The Foundations of a Non-Naive Mathematics

[sparkster]

Dear ex-xian,

What exactly is your problem, can't you understand my axiomatic system?
https://www.physicsforums.com/showpost.php?p=272134&postcount=278

Cant you understand my original point of view here:
http://www.geocities.com/complementarytheory/PZstar.pdf

Is this the reason that you so aggressive and jumpy?

Dear Lama/Organic/Dorian/Math Crank/Person badly in need of a math class,

What exactly is your problem, can't you understand questions that are asked and you've agreed to answer?
https://www.physicsforums.com/showpost.php?p=275215&postcount=312

Can't you understand my original question and remember your promise to demonstarte your knowledge of "standard" math?
https://www.physicsforums.com/showpost.php?p=275118&postcount=303

Is this the reason that you are so defensive and appear so ignorant?

 

[Lama]

CrankFan,

Can you write more to the point?

For example please look at https://www.physicsforums.com/showpost.php?p=272134&postcount=278

 

[Lama]

Is this the reason that you are so defensive and appear so ignorant?

If you refuse to read my papers, where I clearly show that I understand standard Math before I air my view about it, then it is your problem.

For example read my dialog with kaiser soze:

Some dialog:

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

Doron:

I think that we do not understand each other.

I gave you MY definiton of the limit concept.

Now, please give the standard definition for this concept.

After you give the standard definition, then we shall compare between
the two approaches.

Any way do you agree with http://mathworld.wolfram.com/Limit.html definition?

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

kaiser:

off course I agree with this definition. I meant for you to provide the defintion for the limit of S(n), no need delta epsilon at this point. A limit can be defined using epsilon and S(n). At any case, I am not interested in your definitions at the moment. I need to be convinced that you understand and know how to use the fundamental "conventional" mathematical defintions before we can move on to your definitions.

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

Doron:

Ok, the main persons in modern Math that are related to the so called rigorous definition of the limit concept are Cauchy and Weierstrass.

Cauchy said:" When some sequence of values that are related one after the other to the same variable, are approaching to some constant, in such a way that they will be distinguished from this constant in any arbitrary smaller sizes that are chosen by us, then we can say that this constant is the limit of these infinitely many values that approaching to it."

Weierstrass took this informal definition and gave this rigorous arithmetical definition:

The sequence S1,S2,S3, … ,Sn, ... is approaching to (limit) S if for any given positive and arbitrary small number (e > 0) we can find a matched place (N) in the sequence, in such a way that the absolute value S-Sn (|S-Sn|) become smaller then any given epsilon, starting from this particular place in the sequence
(|S-Sn| < e for any N < n).

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

kaiser:

Very good! now based on the definition you provided, which is a correct mathematical definition please find out the limit of the following sequence:

0.9,0.99,0.999,0.9999,0.99999,...

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

Doron:

-------post #190

Now please listen to what I have to say.

First please read http://www.geocities.com/complementarytheory/9999.pdf
(which is also related to your question) before we continue.

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

Doron:

-------post #191

I disagree with the intuitions of Weierstrass, Cauchy, Dedekind, Cantor and other great mathematicians that developed the current mathematical methods, which are dealing with the Limit and the Infinity concepts.

And my reason is this:

No collection of infinitely many elements that can be found in infinitely many different scales, can have any link with some given constant, in such a way that it will be considered as a limit of the discussed collection.

In short, Nothing is approaching from the collection to the given constant, as can be clearly seen in my sports car analogy at page 2 of http://www.geocities.com/complementarytheory/ed.pdf

Take each separate position of the car, then compare it to zero state and you can clearly see that nothing is approaching to zero state.

Therefore no such constant can be considered as a limit of the above collection.

It means that if the described collection is A and the limit is B, then the connection between A,B cannot be anything but A_XOR_B.

So here is again post #184:

Since I am not a professional mathematician, my best definition at this stage is:

A Limit is any arbitrary well-defined element, where no collection of well-defined infinitely many elements can reach it.

It means that if A is the collection of infinitely many elements and B is the limit, then we can reach B only if we leap from A to B and vise versa.

By using the word "leap" we mean that we have a phase transition from state A to state B.

There is no intermediate state that smoothly links between A,B states therefore we cannot define but a A_XOR_B relations between A, B states.

A collection A is incomplete if infinitely many elements of it cannot reach some given limit, or if no limit is given.

From the above definition we can understand that no collection of infinitely many elements is a complete collection, and therefore no universal quantification can be related to it.

If you disagree with me, then please define a smooth link (without “leaps”) between A,B states.

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

Doron:

-------post #192

'Any x’ is not ‘All x’​

By inconsistent system we can "prove" what ever we want with no limitations
but then our "proofs" are inconsistent.

A consistent system is based on a finite quantity of well-defined axioms, but then we can find in it statements which are well-defined by the consistent system but they cannot be proven by the current axioms of this system, and we need to add more axioms in order to prove these statements.

So any consistent system is limited by definition and any inconsistent system is not limited by definition.


Let us examine the universal quantification 'all'.

As I see it, when we use 'all' it means that everything is inside our domain and if our domain is infinitely many elements, even if they are limited by some common property, the whole idea of "well-defined" domain of infinitely many elements is an inconsistent idea.

For example:

Someone can say that [0,1] is an example of a well-defined domain, which is also a collection of infinitely many elements, but any examined transition from the internal collection of the infinitely many elements to 0 or 1, cannot be anything but a phase transition that terminates the state of infinitely many smeller states of the collection of the infinitely many elements, and we have in our hand a finite collection of different scales and 0 or 1.

In short, the well-defined ‘[0’ or ‘1]’ values and a collection of infinitely many elements that existing between them, has a XOR-like relations that prevents from us to keep the property of the internal collection as a collection of infinitely many elements, in an excluded-middle reasoning.

Again, it is clearly shown in: http://www.geocities.com/complementarytheory/ed.pdf

Form this point of view a universal quantification can be related only to a collection of finitely many elements.

An example: LIM X---> 0, X*(1/X) = 1

In that case we have to distinguish between the word 'any' which is not equivalent here to the word 'all'.

'any' is an inductive point of view on a collection of infinitely many elements, that does not try to capture everything by forcing a deductive 'all' point of view on a collection of infinitely many X values that cannot reach 0.

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

kaiser:

If you do not see that the limit of the sequence I provided is 1, then you do not understand what a limit is, and therefore can not agree or disagree with its definition.

In loose terms we can say that a sequence has a limit if it is approaching (but never reaching) some conststant. A sequence does not have a limit, if it is not approaching some constant, for example the sequence 1,2,3,4,... does not have a limit, it disperses to infinity.

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

Doron:

1 as the limit of the sequence 0.9,0.99,0.999,0.9999,0.99999,... is based on an ill intuition about a collection of infinitely many elements that can be found in infinitely many different scales, as can be clearly understood by posts #190,#191,#192.

You can show that 1 is really the limit of sequence 0.9,0.99,0.999,0.9999,0.99999,... , only if you can prove that there is a smooth link (without "leaps") between this sequence and 1, which is not based on {0.9,0.99,0.999,0.9999,0.99999,... }_XOR_{1} connection.

Maybe this example can help:

r is circle’s radius.

s' is a dummy variable (http://mathworld.wolfram.com/DummyVariable.html)

a) If r=0 then s'=|{}|=0 --> (no circle can be found) = A

b) If r>0 then s'=|{r}|=1 --> (a circle can be found) = B

The connection between A,B states cannot be but A_XOR_B

Also s' = 0 in case (a) and s' = 1 in case (b), can be described as s'=0_XOR_s'=1.

You can prove that A is the limit of B only if you can show that s'=0_AND_s'=1 --> 1

A collaction of elements, which can be found on many different scales, really approaching to some given constant, only if it has finitely many elements.

 

[Hurkyl]

There's something I've been trying to stress for a long time...

If you're not willing to discuss standard mathematics, you shouldn't bring it up.

 

[Lama]

If you're not willing to discuss standard mathematics, you shouldn't bring it up.

Please explain what do you mean exactly.

 

[hello3719]

Lama, are you a Godel wanabe ? Doesn't mean that if his ideas were accepted around 20 years after his death that yours are of the same type. Until now you didn't even show how to obtain pi in your system. And could you please define a number in english without using symbols ?

 

[kaiser soze]

Lama, by your analogy, your system does not contain "our" system, which is it then?

Kaiser.

 

[Lama]

Lama, by your analogy, your system does not contain "our" system, which is it then?

Kaiser.

By my analogy the whole is more then its parts.

Any this is why I call my natural numbers 'Organic Natural Numbers':

http://www.geocities.com/complementarytheory/ONN1.pdf

http://www.geocities.com/complementarytheory/ONN2.pdf

http://www.geocities.com/complementarytheory/ONN3.pdf

 

[kaiser soze]

Lama, as long as you are happy with it - just be aware that what you are doing does not even remotely resemble mathematics...

Kaiser.

 

[Lama]

Lama, as long as you are happy with it - just be aware that what you are doing does not even remotely resemble mathematics...

Kaiser.

There is no objective and totally external thing to us, which we can call it Mathematics:

Math, in my opinion, is first of all a rigorous agreement that based on language.

Symmetry is maybe the best tool that can be used to measure simplicity, where simplicity is the best platform for stable agreement.

In Any agreement we must be aware to the fact that no model of simplicity is simplicity itself.

This awareness to the difference between x-model and x-itself is the first condition for any stable agreement, because it gives it the ability to be changed.


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.

 

[Lama]

Until now you didn't even show how to obtain pi in your system...

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

Pi is invariant in any arbitrary given scale, but when several scale levels are simultaneously compared, we can clearly see that each circle has a different curvature.

If our system is a circle, then if we want to understand what is a circle, then both its invariant and its variant properties cannot be ignored.

(We also have to be aware to the fact the no circle can be found when Diameter or Perimeter = 0, or Diameter or Perimeter = oo.

In short, our basic approach is to find the gateways between opposite properties, and the best way to do it, is by an including-middle logical reasoning (http://www.geocities.com/complement...y/CompLogic.pdf).


Pi by the standard system:

Pi is also considered as some exact and fixed place in the Real-Line, which is determinated by the exact and fixed positions of 0 and 1 (the length of the radius).

...And could you please define a number in english without using symbols ?

A number is any intercation between our memory and some abstract or non-abstract element(s), where pre-interaction is |{}|(=the cardinal of the Empty set).

 

[kaiser soze]

Lama: "Math, in my opinion, is first of all a rigorous agreement that based on language."

Obviously, this is what YOU think mathematics is.

"Source: The American Heritage® Dictionary of the English Language, Fourth Edition
Copyright © 2000 by Houghton Mifflin Company.
Published by Houghton Mifflin Company. All rights reserved.


mathematics

\Math`e*mat"ics\, n. [F. math['e]matiques, pl., L. mathematica, sing., Gr. ? (sc. ?) science. See Mathematic, and -ics.] That science, or class of sciences, which treats of the exact relations existing between quantities or magnitudes, and of the methods by which, in accordance with these relations, quantities sought are deducible from other quantities known or supposed; the science of spatial and quantitative relations.

Note: Mathematics embraces three departments, namely: 1. Arithmetic. 2. Geometry, including Trigonometry and Conic Sections. 3. Analysis, in which letters are used, including Algebra, Analytical Geometry, and Calculus. Each of these divisions is divided into pure or abstract, which considers magnitude or quantity abstractly, without relation to matter; and mixed or applied, which treats of magnitude as subsisting in material bodies, and is consequently interwoven with physical considerations."

--> not that I fully agree with the above defintion but obviously your "mathematics" is something else than what most of us here think mathematics IS.

Kaiser.

 

[matt grime]

Why would you wish to contact my university? Where have I made any claims that I was in any way representing my unviersity. I've never even named my University in threads to you have I? I certainly don't post from my university email address in forums exactly because I am not in anyway representing them.

Incidentally you might wish to look at www.crank.net and see the legal argument as to the meaning of crank or crackpot.

And I keep telling you that my interest is in making sure you do not spout anything that is false about mathematics.

There are several rebuttals of your (mathematical) claims by me and others, none of which you've addressed.


Moreover, a good example of me correcting your opinions of mathematics must be where I'vew pointed out repeatedly that not all of mathematics is done using "excluded middle reasoning" (though your deduction that it is uses that which you won't allow us), there are plenty of theories that use different logical systems.

Of course if you think it is deficient then you'd better get rid of the computer you use to post this with since (almost) every computer language is just predicate or propositional calculus in disguise.


Here's another one, well, one we've had before:

you think mathematics is not good enough in some sense, and so you start talking about your axioms the real numbers satisfy, but they are not sufficient to define a set of anything, let alone something like the real numbers.
When you say real numbers, you cannot mean R as we know it because that is something constructed by the mathematics that you think is about to collapse because it is wrong. So what are you referring to when you say the real numbers?

You've also not explained what you mean in these axioms for the "reals" what you mean by things like: a point is that which can only be defined by =" and so on, after all the point 0 is the complement of (-inf,0)u(0,inf), moreover by tarski's construct any set defined by a finite number of inequalities may be defined by equalities, but that of course is in the real numbes as we know them (complete totally ordered field), what your version of the reals are is a mystery even now. I told you that bit of mathematics you briefly acknowledged it but didn't explain yourself, or even correct your "axioms", not that we'd expect you to.

 

[Lama]

--> your "mathematics" is something else than what most of us here think mathematics IS.

Dont you see that also your example is first of all an interpretation, which is based on agreement between a group of people.

This is exactly the reason why I say that my point of view is a paradigm-shift of this agreement.

 

[kaiser soze]

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.

 

[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 {.}