The Foundations of a Non-Naive Mathematics

[Lama]

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

Please understand that you did not show any understading of my work.

On the contrary I showed that understand the standard framework.

 

[Deeviant]

Lama, I think you would have more success creating a new religious belief than a new system of math. You skillsets would serve you better in that endeavor.

 

[arildno]

On the contrary I showed that understand the standard framework.

Eeh, whenever you've been pinned down to specifics (for example, in your usage of "tautology" or "="), you have consistently shown a lack of understanding of these terms; you have been using them with completely different meanings than standard usage

 

[Lama]

you have been using them with completely different meanings than standard usage.

Yes, you are right because my framework is based on an included-middle reasoning.

Lama, I think you would have more success creating a new religious belief than a new system of math.

Do you say it after you showed us that you understand what is an included-middle reasoning?

 

[arildno]

Yes, you are right because my framework is based on an included-middle reasoning.

Please explain what included-middle reasoning is, and why it is valid to term it a "reasoning"

 

[Lama]

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

Each one of us can understand the other by using its logical reasoning method.

You use the excluded-middle reasoning and I use an included-middle reasoning, which is deeper, simpler and richer then the excluded-middle reasoning.

In short, excluded-middle reasoning is nothing but a trivial sub-system of included-middle reasoning, and included-middle reasoning cannot be understood from this "spot of reasoning".

 

[arildno]

You are an arrogant idiot. Goodbye

 

[Lama]

Please explain what included-middle reasoning is, and why it is valid to term it a "reasoning"

Please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf and see for yourself not just what is an included-middle reasoning, but how I use it to develop my framework.

You are an arrogant idiot. Goodbye

Why, is it because of something that you have found in my article?

 

[anti_crank]

Each one of us can understand the other by using its logical reasoning method.

You use the excluded-middle reasoning and I use an included-middle reasoning, which is deeper, simpler and richer then the excluded-middle reasoning.

In short, excluded-middle reasoning is nothing but a trivial sub-system of included-middle reasoning, and included-middle reasoning cannot be understood from this "spot of reasoning".

Lama, please try to understand that this is not a valid response to the question you were asked. You have simply made grandiose statements that your reasoning is so much better than the one we've been using for 200 years, but failed to offer a single statement in support of that. Show how it can be "simpler, richer and deeper" than what we have. Show how the excluded middle reasoning is a trivial subsystem of it (which I think is not true as the two systems are contradictory. Feel free to correct me if you can convincingly show otherwise). I don't know whether you realize that absolutely everyone here says simiar things about their pet theories, and such claims always have the opposite effect of what they intend. You may also not realize that people don't like being constantly given reading assignments, since it is your job to make your case through reasonably self-contained posts. You may not be aware that in some forums, this practice is considered spam and such posts are summarily erased.

Call me a bodyguard too if you like, as you've expressed your dislike of my methods before. It is ironic that for one who is having such a hard time making himself understood (which you seem to genuinely want), you have not even tried to understand the method to my madness. As mathematics is not subject to the constraint of having to describe an experimentally accessible world, your system can coexist with any other system, assuming of course it is consistent, satisfies the standards of rigor, and it is useful in that it allows desirable results to be drawn. I am not a mathematician so I am not qualified to judge that, hence I do not want you to waste your time on me. Feel free to put my advice to whatever use you wish.

 

[sparkster]

Yes, you are right because my framework is based on an included-middle reasoning.

What does included middle reasoning have to do with you changing the defintion of tautology?

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

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.

This is circular reasoning! I asked to define finite w/o "standard" mathematics or else concede everything I specified above. You respond by "answering" finite with an axiom that includes the disputed definition.

This is really getting silly.

 

[Lama]

Dear anti-crank,

What I was doing in the last 2 years in this forum is to develop my framework by trying the best I can to explain my non-standard ideas.

Instead of look around you and see what other people are saying, I suggesting you to read by yourself the fruits of my efforts.

You do not need complecated mathematical teachings to understand my work because my work is on the most fundamental concepts of the Langauge of mathematics, and because it is new in cannot be compared with the goals and achievements of the standard framework.

Please read:

http://www.geocities.com/complementarytheory/No-Naive-Math.pdf and all of it links,

and also http://www.geocities.com/complementarytheory/My-first-axioms.pdf.

More general view of my work can be found in:

http://www.geocities.com/complementarytheory/CATpage.html

After you have your personal opinion on my work, I'll be glad to discuss with you about it.

Yours,

Lama

 

[Lama]

This is circular reasoning! I asked to define finite w/o "standard" mathematics or else concede everything I specified above. You respond by "answering" finite with an axiom that includes the disputed definition.

This is really getting silly.

Then you missed the good part of this post where I wrote:

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.

As you see I have two basic states for finite things:

One is for singletons like {},{.},{._.} and {__} sets and the other is for collections of {.}_AND_{._.} elements.

If each {.}_AND_{._.} has a unique notation (for example: '1', 'pi' , (1/3=@) '@', (.99999...=&) '&', ...), then a collection with more the one notation can have finite or infinitely many notations, and we do not need ZF axioms for this.

Also please pay attention that Cantor's second diagonal proof does not hold here because we are using a single and unique notation for any given number.

 

[sparkster]

Then you missed the good part of this post where I wrote:

As you see I have two basic states for finite things:

But you still have never given a defintion for what finite is.

 

[Lama]

But you still have never given a defintion for what finite is.

Dear ex-xian, please show why do we need a special definition for 'finite'
by demonstrate how we can understand 'finite' in more than one way, thank you.

Anyway I can use this:

a) A colloction of more then one element where its lowest bound and its highest bound are included in it.

b) Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found.

 

[sparkster]

Dear ex-xian, please show why do we need a special definition for 'finite'
by demonstrate how we can understand 'finite' in more than one way, thank you.

Anyway I can use this:

a) A colloction of more then one element where its lowest bound and its highest bound are included in it.

b) Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found.

Ok, but what are the "ends of some given element?" What does "beyond them" mean? As Matt pointed out, you're assigned order to sets without specifying what dicates that order.

 

[Lama]

you're assigned order to sets without specifying what dicates that order

Instead of repeating Matt's point of view, all you have to do is to read my papers, and then you will discover that my elements ({} and {__} are excluded) are {.}_AND_{._.}, where {.}_AND_{._.} are at least Multiset_AND_Set which is ordered by its own internal symmetry degrees, when a given quantity remains unchanged.

But if we return to standard Math where we have Set_XOR_Multiset (and any fundamental concept is based only on quantity), then in the case of a multiset {1,1,1,1,1} for example, we cannot order its content, but even in this case, it has ends that beyond them it cannot be found.

For example: its lowest bound is '{1,' and its highest bound is ',1}’.

In the case of a "normal" set, where any member is a unique member, the lowest bound is '{x1,' and the highest bound is ',xn}' and we don't care about the internal order of the set, so as you see in both cases the standard point of view remains the same, which means, we care only about the cardinality of some set or multiset contents, and we do not care about their order.

So as you see we can use:

1) Peano axioms of the natural numbers (where a definition of a set does not exist).

2) then we can use the definition of a set as:

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

3) And then we can use:

“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” (and this is an example that clearly shows that 'finite' in standard Math is based only on 'quantity').

Let us check again my definition for 'finite':

a) A singleton or a collection of more then one element where its lowest bound and its highest bound are included in it.

b) Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found.

As you see my definition is stronger, simpler and richer then the quantity-only definition, because it works for both quantity and structure.

Ok, but what are the "ends of some given element?" What does "beyond them" mean? As Matt pointed out, you're assigned order to sets without specifying what dicates that order.

We can do this endless game of questions also to:


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

(What is: empty, set, every, member?)

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.

(What is: set, proposition?)

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

(What is: set?)

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

(What is: set, union?)

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”

(What is: set, nonempty, intersection?)

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.

(What is: subset?)

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

(What is: set?)

8. No set is a member of itself.

(What is: set, member, itself?)

And in a very short time we get questions like:

What is a definition?

What is What is?

And so on ...

So, as you see ex-xian, any agreement is always based on some arbitrary decision between some community of people that stop the game of endless questions end start to work with each other on the basis of this agreement.

And the Community of Mathematician is not a special community.

 

[matt grime]

Dear ex-xian, please show why do we need a special definition for 'finite'
by demonstrate how we can understand 'finite' in more than one way, thank you.

Anyway I can use this:

a) A colloction of more then one element where its lowest bound and its highest bound are included in it.

b) Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found.

But this is rubbish. [0,1] is a set with an infinite cardinality (using the words properly, and we cannot use your definitions becuase they make no sense), that "contains its end points". And this would only be something that applies to sets that have "ends" which is stupid. (the collection of all isomorphism classes of groups of order 6 is finite, but doesn't have any bounds)

 

[Lama]

(the collection of all isomorphism classes of groups of order 6 is finite, but doesn't have any bounds)

Ho, yes they have, you cannot find the 7 of them and you cannot find 0 of them, but you can find 1 to 6 of them.

[0,1] is a set with an infinite cardinality

Only if there are also infinitely many scales.


Please read post #471, thank you.



Also you missed this so I copeid it for you:

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, 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}

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

(*) 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.

 

[matt grime]

More demonstrations of ignorance? wow you're good.

there are exactly TWO isomorphim classes of groups of order 6, now which of them is the lowest bound and which is the highest bound? it is a finite set after all, so it must have them in your logic.

the rest as i think ex-xian said is spamming. you've no need to repost it, it is "vandalism" to do so in some people's opinions.

 

[Lama]

Matt,

Pease read my posts, because I do not see any detailed comments of you on posts #471 and #473.

there are exactly TWO...

So, it is 2 instead of 6, but it does not matter because both of these cardinals are finite cardinals, and in this case you say bye,bye in 0 and 3.

 

[HallsofIvy]

Originally Posted by Matt Grime
(the collection of all isomorphism classes of groups of order 6 is finite, but doesn't have any bounds)

Originally Posted by Lama
Ho, yes they have, you cannot find the 7 of them and you cannot find 0 of them, but you can find 1 to 6 of them.

You have absolutely no idea what Matt Grime is talking about, do you?
You asserted that any finite set must contain upper and lower bounds as part of your definition of "finite". Matt gave an example in which the members of the set are not linearly ordered and so your statement makes no sense for that set.

And please stop responding to every criticism with "read all of ...". The fact that someone points out that your second sentence is nonsense doesn't mean that he hasn't already waded through the whole thing.

You are correct that "You do not need complecated mathematical teachings to understand my work". What we need are actual definitions not hand waving and vague examples of things that seem to you to illustrate an idea.

 

[matt grime]

But the set of iso classes isn't a set of numbers, so it has no ordering in it, it has no way of describing an upper or lower limit of the elements in it. The set is a collection of things, is it finite? You're confunsing a set and its cardinality. The bounds you gave are the least integer strictly greater than the cardinality of the set in question, and the greatest integer strictly less than ... They bound the cardinality, but in no way form a bound for the set which is what your definition requires.

 

[Lama]

Matt and HallsofIvy,

In my system there is no such thing like lower and upper bounds.

In my system I use the terms lowest and highest bounds, which are totally different things.

Since you do not really read my posts in order to understan them, I don't no how we can communicate with each other.

 

[matt grime]

ok, rewrite my post in your head with lowest and highest then. the question still makes sense and the answer doesn't.

the point is quite simple, the only sets you can talk about being finite are those with highest and lowest bounds by your own admission, so this cannot be applied to any set other than something with an ordering and even that makes little sense since i an produce infinite (in the proper sense) sets with highest and lowest memebers

 

[Lama]

Let us check again my definition for 'finite':

a) A singleton or a collection of more then one element where its lowest bound and its highest bound are included in it.

b) Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found.

And there is nothing here, which is connected to order!

As you see my definition is stronger, simpler and richer then the quantity-only definition, because it works for both quantity and structure.

This is a part of post #471, and if you don't read it, as I can clearly understand from what is written in your last post, then you keep talking to yourself.

 

[matt grime]

it doesn't even work for quantity. consider the set underlying the ordinal w+1, then it more than one element in it, and contains a least element and a greatest element, hence ordinals are not allowable in your system, unless you're going to claim w+1 is a finite ordinal?

 

[matt grime]

"Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found"

what on Earth is "it" in that sentence referring to? there must be some order on something otherwise you cannot talk of highest or lowest, so what is ordered?

 

[Lama]

There are two kinds of ordinals in my system, internal and external.

The external ordinals are any change in the quantity of {.}_AND_{._.}.

The internal ordinals are any change in the symmetrical degrees of {.}_AND_{.}, where the cardinality of {.}_AND_{._.} is unchanged.

 

[Lama]

there must be some order on something otherwise you cannot talk of highest or lowest, so what is ordered?

By 'lowest' and 'highest' I mean lowest existence and highest existence of some element, and there is no connection to order here.

 

[matt grime]

you#ve just redefined ordinal then.

"lowest existence" what does that mean? please give me some example where you may talk of things being lowest and highest which is not from some order?

 

[Lama]

Take for example the multiset {1,1,1,1,1}.

It cannot be ordered but cardinals 0 and 6 are the lowest and highest bounds of it (I already gave this example in post #471).

Also the notations '{' and '}' are also the bounds of existence of any concept that we research, isn't it?

 

[matt grime]

Take for example the multiset {1,1,1,1,1}.

It cannot be ordered but cardinals 0 and 6 are the lowest and highest bounds of it (I already gave this example in post #471).

Also the notations '{' and '}' are also the bounds of existence of any concept that we research, isn't it?

what has this post to do with anything? why is 6 a bound of the set? it is a bound for that set which allows repeated elements, but? that is not what your axiom for "finiteness"states, since it requires that the "end" elements are in the set for it tobe finite. what is the first and last element of a set without an order on the elements in it?

psot 482, the question about what the"it" refers to in your axiom is still unanswered

 

[Lama]

Beyond your end you cannot be found anymore, isn't it?

To what "it" do you mean exactly?

what is the first and last element of a set without an order on the elements in it?

I do not care about the internal order of the given set, its end is always determinated by its cardinality.

 

[matt grime]

"Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found"

this is your axiom, so what does the "it" refer to? it's your sentence, presumably you know what the pronoun references.


"I do not care about the internal order of the given set, its end is always determinated by its cardinality"

that is also wrong trivially. {2} and {3} both have cardinality 1, the "end" is not determined by the cardinality. you're confusing a set and its cardinality again.

 

[Lama]

{2} and {3} both have cardinality 1

Yes you are right, therefore the values beyond their ends are cardinals 0 and 2.

"Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found"

It is an element or a collection .

I think that I made a mistake in my previous posts:

The cardinals are not just the ends of a given finite set, but they are also the values that beyond its ends.

I am really Sorry about the confusion.


Here it is again my definition of 'finite' to clarify this mistake:

a) A singleton or a collection of more then one element where its lowest bound and its highest bound are included in it.

b) Lowest bound and highest bound are the ends of some given element (or a collection of more than one element), where beyond them it cannot be found.