The Foundations of a Non-Naive Mathematics

[Lama]

You know what Matt?

Since you cannot see the connections between my axiomatic system and the standard system, let us put aside any standard system defititions, and let us look only on my axiomatic system.

Here it is again, and this time try to look at it as a new Axiomatic system (unless you find some connections to another and already known Axiomatic system):

Here is again a list of my axioms, which are related to R:

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.

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

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).

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 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 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.

The Axiom of the unreachable weak limit:
No input can be found by {} which stands for Emptiness.

The Axiom of the unreachable strong limit:
No input can be found by {__} which stands for Fullness.

The Axiom of potentiality:
p {.} is a potential Emptiness {}, where s {._.} is a potential Fullness {__}.

The Axiom of phase transition:
a) There is no Urelement between {} and {.}.
b) There is no Urelement between {.} and {._.}.
c) There is no Urelement between {._.} and {__}.

Urelement (http://mathworld.wolfram.com/Urelement.html).


The axiom of abstract/representation relations:
There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.


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

Tautology means x is itself or x=x.

Singleton set is http://mathworld.wolfram.com/SingletonSet.html .

Multiset is http://mathworld.wolfram.com/Multiset.html .

Set is http://mathworld.wolfram.com/Set.html .

(By the way the diagrams in my papers are also proofs without words http://mathworld.wolfram.com/ProofwithoutWords.html )



The Axiom of the paradigm-shift:

Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current system.



Let us stop here to get your remarks.

 

[Lama]

So you agree that your "mathematics" is not ours, since you are making out your own "agreement" as you go along. The problem is that no one seems to agree with you. This is exactly what you are doing wrong, you continue to use the same terms we use without proving that (the terms you are using) they are the same (as ours).

Kaiser.

Dear Kaizer soze, the difference between you and me is in the level of intuition/reasoning gentle interactions, where Axioms come form.

This level cannot be learned by any external method, which means, that you understand directly from within you something, or you don't.

My intuition/reasoning internal interaction goes deeper, in my opinion, then the common intuition/reasoning internal interaction.

Please read the following to get some perspective about what I wrote above:


When two violins in the same room are tuned with each other, if we play on one of them we find that the strings of the other violin are also vibrate.

Now let as say that intuition is our tuned instrument, and if a person expresses its intuitions by developing a way of thinking, the people that embrace this way of thinking probably share the same intuitions.

On the basis of these common intuitions a community can be established.

Let us say that this community is the first organization that deals with some part of the human knowledge, so in these early stages this community has no comparators on this part of the human knowledge.

Quickly this community becomes the most developed organization, which holds this part of the human knowledge, and other parts of human civilization look at this organization as the one and only one possible intuition which standing in the basis of a one and only one way (school) of thought (and I am not talking about variations, which are actually different brunches of the same way of thought, or the same school of thought if you like).

2500 years are passing and this school of thought survives because of two main reasons:

1) This way of thought was fitting to the needs of the human civilization along these 'slow' (linear) years.

2) Any other alternative intuitions (if they where at all) where put aside because:

a) They where not useful in their time.

b) And if they where useful and also a real alternative to the current school, then the current school used its power and money to block this alternative intuition by forcing its educational methods on the public.

We have to understand that intuitions cannot be learned, but a lot of external power can distort them until they lost their ability to be the source of a new school of thought.


The 120 century is the time where our civilization moved from linear time to non-linear time.

In this time the power of few holds the destiny of our civilization, and most of their power is based on the technical abilities that where developed by this school of thought, that was established 2500 years ago.

But our technical achievements, which are not balanced by another ways of thought, are like a government with no opposite.

We have learned that evolution needs diversity; otherwise we quickly get a dead planet.

The field of evolution in our non-linear time splits to "hardwhere" and "softwhere" parallel paths, where the hardwhere side is our technology and the softwhere side is our morality.

We can clearly see that there is no balance between the levels of these two paths, and this lack of balance in a non-linear time can quickly lead us to a dead-end street.

Therefore I think that we have to do the best we can to find the balance between our morality level and our technical abilities.

The first place that binds both paths is the language of mathematics.

In my opinion people how learn this powerful language, must first of all to develop their moral abilities by opening themselves to another intuitions which are not their intuitions and let them flourish in their communities.

By this way we develop our tolerance and learn how to live side by side, and if other intuitions are better then our intuition in this period of time, we do our best to help them flourish instead of trying our best to shut them down.

And we have the motivation to do that because we understand that we are all in the same boat.

My intuitions and ideas about the language of mathematics are different then the standard school of mathematics.

But in my opinion the most important difference, which I think fits to our non-linear time (more then the standard school) is that I include the mathematician cognition's ability to develop Math as a part of the mathematical research.

By this self-reference attitude I hope to develop the gateway that can connect between our moral abilities to our technical abilities.

 

[matt grime]

"Here is again a list of my axioms, which are related to R:"

so is this supposed to define R, or is R already defined, if so explain what YOU think it is, and explain please what related to means in this context.

"A singleton set p that can be defined only by tautology ('='), where p has no internal parts."

this makes little sense, so post an example.

"p and s cannot be defined by each other"

again makes no sense, so post an exmaple of "p" and demonstrate that it does not define "s", that is prove that no definition exists, or are you not using "cannot" to mean that?


"Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set."

number? what number? multiset not been defined?

"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."


what is a scale factor?

"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."

what's a number? i mean that sincerely. since you've not offered an alternative definition of the set of numbers, one cannot conclude anything since the only ones we know about are those defined by a system that you reject.

"There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them."

that is not an axiom, that is your opinion, and one with no backing. It's a niuce sentiment, but does not in itself add anything to an axiomatic system.

"Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current system."

what is "well defined" in this context? you've not offered here any way of defining what is and what isn't a set. this has no place in the "axioms" above.

You've still not said what R is. What do you mean by R. YOU, not US.

 

[Lama]

What shall we do to communicate with each other?

I think that we have to go step by stap so first let us look only on my two first definitions:

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.
-------------------------------------------------------------------------

Tautology means x is itself or x=x.

Singleton set is http://mathworld.wolfram.com/SingletonSet.html .

Multiset is http://mathworld.wolfram.com/Multiset.html .

Set is http://mathworld.wolfram.com/Set.html .

No internal parts means: Indivisible.

 

[Hurkyl]

Before we move on, do you acknowledge by using the standard definitions for sets and multisets that your theory must include all standard theorems about such things? Theorems including \aleph_0 + 1 = \aleph_0, infinite sets with different cardinalities, infinite sets have the same cardinality as some of their subsets, and that there can be a 1-1 bijection between an infinite set and some of its subsets.

(of course, several of those statements mean exactly the same thing)

 

[kaiser soze]

Lama,

I will say it again, what you are doing is not mathematics as we know it - mathematics has nothing to do with intuitions, moreover, sometimes our intuitions prevent us from understanding mathematical facts.

Kaiser.

 

[Lama]

Before we move on, do you acknowledge by using the standard definitions for sets and multisets that your theory must include all standard theorems about such things? Theorems including , infinite sets with different cardinalities, infinite sets have the same cardinality as some of their subsets, and that there can be a 1-1 bijection between an infinite set and some of its subsets.

(of course, several of those statements mean exactly the same thing)

No I take only the most basic definitions of a Set and a Multiset:


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.

 

[Lama]

Lama,

I will say it again, what you are doing is not mathematics as we know it - mathematics has nothing to do with intuitions, moreover, sometimes our intuitions prevent us from understanding mathematical facts.

Kaiser.

I am talikng about intuitions/reasoning interactions.

Please read carefully again what I wrote to you in:

https://www.physicsforums.com/showpost.php?p=275808&postcount=332

 

[matt grime]

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

so you're doing your version of naive set theory then.

you haven' posted an example of a point being defined according to your axioms, nor have you said what the set R is in your opinion.

Once more yo'ure ignoring the mathematics.

 

[kaiser soze]

Lama, a computer has no intution, yet it is perfectly able to perform calculations based on mathematical reasoning and constructs. Moreover, every computer is guaranteed to have the same result for a given mathamtical calculation.

Kaiser.

 

[Lama]

so you're doing your version of naive set theory then.

No Matt this is the standard basic definition for a Set.

See for yourself : http://mathworld.wolfram.com/Set.html.

At this stage of our dialog let us put aside R Collection.

Please let us concentrate only on:

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.


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

Tautology means x is itself or x=x.

Singleton set is http://mathworld.wolfram.com/SingletonSet.html .

Multiset is http://mathworld.wolfram.com/Multiset.html .

Set is http://mathworld.wolfram.com/Set.html .

No internal parts means: Indivisible.

 

[Lama]

Lama, a computer has no intution, yet it is perfectly able to perform calculations based on mathematical reasoning and constructs. Moreover, every computer is guaranteed to have the same result for a given mathamtical calculation.

Kaiser.

A computer is nothing but a blind electro mechanic tool, which is no more than a dynamic mirror of your intuition/reasoning interactions reasoning.

In short, it is the image of your own intuition/reasoning interactions.

 

[matt grime]

we cannot really put aside R, since we are talking about numbers, and elements of R and intervals, and it is the R that is the key thing in the errors in your ideas.

give an example of a singleton set then defined only by '=', demonstrating that it cannot be defined by some other means, whatever those means might be, moreover, explain what it means for a set to have no internal parts, ie a set to be indivisible. do you mean "is not the empty set, and if written as the disjoint union of two other sets then one of them is the empty set"? If so, why not say so more clearly rather than inventing new meanings for words?

NB you really ought to stop relying on mathworld and treating it as if its general explanations are actually the definitions of the things you don't understand.

 

[Lama]

we cannot really put aside R

You are right, but before we get R I want to be sure that you understand my first two definitions, so at this stage, ask your questions only about them, thank you.

 

[kaiser soze]

Lama,

Intuition is individual - mathematics is universal.

Kaiser.

 

[matt grime]

I have asked questions. I make it three times in three posts:
quoting myself:

give an example of a singleton set then defined only by '=', demonstrating that it cannot be defined by some other means, whatever those means might be (something that you need to explain as well), moreover, explain what it means for a set to have no internal parts, ie a set to be indivisible. do you mean "is not the empty set, and if written as the disjoint union of two other sets then one of them is the empty set"? If so, why not say so more clearly rather than inventing new meanings for words?

 

[Lama]

Lama,

Intuition is individual - mathematics is universal.

Kaiser.

Mathematics is the interactions between the individual/universal, local/global, intuition/reasoning, induction/deduction, integral/differential, 0/1, Emptiness/Fullness, ...

In short, it is based on an Included-middle framework, which is based on a dialog between at least two opposite concepts.

Thats why it is first of all a language.

 

[Lama]

give an example of a singleton set then defined only by '='

{.} is a clear and simple example of it.

 

[matt grime]

But how is that defined by '=', and only by '=', where apparently you mean that '=' means tautology. Moreover, what is that set? The set which contains a full stop? Moreover, why is it not also "defined" as {{.}u{,}}n{{.}u{:}} where I assign arbitrary and distinct meanings to the symbols . , and :

 

[Lama]

But how is that defined by '=', and only by '=', where apparently you mean that '=' means tautology. Moreover, what is that set? The set which contains a full stop? Moreover, why is it not also "defined" as {{.}u{,}}n{{.}u{:}} where I assign arbitrary and distinct meanings to the symbols . , and :

The axiom of abstract/representation relations:
There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.

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

By these two axioms, the result cannot be but {.}

 

[matt grime]

You've still not explained what {.} means, but we'll presume that it is an arbitrary set of cardinality 1, let us take an example: we can all agree on the finite cardinals, and their existence. So {1} is a set with one element. Demonstrate that this set can ONLY be defined by a tautology, further, explain how the axiom of abstract/representation relations is used in the implication you claim (it is not obvious), moreoever, define what an internal part is. also explain why {1,2}n{1,3} does not also define a singleton set, or is that a tautology? If so please explain what you think tautology is, and give an example defining some set that is not a tautologous statement, and which must therefore be a non-singleton set.

 

[Lama]

Why do you ignore the name of the axiom?


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


Tautology means x is itself or x=x.

Singleton set is http://mathworld.wolfram.com/SingletonSet.html .

Multiset is http://mathworld.wolfram.com/Multiset.html .

Set is http://mathworld.wolfram.com/Set.html .

No internal parts means a Urelement (http://mathworld.wolfram.com/Urelement.html).

also explain why {1,2}n{1,3} does not also define a singleton set

At this first stage we cannot cannot talk about number > 0, beause in my system we need at leaset two types of Urelements to define a number, which is not |{}|.

 

[matt grime]

Yes, look at your own axiom. You must now prove that the set {.}, whatever that may be can ONLY be defined by a tautologous statement. However it is not clear what it means for something to be defined as a tautology, perhaps you would care to explain.

Fine, you don't like numbers, just pretend the symbols 1,2,3 are distinct objects, that they are numbers is not important, I was just trying to offer some example with elements we could be fairly sure existed.

 

[Lama]

However it is not clear what it means for something to be defined as a tautology

Tautology means x is itself or x=x.

At this stage, by the definition of a point, all we have is {.}={.}

I was just trying to offer some example with elements we could be fairly sure existed

We will get them, please be patient.

{.} is only one of two different building-blocks that we need before we can define a number, which is not |{}|.

Do you have something to say before we continue to the next definition?

 

[kaiser soze]

Lama:Mathematics is the interactions between the individual/universal, local/global, intuition/reasoning, induction/deduction, integral/differential, 0/1, Emptiness/Fullness, ...


Prove it! In mathematics, we have axioms, definitions and proofs.

Kaiser.

 

[Lama]

Prove it! In mathematics, we have axioms, definitions and proofs.

Kaiser.

Please tell us how do we get our Axioms and definitions?

 

[kaiser soze]

How? by thinking about them, then stating them in a coherent fashion.

Kaiser.

 

[hello3719]

Pi = the relations between the perimeter and the diameter of a circle.

Well give me a way to find an approximation to this number using your system.

 

[matt grime]

Tautology means x is itself or x=x.

but this is self referential, or at least inconsistent because you defined '=' to mean tautology. but what is x? is it a proposition or a set, or something else? since you've not bothered to say we can only guess, try saying a singleton set p is defined tautologically when.. and then use some statement about p.

since we're talking about sets, you are saying that a singleton set is defined by p=p, where p is a set. but every set is equal to itself.

 

[Lama]

Matt,

Again:

The axiom of abstract/representation relations:
There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.

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

By these two axioms, the result (our singleton) cannot be but {.}