Deeper than primes

[jsfisher]

In other words, I am right.

You simply do not wish to understand things that are not derived from "How to define and\or use"? question.

As a result you do not get that your reasoning is nothing but an arbitrary deremination of some snapshot of an "hand-waving".

Is that really what you got out of all this? Ok. Have a nice life.

By the way, you are still avoiding the question. What's the difference between 1 and 0.999..., then?

 

[doronshadmi]

you are using this (incorrect) assessment to avoid answering the question for yourself. What do you get for the difference?

0.111...[base 2] < 0.222...[base 3] ... < 0.999...[base 10] ... < ... < ... 1

 

[doronshadmi]

Is that really what you got out of all this? Ok. Have a nice life.

Accept the corrections (and thank you very much for them), you did not try at all to read and undertand http://www.internationalskeptics.com/forums/showpost.php?p=4767578&postcount=3316 , beyond your agreed determinations.

 

[jsfisher]

0.111...[base 2] < 0.222...[base 3] ... < 0.999...[base 10] ... < ... < ... 1

You still are evading the answering the question. It asked for the difference between 1 and 0.999....

 

[jsfisher]

Accept the corrections (and thank you very much for them), you did not try at all to read and undertand http://www.internationalskeptics.com/forums/showpost.php?p=4767578&postcount=3316 , beyond your agreed determinations.

You didn't present any "determinations" that changed the answer. The question was always about an infinite geometric series. The presumed origin of the terms, or the origin of the origin of the terms, doesn't change the series under inspection.

 

[doronshadmi]

I have a request. Please do not reply to any part of this post, before you read all of it, thank you.


Ok, let us see what we got from the last very interesting dialog with jsfisher.

Jsfisher is, without a doubt a perfectly trained and skilful expert of serial thinking (also known as analysis) that was educated for many years to research and understand things by using one and only one question, which is:

"How (by using only serial thinking (also known as analysis)) to define and\or use?"

But like any expert of a one and only one thinking style, this is his strength and this is his weakness.

From one hand he will immediately find any mistake that may exist in the limits of his expertise (you can see his fine corrections of my last posts, along this last dialog), but on the other hand he misses the deeper understanding that can be derived from a more comprehensive view of the researched (which transcendent the limits of his expertise).

Our last discussion was about the relations between the Q sequence (0.9[base 10], 0.99[base 10], 0.999[base 10], …) (that each one of its elements is obviously < 1) and the Q sequence (0.9[base 10], 0.09[base 10], 0.009[base 10], …) (that each one of its elements is obviously < 1).

In both cases we deal with a non-finite sequence of finite Q members that each one of them is obviously < 1.

I asked this question:

Why in the case of (0.9[base 10], 0.99[base 10], 0.999[base 10], …) it is obvious that we always < 1, but in the case of the sum of (0.9[base 10], 0.09[base 10], 0.009[base 10], …) we get the result = 1?

Jsfisher, by using his expertise, used this argument:

Every element of your second set {0.9, 0.99, 0.999, ...} is the sum of a finite sequence of elements from your first set {0.9, 0.09, 0.009, ...}.

(Remark: what jsfisher calls "second set" in his argument, is the first set in this post, and what jsfisher calls "first set" in his argument, is the second set in this post) .

By following his argument I wrote post http://www.internationalskeptics.com/forums/showpost.php?p=4767578&postcount=3316 (this version is the result of several important corrections of jsfisher, that help me to write it according to the required term of jsfisher's expertise).

Jsfisher's reply to this post was:

All this hand-waving with sequences derived from other sequences is irrelevant. The original answer stands.

First let us see jsfisher's original answer:

Every element of your second set {0.9, 0.99, 0.999, ...} is the sum of a finite sequence of elements from your first set {0.9, 0.09, 0.009, ...}.

The value 1 is the sum of an infinite sequence of elements. Big difference between sums over finite and infinite sequences.

By the way, you may notice that even though 1 does not appear as an element of the second set (in either 1 or 0.999... form), it is the limit of the sequence represented by the set. This exactly parallels the summations. Even though 1 is not the sum of any finite sequence from the first set, it is the limit of those sums.

As I said, jsfisher choose (without any given reason) to ignore http://www.internationalskeptics.com/forums/showpost.php?p=4767578&postcount=3316 content.

Instead of dealing with it, he wrote these posts:

Post 1:

The sum is the result of infinitely many addends where each added value is finite.

So why do you think that inifintely many finite values can reach their limit?

I think that because I have quite the understanding of Mathematics, including how limits work. By Standard Mathematics (which you have already conceded as the domain of discourse, here), it is a geometric series of the form:

[latex]$$ S = a + ar + ar^2 + ar^3 + ... $$[/latex]

The series converges to a value given by:

[latex]$$ S = \sum_{n \ge 0} ar^n = \frac{a}{1 - r} $$[/latex]

Simple substitution with a=0.9 and r = 0.1 gives S = 0.9/(1-0.1) = 0.9/0.9 = 1.

This post is again, a string of agreed notations, that is based on one and only one question, which is:

"How (by using only serial thinking (also known as analysis)) to define and\or use?"

where this is the only agreed fundamental question of jsfisher's community of experts.

Post 2:

If you can't accept that 0.999... is equal to 1, then please tell us what the difference between the two is. That is, what is 1 - 0.999..., and please restrict you response to standard mathematics constructs.

(When I do the subtraction, I get 0.000..., which for all practical purposes is just 0.)

My answer to post 2 was:

(When I do the subtraction, I get 0.000..., which for all practical purposes is just 0.)

It is possible because you already say that 1=0.999... so your "practical purposes" is based on circular reasoning.

In post 2 jsfisher asked an important question, which is:

If you can't accept that 0.999... is equal to 1, then please tell us what the difference between the two is. That is, what is 1 - 0.999..., and please restrict you response to standard mathematics constructs.

but he had a condition, which is:

… please restrict you response to standard mathematics constructs.

In other words "I will not accept anything that is not based on standard mathematics constructs".

My answer to this question (I have to admit that I did not immediately answered to this jsfisher's particular post) was:

0.111...[base 2] < 0.222...[base 3] ... < 0.999...[base 10] ... < ... < ... 1

Jsfisher's reply to my answer was:

You still are evading the answering the question. It asked for the difference between 1 and 0.999....

Here is my reply to this jsfisher's post:

1) Jsfisher missed the general conclusion that is derived from the non-finite base\value sequence (where 0.999...[base 10] is nothing but some particular case of it).

2) If we use "a" in base 11 , in the same manner as "9" is used in base 10 then:

0.999…[base 10] < 0.aaa…[base 11] < 1

3) The general case of this is:

For any given base n>1 and any 0.kkk…[base n] (where k=n-1) there is 0.nnn…[base n+1] such that 0.kkk…[base n] < 0.nnn…[base n+1] < 1

The typical response of experts like jsfisher to (3) is:

No, these are different representations (numerals) of the same number (1 in this case).

You do not distinguish between numerals and numbers.

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

At this point we get an unclosed gap between jsfisher's reasoning and OM's reasoning, which can be closed only by a paradigm-shift about the base\value form.

By OM, the base\value form is a kosher number exactly as 1 is a kosher number.

By OM the base\value form is called a non-local number, and for further reading about this number, please read at least page 11 of http://www.geocities.com/complementarytheory/OMPT.pdf .

By OM, the Q sequence (0.9[base 10], 0.99[base 10], 0.999[base 10], …) and the Q sequence (0.9[base 10], 0.09[base 10], 0.009[base 10], …) are actually two aspects of a one thing, called a non-local number.

For technical considerations, let us use a more convenient example (that is based on the non-local number 0.111…[base 2]) in order to demonstrate the following:

This is a rigorous expression of a geometric series in terms of OM, exactly as a geometric series of the form

[latex]$$ S = a + ar + ar^2 + ar^3 + ... $$[/latex]

The series converges to a value given by:

[latex]$$ S = \sum_{n \ge 0} ar^n = \frac{a}{1 - r} $$[/latex]

is a rigorous expression in terms of Standard Math.

But unlike Standard Math, OM explicitly shows that the Q sequence (0.1[base 2], 0.11[base 2], 0.111[base 2], …) and the Q sequence (0.1[base 2], 0.01[base 2], 0.001[base 2], …) are actually two aspects of a one thing, called a non-local number, as follows:

The non-finite case of 0.111…[base 2] is clearly shown by the bolded double line, and it is easily shown that 0.222…[base 3] is a non-local number between 0.111…[base 2] and 1, such that 0.111…[base 2] < 0.222…[base 3] < 0.999…[base 10] < 0.aaa…[base 11] < … < … 1 .

By OM expression of a geometric series, it is also rigorously shown that the Q sequence (0.1[base 2], 0.11[base 2], 0.111[base 2], …) and the Q sequence (0.1[base 2], 0.01[base 2], 0.001[base 2], …) are (respectively) "horizontal" and "vertical" aspects of the same mathematical object, called the non-local number.

EDIT:

OM is based on deeper notions than Standard Math, because its foundations transcendent the single question ("How (by using only serial thinking (also known as analysis)) to define and\or use?") of Standard Math.

In order to realize this fact, one has to read all of http://www.geocities.com/complementarytheory/OMPT.pdf before he airs his view about any part of it.

Furthermore, one cannot understand any part of http://www.geocities.com/complementarytheory/OMPT.pdf if his reasoning is based only on "How (by using only serial thinking (also known as analysis)) to define and\or use?" question.

( About jsfisher's basic attitude about this thread, please see http://www.internationalskeptics.com/forums/showpost.php?p=4753962&postcount=3196 )

 

[jsfisher]

Doron,

If you really want a dialog, don't be so condescending and insulting.

 

[doronshadmi]

Doron,

If you really want a dialog, don't be so condescending and insulting.

It is a fantasy of doronetics, formed out of ignorance, inconsistency, and contradiction.

This is insulting, exactly because the reason for this is based on

there is no reason for me to define my terms when the usage is in complete agreement with accepted definitions.

which is condescending.

 

[doronshadmi]

jsfisher and The Man,

I actually wish to know your real names, in order to add you to http://www.geocities.com/complementarytheory/OMPT.pdf Acknowledgements.

I have no doubt that I had no chance to develop it to its current stage, without you.

 

[The Man]

jsfisher and The Man,

I actually wish to know your real names, in order to add you to http://www.geocities.com/complementarytheory/OMPT.pdf Acknowledgements.

I have no doubt that I had no chance to develop it to its current stage, without you.

Forget it, if we were actually contributors you would have been listening to what we have been saying. This fantasy of yours that ignoring the ‘standard math’ you claim to be supplanting and what people tell you about it somehow “develops” your notions is certainly not one I will indulge you in. The ignorance is entirely yours Doron, if you want to proclaim that ignorance as some ‘developed theory’ that is your choice and you can place all the ‘credit’ and responsibility exactly where it belongs, on yourself.

ETA:

If you want to be honest in your acknowledgements and give credit to the efforts of all those involved, then I would suggest the following statement.

“At various times and on numerous forums people have tried to explain ‘standard math’ to me and I would like to sincerely acknowledge and thank them all for their efforts. However, my theories are entirely dependent on misinterpreting ‘standard math’ so actually understanding ‘standard math’ would be counterproductive to the development of my theory.”

 

[doronshadmi]

Forget it, if we were actually contributors you would have been listening to what we have been saying. This fantasy of yours that ignoring the ‘standard math’ you claim to be supplanting and what people tell you about it somehow “develops” your notions is certainly not one I will indulge you in. The ignorance is entirely yours Doron, if you want to proclaim that ignorance as some ‘developed theory’ that is your choice and you can place all the ‘credit’ and responsibility exactly where it belongs, on yourself.

ETA:

If you want to be honest in your acknowledgements and give credit to the efforts of all those involved, then I would suggest the following statement.

“At various times and on numerous forums people have tried to explain ‘standard math’ to me and I would like to sincerely acknowledge and thank them all for their efforts. However, my theories are entirely dependent on misinterpreting ‘standard math’ so actually understanding ‘standard math’ would be counterproductive to the development of my theory.”

http://www.internationalskeptics.com/forums/showpost.php?p=4767637&postcount=3320

 

[The Man]

http://www.internationalskeptics.com/forums/showpost.php?p=4767637&postcount=3320

What? So your argument against my assertions that you simply dismiss and ignore the contributions of those you claim you want to acknowledge, is a post where you simply claim you will dismiss and ignore the contributions of others because you feel it is limited by what you call being “derived from "How to define and\or use"? question” and “an arbitrary deremination of some snapshot of an "hand-waving"”? The cup of irony overfloweth, again.

 

[sympathic]

What? So your argument against my assertions that you simply dismiss and ignore the contributions of those you claim you want to acknowledge, is a post where you simply claim you will dismiss and ignore the contributions of others because you feel it is limited by what you call being “derived from "How to define and\or use"? question” and “an arbitrary deremination of some snapshot of an "hand-waving"”? The cup of irony overfloweth, again.

He thinks that the rejections actually help him clarify his views, by using other terms/graphics. His basic view stays the same, and you summed it up quite accurately.

 

[doronshadmi]

He thinks that the rejections actually help him clarify his views, by using other terms/graphics. His basic view stays the same, and you summed it up quite accurately.

Category theory also uses "graphics", so what is exactly your point?

 

[zooterkin]

What's the immediate predecessor of 7.3? What's the definition of 'crisp', and an example crisp id?

 

[The Man]

What's the immediate predecessor of 7.3? What's the definition of 'crisp', and an example crisp id?

Oh come on zooterkin, we all know you’re just looking for an acknowledgement is Doron's PDF. That certainly seems more likely then you getting any kind of substantial response to those questions.

 

[The Man]

He thinks that the rejections actually help him clarify his views, by using other terms/graphics. His basic view stays the same, and you summed it up quite accurately.

Thanks, I just whish instead of simply wanting to put some acknowledgement of others into his 'paper', Doron actually acknowledged what those others have been telling him.

 

[sympathic]

Thanks, I just whish instead of simply wanting to put some acknowledgement of others into his 'paper', Doron actually acknowledged what those others have been telling him.

I think this is unlikely.

 

[jsfisher]

This is insulting...condescending.

Yes, but I wasn't trying to engage you in a dialog on a topic of my choosing. If you want a favor from someone, it is best not to punch him first.

jsfisher and The Man,

I actually wish to know your real names, in order to add you to http://www.geocities.com/complementarytheory/OMPT.pdf Acknowledgements.

I have no doubt that I had no chance to develop it to its current stage, without you.

No, thank you.

 

[doronshadmi]

Yes, but I wasn't trying to engage you in a dialog on a topic of my choosing. If you want a favor from someone, it is best not to punch him first.

Engage?? Favor??

Are these the words that guided you along this long dialog?

No wonder that is what you get.