The Foundations of a Non-Naive Mathematics

[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 {.}

 

[hello3719]

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 {.}

Not true. The first axiom as you call it is very subjective so you can conclude that the singleton can only be {.} but it isn't a universal conclusion.

 

[Lama]

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

You can continue to use the base value expansion method, but now you can use more than one information form for each given quantity along the decimals,
and also you have the knowledge of your unique path in the fractal, which is standing in the basis of some particular base value, for example please look at http://www.geocities.com/complementarytheory/9999.pdf pages 3,4 and also in http://www.geocities.com/complementarytheory/ONN3.pdf page 20.

Thank you.

 

[matt grime]

That isn't a proof, where is there any deduction? the first "axiom" is not used at all, and you've not proven that {.}, whatever that might be, cannot be defined by some other "rule", principally because you've not said what it means to "define" something. You may say we are going round in circles, but you haven't actually said anything logically sound yet. You see that word you keep writing in BOLD? well, you've never actually used that part of the axiom or shown it to be satisfied by a singleton set.

So, the conclusion we can reach is that singelton sets as we know them from your links to wolfram are not what you mean, unless you can show that the singleton set as we know it satisfies that axiom. Thus we are left to conclude that only sets which are defined only be this alleged "tautology" are permissible in your theory. Otherwise, once more, you are assuming that sound mathematical objects must satisfy your axioms without proof.

This begs the question of what exactly a tautology is still. you keep saying "x is itself, or x=x" but don't say what type of thing "x" is.

 

[Lama]

Not true. The first axiom as you call it is very subjective so you can conclude that the singleton can only be {.} but it isn't a universal conclusion.

Yes I agree with you, but there is another definition which complementing the picture which is:

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.

That isn't a proof...

Axioms are not proven.

 

[matt grime]

You keep repeating this as though the definition of a singelton set from wolfram is equivalent to yours, and i keep asking you t prove this and you keep posting the same non-argument. and you still don't say what kind of thing x is, it must be a proposition, so what proposition is it that allows you to conclude {.} is a a singelton set is a point in your system? You can't mix and match, come on, what is the proposition which is tautologous?

 

[Lama]

That isn't a proof...

Axioms are not proven.

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.

 

[matt grime]

I'm not asking you to prove an axiom I'm asking you to show that a singleton set (as understood by wolfram) satisfies your axioms, that is can be proven to only be "defined" by a tautology, what this alleged tautology is, even if your singelton sets aren't those as defined by wolfram, and to explain what you mean by "define" what is a definition, what is the allowable syntax and so on. Tautologies are applied to propositions, so what is that proposition?

 

[matt grime]

and we're now onto something strange because we must now decide what you mean by a "point"

let us pretend we accept your notions, continue.

 

[matt grime]

actually forget it, I'm bored for today, until you can explain what the proposition is that allows you to say {.} is a point, what kind of thing . is for that matter, and so on, I'll be off doing some real maths.

 

[Lama]

Tautology:
x implies x (An example: seuppose 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).


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

{.} is both a Singleton set and a Urelement.

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

(*) only by tautology means: the minimal possible existence of a non-empty set.

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

 

[Lama]

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.

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

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

Kaiser.

And what internal properties do you use during the thinking process?

Can you be sure that you do not use both intuition and reasoning?

If you ask me then you cannot get far without using also you intuition, in order to develop a meaningful thing in Math.

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

 

[kaiser soze]

Lama, you are naive to think that mathematics is based on intuition. You can develop perferctly valid (and maybe applicable) mathematical constructs even with the absolute non-intuitive foundations as long as your constructs are consistent. There are even complete mathematical systems developed with no intuition at all - they were developed by software.

Kaiser.

 

[Lama]

Lama, you are naive to think that mathematics is based on intuition. You can develop perferctly valid (and maybe applicable) mathematical constructs even with the absolute non-intuitive foundations as long as your constructs are consistent. There are even complete mathematical systems developed with no intuition at all - they were developed by software.

Kaiser.

No software can develop anything without some user that starting the whole process by entering the consistent axiomatic system.

Furthermore, the whole process is closed under the axiomatic system (that was first developed by some user) and can be understood only by the group of people who knows the rules.

Please show me a one case where a computer invented its own consistant axiomatic system, which is not trivial.

And if you can find such a system it means that your reasoning itself is a trivial thing if a "blind" thing like a computer can develop it.

So, what exactly is your point here?

Please correct me if I did not understand you, but as I see it, your approach leading me to understand that the language of Mathematics is no more then a mechanic process, where creativity is not needed.

you are naive to think that mathematics is based on intuition

On both intuition_AND_reasoning

Please do not omit again reasoning.

 

[sparkster]

Tautology:
x implies x (An example: seuppose 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).

So when you write "x = x" as an example of a tautology, what you really mean is "x -> x" or "if x then x?"

Also, you state that a point can only be defined by a tautology. How does {p, x} n {p, q} not define the point p (where "n" is an intersection)?

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

{.} is both a Singleton set and a Urelement.

By the definitions you given, nothing can be a singelton set and an urelement. A singleton set has "exactly one element" and an urelement "contains no elements."

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

(*) only by tautology means: the minimal possible existence of a non-empty set.

Since a set with one and only one element is "the minimal possible existence of a non-empty set," you're saying that a point is a single set that is a singleton set. Wouldn't it just be easier to say a point is a set with exactly one element instead of resorting to big words to try to make yourself sound smarter?

 

[Lama]

So when you write "x = x" as an example of a tautology, what you really mean is "x -> x" or "if x then x?"

Also, you state that a point can only be defined by a tautology. How does {p, x} n {p, q} not define the point p (where "n" is an intersection)?

Very simple, x is a self-evident true.

By the definitions you given, nothing can be a singelton set and an urelement. A singleton set has "exactly one element" and an urelement "contains no elements."

"exactly one element that contains no elements" is an indivisible one element.

If it is only a one element (a singleton) but not also a urelement, then it can be also 1 xor 2 xor 3 xor ... where each one of them is divisible, and in this case we cannot get our self-evident_true indivisible {.} .

Since a set with one and only one element is "the minimal possible existence of a non-empty set," you're saying that a point is a single set that is a singleton set. Wouldn't it just be easier to say a point is a set with exactly one element instead of resorting to big words to try to make yourself sound smarter?

"exactly one element that contains no elements" is an indivisible one element.

And if it can be defind only by self-evident true (which means that no other property like '<' or '>' can be related to it) then we cannot get anything but {.}, and we have our point.

 

[sparkster]

<Meaningless gibberish>

Congratulations. You've again managed to not answer a question that's been posted.

 

[Lama]

No dear ex-xian,

Your aggressive attitude simply blocking your ability to see (understand) fine things.

 

[sparkster]

No dear ex-xian,

Your aggressive attitude simply blocking your ability to see (understand) fine things.

Whatever. I realize it's much easier to accuse those who disagree with you of having an insurmountable bias, but it doesn't make anyone take you seriously. Everytime anyone finds problems with your posts you just change the subject, post something unrelated, or ignore the questions. It's a pattern you've developed and if you can't see it, I really pity you.

 

[Lama]

I am sorry for you ex-xian, I gave you my full answer to your questions.

You are the one who wrote back "<Meaningless gibberish>" instead of clearly show why you disagree with my answers to your previous post.

"<Meaningless gibberish>" is not a basis for any meaningful dialog between two persons, so please be more serious when you reply if you really want to understand someone's point of view, thank you.

 

[kaiser soze]

Lama:So, what exactly is your point here?

We are drifting off mathematical grounds. My point is that mathematics is universal, it does not rely on individual intuition. Sometimes intuition helps us understand and discover mathematical constructs, and sometimes it prevents us from understanding/discovering them.

If you think mathematics depends on intuition then prove that it does not exist without it - you will find this an impossible task.

Kaiser.

 

[Lama]

My point is that mathematics is universal

Please define 'universal', because maybe we have different interpretations about this word.

My definition for universal is:

The common source of abstract or non-abstract opposites.

This common source is the invariant super-symmetry, which is the gateway of interactions between opposites in non-destructive ways.

And in my opinion, the Langauge of Mathematics is the art of non-destructive interactions.

Please read http://www.geocities.com/complementarytheory/RTD.pdf to understand my point of view, thank you.

 

[matt grime]

Doron, you'll be pleased to know that I@ve figured out some of my confusion. I haven't read back to check since my last post, and I suspect anything you've posted will be made unnecessary by this:

So, a "point", p, is a set (singleton), that can only be defined by a tautology. Ok, I'll accept that as a definition, fine. You generally denote a "point" as {.}, which is by definition something that can only be defined by a tautology. Now, the question is can you provide me with a model of this that we can understand, ie some mathematical object that we all can recognize. And could you then give me the alleged tautology that defines it and can be defined no other way. It's all very well having a statement along the lines X is a degree three extension of Q that is the splitting field for x^2-2, but no such X actually exists, as can be proven. So, is there actually anything that we can understand that is a "point"? Presumably, since it arose there, points in R are defined only by a tautology, and are singleton sets, so show me how, say, sqrt(2) is defined by a tautology.

In fact just give me an example of what it means to be defined by a tautology, since a tautology involves propositions, and objects in general aren't propositions.

 

[Lama]

Fist let us write again our last post:

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).


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.

(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.

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.


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

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

 

[matt grime]

I know what a tautology is, however, please give an example of a *set* defined by a tautology, and a definition of a set that is not a tautology, so that we might understand what these look like. A tautology is a proposition that is always true regardless of the truth values of its components. How does that have anything to do with a set? How do we know that these are not vacuous statements?

What differentiates a general "singleton set" from one that is also a "point"? Give me an example of a singleton set that is not a point, give me an example of a singleton set that is a point.

Simple requests you keep ignoring? The best you've done is offer a circular argument, saying a point is {.} where {.} is a point. Well, what is {.}? Oh, the minimal structure etc. prove that such a thing exists from the axioms, or is it an axiom that such a thing exists? What is the reason behind this axiom if it is such?


Obviously your definition of an interval does not agree with the proper one, hence any conclusions you draw from your reasoning are not applicable to real mathematics

 

[Lama]

Obviously your definition of an interval does not agree with the proper one, hence any conclusions you draw from your reasoning are not applicable to real mathematics

By my definition of an interval we define the indivisible singleton set, which is not defined only by tautology, and then you can understand what is {.} and also what is {._.}.

Please keep reading all of my previous post (until the end of it) if you want to understand how my axiomatic system is related to R.