The Martian and the Earthman - and the limit concept

[Organic]

Martian mathematician visiting Earth, and he wishes to know how an Earthman mathematician understanding the limit concept.

After couple of hours of communication we get this situation:

1) a not= b

Earthman: By my way (1) is a hypothesis.

Martian: By my way (1) is an invariant state.

2) abs(a-b)=d < e > 0

Earthman: a) By my way you compare d to set S that includes in it all R members > 0. In this case d<d is impossible; therefore d must be = 0 --> a=b

Earthman: b) Another version of my way is to say that e=d/2 but then |a-b|=d AND |a-b|<d/2 which is impossible; therefore a=b.

Martian: e and d relation remaining unchanged in any arbitrary scale that you choose, which means: d is always smaller then e but greater than 0. It means that e=d/2 is impossible because e > d/n > 0.

Martian: S is an open collection (has infinitely many elements) therefore cannot be completed by definition. Only finite collection can be a complete collection. Therefore there is no such thing like S which includes all r > 0.

Earthman: How can a set be not completed? For example: please show us n which is not in N.

Martian: Natural numbers do not exist because of the existence of N, but because of the axioms that define them, N is only the name of the container that its content is infinitely many elements that can never be completed, and defined by the proper axiomatic system.

Earthman: e and d are fixed values.

Martian: e and d are variables, and both of them always greater 0, which means both e and d are changeable but the proportion of e>d>0 holds in any arbitrary scale.


Options:

a) Earthman's method is the right method.

b) Martian's method is the right method.

c) There is no one right method; therefore both methods are reasonable methods.


Please choose one of the options or add your own option, but in both cases please tell us why are you choosing or adding an option?


Thank you,

Organic

 

[pig]

are you trying to say that if martians existed, they would be idiots?

 

[Organic]

Pig,

Do you understand the Martian point of view?

You can do it by explain it to us.

 

[pig]

first, the 2) isn't really stated in the right way.

but i will still try to explain why the proof still works withing the martian's point of view..

you say:

"Martian: e and d relation remaining unchanged in any arbitrary scale that you choose, which means: d is always smaller then e but greater than 0. It means that e=d/2 is impossible because e > d/n > 0."

now the Martian continues:

"Martian: so, we can't choose e=d/2 because it is impossible. but by matt's hypothesis, we MUST be able to chose ANY number > 0 as e.

since according to 1) d/2 must be > 0, then according to 2) we MUST be able to choose it as e, and we can't because according to 2) it is impossible.

we have a contradiction, therefore 1. and 2. cannot both be true for given a and b."

 

[Organic]

Pig,

First please write (1) and (2) as you see it, and then I'll reply.

Thank you.

 

[matt grime]

That is a perfect answer, pig; somehow I don't think it will be accepted.

Moreover, Organic, if you continue to use the word complete and the quantifier for all in this manner then you will never get a thread to stick in the mathematics forum.

Simple proposition:

n(n+1) is even for all n in the set of natural numbers

by your logic that statement is meaningless as I've said it is true for all things in an infinite set. You do understand that the negation of "for all" is "there exists"? And thus there is a minimal natural number m for which that statement would fail to be true (if it were false), but then one of m and m+1 is even... contradiction... thus it is true for all n in N... where does that go wrong? Apart from being a completely unnecessary proof by contradiction that can be written as proof by construction.

 

[Organic]

since according to 1) d/2 must be > 0, then according to 2) we MUST be able to choose it as e, and we can't because according to 2) it is impossible.

All you did is to force Earthman method on Martian method, which means that you don't understand the Martian method.

Let us write again the two different points of view, and we shall show how you ignore the Martian method and forcing Earthman method on it.

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

1) a not= b

Earthman: By my way (1) is an hypothesis.

Marsian: By my way (1) is an invariant state.

2) abs(a-b)=d < e > 0

Earthman: a) By my way you compare d to set S that includes in it all R members > 0. in this case d<d is impossible; therefore d must be = 0 --> a=b

Earthman: b) Another version of my way is to say that e=d/2 but then |a-b|=d AND |a-b|<d/2 which is impossible; therefore a=b.

Marsian: e and d relation remaining unchanged in any arbitrary scale that you choose, which means: d is always smaller then e but greater than 0. It means that e=d/2 is impossible because e > d/n > 0.

Marsian: S is an open collection (has infinitely many elements) therefore cannot be completed by definition. Only finite collection can be a complete collection. Therefore there is no such thing like S which includes all r > 0.

Earthman: How a set cannot be completed? for example: please explain how N is not a complete collection of all n's by showing me n which is not in N.

Marsian: Natural numbers do not exist because of the existence of N, but because of the axioms that define them, N is only the name of the container that its content is infinitely many elements that can never be completed, and defined by the proper axiomatic system.

Earthman: e and d are fixed values.

Marsian: e and d are variables, and both of them are always greater than 0.

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

Now pig, let us examine your forcing method:

Martian: so, we can't choose e=d/2 because it is impossible. but by Matt's hypothesis, we MUST be able to chose ANY number > 0 as e.

Matt's hypothesis in Earthman's mathod, for example:

1) a not= b

Earthman: By my way (1) is an hypothesis (= Matt's method).

Marsian: By my way (1) is an invariant state because:

e and d relation remaining unchanged in any arbitrary scale that you choose, which means: d is always smaller then e but greater than 0. It means that e=d/2 is impossible because e > d/n > 0.


So as you see, you simply ignore the Martian method and forcing Earthman method.

My heart with you if by forcing your point of view you naturally ignore another point of view on some examined concept.

I can call it "raping", but Math ideas cannot be raped and if you don't understand this then you are in a big trouble.

Also your "raping" method ignores this:

Earthman: e and d are fixed values.

Martian: e and d are variables, and both of them always greater 0, which means both e and d are changeable but the proportion of e>d>0 holds in any arbitrary scale.

 

[matt grime]

The martian method requires us to know that the proposition that the Earthman's method proves is true. As all the Earthman is doing is proving the result (I am he so I should know) he cannot know that the Martian's view has any truth in it. Once he proves the proposition the earthman knows that one cannot have a=\=b and the premise being true, until such time as he's done that he may make no such deductions.

 

[Organic]

So, please choose:

Options:

a) Earthman's method is the right method.

b) Martian's method is the right method.

c) There is no one right method; therefore both methods are reasonable methods.

Please choose one of the options or add your own option, but in both cases please tell us why are you choosing or adding an option?

 

[Organic]

n(n+1) is even for all n in the set of natural numbers

even or odd are not fundamental properties of n members existence, because they are based on devision operation, which its results belong to Q.

And from Q point of view we can see that prime numbers are the special case of n/n_AND_n/1.

But let us say that we are not taking even as fundamental property of n members existence.

In this case n(n+1) define the invariant state of any n which is even, over infinitely many N members, exactly as the proportion e>d>0 does not changed over infinitely many scales, that can never be completed.

So as you see, Earthman "true" is only one of many possibilities.

Moreover, Organic, if you continue to use the word complete and the quantifier for all in this manner then you will never get a thread to stick in the mathematics forum.

I do not care anymore about the current professional mathematician's society,
I clearly understand now that there is no big difference between this society and organizations like the middle age inquisition, when we examine their abilities to grasp new ideas and let them be part of their world.

 

[matt grime]

Divisiblity is not dependent on Q, Organic, and you're just demonstrating once more that you do not understand, or have any knowledge of, any mathematics, one more nail in the coffin. Did I mention division? No, divisibility. They are different: one is a property the other an operation.

Add another 30 points to the crackpot index for citing the inquisition.

 

[Organic]

Divisiblity is not dependent on Q, Organic, and you're just demonstrating once more that you do not understand, or have any knowledge of, any mathematics, one more nail in the coffin. Did I mention division? No, divisibility. They are different: one is a property the other an operation.

Add another 30 points to the crackpot index for citing the inquisition.

Who said that Divisiblity depends on Q?

I said that division operation is not fundamental property for Natural numbers existence, and since even and odd are defined by division there is nothing here, which is fundamental for N members, no more no less.

But since you don't read (as usual) all what someone writes, before you response, then all you understand is no more than the shadow of youself.

 

[matt grime]

"even or odd are not fundamental properties of n members existence, because they are based on devision operation, which its results belong to Q"

that is what you wrote, and what i based my answer upon.

here is what divisibilty means (see not a single point of reference to division)

p is divisible by q if there is an r with p=qr. Not a rational number in sight, not a single mention of division. see?

evenness is equivalent to divisibilty by 2. no mention of division at all in the slightest.

 

[Organic]

evenness is equivalent to divisibilty by 2

Q members are the rational numbers, and "rational" is based on the word "ratio", which means that any Q member defined by p/q.

In that case evennes defined by n/2, which is the ratio between n and 2.

But again, you did not get to the main point which is:

Let us say that we are not taking even as fundamental property of n members existence.

In this case n(n+1) define the invariant state of any n which is even, over infinitely many N members, exactly as the proportion e>d>0 does not changed over infinitely many scales, that can never be completed.

So as you see, Earthman "true" is only one of many possibilities.

 

[matt grime]

erm, it is not necessary to define the rational numbers in order to define divisibilty and evenness. but why would I expect someone who thinks that we must accept all possibilities to accept that he may be wrong in thinking he knows there is a unique way to define things? slightly inconsistent and selfcontradictory there organic, crank the crackpot index another few notches.

 

[Organic]

but why would I expect someone who thinks that we must accept all possibilities to accept that he may be wrong in thinking he knows there is a unique way to define things?

Matt,

1) A one true is based (in general) on western philosophy.

2) Many possibilities are based (in general) on eastern philosophy.

Complementary Logic ( http://www.geocities.com/complementarytheory/BFC.pdf ) tries to finds the ways to associate between (1) and (2) and by this approach, it opens a gate to complexity:

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

The heart of life is the open dialog that exists between different things.

Shortly speaking, life without a dialog (and by using the word dialog, I mean to the general meaning of any mutual influence between different things) is the beginning of their end.

The current Professional Mathematicians Society using philosophical methods that are based on (1), and by this approach, the dialog between different mathematical branches or between each branch to real life complexity, is barely existing.

Without an open dialog, the technical power of Math language quickly brings us to the end of a one-way street.


The axiom of a Dialog:

"Let D be a Dialog, where D is any mutual influence between A and not A."

 

[kaiser soze]

Matt,

Why do you even bother arguing with Organic? Can't you see that his actions are not
Motivated by curiosity or pure interest, instead they are motivated by beliefs. You can
Not argue with beliefs. Arguing with Organic, is like arguing with an orthodox person about the existence of god. It is Organic's BELIEF that mathematicians are wrong, and for that matter, that he KNOWS mathematics and practicing it. It is his belief that the he has found a "new type" of infinity, and that this is the "right one". Arguing with Organic is a just a waste of time, since he does not really want to learn or do anything practical with his ideas.

 

[pig]

organic, do you find this proof incorrect:

I want to prove:

0. If Arnold is fatter than every hippopotamus, then Arnold is not a hippopotamus.

Proof:

Assume Arnold is a hippopotamus.
1. Arnold is fatter than every hippopotamus.
2. Arnold is a hippopotamus.

Since Arnold is fatter than every hippopotamus, and he is a hippopotamus, Arnold is fatter than himself. This cannot be true, so 1. and 2. cannot both be true. Therefore, it is true that "If 1, then not 2", QED (0 is exactly that).

I think it is clear why this is true, and that if Arnold is a hippopotamus, he can be fatter than every OTHER hippopotamus at best. Also, if he is fatter than every single hippopotamus, then he surely isn't one of them himself.

If you do agree with this proof, please replace "Arnold" with "|a-b|", "fatter" with "smaller" and "hippopotamus" with "number>0".

 

[matt grime]

I know, kaiser, but it's at least interesting to point out the inconsistencies in his arguments (he has at least one acolyte, choosing a word not at random). His latest one, in large blue type above (which may disappear at any point) is that he wants A and not A to be in a dialog(ue) with something, which by his own logic is not allowed, because it is meaningless to talk about the possibility of having two mutually exclusive propostions both true.

 

[Organic]

he wants A and not A to be in a dialog(ue) with something, which by his own logic is not allowed,

Shortly speaking, you don't understand Complementary Logic because it is based on A_AND_not_A products.

because it is meaningless to talk about the possibility of having two mutually exclusive propostions both true

So as I said before, Matt Grime cannot see beyond Boolean Logic or Fuzzy Logic, and he also cannot understand that these two methods are based on the one true method of the western philosophy.

More then that, he is afraid to open himself to the idea that fundamental math concepts, can be changed when they are examined from different point of views.

Shortly speaking, I invite Matt, kaiser soze, pig and any other person, to an open dialog on any fundamental Math language concept.

Kaizer please pay attention that I speak straightly to you, but you choose to speak with Matt about me, can you tell us why?

kaiser soze, if some one has another point of view which is different fron the conventional point of view and he call you to an open dialog on any fundamental Math concept, if you have no guts for this kind of an open dialog, then anyone who is looking now on this post, does not have to guess who is the orthodox and who is the open minded here.

 

[pig]

Organic, you call your claim that a and not a can both be true "open minded". I call it "lack of common sense".

If I told someone that I believe Earth is carried on the back of a giant pink elephant, and the person told me I was wrong, would that make him or her "orthodox"?

It is good to be ready to accept new things, but not stupid things.

I do not wish to offend you, I understand that just like I can say that you limit yourself by refusing to accept logic, you can say that we are confined by our logic, but from my point of view, "a and not a" is a giant pink elephant :)

 

[matt grime]

But, Organic, you explicitly stated that we are n ot allowed to have these two invariant states to do with abs values because they are meaningless as they cannot simultaneously happen, now you want to have a dialog(ue) with both A and Not A, surel that must be meaningless nonsense too. Well, if you wrote it it's got a better chance than average.

 

[Organic]

Pig look at this:


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


Option 1 (by Pig):

If Arnold (|a-b|) is smaller_than (<) any hippopotamus (e > 0)
then Arnold (|a-b|) is not an hippopotamus (|a-b| = 0).

Proof:

Let us say that Arnold d (|a-b|) is a hippopotamus (|a-b| > 0).

1. Arnold (|a-b|) is smaller_than (<) every hippopotamus (e > 0).
2. Arnold (|a-b|) is a hippopotamus (|a-b| > 0).

Since Arnold (|a-b|) is smaller_than (<) every hippopotamus (e > 0), and he is a hippopotamus (|a-b| > 0), Arnold (|a-b|) is smaller_than (<) himself (e is a notation for any hippopotamus including Arnold --> |a-b|<|a-b|) that cannot be true, so 1. and 2. cannot both be true.

Therefore, it is true that "If 1, then not 2 --> |a-b|=0", QED.

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

Option 2 (by Organic):

If Arnold (|a-b|) is smaller_than (<) any hippopotamus (e > 0)
then Arnold (|a-b|) is an hippopotamus (|a-b| > 0).

Proof:

Let us say that Arnold (|a-b|) is not a hippopotamus (|a-b| = 0).

1. Arnold (|a-b|) is smaller_than (<) every hippopotamus (e > 0).
2. Arnold (|a-b|) is not a hippopotamus (|a-b| = 0).

Since Arnold (|a-b|) is smaller_than (<) every hippopotamus (e > 0), and he is not a hippopotamus (|a-b| = 0), Arnold (|a-b|) is smaller_than (<) himself (|a-b| < 0) that cannot be true, so 1. and 2. cannot both be true.

Therefore, it is true that "If 1, then not 2 --> |a-b|>0", QED.

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

But pig, this is not the main point here.

The main point here is this:

1) a not= b

Earthman: By my way (1) is an hypothesis.

Marsian: By my way (1) is an invariant state.

By Marsian point of view a not= b is not an hypothesis but a fact, which is totally different point of view of (1).

 

[matt grime]

Your option two is patently and obviously false.

 

[pig]

1. Arnold (|a-b|) is smaller_than (<) every hippopotamus (e > 0).
2. Arnold (|a-b|) is not a hippopotamus (|a-b| = 0).

Since Arnold (|a-b|) is smaller_than (<) every hippopotamus (e > 0), and he is not a hippopotamus (|a-b| = 0), Arnold (|a-b|) is smaller_than (<) himself (|a-b| < 0) that cannot be true, so 1. and 2. cannot both be true.

Therefore, it is true that "If 1, then not 2 --> |a-b|>0", QED.

Sorry but this is not right.

look at this (i removed the brackets):

"Since Arnold is smaller_than every hippopotamus, and he is not a hippopotamus, Arnold is smaller_than himself "

It is simply not the right conclusion. It's like writing "Since all elephants are red, and I am not an elephant, I am red".

 

[Organic]

Yes Matt and pig,

You are both absolutely right, option 2 does not hold because in this case we also get |a-b|=0 as the right result, simply because if Arnold (|a-b|) is not a hippopotamus (|a-b|=0) then we cannot conclude that |a-b|<0; therefore |a-b|=0.

But, again this is the "true" iff |a-b| < e > 0 and |a-b| > 0, is an hypothesis ( the Earthman point of view).

By the Martian point of view, |a-b| < e > 0 and |a-b| > 0 --> e > |a-b| > 0 is a given fact and not an hypothesis.

What I wanted to show here is that you simply ignore any other possibility that can look at
|a-b| < e > 0 and |a-b| > 0 not as a hypothesis but as a given fact (the Martian point of view).

look at this (i removed the brackets):

"Since Arnold is smaller_than every hippopotamus, and he is not a hippopotamus, Arnold is smaller_than himself "

It is simply not the right conclusion. It's like writing "Since all elephants are red, and I am not an elephant, I am red".

You are right pig but this time let us try to examin what you wrote from a more global point of view, like this:

You wrote:

It is simply not the right conclusion. It's like writing "Since all elephants are red, and I am not an elephant, I am red".

Now look at this:

It is simply not the right conclusion. It's like writing "Since all Erthmen mathemathicians take |a-b| < e > 0 and |a-b| > 0 as an hypothesis, and I am a Martian mathemathician, I also take |a-b| < e > 0 and |a-b| > 0 as an hypothesis".

Shortly speaking, I have a very fundamental question, which is:

Why are you so happy to eliminate Mathematical universes which includes infinitely many non-trivial complex information forms that can be systematically explored by Math Language.

Do you really prefer to keep some trivial logical methods as your one and only one true, and through this approach to prevent from yourselfers any chance to be exposed to a non-trivial complexity?

A little piece of these interesting mathematical universes can be shown here:

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

IF YOUR APPROACH IS TO WIN THE BATTLE BUT TO LOSE THE WAR, THEN WHY DO YOU THINK THAT EVERYBODY MUST AGREE WITH THIS ILLOGICAL POINT OF VIEW?

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

Organic, you call your claim that a and not a can both be true "open minded". I call it "lack of common sense".

If I told someone that I believe Earth is carried on the back of a giant pink elephant, and the person told me I was wrong, would that make him or her "orthodox"?

It is good to be ready to accept new things, but not stupid things.

I do not wish to offend you, I understand that just like I can say that you limit yourself by refusing to accept logic, you can say that we are confined by our logic, but from my point of view, "a and not a" is a giant pink elephant :)

If you really read and try to understand this http://www.geocities.com/complementarytheory/BFC.pdf
Then you will find that Boolean Logic and Fuzzy Logic are proper sub-systems of Complementary Logic.

For trivial facts (by the way, your example is trivial only for the last 400 years) you have trivial solutions, but they do not have enough power to deal with real complexity with high levels of uncertainty, where uncertainty is a natural property (therefore "wellcome") of the examined elements.

 

[kaiser soze]

Dear Organic,

I did not address you in my previous post to Matt, since I do not wish to argue with you. As I wrote to Matt, I think your opinions are driven by beliefs and I do not with to argue about beliefs.

As to the issue of being open minded, I think that by refusing to listen to others and really LEARN the fundamentals of mathematics (using conventional methods and interpretations) before dealing with such complex issues as infinity and set theory foundations, you are the one being fixated and close minded!

 

[Organic]

Dear kaiser soze,

First, thank you for your reply.

I think that by refusing to listen to others and really LEARN the fundamentals of mathematics (using conventional methods and interpretations) before dealing with such complex issues as infinity and set theory foundations, you are the one being fixated and close minded!

Let us go step by step.

If you have the time to look at my last post that I sent to Matt and Pig, then please follow our dialog, and reply your detailed remarks and insights.

Thank you,

Organic

 

[kaiser soze]

Dear Organic,

Why should I make the effort? You have provided us with no good reason as to why we should try and understand your language. A dialog is a usually carried out when the ones participating in it either speak the same language or understand each others language(s). Clearly you do not speak in the language mathematicians use (and don't ask me to show you how or where since, many of us have tried to do so in the past, but you simply chose to ignore them, claiming that they do not understand you).

I will be more than happy to discuss your ideas with you, after you have:

1. demonstrated that you understan fundamental methematical concepts and techniques.
2. Or: provide us with a good reason as to why we should bother (and just saying that a richer world can be discovered is not enough - you need to explictly show us mathematical problems that can be addressed better or at all using your language).

Kaizer soze.

 

[Organic]

Dear kaiser soze,

First I am glad to know that you a one of many mathematicians that really tried to understand my point of views on fundamental concepts of Math Language.

1. demonstrated that you understan fundamental methematical concepts and techniques.

I asked if you can give your detailed remarks by following the dialog between Matt, Pig and me.

I think it will be a good starting point to open a dialog between us.

What do you think?