Deeper than primes

[Skeptic]

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.

 

[doronshadmi]

But all that is pointless, since what is important is not if one can invent such a model, but whether it is important.

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.

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.

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

There probably are some inconsistencies here -- but there are also in Doron's "Organic Mathematics".

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 .

 

[jsfisher]

No: Distinction refers to the amount of levels a thing is distinct.

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.

 

[doronshadmi]

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.

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 .

 

[doronshadmi]

Why isn't (AB, AC, BC) among the possibilities for 3?

Why isn't (AB, ABC, ABC) among the possibilities for 3?

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.

 

[The Man]

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

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.

 

[doronshadmi]

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.

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.

 

[ddt]

ddt,

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

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.

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.

Do you want to write a common paper with me
about the formula and your algorithm.

I'm not really sure if my algorithm for generating partitions is very interesting. I don't even know if it's new . For the rest, I just implemented your algorithm and put some thought into what intermediate results are worth caching (see Memoization^WP).

 

[doronshadmi]

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.

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

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?

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 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).

 

[The Man]

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.

Indeed a bizarre interpretation of the word 'ecosystem’.

 

[The Man]

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).

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”.

 

[Apathia]

ok you got it.

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

can you imagine his answer ?

Moshe

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]

Indeed a bizarre interpretation of the word 'ecosystem’.

Your interpretation is bizarre.

 

[zooterkin]

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

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

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?

 

[jsfisher]

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, 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).

 

[jsfisher]

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

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.

 

[doronshadmi]

Again showing that you do not include distinctions based simply on ordering does not answered the question as to why such distinctions are excluded.

I include distinctions based simply on ordering.

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”.

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.

 

[doronshadmi]

MosheKlein,

While you are thinking about (AB, BC, AC) and (AB, ABC, ABC) as missing distinctions for 3,

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

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, 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).

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.

 

[doronshadmi]

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?

Please look at http://www.internationalskeptics.com/forums/showpost.php?p=4852892&postcount=4112 in order to understand 3 as a sub-form of 4 in 4=(3)+1 case.

 

[The Man]

I include distinctions based simply on ordering.

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.

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.

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.

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.

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”.

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

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.