your axioms are rather clever in that part of them is that anything you choose to exist exists without proof.

Yes Matt isn't it beautiful and simple?

1) A thing is always consistent with itself, otherwise we cannot find it.

2) Since we can find it we do not need to prove its existence.

3) Now all we need is to find interesting and useful ways of interactions between our tautologies.

4) The definitions of our used concepts help us to construct some agreement that can help us to work together, but we have to be aware that this common agreement is only a tool for the mathematical creation itself, which is always a unique creation of the person which creates it, and it is never a technical tuning of the agreement.

The mathematical unique creation changes the agreement, because it is an open thing by nature, where any agreement is a closed thing by nature that has to fit itself to Math development when time comes.

During the last 2500 years we can see that meaningful developments in Math always changed the common agreement, where persons that had no creative minds where forced to change their common agreements in order to fit themselves to the new development.

And why they where forced to do it?

Because this development was not their creation, and this creation broke their agreement, which is the only playground for non-creative persons.

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

Edit:

I want to add some notes about '=' notation which is in my system is first of all used for tautology, where tautology in my system is the identity of a thing to itself.

But I also continue to use this symbol in the standard way for example:

4=2+2 but this equation is different then 4=4 and 2+2=2+2 which are tautologies

(for eample:

4=0_______4 ; 2=0___2 ; 2+2=0___2 + 0___2 ).


By 4=2+2 we clearly mean that we care only about the quantity,

because it is clearly understood that (0_______4) is not (0___2 + 0___2) by tautology.

Actually we use quantity as a "glue" to connect different tautologies to each other, for example:

For example, let us represent the variations of cardinals(*) 2,3,4:

Let Redundancy be more then one copy of the same value can be found.

Let Uncertainty be more than one unique value can be found.

Let XOR be #

Let a=0,b=1,c=2,d=3 then we get:

    b  b                                        
    #  #                                        
   {a, a,  {a, b}                               
    .  .    .  .                                
    |  |    |  |                                
    |__|_   |__|                                
    |       |                                   
                                                
    {x,x}  {{x},x}                              
                                                
                                                
                                 
                                                
                                                
     c  c  c                                    
     #  #  #                                    
     b  b  b          b  b                      
     #  #  #          #  #                      
    {a, a, a,}       {a, a, c}       {a, b, b}  
     .  .  .          .  .  .         .  .  .   
     |  |  |          |  |  |         |  |  |   
     |  |  |          |__|_ |         |__|_ |   
     |  |  |          |     |         |     |   
     |__|__|_         |_____|         |_____|   
     |                |               |         
     |                |               |         
    {{x,x,x}         {{x,x},x}       {{x},x},x}

              
                [COLOR=Red][B]Uncertainty[/B][/COLOR]
  <-[B][COLOR=Blue]Redundancy[/COLOR][/B]->^
    d  d  d  d  |
    #  #  #  #  |
    c  c  c  c  |
    #  #  #  #  |
    b  b  b  b  |
    #  #  #  #  |
   {a, a, a, a} V   {a, b, c, d}
    .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |
    |  |  |  |       |__|  |  |
    |  |  |  |       |     |  | <--(Standard Math language uses only 
    |  |  |  |       |_____|  |     this no-redundancy_
    |  |  |  |       |        |     no-uncertainty_symmetry)
    |__|__|__|_      |________|
    |                |
    ={x,x,x,x}       ={{{{x},x},x},x}



============>>>

                [COLOR=Red][B]Uncertainty[/B][/COLOR]
  <-[B][COLOR=Blue]Redundancy[/COLOR][/B]->^
    d  d  d  d  |          d  d             d  d
    #  #  #  #  |          #  #             #  #        
    c  c  c  c  |          c  c             c  c
    #  #  #  #  |          #  #             #  #   
    b  b  b  b  |    b  b  b  b             b  b       b  b  b  b
    #  #  #  #  |    #  #  #  #             #  #       #  #  #  #   
   {a, a, a, a} V   {a, a, a, a}     {a, b, a, a}     {a, a, a, a}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |  |  |  |       |__|_ |  |       |__|  |  |       |__|_ |__|_
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |  |  |  |       |     |  |       |     |  |       |     |
    |__|__|__|_      |_____|__|_      |_____|__|_      |_____|____
    |                |                |                |
    {x,x,x,x}        {{x,x},x,x}      {{{x},x},x,x}    {{x,x},{x,x}}     
 
                                      c  c  c
                                      #  #  #      
          b  b                        b  b  b          b  b
          #  #                        #  #  #          #  #         
   {a, b, a, a}     {a, b, a, b}     {a, a, a, d}     {a, a, c, d}
    .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
    |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
    |__|  |__|_      |__|  |__|       |  |  |  |       |__|_ |  |
    |     |          |     |          |  |  |  |       |     |  |
    |     |          |     |          |__|__|_ |       |_____|  |
    |     |          |     |          |        |       |        |
    |_____|____      |_____|____      |________|       |________|
    |                |                |                |
    {{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x}      {{{x,x},x},x}
 
    a, b, c, d}
    .  .  .  .
    |  |  |  |
    |__|  |  |
    |     |  | <--(Standard Math language uses only this
    |_____|  |     no-redundancy_no-uncertainty_symmetry)
    |        |
    |________|
    |    
    {{{{x},x},x},x}

 

[CrankFan]

Lama,

What makes you so certain that you aren't the one who is misunderstanding?

Why are you so insistent that all of the people who have criticized your presentation for being ambiguous aren't fully grasping it, and need to see a repost of one of your papers? Maybe it's the other way around, maybe you aren't entirely grasping the nature of their criticism. Maybe you need to work harder to understand criticism of your work and stop automatically dismissing it.

If you are sincerely interested in communicating your ideas to people in this forum then I think the best approach for you is to find some rigorous mathematics texts on a subject you think you would enjoy and then study those texts in earnest. Follow all of the proofs, do all of the exercises, suppress your ego and seek help from people when you are genuinely stuck. Be sincere in your approach to learn and understand it.

If you do this, you will learn by example (of the author) and through practice (completing the exercises) what is roughly considered to be an acceptable definition and proof. Then, you can utilize this knowledge to describe your ideas so that they can be understood by your target audience on this forum.