Deeper than primes

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Why do these folks think that creating a set of axioms and thus defining a "new field" of mathematics is a big deal? Here, let me do it:

SKEPTICAL MATHEMATICS

0. A set of primitive objects.
1. There is one binary operation the Skeptian (S).
2. There is a one-place relationship, "successor" (+1)
3. If xSy, then ySx.
4. For every x, xS(x+1) is ((x+1)+1)+1
5. For every x and y, ySx is (y+1)+1.
6. The rule of Skepiticism: (x+1)S(y+1) = y.

There. There probably are some inconsistencies here -- but there are also in Doron's "Organic Mathematics". Assuming, however, that there aren't any, or that they can be straightened out... so what?

Can I now demand that as well as set theory, group theory, algebra, etc., math departments will teach "Skeptic Mathematics" theory, to consider the theorems that follow from these axioms?

Well, I can demand, but it would be silly. The point is, it is the easiest thing in the world to invent a set of axioms, and call the theorems that follow from them "Organic Mathematics" or "New Mathematics" or "Dononian Mathematics" or "Skeptical Mathematics". There is, in fact, an infinite number (and a very large infinite, if you know what I mean) of possible mathematical models.

But all that is pointless, since what is important is not if one can invent such a model, but whether it is important. Quite apart from the fact that Doron's terms are ill-defined and his set of theorems of "Organic Mathematics" are, it seem, self-contradictory or meaningless, even if such mistakes could be sorted out it would still be totally trivial and without the least importance to mathematics as a whole.
 
Skeptic said:
But all that is pointless, since what is important is not if one can invent such a model, but whether it is important.
Click to expand...

In order to define importance, there must be a bridge between the observer and the observed, and this is what OM is all about.

OM is a one organism where the knower and the known are its organs.

Skeptic said:
SKEPTICAL MATHEMATICS

0. A set of primitive objects.
1. There is one binary operation the Skeptian (S).
2. There is a one-place relationship, "successor" (+1)
3. If xSy, then ySx.
4. For every x, xS(x+1) is ((x+1)+1)+1
5. For every x and y, ySx is (y+1)+1.
6. The rule of Skepiticism: (x+1)S(y+1) = y.
Click to expand...

"SKEPTICAL MATHEMATICS" (if it is consistant) is nothing but a serial case (according to 2.) of OM.

Skeptic said:
There probably are some inconsistencies here -- but there are also in Doron's "Organic Mathematics".
Click to expand...
Show it in OM. At least we shall see if you first get it.

Please do your best. OM can be found in http://www.geocities.com/complementarytheory/OMPT.pdf .
 
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doronshadmi said:
No: Distinction refers to the amount of levels a thing is distinct.
Click to expand...

Perhaps you and MosheKlein should compare notes on this.

Also, you still haven't addressed my question about the actual meaning of (A, B, ABCD, ABCD). You continue to copy and paste from MosheKlein's presentation regarding how you came up with it; nothing on meaning.
 
jsfisher said:
Perhaps you and MosheKlein should compare notes on this.

Also, you still haven't addressed my question about the actual meaning of (A, B, ABCD, ABCD). You continue to copy and paste from MosheKlein's presentation regarding how you came up with it; nothing on meaning.
Click to expand...

I am the inventor of ONs, so if you wish to ask something about them, I am the address.

(A, B, ABCD, ABCD) is the (((1), 1), 4, 4) case under partiton (2+1+1) of n=4.

This time please try to grasp http://www.internationalskeptics.com/forums/showpost.php?p=4853642&postcount=4139 .
 
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jsfisher said:
Why isn't (AB, AC, BC) among the possibilities for 3?

Why isn't (AB, ABC, ABC) among the possibilities for 3?
Click to expand...

These cases are not consistent with the recursive structure, where each form of some ON = n>1 (the form of partition(1+1+1,+...) is excluded) is the result of previous ON's form, which is based on any given k=1 to n-1 , where a form is a superposition or distinct states between k=1 to n-1 ids (where each previous ON's form is taken as a whole).

For example (AB, ABC, ABC) is based on (2,3,3), and it is is not a form of the minimal case of an ON that is based on forms of previous ONs (where each previous ON's form is taken as a whole).

Here is the minimal case of an ON that is based on previous forms of ONs (where each previous ON's form is taken as a whole), for n=3:

n=3

(1+1+1)
(3,3,3) (this form of ON3 is not based on previous ONs)

(2+1)
((2,2),1) (this form of ON3 is based on the non-distinct case of ON2)
(((1),1),1) (this form of ON3 is based on the distinct case of ON2)

As you see (2,3,3) is not one of these forms, and so is (2,2,2) (= the case of (AB, AC, BC) ).

EDIT:

Actually your construction of Organic Numbers is not based on previous ONs as a whole. Instead you take also parts of the previous ONs and construct some form of the next ON.

For example:

You take a part of (AB, AB) = (2,2) that is the non-distinct case of n=2 and construct the form (AB, ABC, ABC) = (2,3,3) of your construction of n=3.

You can do that of course, as long as your construction is consistent, but also in this case Distinction is used as a main principle,
and this is the important notion here.

Also let us write the definition of Distinction in a better English:

Distinction refers to the amount of the distinct levels of a thing.

Be aware that by this definition also -for example- (ABCD) = (4) = (a superposition of 4 ids of a singleton) is possible, and open a new universe which is beyond my first introduction of Organic Numbers.
 
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MosheKlein said:
My first and intuitive answer to you is that ordering of distinction in significant in serial observation like {......} and not significant in parallel observation of {____} is that help you in some sense ?

Moshe:boggled:
Click to expand...


Well actually no, the question wasn’t how ordering distinctions differ from associative distinctions in OM, but why are they excluded. So really you have just switched the question to why are serial distinctions excluded when, particularly for larger numbers, there are going to be more ordering (serial) distinctions then associative (parallel) distinctions? Again I must say that I am glad to see someone willing and capable of giving clear and concise answers on the subject, but unfortunately it did not answer the question asked.
 
The Man said:
I’ll say this for you Doron you have at least one unflappable consistency, your knowledge of biology and ecosystems is right on par with your knowledge of math, physics, philosophy, ethics, morality and any of the other topics that you display absolutely no understanding of or at best a very rudimentary and naive understanding of and at worst a complete confabulation that exists only in your mind.
Click to expand...
You have a naïve understanding of the Organic view of the ecosystem.

The non-naïve understanding of the Organic view is not clearly cut to different disciplines like math, physics, philosophy, ethics, biology, etc … as your western school of thought does for the past 3000 years.

I am talking about an ecosystem where disciplines like math, physics, philosophy, ethics, biology, etc … are organs of a one Organism, called the ecosystem.

Under this ecosystem the knower and the known are its organs and so are the relations between the knower and the known (weather these organs are abstract or not).

In general, the ecosystem is both abstract and non-abstract one environment, and Organic Numbers are a model of this one Organism, where Distinction, Non-locality and Locality are main principles of it, no matter what disciplines are considered.

In our researchable ecosystem nothing is totally connected and nothing is totally isolated, and this is exactly what makes it a one organism.
 
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jsfisher said:
ddt,

www.site.uottawa.ca/~ivan/F49-int-part.pdf
Click to expand...

ddt said:
Yep, that's the one I found. I implemented algorithm ZS1, and it was slower than my own :(

First, some Java technicalities: I implemented it using the Iterator interface. That means you have to make two methods:
1) hasNext() which reports if there's a next element in the list;
2) next() which actually gives the next element.
I put all the actual calculation in hasNext(); if it finds a next partition, it reports 'true', but it also has that next partition prepared and well. The implementation of next() only entails returning that partition, and two checks before that if one was found or if we first have to find the next one (in case next() is called twice in a row without a hasNext() in between).

So color me skeptical when the built-in Java profiling said he spent equal amount in both methods. Then I put in time counters myself in the code, and this is what it reported for a run calculating partitions of 75:
time in hasNext: 10828831745 8118265
time in next: 10659977218 8118263​
The first number is the nanoseconds, the second the number of calls.

My implementation of the ZS1 algorithm from the paper gives:
time in convert: 10953318714 8118263
time in hasNext: 10293843313 8118265
time in next: 10072671076 8118263​
The extra method 'convert' is needed to convert the result to the right format.
For the partition 4 = 2 + 1 + 1, the ZS1 algorithm gives the list (2, 1, 1), and I have to convert that to the list (2, 1, 0, 0). As you see, that costs equal time to any of the other two methods.

My conclusion of this is, that the time needed for the actual calculations are trifle compared to the overhead of the function calls. It would be a nice experiment to rewrite the stuff in C or C++ to see if that works better :).

Oh, and I already threw out most of the other overhead. The partitions I return are simple integer arrays, no fancy objects.
Click to expand...

To follow up on this: I got intrigued about those numbers and rewrote both partition algorithms in C. Guess what? Profiling turned out that the program spent more than 75% of its time in converting a partition from its 'member list' representation (from the above example: (2, 1, 1) to a 'frequency list' representation (2, 1, 0, 0). So I took a good look at the ZS1 algorithm, and managed to first make that conversion incremental, and secondly to do away with the 'member list' representation entirely. Pitting the four versions again each other:
(1) my own algorithm
(2) ZS1 with a naive conversion
(3) ZS1 with incremental conversion
(4) ZS1 using only a frequency list representation
for the partitions of 100 gives a runtime of about 4 seconds for (1), (3) and (4), and 22 seconds for (2).

Implementing the whole OR algorithm in C (using the GNU MP library for the bignums) gives a runtime of 18 minutes for OR(100) using version (4), and backporting it to Java gives a runtime of 38 minutes for OR(100).

I did some other Java-related optimizations on the Java stuff (like making members final), but I'm still not sure what caused the profiling numbers I posted earlier.

My current implementations of (1) and (4) are competitive in C, at least up to n=150, whereas (4) beats (1) in Java hands-down - I guess that has to do with using two arrays in (1) compared to only one in (4), so the penalty is in the bounds checking.

MosheKlein said:
Do you want to write a common paper with me
about the formula and your algorithm.
Click to expand...

I'm not really sure if my algorithm for generating partitions is very interesting. I don't even know if it's new :blush:. For the rest, I just implemented your algorithm and put some thought into what intermediate results are worth caching (see MemoizationWP).
 
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The Man said:
Well actually no, the question wasn’t how ordering distinctions differ from associative distinctions in OM, but why are they excluded. So really you have just switched the question to why are serial distinctions excluded when, particularly for larger numbers, there are going to be more ordering (serial) distinctions then associative (parallel) distinctions? Again I must say that I am glad to see someone willing and capable of giving clear and concise answers on the subject, but unfortunately it did not answer the question asked.
Click to expand...

http://www.internationalskeptics.com/forums/showpost.php?p=4856124&postcount=4164

http://www.internationalskeptics.com/forums/showpost.php?p=4856162&postcount=4165

http://www.internationalskeptics.com/forums/showpost.php?p=4854660&postcount=4151

http://www.internationalskeptics.com/forums/showpost.php?p=4854737&postcount=4153

The Man said:
So really you have just switched the question to why are serial distinctions excluded when, particularly for larger numbers, there are going to be more ordering (serial) distinctions then associative (parallel) distinctions?
Click to expand...
Nothing was switched simply because ONs (for example, my first introduction of them in http://www.internationalskeptics.com/forums/showpost.php?p=4852892&postcount=4112 ) are not less then parallel (what you call associative) OR serial (what you call ordering) things, exactly because Distinction is used as a main principle here.

EDIT:

Take for example my first n=4 of my first introduction of OMs:

ON4.jpg


ON4 can be taken:

1) At once (all its form are taken at once (in this case parallel observation is used).
2) As an ordered forms of different states of distinction (in this case Distinction is observed serially).
3) As any combination of Parallel\Serial Distinction, that is base on some ON4 form (which is not fully parallel or fully serial case).

(1) Is based on the non-distinct form of ON4 (each form is global\local state of a given ON).

(2) Is based on the distinct form of ON4 (each form is global\local state of a given ON).

(3) Is based on any given form (if exists) that is not non-distinct AND not distinct (each form is global\local state of a given ON).
 
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doronshadmi said:
You have a naïve understanding of the Organic view of the ecosystem.

The non-naïve understanding of the Organic view is not clearly cut to different disciplines like math, physics, philosophy, ethics, biology, etc … as your western school of thought does for the past 3000 years.

I am talking about an ecosystem where disciplines like math, physics, philosophy, ethics, biology, etc … are organs of a one Organism, called the ecosystem.

Under this ecosystem the knower and the known are its organs and so are the relations between the knower and the known (weather these organs are abstract or not).

In general, the ecosystem is both abstract and non-abstract one environment, and Organic Numbers are a model of this one Organism, where Distinction, Non-locality and Locality are main principles of it, no matter what disciplines are considered.

In our researchable ecosystem nothing is totally connected and nothing is totally isolated, and this is exactly what makes it a one organism.
Click to expand...

Indeed a bizarre interpretation of the word 'ecosystem’.
 
doronshadmi said:
http://www.internationalskeptics.com/forums/showpost.php?p=4856124&postcount=4164

http://www.internationalskeptics.com/forums/showpost.php?p=4856162&postcount=4165

http://www.internationalskeptics.com/forums/showpost.php?p=4854660&postcount=4151

http://www.internationalskeptics.com/forums/showpost.php?p=4854737&postcount=4153


Nothing was switched simply because ONs (for example, my first introduction of them in http://www.internationalskeptics.com/forums/showpost.php?p=4852892&postcount=4112 ) are not less then parallel (what you call associative) OR serial (what you call ordering) things, exactly because Distinction is used as a main principle here.

EDIT:

Take for example my first n=4 of my first introduction of OMs:

[qimg]http://www.geocities.com/complementarytheory/ON4.jpg[/qimg]

ON4 can be taken:

1) At once (all its form are taken at once (in this case parallel observation is used).
2) As an ordered forms of different states of distinction (in this case Distinction is observed serially).
3) As any combination of Parallel\Serial Distinction, that is base on some ON4 form (which is not fully parallel or fully serial case).

(1) Is based on the non-distinct form of ON4 (each form is global\local state of a given ON).

(2) Is based on the distinct form of ON4 (each form is global\local state of a given ON).

(3) Is based on any given form (if exists) that is not non-distinct AND not distinct (each form is global\local state of a given ON).
Click to expand...


Again showing that you do not include distinctions based simply on ordering does not answered the question as to why such distinctions are excluded. Similarly why are integers excluded or real numbers, as even more distinctions would then be available? For a notion where you claim “Distinction is used as a main principle here” the primary distinction of your notion is just its arbitrary method of considering distinctions.

You refer to “minimal case” or “minimal construction” in some of those posts you likned, however the minimal “case”, “construction” and representation of “4” is simply “4”. As I have said before you are simply engaged in term rewriting and although there are rules for rewriting terms is simpler forms (or at least one rule: fewer terms) there are none for rewriting them in more complex forms. Your entire notion is simply your arbitrary selection of your own personal rules for writing terms in a more complex form while excluding other entirely distinct and somewhat distinct forms and still attempting to claim ““Distinction is used as a main principle here”.
 
MosheKlein said:
ok you got it.

I will ask Anthony: "what do you see on the table?"

can you imagine his answer ?

Moshe:blush:
Click to expand...

Thank you Moshe.

Now we're assuming some things, of course.

1.) There are three quarters on the table. (Again "quarter is the name of an American coin value of 25 cents.)

2.) They are new coins and look alike. You'd have to look very hard to tell them apart.

3.) First off they are in a neat little row.

4.) I began with the suggestion that Athony count the coins.
You are asking him what he sees. So let's go there.

I imagine Anthony knows what quarters are and has seen them in action.
His answer could likely be, "A row of quarters." And he mighty just tell us "three quarters" without our asking.

But If our Anthony has never before seen quarter in his life, then there's a row of round things.
An if he doesn't know counting words, he's not going to tell us there are three round things.

What does he see?
Perhaps a row of individual objects.
Perhaps a row of duplicates of the same thing.

Shall we make something of that?

Maybe I shouldn't have chosen coins that look alike, but seperate fruit items.
like an orange, an apple, and a banana.

Anthony says:
"I see an orange, an apple, and a Banana."
or
"I see some fruit."



(Just a note:
Objects
Duplicates
Both are plural.
Both are its of third person discourse.
Anthony might "see" something entirely different, if he weren't reporting things he saw.)


Now you'll begin correcting me about what Anthony might have seen.
Please do. That's going to tell me a lot.
 
doronshadmi said:
Take for example my first n=4 of my first introduction of OMs:

[qimg]http://www.geocities.com/complementarytheory/ON4.jpg[/qimg]
Click to expand...

Could you explain the 4= (3) + 1 line? Where does the 3 come from in the second and third figures on that line? And is there not a figure missing, like the first on the line, but without the horizontal line linking the first three dots extending beyond the upward line to the third dot?
 
MosheKlein,

While you are thinking about (AB, BC, AC) and (AB, ABC, ABC) as missing distinctions for 3, pardon me for adding more fuel to the fire: For both 1 and 2, you invoke special rules.

For 1, you use the partition 1=1 to get 1 distinction. However, 1=1 is one of those n=n cases you said to ignore. You are therefore applying a special rule for 1 to get Or(1)=1. (Interestingly, too, you seldom show one of those stick drawings for the distinctions of 1 case like you do for all the other numbers.)

For 2, you omit the 2=2 partition (as you should, at least according to your rules), but somehow you generate two distinctions from the 2=1+1 partition. For all other numbers, the 1+1+...+1 case yields only one distinction. You are therefore applying a special rule for 2 to get Or(2)=2.

These very same inconsistencies exist in your formula for Or(n), too. You "define" Or(1) to be 1 and Or(2) to be 2. However, if you were to just calculate Or(1) from your formula directly, ignoring the defined value, you get Or(1)=0. So, on the one hand the definition for Or(1) is unnecessary, but on the other hand the definition you provided is inconsistent with the formula. You are therefore introducing a special rule for 1 to get Or(1)=1.

Similarly, if we were to accept Or(1)=1 and calculate Or(2) directly from the formula, we get Or(2)=1. Again, the definition for Or(2) is unnecessary, but the definition you provided is inconsistent with the formula. You are therefore introducing a special rule for 2 to get Or(2)=2.


What does all this mean? Well, from my point of view it means what was supposed to be a simple, elegant observation about the structure of the positive integers starts out with two inconsistencies followed by the wrong answer (for 1, 2, and 3, respectively).
 
Apathia said:
...
Anthony says:
"I see an orange, an apple, and a Banana."
or
"I see some fruit."
...
Click to expand...


It is an interesting question. For many pre-K youngsters, the answer to "What do you see?" would be just one of the fruit as a one-word response. "What else?" would get you the other two, but only one at a time.

For post-K youngsters, the compound answer would be more likely than the single fruit response, but I wouldn't think you would get the "I see some fruit" response (without prompting) until about middle school.

For the typical kindergartener, though, I'd guess most would respond with just one of the fruit.

YMMV. :)
 
The Man said:
Again showing that you do not include distinctions based simply on ordering does not answered the question as to why such distinctions are excluded.
Click to expand...
I include distinctions based simply on ordering.

The Man said:
You refer to “minimal case” or “minimal construction” in some of those posts you likned, however the minimal “case”, “construction” and representation of “4” is simply “4”.
Click to expand...

4 is the amount of a thing. The amount of a thing needs elements and something that gather then into a one thing. I call the gathered "the local aspect of that thing" and I call the gatherer ""the non-local aspect of that thing", for example in lisp language expression (+ 1 2 76.890 pi) "+" is the non-local aspect and "1 2 76.890 pi" is the local aspect.

As about Distinction, this is the property of the gathered things, and 4 can be a singleton id or and amount of several gathered things, where 4 does not provide any information about their distinction.

My minimal introduction of Distinction goes beyond 4 as a singleton id, and beyond the case where the gathered elements are clearly distinct.
 
jsfisher said:
MosheKlein,

While you are thinking about (AB, BC, AC) and (AB, ABC, ABC) as missing distinctions for 3,
Click to expand...
Already answerd in http://www.internationalskeptics.com/forums/showpost.php?p=4856162&postcount=4165 .

jsfisher said:
For 1, you use the partition 1=1 to get 1 distinction. However, 1=1 is one of those n=n cases you said to ignore. You are therefore applying a special rule for 1 to get Or(1)=1. (Interestingly, too, you seldom show one of those stick drawings for the distinctions of 1 case like you do for all the other numbers.)

For 2, you omit the 2=2 partition (as you should, at least according to your rules), but somehow you generate two distinctions from the 2=1+1 partition. For all other numbers, the 1+1+...+1 case yields only one distinction. You are therefore applying a special rule for 2 to get Or(2)=2.

These very same inconsistencies exist in your formula for Or(n), too. You "define" Or(1) to be 1 and Or(2) to be 2. However, if you were to just calculate Or(1) from your formula directly, ignoring the defined value, you get Or(1)=0. So, on the one hand the definition for Or(1) is unnecessary, but on the other hand the definition you provided is inconsistent with the formula. You are therefore introducing a special rule for 1 to get Or(1)=1.

Similarly, if we were to accept Or(1)=1 and calculate Or(2) directly from the formula, we get Or(2)=1. Again, the definition for Or(2) is unnecessary, but the definition you provided is inconsistent with the formula. You are therefore introducing a special rule for 2 to get Or(2)=2.


What does all this mean? Well, from my point of view it means what was supposed to be a simple, elegant observation about the structure of the positive integers starts out with two inconsistencies followed by the wrong answer (for 1, 2, and 3, respectively).
Click to expand...

Moshe's formula gets on stage only for any n>2 ( see http://www.internationalskeptics.com/forums/showpost.php?p=4839507&postcount=3906 ).

Please look at the basis of my first indoduction of ONs in http://www.internationalskeptics.com/forums/showpost.php?p=4850095&postcount=4039, that includes also the reason of why n>2+0 is ignored in Moshe's formula.
 
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doronshadmi said:
I include distinctions based simply on ordering.
Click to expand...

Then ABC, ACB, BCA, BAC, CBA and CAB are all distinct orderings of an A,B,C association and are “distinctions based simply on ordering” that you must “include”. Unless as usual you are simply claiming that you only “include distinctions based simply on ordering” that you have arbitrarily selected.

doronshadmi said:
4 is the amount of a thing. The amount of a thing needs elements and something that gather then into a one thing. I call the gathered "the local aspect of that thing" and I call the gatherer ""the non-local aspect of that thing", for example in lisp language expression (+ 1 2 76.890 pi) "+" is the non-local aspect and "1 2 76.890 pi" is the local aspect.
Click to expand...

4 is simply a representation of a value, anything else that you ascribe to it such as “elements”, “gathering”, “local” or “non-local” are simply your arbitrary ascriptions and not any intrinsic aspect of that value.


doronshadmi said:
As about Distinction, this is the property of the gathered things, and 4 can be a singleton id or and amount of several gathered things, where 4 does not provide any information about their distinction.
Click to expand...

So once again it is about arbitrary “Distinction” or distinctions that you simply make up in your own mind as confirmed by your claim “where 4 does not provide any information about their distinction”.


doronshadmi said:
My minimal introduction of Distinction goes beyond 4 as a singleton id, and beyond the case where the gathered elements are clearly distinct.
Click to expand...

No your “minimal introduction of Distinction goes beyond” any information provided by the simple representation of the value 4. Thus you have to make up your own arbitrary distinctions that you now claim go even “beyond the case where the gathered elements are clearly distinct”. So even when specific and applicable distinctions are clearly given or are not applicable and perhaps even specifically denied you just make up your own arbitrary distinctions anyway.
 
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The Man said:
Well actually no, the question wasn’t how ordering distinctions differ from associative distinctions in OM, but why are they excluded. So really you have just switched the question to why are serial distinctions excluded when, particularly for larger numbers, there are going to be more ordering (serial) distinctions then associative (parallel) distinctions? Again I must say that I am glad to see someone willing and capable of giving clear and concise answers on the subject, but unfortunately it did not answer the question asked.
Click to expand...

Hi Man,

I have some difficulties to understand your question
Maybe because of my English..
Can we concentrate on a specific example?
for example the case 4=2+2 bring to 3 different distinction since we delete one case
Because of symmetry


Can this be related to your question?

Best
Moshe:boggled:
 
ddt said:
I'm not really sure if my algorithm for generating partitions is very interesting. I don't even know if it's new :blush:. For the rest, I just implemented your algorithm and put some thought into what intermediate results are worth caching (see MemoizationWP).
Click to expand...

Hi ddt,

Following your computation Or(n) : n=1..100
is equal to the sequences A5069.. ( don't remember it's number)
So we can present in the paper an nice open problem
if it will continue forever , What do you think ?

Moshe:con2:
 
Apathia said:
Thank you Moshe.

Now we're assuming some things, of course.

1.) There are three quarters on the table. (Again "quarter is the name of an American coin value of 25 cents.)

2.) They are new coins and look alike. You'd have to look very hard to tell them apart.

3.) First off they are in a neat little row.

4.) I began with the suggestion that Athony count the coins.
You are asking him what he sees. So let's go there.

I imagine Anthony knows what quarters are and has seen them in action.
His answer could likely be, "A row of quarters." And he mighty just tell us "three quarters" without our asking.

But If our Anthony has never before seen quarter in his life, then there's a row of round things.
An if he doesn't know counting words, he's not going to tell us there are three round things.

What does he see?
Perhaps a row of individual objects.
Perhaps a row of duplicates of the same thing.

Shall we make something of that?

Maybe I shouldn't have chosen coins that look alike, but seperate fruit items.
like an orange, an apple, and a banana.



(Just a note:
Objects
Duplicates
Both are plural.
Both are its of third person discourse.
Anthony might "see" something entirely different, if he weren't reporting things he saw.)


Now you'll begin correcting me about what Anthony might have seen.
Please do. That's going to tell me a lot.
Click to expand...

Hi Apathia,

I can see that you see two children's Anthony and Michel everyone say something else so let make the distinction between them.

Michel says : "I see a fruit."
Anthony says: "I see an orange, an apple, and a Banana."

So I can asked them both : Do you see the same thing ?

What do you think can be there answer ?


Best
Moshe:c2:
 
jsfisher said:
MosheKlein,

While you are thinking about (AB, BC, AC) and (AB, ABC, ABC) as missing distinctions for 3, pardon me for adding more fuel to the fire: For both 1 and 2, you invoke special rules.

For 1, you use the partition 1=1 to get 1 distinction. However, 1=1 is one of those n=n cases you said to ignore. You are therefore applying a special rule for 1 to get Or(1)=1. (Interestingly, too, you seldom show one of those stick drawings for the distinctions of 1 case like you do for all the other numbers.)

For 2, you omit the 2=2 partition (as you should, at least according to your rules), but somehow you generate two distinctions from the 2=1+1 partition. For all other numbers, the 1+1+...+1 case yields only one distinction. You are therefore applying a special rule for 2 to get Or(2)=2.

These very same inconsistencies exist in your formula for Or(n), too. You "define" Or(1) to be 1 and Or(2) to be 2. However, if you were to just calculate Or(1) from your formula directly, ignoring the defined value, you get Or(1)=0. So, on the one hand the definition for Or(1) is unnecessary, but on the other hand the definition you provided is inconsistent with the formula. You are therefore introducing a special rule for 1 to get Or(1)=1.

Similarly, if we were to accept Or(1)=1 and calculate Or(2) directly from the formula, we get Or(2)=1. Again, the definition for Or(2) is unnecessary, but the definition you provided is inconsistent with the formula. You are therefore introducing a special rule for 2 to get Or(2)=2.


What does all this mean? Well, from my point of view it means what was supposed to be a simple, elegant observation about the structure of the positive integers starts out with two inconsistencies followed by the wrong answer (for 1, 2, and 3, respectively).
Click to expand...

Hi jsfisher,

I don't understand the problem
every recursion need initialization
For example the Fibonacci sequences

1,1,2,3,5,8,13

you define :

a1=1
a2=1

you don't have to explain the beginning

or(1)=1
or(2)=2

am I wrong ?

Moshe:con2:
 
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MosheKlein said:
Hi jsfisher,

I don't understand the problem
every recursion need initialization
For example the Fibonacci sequences

1,1,2,3,5,8,13

you define :

a1=1
a2=1

you don't have to explain the beginning

or(1)=1
or(2)=2

am I wrong ?

Moshe:con2:
Click to expand...
Hi Moshe, :idea:

There is a simple explanation of why your function gets on stage only for any n>2.

1) Your function ignores n+0 case, so it does not work for partition 1+0, which is the only partition of n=1

2) Your function does not work in partition 1+1 of n=2 because partition 1+1 is used for both the non-distinct case (AB,AB)=(2,2) and for the distinct case (A,B)=((1),1):

n=2

(1+1)
(AB,AB)=(2,2)
(A,B)=((1),1)


3) Your function works for any n>2 because only from n=3 to ∞ the partition of the form (1+1+1+…) is clearly used only for the non-distinct form for any n>2, for example, in n=3:

n=3

(1+1+1)
(ABC,ABC,ABC)=(3,3,3) (the non-distinct case of n=3)

(2+1)
(AB,AB,C)=((2,2),1)
(A,B,C)=(((1),1),1)
 
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The Man said:
4 is simply a representation of a value,
Click to expand...
I am talking about:

1) 4 as a cardinal of a collection, that does not provide any information about the level of distinction of the elements of that collection.

2) 4 as a distinct value (what you call "4 is simply a representation of a value").


The Man said:
Then ABC, ACB, BCA, BAC, CBA and CAB are all distinct orderings of an A,B,C association and
are “distinctions based simply on ordering” that you must “include”.
Click to expand...

(A,B,C)=(((1),1),1)
(ABC,ABC,ABC)=(3,3,3) and it is a superposition of 3 ids and not a permutation of 3 ids.

A permutation of 3 ids is based on the particular distinct case (A,B,C)=(((1),1),1) of ON3, and on this basis we have ABC, ACB, BCA, BAC, CBA and CAB premutation.

In general, the minimal case of ON3 is:

(ABC,ABC,ABC)=(3,3,3) (superposition of 3 ids)
(AB,AB,ABC)=((2,2),1) (superposition of 2 ids within n=3)
(A,B,C)=(((1),1),1) (distinct ids of n=3)

where the order of the symbols has no significance.



The Man said:
Unless as usual you are simply claiming that you only “include distinctions based simply on ordering” that you have arbitrarily selected.
Click to expand...

The Man I am sorry to tell you that after more than 4000 posts you simply do not understand Organic Numbers and how they use Distinction.

You do not understand how uncertainty and redundancy are used by them, in order to determine the ids of the gathered elements.

There is no use to continue the dialog with you on this case, because all you wish to do is to show what is wrong in my head, instead of using your head in order to understand what you read.

Your kind of criticism is sterile and boring, and any further reply to you (as long as you continue this style of criticism) is a waste of energy.
 
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MosheKlein said:
Hi jsfisher,

I don't understand the problem
every recursion need initialization
For example the Fibonacci sequences

1,1,2,3,5,8,13

you define :

a1=1
a2=1

you don't have to explain the beginning

or(1)=1
or(2)=2

am I wrong ?
Click to expand...


Yes, you are wrong.

The Fibonacci sequence can be defined by:

F(1) = 1
F(2) = 1
F(n+2) = F(n+1) + F(n)

The first two parts of this definition are necessary to the recursion. It cannot work without it. Moreover, there is nothing in the definition that makes one part contradict any other part.

For your definition of Or(n), you included unnecessary components. Worse, those unnecessary component contradict other components. Your definition for Or(n) is not self-consistent.

For Or(1), your formula devolves to a summation over an empty set and requires no explicit definition for value. The definition for Or(1), therefore, is unnecessary. Worse, the definition doesn't agree with the formula.

Also, the Fibonacci sequence recursion requires two "starting values" because of its formulation. Your recursion for Or(n), just by its form, doesn't require two starting values.

To be proper, your starting values for a recursion must be (1) necessary and (2) consistent. Yours are neither.
 
doronshadmi said:
(A,B,C)=(((1),1),1)
(ABC,ABC,ABC)=(3,3,3) and it is a superposition of 3 ids and not a permutation of 3 ids.

A permutation of 3 ids is based on the particular distinct case (A,B,C)=(((1),1),1) of ON3, and on this basis we have ABC, ACB, BCA, BAC, CBA and CAB premutation.

In general, the minimal case of ON3 is:

(ABC,ABC,ABC)=(3,3,3) (superposition of 3 ids)
(AB,AB,ABC)=((2,2),1) (superposition of 2 ids within n=3)
(A,B,C)=(((1),1),1) (disdinct ids of n=3)

where the order of the symbols has no significance.
Click to expand...

By labeling “(((1),1),1)” as “(A,B,C)” it is distinctions that you have established yourself. The orders of the symbols has significance by those distinctions you just established yourself. That fact that you simply ignore this particular type of distinction only goes to further demonstrate the arbitrary nature of your so called distinctions


doronshadmi said:
The Man I am sorry to tell you that after more than 4000 posts you simply do not understand Organic Numbers and how they use Distinction.
Click to expand...

After 4000 posts Doron you still do not know what it is you are doing by individually labeling your “1”s as either A, B , or C. You gave those indpendent distinctions which gives ordering distinctions like ABC, BAC and CAB validity. That you simply ignore those ordering and independent distinctions you gave them yourself only goes to show the insignificance of your so called distinctions even just to yourself.

doronshadmi said:
You do not understand how uncertainty and redundancy are used by them, in order to determine the ids of the gathered elements.
Click to expand...

You still do not understand that it is all a bunch of arbitrary nonsense that you specifically admitted in your last post that it is not from any information provided by the “number”

doronshadmi said:
There is no use to continue the dialog with you on this case, because all you wish to do is to show what is wrong in my head, instead of using your head in order to understand what you read.

Your kind of criticism is sterile and boring, and any further reply to you (as long as you continue this style of criticism) is a waste of energy.
Click to expand...

We have all heard that before Doron.
 
doronshadmi said:
Moshe's formula gets on stage only for any n>2.
Click to expand...

You have already told us you don't understand MosheKlein's formula, and I am confident you were truthful in that statement. I am also confident you don't understand the inadequacy of your explanation.
 
The Man said:
By labeling “(((1),1),1)” as “(A,B,C)” it is distinctions that you have established yourself
Click to expand...

No, it is not:

(((1),1),1) is the general form for (A,B,C), (A,C,B), (B,C,A), (B,A,C), (C,B,A) and (C,A,B) and you simply do not get it.

The "1"+"()" simply represent a unique id, no matter what value it has.
 
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jsfisher said:
It is an interesting question. For many pre-K youngsters, the answer to "What do you see?" would be just one of the fruit as a one-word response. "What else?" would get you the other two, but only one at a time.

For post-K youngsters, the compound answer would be more likely than the single fruit response, but I wouldn't think you would get the "I see some fruit" response (without prompting) until about middle school.

For the typical kindergartener, though, I'd guess most would respond with just one of the fruit.

YMMV. :)
Click to expand...

Your right. The kid would name whichever one his or her attention was upon.

Of course I'm going on a fishing trip with Moshe here.
I want to see just what bait he's going to put on the line, and what answers he thinks are significant.
It will get me to what's concrete to him about Organic Numbers.
Then, hopefully, I'll be able to communicate with him about the concept.
 
MosheKlein said:
Hi Man,

I have some difficulties to understand your question
Maybe because of my English..
Can we concentrate on a specific example?
for example the case 4=2+2 bring to 3 different distinction since we delete one case
Because of symmetry


Can this be related to your question?

Best
Moshe:boggled:
Click to expand...


No problem let my try to make it clearer. For the value 4 it is apparently being claimed that this must be 4 ‘things’ which may or may not be distinct from each other. So we label these things as A, B, C and D, giving the each distinct identifications. As there is only 1 A, 1 B 1 C and 1 D in this case of complete distinction then when we sum these things we get 4 in total. At lower levels of distinction we may have 2 As 1 B and 1 C but still only 4 total items. The problem is ordering is another type of distinction that specifically comes into play the more independent those individual distinctions become. In fact ordering distinctions become the most significant in the first instance I gave of complete distinctions. Now ordering is not that critical when simply discussing math or specifically summation, but that is not what we are talking about here. We are talking about things, specifically potentially uniquely identifiable things, and in that respect ordering can be an important if not essential factor. Let’s add some very specific identification to those things and see what we get when we change ordering. If we take 1 slab of butter, 1 slice bread, 1 slice of cheese and 1 application of heat we have lunch as a grilled cheese sandwich. If we gather, combine or add these things together then we are ready to eat. However if we do not gather them in a suitable order we do not end with the same “total” (lunch) but still have brought together 4 items in total. If we add the heat to the butter or cheese with out having gathered the bread first we just end up with mess to clean up and our elements go into the trash instead of in our stomachs. Other orderings might not be as unpalatable as this but still will not result in the same total of a grilled cheese sandwich. The proper ordering would be gather beard, add butter to bread add cheese then add heat. We can reverse the butter and cheese ordering without much consequence, so some identities can require more specific or singular ordering while others do not.


If you are just going to start assigning distinctions to possible elements of 4 then you must consider all the ramification of such assignment otherwise that assignment is just arbitrary and insignificant. If you are going to specifically base such assignments on the possibility or insistence that they must represent real things then again the consequences of complete consideration are required invoked. Otherwise there is absolutely no point in making such distinctions and any 1 is no different then any other 1. When you do make such distinctions particularly such that some 1 is some how different then some other 1 then ordering distinctions must come into play otherwise you are simply claiming that this particular 1 is no different then that particular 1 and your ascribed distinctions have absolutely no meaning.
 
doronshadmi said:
No, it is not:

(((1),1),1) is the general form for (A,B,C), (A,C,B), (B,C,A), (B,A,C), (C,B,A) and (C,A,B) and you simply do not get it.

The "1"+"()" simply represent a unique id, no matter what value it has.
Click to expand...

Doron your “general form” again only demonstrates that you ignore your ascribed A, B and C distinctions and the ordering distinctions that creates. You might as well just make it (A,A,A) if you are going to simply ignore your own distinctions.


Only 15 minutes from my last post after you said.


doronshadmi said:
There is no use to continue the dialog with you on this case, because all you wish to do is to show what is wrong in my head, instead of using your head in order to understand what you read.

Your kind of criticism is sterile and boring, and any further reply to you (as long as you continue this style of criticism) is a waste of energy.
Click to expand...

I think that has got to be my best time yet in getting you to reply after you saying you would not.
 
doronshadmi said:
The amount of a thing needs elements and something that gather then into a one thing. I call the gathered "the local aspect of that thing" and I call the gatherer ""the non-local aspect of that thing", for example in lisp language expression (+ 1 2 76.890 pi) "+" is the non-local aspect and "1 2 76.890 pi" is the local aspect.
Click to expand...
So the 'non-local aspect' is the operator and the 'local aspect' is the operands... so why not use conventional nomenclature? Operators and operands are well defined and well understood. Local and non-local are not, in this context. Or perhaps you're just trying to hitch a ride on the idea of quantum 'non-locality' by using the same terminology (for something quite different) ?
 
How curious.

Using the meaning of distinction MosheKlein conveyed in his recent presentation in Sweden, I've tried to generate by hand the distinctions for the first 4 positive integers. My results are presented below. Since hand-generation is error-prone, by confidence in the correctness and completeness of these is low; however, I do note the following:

For 1, there is 1! distinction.
For 2, there are 2! distinctions.
For 3, there are 3! distinctions.
For 4, there are 4! distinctions.

I claim this observation, although highly tentative, is far more interesting than much else we have seen in this thread.

Code: 
1:
(A)

2:
(A,  B)
(AB, AB)

3:
(A,   B,   C)
(A,   BC,  BC)
(AB,  ABC, ABC)
(AB,  AC,  BC)
(AB,  AC,  ABC)
(ABC, ABC, ABC)

4:
(A,    B,    C,    D)
(A,    B,    CD,   CD)
(A,    BC,   BCD,  BCD)
(A,    BC,   CD,   BD)
(A,    BC,   CD,   BCD)
(A,    BCD,  BCD,  BCD)
(AB,   AB,   CD,   CD)
(AB,   ABC,  ABCD, ABCD)
(AB,   ABC,  ABD,  ACD)
(AB,   ABC,  ABD,  ABCD)
(AB,   ABC,  ACD,  BCD)
(AB,   ABD,  ACD,  BCD)
(AB,   ABCD, ABCD, ABCD)
(AB,   BC,   ABCD, ABCD)
(AB,   BC,   ACD,  ACD)
(AB,   BC,   CD,   AC)
(AB,   CD,   ABC,  ABD)
(AB,   CD,   ABCD, ABCD)
(ABC,  ABC,  ABCD, ABCD)
(ABC,  ABC,  ABD,  ABD)
(ABC,  ABCD, ABCD, ABCD)
(ABC,  ABD,  ACD,  ABCD)
(ABC,  ABD,  ABCD, ABCD)
(ABCD, ABCD, ABCD, ABCD)
 
The Man said:
Doron your “general form” again only demonstrates that you ignore your ascribed A, B and C distinctions and the ordering distinctions that creates. You might as well just make it (A,A,A) if you are going to simply ignore your own distinctions.
Click to expand...

You can do that, but (A,A,A) (which is, by the way, redundancy with no uncertainty) is not a form of the minimal introduction of ONs, for example:

Let us see (((1),1),1) general form of the minimal introduction of ONs:

Code: 
A *  .  .                                                                               
  |  |  |                                                                               
B .  *  . = (A,B,C) , (C,B,A)                                                           
  |  |  |                                                                               
C .__.__*                                                                               
                                                                                        
                                                                                        
A .  *  .                                                                               
  |  |  |                                                                               
B *  .  . = (B,A,C) , (C,A,B)                                                           
  |  |  |                                                                               
C .__.__*                                                                               
                                                                                        
                                                                                        
A *  .  .                                                                               
  |  |  |                                                                               
B .  .  * = (A,C,B) , (B,C,A)                                                           
  |  |  |                                                                               
C .__*__.                                                                               
                                                                                        
                                                                                        
                                                                                        
You can play another game, where redundancy with no uncertainty is valid, 
for example:   
                                                                                        
                                                                                        
A *  *  *                                                                               
  |  |  |                                                                               
B .  .  . = (A,A,A)                                                                     
  |  |  |                                                                               
C .__.__.

In my first introduction of ONs ucretainty and redundancy levels are equal, for example:

0
icmfig1-1.jpg

1
icmfig1-2.jpg

2
icmfig1-3.jpg

3
icmfig1-4.jpg

4 and 5
icmfig2.jpg


Uncertainty\Redundancy matrix:
icmfig3.jpg


In other words The Man, it will be much more fruitful if you start to think abstract and general, instead of use your energy in order to show what is wrong in my head.
 
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jsfisher said:
How curious.

Using the meaning of distinction MosheKlein conveyed in his recent presentation in Sweden, I've tried to generate by hand the distinctions for the first 4 positive integers. My results are presented below. Since hand-generation is error-prone, by confidence in the correctness and completeness of these is low; however, I do note the following:

For 1, there is 1! distinction.
For 2, there are 2! distinctions.
For 3, there are 3! distinctions.
For 4, there are 4! distinctions.

I claim this observation, although highly tentative, is far more interesting than much else we have seen in this thread.

Code: 
1:
(A)

2:
(A,  B)
(AB, AB)

3:
(A,   B,   C)
(A,   BC,  BC)
(AB,  ABC, ABC)
(AB,  AC,  BC)
(AB,  AC,  ABC)
(ABC, ABC, ABC)

4:
(A,    B,    C,    D)
(A,    B,    CD,   CD)
(A,    BC,   BCD,  BCD)
(A,    BC,   CD,   BD)
(A,    BC,   CD,   BCD)
(A,    BCD,  BCD,  BCD)
(AB,   AB,   CD,   CD)
(AB,   ABC,  ABCD, ABCD)
(AB,   ABC,  ABD,  ACD)
(AB,   ABC,  ABD,  ABCD)
(AB,   ABC,  ACD,  BCD)
(AB,   ABD,  ACD,  BCD)
(AB,   ABCD, ABCD, ABCD)
(AB,   BC,   ABCD, ABCD)
(AB,   BC,   ACD,  ACD)
(AB,   BC,   CD,   AC)
(AB,   CD,   ABC,  ABD)
(AB,   CD,   ABCD, ABCD)
(ABC,  ABC,  ABCD, ABCD)
(ABC,  ABC,  ABD,  ABD)
(ABC,  ABCD, ABCD, ABCD)
(ABC,  ABD,  ACD,  ABCD)
(ABC,  ABD,  ABCD, ABCD)
(ABCD, ABCD, ABCD, ABCD)
Click to expand...

jsfisher,

This is simply great!! :clap:

Are forms like:

(A,A), (B,B)

(A,A,A), (B,B,B), (C,C,C)

etc.. are also valid in your ONs game?

I think that if you use Uncertainty\Redundancy matrix ( shown in http://www.internationalskeptics.com/forums/showpost.php?p=4859114&postcount=4198 )
you can improve your ONs game(s).
 
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MosheKlein said:
Hi ddt,

Following your computation Or(n) : n=1..100
is equal to the sequences A5069.. ( don't remember it's number)
So we can present in the paper an nice open problem
if it will continue forever , What do you think ?

Moshe:con2:
Click to expand...

The sequence number is A056198. :) Well, I've only shown that the first 30 numbers are the same. Just the conjecture that they're the same seems not very appealing - at least to me. Somewhere, there's gotta be a proof for that. I haven't taken a close look at the definition of the sequence (and I don't have Maple so I can't try that one out), but it says:
Recurrence suggested by that for A000669
Click to expand...
and the latter is a graph problem. I have no idea how it is "suggested" from A000669, but with the combinatorial nature of the OR definition, I wouldn't be surprised if it's not already simply in there.

BTW, I've also implemented algorithm ZS2 for generating partition lists from the paper jsfisher mentioned. It blows ZS1 out of the water with roughly a factor 2 :D. Execution times for a C, C++ and Java variant are nearly identical: around 2 seconds for n=100. Execution time for OR(100) is down to ca. 40 minutes :). Clearly, generating the list of partitions isn't the bottleneck (any more).
 
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